[{"data":1,"prerenderedAt":17800},["ShallowReactive",2],{"uc-internet-shopping":3},{"en":4,"de":9158},{"id":5,"title":6,"body":7,"description":9150,"extension":9151,"faq":9152,"meta":9153,"navigation":7921,"path":9154,"seo":9155,"stem":9156,"__hash__":9157},"content\u002Fuse-cases\u002Finternet-shopping.md","Internet-shopping optimization problem",{"type":8,"value":9,"toc":9141},"minimark",[10,35,52,65,70,75,78,1262,1266,1285,2587,3124,3128,3240,5921,5924,7861,7865,7875,8138,8143,8147,8150,8209,8341,8535,8538,9010,9013,9060,9063,9066,9118,9121,9137],[11,12,13,14,18,19,22,23,30,31,34],"p",{},"Let us take a deep dive into the so-called \"internet-shopping optimization problem\", also known as \"supplier selection and order allocation\". This is a very common problem that occurs basically whenever a large number of items needs to be ordered from a large number of suppliers. As it involves ",[15,16,17],"em",{},"integer"," quantities, it is usually formulated as a ",[15,20,21],{},"mixed-integer linear problem"," (MILP). Unfortunately, MILPs are known to be ",[24,25,29],"a",{"href":26,"rel":27},"https:\u002F\u002Fen.wikipedia.org\u002Fwiki\u002FNP-hardness",[28],"nofollow","NP-hard"," - which more or less says that they can be ",[15,32,33],{},"very"," hard to solve.",[11,36,37,38,43,44,47,48,51],{},"You can replicate all results by running ",[24,39,42],{"href":40,"rel":41},"https:\u002F\u002Fgithub.com\u002FQuicopt\u002FTutorials\u002Fblob\u002Fmain\u002Fuse-cases\u002Finternet-shopping.ipynb",[28],"this notebook",". Note that this will execute the open-source solver CP-SAT ",[15,45,46],{},"on your machine"," while Quicopt runs on our ",[15,49,50],{},"API server",". This means the hardware is not identical, which it should be for a strictly rigorous comparison.",[11,53,54,55,59,60,64],{},"Even if you're running CP-SAT on CPUs faster than our test server, you should still notice that ",[56,57,58],"strong",{},"Quicopt is a lot faster!"," On top of that, CP-SAT uses 8 cores while Quicopt uses just one! (CP-SAT's number of cores can be controlled by adjusting ",[61,62,63],"code",{},"WORKERS",", see below. Finally, be mindful of the fact that CP-SAT operates non-deterministically, i.e. runtimes will fluctuate quite a lot, as will solution quality.)",[66,67,69],"h2",{"id":68},"the-problem-as-a-milp","The problem as a MILP",[71,72,74],"h3",{"id":73},"variables","Variables",[11,76,77],{},"The variables of our optimization problems are given in the following table:",[79,80,81,100],"table",{},[82,83,84],"thead",{},[85,86,87,91,94,97],"tr",{},[88,89,90],"th",{},"symbol",[88,92,93],{},"values",[88,95,96],{},"meaning",[88,98,99],{},"cost it 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ships for free no\u002Fyes",[106,1159,1160,1261],{},[109,1161,1163,1187],{"className":1162},[112],[109,1164,1166],{"className":1165},[116],[118,1167,1168],{"xmlns":120},[122,1169,1170,1184],{},[125,1171,1172,1175],{},[652,1173,1174],{},"−",[267,1176,1177,1179,1181],{},[131,1178,271],{},[131,1180,862],{},[275,1182,1183],{},"ship",[138,1185,1186],{"encoding":140},"-C^{\\text{ship}}_k",[109,1188,1190],{"className":1189,"ariaHidden":146},[145],[109,1191,1193,1197,1200],{"className":1192},[150],[109,1194],{"className":1195,"style":1196},[154],"height:1.2683em;vertical-align:-0.3013em;",[109,1198,1174],{"className":1199},[159],[109,1201,1203,1206],{"className":1202},[159],[109,1204,271],{"className":1205,"style":297},[159,163],[109,1207,1209],{"className":1208},[168],[109,1210,1212,1252],{"className":1211},[172,173],[109,1213,1215,1249],{"className":1214},[177],[109,1216,1219,1231],{"className":1217,"style":1218},[181],"height:0.967em;",[109,1220,1222,1225],{"style":1221},"top:-2.3987em;margin-left:-0.0715em;margin-right:0.05em;",[109,1223],{"className":1224,"style":190},[189],[109,1226,1228],{"className":1227},[194,195,196,197],[109,1229,862],{"className":1230,"style":905},[159,163,197],[109,1232,1234,1237],{"style":1233},"top:-3.1809em;margin-right:0.05em;",[109,1235],{"className":1236,"style":190},[189],[109,1238,1240],{"className":1239},[194,195,196,197],[109,1241,1243],{"className":1242},[159,197],[109,1244,1246],{"className":1245},[159,338,197],[109,1247,1183],{"className":1248},[159,197],[109,1250,206],{"className":1251},[205],[109,1253,1255],{"className":1254},[177],[109,1256,1259],{"className":1257,"style":1258},[181],"height:0.3013em;",[109,1260],{}," (a saving)",[71,1263,1265],{"id":1264},"objective-function","Objective function",[11,1267,1268,1269,1272,1273,1276,1277,1280,1281,1284],{},"The problem has four \"levels\": ",[56,1270,1271],{},"items"," to buy, ",[56,1274,1275],{},"variants"," of each item, ",[56,1278,1279],{},"offers"," for each variant, and finally the ",[56,1282,1283],{},"suppliers"," offering the offers. The cost function has two main terms:",[109,1286,1289],{"className":1287},[1288],"katex-display",[109,1290,1292,1522],{"className":1291},[112],[109,1293,1295],{"className":1294},[116],[118,1296,1298],{"xmlns":120,"display":1297},"block",[122,1299,1300,1519],{},[125,1301,1302,1334,1337,1354,1371,1375,1383,1386,1392,1394,1397,1399,1416,1419,1427,1429,1435,1437,1445,1447,1453,1456,1458,1460,1462,1464,1477,1479,1487,1489,1495,1497,1505,1507,1513,1515],{},[1303,1304,1305,1313],"munder",{},[125,1306,1307,1310],{},[131,1308,1309],{},"min",[652,1311,1312],{},"⁡",[125,1314,1315,1318,1320,1322,1324,1326,1328,1330,1332],{},[131,1316,133],{"mathvariant":1317},"bold-italic",[652,1319,654],{"separator":146},[131,1321,374],{"mathvariant":1317},[652,1323,654],{"separator":146},[131,1325,578],{"mathvariant":1317},[652,1327,654],{"separator":146},[131,1329,859],{"mathvariant":1317},[652,1331,654],{"separator":146},[131,1333,862],{"mathvariant":1317},[275,1335,1336],{},"  ",[1303,1338,1339,1342],{},[652,1340,1341],{},"∑",[125,1343,1344,1347,1350],{},[131,1345,1346],{},"i",[652,1348,1349],{},"∈",[131,1351,1353],{"mathvariant":1352},"script","I",[1303,1355,1356,1358],{},[652,1357,1341],{},[125,1359,1360,1362,1364],{},[131,1361,136],{},[652,1363,1349],{},[128,1365,1366,1369],{},[131,1367,1368],{"mathvariant":1352},"V",[131,1370,1346],{},[652,1372,1374],{"fence":146,"stretchy":146,"minsize":1373,"maxsize":1373},"1.8em","(",[267,1376,1377,1379,1381],{},[131,1378,271],{},[131,1380,136],{},[275,1382,277],{},[275,1384,1385],{}," ",[128,1387,1388,1390],{},[131,1389,133],{},[131,1391,136],{},[275,1393,1336],{},[652,1395,1396],{},"+",[275,1398,1336],{},[1303,1400,1401,1403],{},[652,1402,1341],{},[125,1404,1405,1407,1409],{},[131,1406,377],{},[652,1408,1349],{},[128,1410,1411,1414],{},[131,1412,1413],{"mathvariant":1352},"O",[131,1415,136],{},[652,1417,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},"1.2em",[267,1420,1421,1423,1425],{},[131,1422,271],{},[131,1424,377],{},[275,1426,768],{},[275,1428,1385],{},[128,1430,1431,1433],{},[131,1432,578],{},[131,1434,377],{},[652,1436,1396],{},[267,1438,1439,1441,1443],{},[131,1440,271],{},[131,1442,377],{},[275,1444,486],{},[275,1446,1385],{},[128,1448,1449,1451],{},[131,1450,374],{},[131,1452,377],{},[652,1454,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},")",[652,1457,1455],{"fence":146,"stretchy":146,"minsize":1373,"maxsize":1373},[275,1459,1336],{},[652,1461,1396],{},[275,1463,1336],{},[1303,1465,1466,1468],{},[652,1467,1341],{},[125,1469,1470,1472,1474],{},[131,1471,862],{},[652,1473,1349],{},[131,1475,1476],{"mathvariant":1352},"K",[652,1478,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,1480,1481,1483,1485],{},[131,1482,271],{},[131,1484,862],{},[275,1486,974],{},[275,1488,1385],{},[128,1490,1491,1493],{},[131,1492,859],{},[131,1494,862],{},[652,1496,1174],{},[267,1498,1499,1501,1503],{},[131,1500,271],{},[131,1502,862],{},[275,1504,1183],{},[275,1506,1385],{},[128,1508,1509,1511],{},[131,1510,1067],{},[131,1512,862],{},[652,1514,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[131,1516,1518],{"mathvariant":1517},"normal",".",[138,1520,1521],{"encoding":140},"\\min_{\\boldsymbol{z}, \\boldsymbol{x}, \\boldsymbol{n}, \\boldsymbol{u}, \\boldsymbol{k}}\\; \\sum_{i \\in \\mathcal{I}}\\sum_{v \\in \\mathcal{V}_i}\\Bigl(C^{\\text{choice}}_v\\, z_v \\;+\\; \\sum_{o \\in \\mathcal{O}_v}\\bigl(C^{\\text{pack}}_o\\, n_o + C^{\\text{use}}_o\\, x_o\\bigr)\\Bigr) \\;+\\; \\sum_{k \\in \\mathcal{K}}\\bigl(C^{\\text{fee}}_k\\, u_k - C^{\\text{ship}}_k\\, f_k\\bigr).",[109,1523,1525,1942,2157,2293,2471],{"className":1524,"ariaHidden":146},[145],[109,1526,1528,1532,1643,1647,1650,1713,1716,1814,1823,1882,1885,1925,1928,1932,1936,1939],{"className":1527},[150],[109,1529],{"className":1530,"style":1531},[154],"height:2.5444em;vertical-align:-1.3944em;",[109,1533,1537],{"className":1534},[1535,1536],"mop","op-limits",[109,1538,1540,1634],{"className":1539},[172,173],[109,1541,1543,1631],{"className":1542},[177],[109,1544,1547,1620],{"className":1545,"style":1546},[181],"height:0.6679em;",[109,1548,1550,1554],{"style":1549},"top:-2.3479em;margin-left:0em;",[109,1551],{"className":1552,"style":1553},[189],"height:3em;",[109,1555,1557],{"className":1556},[194,195,196,197],[109,1558,1560,1571,1574,1583,1586,1595,1598,1607,1610],{"className":1559},[159,197],[109,1561,1563],{"className":1562},[159,197],[109,1564,1566],{"className":1565},[159,197],[109,1567,133],{"className":1568,"style":1570},[159,1569,197],"boldsymbol","margin-right:0.0421em;",[109,1572,654],{"className":1573},[687,197],[109,1575,1577],{"className":1576},[159,197],[109,1578,1580],{"className":1579},[159,197],[109,1581,374],{"className":1582},[159,1569,197],[109,1584,654],{"className":1585},[687,197],[109,1587,1589],{"className":1588},[159,197],[109,1590,1592],{"className":1591},[159,197],[109,1593,578],{"className":1594},[159,1569,197],[109,1596,654],{"className":1597},[687,197],[109,1599,1601],{"className":1600},[159,197],[109,1602,1604],{"className":1603},[159,197],[109,1605,859],{"className":1606},[159,1569,197],[109,1608,654],{"className":1609},[687,197],[109,1611,1613],{"className":1612},[159,197],[109,1614,1616],{"className":1615},[159,197],[109,1617,862],{"className":1618,"style":1619},[159,1569,197],"margin-right:0.0185em;",[109,1621,1623,1626],{"style":1622},"top:-3em;",[109,1624],{"className":1625,"style":1553},[189],[109,1627,1628],{},[109,1629,1309],{"className":1630},[1535],[109,1632,206],{"className":1633},[205],[109,1635,1637],{"className":1636},[177],[109,1638,1641],{"className":1639,"style":1640},[181],"height:0.8882em;",[109,1642],{},[109,1644],{"className":1645,"style":1646},[691],"margin-right:0.2778em;",[109,1648],{"className":1649,"style":692},[691],[109,1651,1653],{"className":1652},[1535,1536],[109,1654,1656,1704],{"className":1655},[172,173],[109,1657,1659,1701],{"className":1658},[177],[109,1660,1663,1688],{"className":1661,"style":1662},[181],"height:1.05em;",[109,1664,1666,1670],{"style":1665},"top:-1.8557em;margin-left:0em;",[109,1667],{"className":1668,"style":1669},[189],"height:3.05em;",[109,1671,1673],{"className":1672},[194,195,196,197],[109,1674,1676,1679,1683],{"className":1675},[159,197],[109,1677,1346],{"className":1678},[159,163,197],[109,1680,1349],{"className":1681},[1682,197],"mrel",[109,1684,1353],{"className":1685,"style":1687},[159,1686,197],"mathcal","margin-right:0.0738em;",[109,1689,1691,1694],{"style":1690},"top:-3.05em;",[109,1692],{"className":1693,"style":1669},[189],[109,1695,1696],{},[109,1697,1341],{"className":1698},[1535,1699,1700],"op-symbol","large-op",[109,1702,206],{"className":1703},[205],[109,1705,1707],{"className":1706},[177],[109,1708,1711],{"className":1709,"style":1710},[181],"height:1.3217em;",[109,1712],{},[109,1714],{"className":1715,"style":692},[691],[109,1717,1719],{"className":1718},[1535,1536],[109,1720,1722,1805],{"className":1721},[172,173],[109,1723,1725,1802],{"className":1724},[177],[109,1726,1728,1792],{"className":1727,"style":1662},[181],[109,1729,1730,1733],{"style":1665},[109,1731],{"className":1732,"style":1669},[189],[109,1734,1736],{"className":1735},[194,195,196,197],[109,1737,1739,1742,1745],{"className":1738},[159,197],[109,1740,136],{"className":1741,"style":201},[159,163,197],[109,1743,1349],{"className":1744},[1682,197],[109,1746,1748,1752],{"className":1747},[159,197],[109,1749,1368],{"className":1750,"style":1751},[159,1686,197],"margin-right:0.0822em;",[109,1753,1755],{"className":1754},[168],[109,1756,1758,1783],{"className":1757},[172,173],[109,1759,1761,1780],{"className":1760},[177],[109,1762,1765],{"className":1763,"style":1764},[181],"height:0.3281em;",[109,1766,1768,1772],{"style":1767},"top:-2.357em;margin-left:-0.0822em;margin-right:0.0714em;",[109,1769],{"className":1770,"style":1771},[189],"height:2.5em;",[109,1773,1777],{"className":1774},[194,1775,1776,197],"reset-size3","size1",[109,1778,1346],{"className":1779},[159,163,197],[109,1781,206],{"className":1782},[205],[109,1784,1786],{"className":1785},[177],[109,1787,1790],{"className":1788,"style":1789},[181],"height:0.143em;",[109,1791],{},[109,1793,1794,1797],{"style":1690},[109,1795],{"className":1796,"style":1669},[189],[109,1798,1799],{},[109,1800,1341],{"className":1801},[1535,1699,1700],[109,1803,206],{"className":1804},[205],[109,1806,1808],{"className":1807},[177],[109,1809,1812],{"className":1810,"style":1811},[181],"height:1.3944em;",[109,1813],{},[109,1815,1818],{"className":1816},[1817],"mopen",[109,1819,1374],{"className":1820}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means that for every item ",[109,2591,2593,2606],{"className":2592},[112],[109,2594,2596],{"className":2595},[116],[118,2597,2598],{"xmlns":120},[122,2599,2600,2604],{},[125,2601,2602],{},[131,2603,1346],{},[138,2605,1346],{"encoding":140},[109,2607,2609],{"className":2608,"ariaHidden":146},[145],[109,2610,2612,2616],{"className":2611},[150],[109,2613],{"className":2614,"style":2615},[154],"height:0.6595em;",[109,2617,1346],{"className":2618},[159,163]," and every variant ",[109,2621,2623,2636],{"className":2622},[112],[109,2624,2626],{"className":2625},[116],[118,2627,2628],{"xmlns":120},[122,2629,2630,2634],{},[125,2631,2632],{},[131,2633,136],{},[138,2635,136],{"encoding":140},[109,2637,2639],{"className":2638,"ariaHidden":146},[145],[109,2640,2642,2645],{"className":2641},[150],[109,2643],{"className":2644,"style":247},[154],[109,2646,136],{"className":2647,"style":201},[159,163]," we pay ",[109,2650,2652,2671],{"className":2651},[112],[109,2653,2655],{"className":2654},[116],[118,2656,2657],{"xmlns":120},[122,2658,2659,2669],{},[125,2660,2661],{},[267,2662,2663,2665,2667],{},[131,2664,271],{},[131,2666,136],{},[275,2668,277],{},[138,2670,280],{"encoding":140},[109,2672,2674],{"className":2673,"ariaHidden":146},[145],[109,2675,2677,2680],{"className":2676},[150],[109,2678],{"className":2679,"style":290},[154],[109,2681,2683,2686],{"className":2682},[159],[109,2684,271],{"className":2685,"style":297},[159,163],[109,2687,2689],{"className":2688},[168],[109,2690,2692,2729],{"className":2691},[172,173],[109,2693,2695,2726],{"className":2694},[177],[109,2696,2698,2709],{"className":2697,"style":310},[181],[109,2699,2700,2703],{"style":313},[109,2701],{"className":2702,"style":190},[189],[109,2704,2706],{"className":2705},[194,195,196,197],[109,2707,136],{"className":2708,"style":201},[159,163,197],[109,2710,2711,2714],{"style":325},[109,2712],{"className":2713,"style":190},[189],[109,2715,2717],{"className":2716},[194,195,196,197],[109,2718,2720],{"className":2719},[159,197],[109,2721,2723],{"className":2722},[159,338,197],[109,2724,277],{"className":2725},[159,197],[109,2727,206],{"className":2728},[205],[109,2730,2732],{"className":2731},[177],[109,2733,2735],{"className":2734,"style":351},[181],[109,2736],{}," if the variant is chosen, and also for every\nused offer ",[109,2739,2741,2754],{"className":2740},[112],[109,2742,2744],{"className":2743},[116],[118,2745,2746],{"xmlns":120},[122,2747,2748,2752],{},[125,2749,2750],{},[131,2751,377],{},[138,2753,377],{"encoding":140},[109,2755,2757],{"className":2756,"ariaHidden":146},[145],[109,2758,2760,2763],{"className":2759},[150],[109,2761],{"className":2762,"style":247},[154],[109,2764,377],{"className":2765},[159,163]," of that variant we pay ",[109,2768,2770,2789],{"className":2769},[112],[109,2771,2773],{"className":2772},[116],[118,2774,2775],{"xmlns":120},[122,2776,2777,2787],{},[125,2778,2779],{},[267,2780,2781,2783,2785],{},[131,2782,271],{},[131,2784,377],{},[275,2786,768],{},[138,2788,771],{"encoding":140},[109,2790,2792],{"className":2791,"ariaHidden":146},[145],[109,2793,2795,2798],{"className":2794},[150],[109,2796],{"className":2797,"style":290},[154],[109,2799,2801,2804],{"className":2800},[159],[109,2802,271],{"className":2803,"style":297},[159,163],[109,2805,2807],{"className":2806},[168],[109,2808,2810,2847],{"className":2809},[172,173],[109,2811,2813,2844],{"className":2812},[177],[109,2814,2816,2827],{"className":2815,"style":310},[181],[109,2817,2818,2821],{"style":313},[109,2819],{"className":2820,"style":190},[189],[109,2822,2824],{"className":2823},[194,195,196,197],[109,2825,377],{"className":2826},[159,163,197],[109,2828,2829,2832],{"style":325},[109,2830],{"className":2831,"style":190},[189],[109,2833,2835],{"className":2834},[194,195,196,197],[109,2836,2838],{"className":2837},[159,197],[109,2839,2841],{"className":2840},[159,338,197],[109,2842,768],{"className":2843},[159,197],[109,2845,206],{"className":2846},[205],[109,2848,2850],{"className":2849},[177],[109,2851,2853],{"className":2852,"style":351},[181],[109,2854],{}," per pack bought as well as the\nfixed cost ",[109,2857,2859,2878],{"className":2858},[112],[109,2860,2862],{"className":2861},[116],[118,2863,2864],{"xmlns":120},[122,2865,2866,2876],{},[125,2867,2868],{},[267,2869,2870,2872,2874],{},[131,2871,271],{},[131,2873,377],{},[275,2875,486],{},[138,2877,489],{"encoding":140},[109,2879,2881],{"className":2880,"ariaHidden":146},[145],[109,2882,2884,2887],{"className":2883},[150],[109,2885],{"className":2886,"style":499},[154],[109,2888,2890,2893],{"className":2889},[159],[109,2891,271],{"className":2892,"style":297},[159,163],[109,2894,2896],{"className":2895},[168],[109,2897,2899,2936],{"className":2898},[172,173],[109,2900,2902,2933],{"className":2901},[177],[109,2903,2905,2916],{"className":2904,"style":518},[181],[109,2906,2907,2910],{"style":313},[109,2908],{"className":2909,"style":190},[189],[109,2911,2913],{"className":2912},[194,195,196,197],[109,2914,377],{"className":2915},[159,163,197],[109,2917,2918,2921],{"style":325},[109,2919],{"className":2920,"style":190},[189],[109,2922,2924],{"className":2923},[194,195,196,197],[109,2925,2927],{"className":2926},[159,197],[109,2928,2930],{"className":2929},[159,338,197],[109,2931,486],{"className":2932},[159,197],[109,2934,206],{"className":2935},[205],[109,2937,2939],{"className":2938},[177],[109,2940,2942],{"className":2941,"style":351},[181],[109,2943],{},". The second sum is over the\nsuppliers: the charge ",[109,2946,2948,2967],{"className":2947},[112],[109,2949,2951],{"className":2950},[116],[118,2952,2953],{"xmlns":120},[122,2954,2955,2965],{},[125,2956,2957],{},[267,2958,2959,2961,2963],{},[131,2960,271],{},[131,2962,862],{},[275,2964,974],{},[138,2966,977],{"encoding":140},[109,2968,2970],{"className":2969,"ariaHidden":146},[145],[109,2971,2973,2976],{"className":2972},[150],[109,2974],{"className":2975,"style":987},[154],[109,2977,2979,2982],{"className":2978},[159],[109,2980,271],{"className":2981,"style":297},[159,163],[109,2983,2985],{"className":2984},[168],[109,2986,2988,3025],{"className":2987},[172,173],[109,2989,2991,3022],{"className":2990},[177],[109,2992,2994,3005],{"className":2993,"style":310},[181],[109,2995,2996,2999],{"style":1008},[109,2997],{"className":2998,"style":190},[189],[109,3000,3002],{"className":3001},[194,195,196,197],[109,3003,862],{"className":3004,"style":905},[159,163,197],[109,3006,3007,3010],{"style":325},[109,3008],{"className":3009,"style":190},[189],[109,3011,3013],{"className":3012},[194,195,196,197],[109,3014,3016],{"className":3015},[159,197],[109,3017,3019],{"className":3018},[159,338,197],[109,3020,974],{"className":3021},[159,197],[109,3023,206],{"className":3024},[205],[109,3026,3028],{"className":3027},[177],[109,3029,3031],{"className":3030,"style":1044},[181],[109,3032],{}," if the supplier is used, less the\nshipping ",[109,3035,3037,3057],{"className":3036},[112],[109,3038,3040],{"className":3039},[116],[118,3041,3042],{"xmlns":120},[122,3043,3044,3054],{},[125,3045,3046],{},[267,3047,3048,3050,3052],{},[131,3049,271],{},[131,3051,862],{},[275,3053,1183],{},[138,3055,3056],{"encoding":140},"C^{\\text{ship}}_k",[109,3058,3060],{"className":3059,"ariaHidden":146},[145],[109,3061,3063,3066],{"className":3062},[150],[109,3064],{"className":3065,"style":1196},[154],[109,3067,3069,3072],{"className":3068},[159],[109,3070,271],{"className":3071,"style":297},[159,163],[109,3073,3075],{"className":3074},[168],[109,3076,3078,3115],{"className":3077},[172,173],[109,3079,3081,3112],{"className":3080},[177],[109,3082,3084,3095],{"className":3083,"style":1218},[181],[109,3085,3086,3089],{"style":1221},[109,3087],{"className":3088,"style":190},[189],[109,3090,3092],{"className":3091},[194,195,196,197],[109,3093,862],{"className":3094,"style":905},[159,163,197],[109,3096,3097,3100],{"style":1233},[109,3098],{"className":3099,"style":190},[189],[109,3101,3103],{"className":3102},[194,195,196,197],[109,3104,3106],{"className":3105},[159,197],[109,3107,3109],{"className":3108},[159,338,197],[109,3110,1183],{"className":3111},[159,197],[109,3113,206],{"className":3114},[205],[109,3116,3118],{"className":3117},[177],[109,3119,3121],{"className":3120,"style":1258},[181],[109,3122],{}," if shipping is free.",[71,3125,3127],{"id":3126},"constraints","Constraints",[11,3129,3130,3131,3167,3168,3203,3204,3239],{},"This problem has many constraints that restrict the decision variables meaningfully. For ",[109,3132,3134,3148],{"className":3133},[112],[109,3135,3137],{"className":3136},[116],[118,3138,3139],{"xmlns":120},[122,3140,3141,3145],{},[125,3142,3143],{},[131,3144,133],{"mathvariant":1317},[138,3146,3147],{"encoding":140},"\\boldsymbol{z}",[109,3149,3151],{"className":3150,"ariaHidden":146},[145],[109,3152,3154,3158],{"className":3153},[150],[109,3155],{"className":3156,"style":3157},[154],"height:0.4444em;",[109,3159,3161],{"className":3160},[159],[109,3162,3164],{"className":3163},[159],[109,3165,133],{"className":3166,"style":1570},[159,1569],", ",[109,3169,3171,3185],{"className":3170},[112],[109,3172,3174],{"className":3173},[116],[118,3175,3176],{"xmlns":120},[122,3177,3178,3182],{},[125,3179,3180],{},[131,3181,374],{"mathvariant":1317},[138,3183,3184],{"encoding":140},"\\boldsymbol{x}",[109,3186,3188],{"className":3187,"ariaHidden":146},[145],[109,3189,3191,3194],{"className":3190},[150],[109,3192],{"className":3193,"style":3157},[154],[109,3195,3197],{"className":3196},[159],[109,3198,3200],{"className":3199},[159],[109,3201,374],{"className":3202},[159,1569]," and ",[109,3205,3207,3221],{"className":3206},[112],[109,3208,3210],{"className":3209},[116],[118,3211,3212],{"xmlns":120},[122,3213,3214,3218],{},[125,3215,3216],{},[131,3217,578],{"mathvariant":1317},[138,3219,3220],{"encoding":140},"\\boldsymbol{n}",[109,3222,3224],{"className":3223,"ariaHidden":146},[145],[109,3225,3227,3230],{"className":3226},[150],[109,3228],{"className":3229,"style":3157},[154],[109,3231,3233],{"className":3232},[159],[109,3234,3236],{"className":3235},[159],[109,3237,578],{"className":3238},[159,1569],", we have the following five:",[79,3241,3242,3260],{},[82,3243,3244],{},[85,3245,3246,3249,3251,3254,3257],{},[88,3247,3248],{},"row",[88,3250],{},[88,3252,3253],{},"for every",[88,3255,3256],{},"known as",[88,3258,3259],{},"in words",[101,3261,3262,3691,4990,5256,5523],{},[85,3263,3264,3267,3630,3685,3688],{},[106,3265,3266],{},"(1)",[106,3268,3269],{},[109,3270,3272,3333],{"className":3271},[112],[109,3273,3275],{"className":3274},[116],[118,3276,3277],{"xmlns":120},[122,3278,3279,3330],{},[125,3280,3281,3292,3294,3300,3303,3309,3311,3322,3324],{},[267,3282,3283,3286,3288,3290],{},[131,3284,3285],{},"N",[131,3287,377],{},[131,3289,1309],{},[652,3291,1312],{},[275,3293,1385],{},[128,3295,3296,3298],{},[131,3297,374],{},[131,3299,377],{},[652,3301,3302],{},"≤",[128,3304,3305,3307],{},[131,3306,578],{},[131,3308,377],{},[652,3310,3302],{},[267,3312,3313,3315,3317,3320],{},[131,3314,3285],{},[131,3316,377],{},[131,3318,3319],{},"max",[652,3321,1312],{},[275,3323,1385],{},[128,3325,3326,3328],{},[131,3327,374],{},[131,3329,377],{},[138,3331,3332],{"encoding":140},"N^{\\min}_o\\, x_o \\le n_o \\le N^{\\max}_o\\, x_o",[109,3334,3336,3462,3518],{"className":3335,"ariaHidden":146},[145],[109,3337,3339,3343,3410,3413,3453,3456,3459],{"className":3338},[150],[109,3340],{"className":3341,"style":3342},[154],"height:1.0775em;vertical-align:-0.247em;",[109,3344,3346,3350],{"className":3345},[159],[109,3347,3285],{"className":3348,"style":3349},[159,163],"margin-right:0.109em;",[109,3351,3353],{"className":3352},[168],[109,3354,3356,3402],{"className":3355},[172,173],[109,3357,3359,3399],{"className":3358},[177],[109,3360,3363,3375],{"className":3361,"style":3362},[181],"height:0.8305em;",[109,3364,3366,3369],{"style":3365},"top:-2.453em;margin-left:-0.109em;margin-right:0.05em;",[109,3367],{"className":3368,"style":190},[189],[109,3370,3372],{"className":3371},[194,195,196,197],[109,3373,377],{"className":3374},[159,163,197],[109,3376,3377,3380],{"style":325},[109,3378],{"className":3379,"style":190},[189],[109,3381,3383],{"className":3382},[194,195,196,197],[109,3384,3386],{"className":3385},[159,197],[109,3387,3389,3393,3396],{"className":3388},[1535,197],[109,3390,3392],{"className":3391},[197],"m",[109,3394,1346],{"className":3395},[197],[109,3397,578],{"className":3398},[197],[109,3400,206],{"className":3401},[205],[109,3403,3405],{"className":3404},[177],[109,3406,3408],{"className":3407,"style":351},[181],[109,3409],{},[109,3411],{"className":3412,"style":692},[691],[109,3414,3416,3419],{"className":3415},[159],[109,3417,374],{"className":3418},[159,163],[109,3420,3422],{"className":3421},[168],[109,3423,3425,3445],{"className":3424},[172,173],[109,3426,3428,3442],{"className":3427},[177],[109,3429,3431],{"className":3430,"style":182},[181],[109,3432,3433,3436],{"style":410},[109,3434],{"className":3435,"style":190},[189],[109,3437,3439],{"className":3438},[194,195,196,197],[109,3440,377],{"className":3441},[159,163,197],[109,3443,206],{"className":3444},[205],[109,3446,3448],{"className":3447},[177],[109,3449,3451],{"className":3450,"style":213},[181],[109,3452],{},[109,3454],{"className":3455,"style":1646},[691],[109,3457,3302],{"className":3458},[1682],[109,3460],{"className":3461,"style":1646},[691],[109,3463,3465,3469,3509,3512,3515],{"className":3464},[150],[109,3466],{"className":3467,"style":3468},[154],"height:0.786em;vertical-align:-0.15em;",[109,3470,3472,3475],{"className":3471},[159],[109,3473,578],{"className":3474},[159,163],[109,3476,3478],{"className":3477},[168],[109,3479,3481,3501],{"className":3480},[172,173],[109,3482,3484,3498],{"className":3483},[177],[109,3485,3487],{"className":3486,"style":182},[181],[109,3488,3489,3492],{"style":410},[109,3490],{"className":3491,"style":190},[189],[109,3493,3495],{"className":3494},[194,195,196,197],[109,3496,377],{"className":3497},[159,163,197],[109,3499,206],{"className":3500},[205],[109,3502,3504],{"className":3503},[177],[109,3505,3507],{"className":3506,"style":213},[181],[109,3508],{},[109,3510],{"className":3511,"style":1646},[691],[109,3513,3302],{"className":3514},[1682],[109,3516],{"className":3517,"style":1646},[691],[109,3519,3521,3524,3587,3590],{"className":3520},[150],[109,3522],{"className":3523,"style":499},[154],[109,3525,3527,3530],{"className":3526},[159],[109,3528,3285],{"className":3529,"style":3349},[159,163],[109,3531,3533],{"className":3532},[168],[109,3534,3536,3579],{"className":3535},[172,173],[109,3537,3539,3576],{"className":3538},[177],[109,3540,3542,3553],{"className":3541,"style":518},[181],[109,3543,3544,3547],{"style":3365},[109,3545],{"className":3546,"style":190},[189],[109,3548,3550],{"className":3549},[194,195,196,197],[109,3551,377],{"className":3552},[159,163,197],[109,3554,3555,3558],{"style":325},[109,3556],{"className":3557,"style":190},[189],[109,3559,3561],{"className":3560},[194,195,196,197],[109,3562,3564],{"className":3563},[159,197],[109,3565,3567,3570,3573],{"className":3566},[1535,197],[109,3568,3392],{"className":3569},[197],[109,3571,24],{"className":3572},[197],[109,3574,374],{"className":3575},[197],[109,3577,206],{"className":3578},[205],[109,3580,3582],{"className":3581},[177],[109,3583,3585],{"className":3584,"style":351},[181],[109,3586],{},[109,3588],{"className":3589,"style":692},[691],[109,3591,3593,3596],{"className":3592},[159],[109,3594,374],{"className":3595},[159,163],[109,3597,3599],{"className":3598},[168],[109,3600,3602,3622],{"className":3601},[172,173],[109,3603,3605,3619],{"className":3604},[177],[109,3606,3608],{"className":3607,"style":182},[181],[109,3609,3610,3613],{"style":410},[109,3611],{"className":3612,"style":190},[189],[109,3614,3616],{"className":3615},[194,195,196,197],[109,3617,377],{"className":3618},[159,163,197],[109,3620,206],{"className":3621},[205],[109,3623,3625],{"className":3624},[177],[109,3626,3628],{"className":3627,"style":213},[181],[109,3629],{},[106,3631,435,3632],{},[109,3633,3635,3653],{"className":3634},[112],[109,3636,3638],{"className":3637},[116],[118,3639,3640],{"xmlns":120},[122,3641,3642,3650],{},[125,3643,3644,3646,3648],{},[131,3645,377],{},[652,3647,1349],{},[131,3649,1413],{"mathvariant":1352},[138,3651,3652],{"encoding":140},"o 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chosen variant must be covered. Each offer yields ",[109,4353,4355,4375],{"className":4354},[112],[109,4356,4358],{"className":4357},[116],[118,4359,4360],{"xmlns":120},[122,4361,4362,4372],{},[125,4363,4364],{},[267,4365,4366,4368,4370],{},[131,4367,3732],{},[131,4369,377],{},[275,4371,768],{},[138,4373,4374],{"encoding":140},"Q^{\\text{pack}}_o",[109,4376,4378],{"className":4377,"ariaHidden":146},[145],[109,4379,4381,4384],{"className":4380},[150],[109,4382],{"className":4383,"style":290},[154],[109,4385,4387,4390],{"className":4386},[159],[109,4388,3732],{"className":4389},[159,163],[109,4391,4393],{"className":4392},[168],[109,4394,4396,4433],{"className":4395},[172,173],[109,4397,4399,4430],{"className":4398},[177],[109,4400,4402,4413],{"className":4401,"style":310},[181],[109,4403,4404,4407],{"style":3912},[109,4405],{"className":4406,"style":190},[189],[109,4408,4410],{"className":4409},[194,195,196,197],[109,4411,377],{"className":4412},[159,163,197],[109,4414,4415,4418],{"style":325},[109,4416],{"className":4417,"style":190},[189],[109,4419,4421],{"className":4420},[194,195,196,197],[109,4422,4424],{"className":4423},[159,197],[109,4425,4427],{"className":4426},[159,338,197],[109,4428,768],{"className":4429},[159,197],[109,4431,206],{"className":4432},[205],[109,4434,4436],{"className":4435},[177],[109,4437,4439],{"className":4438,"style":351},[181],[109,4440],{}," units and loses ",[109,4443,4445,4465],{"className":4444},[112],[109,4446,4448],{"className":4447},[116],[118,4449,4450],{"xmlns":120},[122,4451,4452,4462],{},[125,4453,4454],{},[267,4455,4456,4458,4460],{},[131,4457,3732],{},[131,4459,377],{},[275,4461,3753],{},[138,4463,4464],{"encoding":140},"Q^{\\text{lost}}_o",[109,4466,4468],{"className":4467,"ariaHidden":146},[145],[109,4469,4471,4474],{"className":4470},[150],[109,4472],{"className":4473,"style":290},[154],[109,4475,4477,4480],{"className":4476},[159],[109,4478,3732],{"className":4479},[159,163],[109,4481,4483],{"className":4482},[168],[109,4484,4486,4523],{"className":4485},[172,173],[109,4487,4489,4520],{"className":4488},[177],[109,4490,4492,4503],{"className":4491,"style":310},[181],[109,4493,4494,4497],{"style":3912},[109,4495],{"className":4496,"style":190},[189],[109,4498,4500],{"className":4499},[194,195,196,197],[109,4501,377],{"className":4502},[159,163,197],[109,4504,4505,4508],{"style":325},[109,4506],{"className":4507,"style":190},[189],[109,4509,4511],{"className":4510},[194,195,196,197],[109,4512,4514],{"className":4513},[159,197],[109,4515,4517],{"className":4516},[159,338,197],[109,4518,3753],{"className":4519},[159,197],[109,4521,206],{"className":4522},[205],[109,4524,4526],{"className":4525},[177],[109,4527,4529],{"className":4528,"style":351},[181],[109,4530],{},". For example, if ",[109,4533,4535,4561],{"className":4534},[112],[109,4536,4538],{"className":4537},[116],[118,4539,4540],{"xmlns":120},[122,4541,4542,4558],{},[125,4543,4544,4552,4555],{},[267,4545,4546,4548,4550],{},[131,4547,3732],{},[131,4549,136],{},[275,4551,3773],{},[652,4553,4554],{},"=",[648,4556,4557],{},"4",[138,4559,4560],{"encoding":140},"Q^{\\text{need}}_v = 4",[109,4562,4564,4636],{"className":4563,"ariaHidden":146},[145],[109,4565,4567,4570,4627,4630,4633],{"className":4566},[150],[109,4568],{"className":4569,"style":290},[154],[109,4571,4573,4576],{"className":4572},[159],[109,4574,3732],{"className":4575},[159,163],[109,4577,4579],{"className":4578},[168],[109,4580,4582,4619],{"className":4581},[172,173],[109,4583,4585,4616],{"className":4584},[177],[109,4586,4588,4599],{"className":4587,"style":310},[181],[109,4589,4590,4593],{"style":3912},[109,4591],{"className":4592,"style":190},[189],[109,4594,4596],{"className":4595},[194,195,196,197],[109,4597,136],{"className":4598,"style":201},[159,163,197],[109,4600,4601,4604],{"style":325},[109,4602],{"className":4603,"style":190},[189],[109,4605,4607],{"className":4606},[194,195,196,197],[109,4608,4610],{"className":4609},[159,197],[109,4611,4613],{"className":4612},[159,338,197],[109,4614,3773],{"className":4615},[159,197],[109,4617,206],{"className":4618},[205],[109,4620,4622],{"className":4621},[177],[109,4623,4625],{"className":4624,"style":351},[181],[109,4626],{},[109,4628],{"className":4629,"style":1646},[691],[109,4631,4554],{"className":4632},[1682],[109,4634],{"className":4635,"style":1646},[691],[109,4637,4639,4643],{"className":4638},[150],[109,4640],{"className":4641,"style":4642},[154],"height:0.6444em;",[109,4644,4557],{"className":4645},[159],"  and  ",[109,4648,4650,4674],{"className":4649},[112],[109,4651,4653],{"className":4652},[116],[118,4654,4655],{"xmlns":120},[122,4656,4657,4671],{},[125,4658,4659,4667,4669],{},[267,4660,4661,4663,4665],{},[131,4662,3732],{},[131,4664,377],{},[275,4666,3753],{},[652,4668,4554],{},[648,4670,662],{},[138,4672,4673],{"encoding":140},"Q^{\\text{lost}}_o = 2",[109,4675,4677,4749],{"className":4676,"ariaHidden":146},[145],[109,4678,4680,4683,4740,4743,4746],{"className":4679},[150],[109,4681],{"className":4682,"style":290},[154],[109,4684,4686,4689],{"className":4685},[159],[109,4687,3732],{"className":4688},[159,163],[109,4690,4692],{"className":4691},[168],[109,4693,4695,4732],{"className":4694},[172,173],[109,4696,4698,4729],{"className":4697},[177],[109,4699,4701,4712],{"className":4700,"style":310},[181],[109,4702,4703,4706],{"style":3912},[109,4704],{"className":4705,"style":190},[189],[109,4707,4709],{"className":4708},[194,195,196,197],[109,4710,377],{"className":4711},[159,163,197],[109,4713,4714,4717],{"style":325},[109,4715],{"className":4716,"style":190},[189],[109,4718,4720],{"className":4719},[194,195,196,197],[109,4721,4723],{"className":4722},[159,197],[109,4724,4726],{"className":4725},[159,338,197],[109,4727,3753],{"className":4728},[159,197],[109,4730,206],{"className":4731},[205],[109,4733,4735],{"className":4734},[177],[109,4736,4738],{"className":4737,"style":351},[181],[109,4739],{},[109,4741],{"className":4742,"style":1646},[691],[109,4744,4554],{"className":4745},[1682],[109,4747],{"className":4748,"style":1646},[691],[109,4750,4752,4755],{"className":4751},[150],[109,4753],{"className":4754,"style":4642},[154],[109,4756,662],{"className":4757},[159],", the row reads ",[109,4760,4762,4794],{"className":4761},[112],[109,4763,4765],{"className":4764},[116],[118,4766,4767],{"xmlns":120},[122,4768,4769,4791],{},[125,4770,4771,4777,4779,4781,4787,4789],{},[128,4772,4773,4775],{},[131,4774,578],{},[131,4776,377],{},[652,4778,1174],{},[648,4780,662],{},[128,4782,4783,4785],{},[131,4784,374],{},[131,4786,377],{},[652,4788,3764],{},[648,4790,4557],{},[138,4792,4793],{"encoding":140},"n_o - 2x_o \\ge 4",[109,4795,4797,4853,4912],{"className":4796,"ariaHidden":146},[145],[109,4798,4800,4804,4844,4847,4850],{"className":4799},[150],[109,4801],{"className":4802,"style":4803},[154],"height:0.7333em;vertical-align:-0.15em;",[109,4805,4807,4810],{"className":4806},[159],[109,4808,578],{"className":4809},[159,163],[109,4811,4813],{"className":4812},[168],[109,4814,4816,4836],{"className":4815},[172,173],[109,4817,4819,4833],{"className":4818},[177],[109,4820,4822],{"className":4821,"style":182},[181],[109,4823,4824,4827],{"style":410},[109,4825],{"className":4826,"style":190},[189],[109,4828,4830],{"className":4829},[194,195,196,197],[109,4831,377],{"className":4832},[159,163,197],[109,4834,206],{"className":4835},[205],[109,4837,4839],{"className":4838},[177],[109,4840,4842],{"className":4841,"style":213},[181],[109,4843],{},[109,4845],{"className":4846,"style":1931},[691],[109,4848,1174],{"className":4849},[1935],[109,4851],{"className":4852,"style":1931},[691],[109,4854,4856,4860,4863,4903,4906,4909],{"className":4855},[150],[109,4857],{"className":4858,"style":4859},[154],"height:0.7944em;vertical-align:-0.15em;",[109,4861,662],{"className":4862},[159],[109,4864,4866,4869],{"className":4865},[159],[109,4867,374],{"className":4868},[159,163],[109,4870,4872],{"className":4871},[168],[109,4873,4875,4895],{"className":4874},[172,173],[109,4876,4878,4892],{"className":4877},[177],[109,4879,4881],{"className":4880,"style":182},[181],[109,4882,4883,4886],{"style":410},[109,4884],{"className":4885,"style":190},[189],[109,4887,4889],{"className":4888},[194,195,196,197],[109,4890,377],{"className":4891},[159,163,197],[109,4893,206],{"className":4894},[205],[109,4896,4898],{"className":4897},[177],[109,4899,4901],{"className":4900,"style":213},[181],[109,4902],{},[109,4904],{"className":4905,"style":1646},[691],[109,4907,3764],{"className":4908},[1682],[109,4910],{"className":4911,"style":1646},[691],[109,4913,4915,4918],{"className":4914},[150],[109,4916],{"className":4917,"style":4642},[154],[109,4919,4557],{"className":4920},[159],", so 6 packs are needed. A second offer used would lose its ",[109,4923,4925,4944],{"className":4924},[112],[109,4926,4928],{"className":4927},[116],[118,4929,4930],{"xmlns":120},[122,4931,4932,4941],{},[125,4933,4934],{},[4935,4936,4937,4939],"msup",{},[131,4938,3732],{},[275,4940,3753],{},[138,4942,4943],{"encoding":140},"Q^{\\text{lost}}",[109,4945,4947],{"className":4946,"ariaHidden":146},[145],[109,4948,4950,4954],{"className":4949},[150],[109,4951],{"className":4952,"style":4953},[154],"height:1.0435em;vertical-align:-0.1944em;",[109,4955,4957,4960],{"className":4956},[159],[109,4958,3732],{"className":4959},[159,163],[109,4961,4963],{"className":4962},[168],[109,4964,4966],{"className":4965},[172],[109,4967,4969],{"className":4968},[177],[109,4970,4972],{"className":4971,"style":310},[181],[109,4973,4974,4977],{"style":325},[109,4975],{"className":4976,"style":190},[189],[109,4978,4980],{"className":4979},[194,195,196,197],[109,4981,4983],{"className":4982},[159,197],[109,4984,4986],{"className":4985},[159,338,197],[109,4987,3753],{"className":4988},[159,197]," as well. For a variant not chosen, the right-hand side is 0, so buying nothing is allowed.",[85,4991,4992,4995,5195,5250,5253],{},[106,4993,4994],{},"(3)",[106,4996,4997],{},[109,4998,5000,5038],{"className":4999},[112],[109,5001,5003],{"className":5002},[116],[118,5004,5005],{"xmlns":120},[122,5006,5007,5035],{},[125,5008,5009,5025,5031,5033],{},[128,5010,5011,5013],{},[652,5012,1341],{},[125,5014,5015,5017,5019],{},[131,5016,136],{},[652,5018,1349],{},[128,5020,5021,5023],{},[131,5022,1368],{"mathvariant":1352},[131,5024,1346],{},[128,5026,5027,5029],{},[131,5028,133],{},[131,5030,136],{},[652,5032,4554],{},[648,5034,657],{},[138,5036,5037],{"encoding":140},"\\sum_{v \\in \\mathcal{V}_i} z_v = 1",[109,5039,5041,5186],{"className":5040,"ariaHidden":146},[145],[109,5042,5044,5048,5134,5137,5177,5180,5183],{"className":5043},[150],[109,5045],{"className":5046,"style":5047},[154],"height:1.1498em;vertical-align:-0.3998em;",[109,5049,5051,5054],{"className":5050},[1535],[109,5052,1341],{"className":5053,"style":3802},[1535,1699,3801],[109,5055,5057],{"className":5056},[168],[109,5058,5060,5126],{"className":5059},[172,173],[109,5061,5063,5123],{"className":5062},[177],[109,5064,5066],{"className":5065,"style":3815},[181],[109,5067,5068,5071],{"style":3818},[109,5069],{"className":5070,"style":190},[189],[109,5072,5074],{"className":5073},[194,195,196,197],[109,5075,5077,5080,5083],{"className":5076},[159,197],[109,5078,136],{"className":5079,"style":201},[159,163,197],[109,5081,1349],{"className":5082},[1682,197],[109,5084,5086,5089],{"className":5085},[159,197],[109,5087,1368],{"className":5088,"style":1751},[159,1686,197],[109,5090,5092],{"className":5091},[168],[109,5093,5095,5115],{"className":5094},[172,173],[109,5096,5098,5112],{"className":5097},[177],[109,5099,5101],{"className":5100,"style":1764},[181],[109,5102,5103,5106],{"style":1767},[109,5104],{"className":5105,"style":1771},[189],[109,5107,5109],{"className":5108},[194,1775,1776,197],[109,5110,1346],{"className":5111},[159,163,197],[109,5113,206],{"className":5114},[205],[109,5116,5118],{"className":5117},[177],[109,5119,5121],{"className":5120,"style":1789},[181],[109,5122],{},[109,5124,206],{"className":5125},[205],[109,5127,5129],{"className":5128},[177],[109,5130,5132],{"className":5131,"style":3883},[181],[109,5133],{},[109,5135],{"className":5136,"style":692},[691],[109,5138,5140,5143],{"className":5139},[159],[109,5141,133],{"className":5142,"style":164},[159,163],[109,5144,5146],{"className":5145},[168],[109,5147,5149,5169],{"className":5148},[172,173],[109,5150,5152,5166],{"className":5151},[177],[109,5153,5155],{"className":5154,"style":182},[181],[109,5156,5157,5160],{"style":185},[109,5158],{"className":5159,"style":190},[189],[109,5161,5163],{"className":5162},[194,195,196,197],[109,5164,136],{"className":5165,"style":201},[159,163,197],[109,5167,206],{"className":5168},[205],[109,5170,5172],{"className":5171},[177],[109,5173,5175],{"className":5174,"style":213},[181],[109,5176],{},[109,5178],{"className":5179,"style":1646},[691],[109,5181,4554],{"className":5182},[1682],[109,5184],{"className":5185,"style":1646},[691],[109,5187,5189,5192],{"className":5188},[150],[109,5190],{"className":5191,"style":4642},[154],[109,5193,657],{"className":5194},[159],[106,5196,5197,5198],{},"item ",[109,5199,5201,5219],{"className":5200},[112],[109,5202,5204],{"className":5203},[116],[118,5205,5206],{"xmlns":120},[122,5207,5208,5216],{},[125,5209,5210,5212,5214],{},[131,5211,1346],{},[652,5213,1349],{},[131,5215,1353],{"mathvariant":1352},[138,5217,5218],{"encoding":140},"i \\in \\mathcal{I}",[109,5220,5222,5241],{"className":5221,"ariaHidden":146},[145],[109,5223,5225,5229,5232,5235,5238],{"className":5224},[150],[109,5226],{"className":5227,"style":5228},[154],"height:0.6986em;vertical-align:-0.0391em;",[109,5230,1346],{"className":5231},[159,163],[109,5233],{"className":5234,"style":1646},[691],[109,5236,1349],{"className":5237},[1682],[109,5239],{"className":5240,"style":1646},[691],[109,5242,5244,5247],{"className":5243},[150],[109,5245],{"className":5246,"style":3681},[154],[109,5248,1353],{"className":5249,"style":1687},[159,1686],[106,5251,5252],{},"exactly-one row (assignment, set partitioning)",[106,5254,5255],{},"Each item has exactly one variant chosen.",[85,5257,5258,5261,5461,5517,5520],{},[106,5259,5260],{},"(4)",[106,5262,5263],{},[109,5264,5266,5305],{"className":5265},[112],[109,5267,5269],{"className":5268},[116],[118,5270,5271],{"xmlns":120},[122,5272,5273,5302],{},[125,5274,5275,5292,5298,5300],{},[128,5276,5277,5279],{},[652,5278,1341],{},[125,5280,5281,5283,5285],{},[131,5282,377],{},[652,5284,1349],{},[128,5286,5287,5289],{},[131,5288,1413],{"mathvariant":1352},[131,5290,5291],{},"e",[128,5293,5294,5296],{},[131,5295,374],{},[131,5297,377],{},[652,5299,3302],{},[648,5301,657],{},[138,5303,5304],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_e} x_o \\le 1",[109,5306,5308,5452],{"className":5307,"ariaHidden":146},[145],[109,5309,5311,5314,5400,5403,5443,5446,5449],{"className":5310},[150],[109,5312],{"className":5313,"style":5047},[154],[109,5315,5317,5320],{"className":5316},[1535],[109,5318,1341],{"className":5319,"style":3802},[1535,1699,3801],[109,5321,5323],{"className":5322},[168],[109,5324,5326,5392],{"className":5325},[172,173],[109,5327,5329,5389],{"className":5328},[177],[109,5330,5332],{"className":5331,"style":3815},[181],[109,5333,5334,5337],{"style":3818},[109,5335],{"className":5336,"style":190},[189],[109,5338,5340],{"className":5339},[194,195,196,197],[109,5341,5343,5346,5349],{"className":5342},[159,197],[109,5344,377],{"className":5345},[159,163,197],[109,5347,1349],{"className":5348},[1682,197],[109,5350,5352,5355],{"className":5351},[159,197],[109,5353,1413],{"className":5354,"style":1984},[159,1686,197],[109,5356,5358],{"className":5357},[168],[109,5359,5361,5381],{"className":5360},[172,173],[109,5362,5364,5378],{"className":5363},[177],[109,5365,5367],{"className":5366,"style":1997},[181],[109,5368,5369,5372],{"style":2000},[109,5370],{"className":5371,"style":1771},[189],[109,5373,5375],{"className":5374},[194,1775,1776,197],[109,5376,5291],{"className":5377},[159,163,197],[109,5379,206],{"className":5380},[205],[109,5382,5384],{"className":5383},[177],[109,5385,5387],{"className":5386,"style":1789},[181],[109,5388],{},[109,5390,206],{"className":5391},[205],[109,5393,5395],{"className":5394},[177],[109,5396,5398],{"className":5397,"style":3883},[181],[109,5399],{},[109,5401],{"className":5402,"style":692},[691],[109,5404,5406,5409],{"className":5405},[159],[109,5407,374],{"className":5408},[159,163],[109,5410,5412],{"className":5411},[168],[109,5413,5415,5435],{"className":5414},[172,173],[109,5416,5418,5432],{"className":5417},[177],[109,5419,5421],{"className":5420,"style":182},[181],[109,5422,5423,5426],{"style":410},[109,5424],{"className":5425,"style":190},[189],[109,5427,5429],{"className":5428},[194,195,196,197],[109,5430,377],{"className":5431},[159,163,197],[109,5433,206],{"className":5434},[205],[109,5436,5438],{"className":5437},[177],[109,5439,5441],{"className":5440,"style":213},[181],[109,5442],{},[109,5444],{"className":5445,"style":1646},[691],[109,5447,3302],{"className":5448},[1682],[109,5450],{"className":5451,"style":1646},[691],[109,5453,5455,5458],{"className":5454},[150],[109,5456],{"className":5457,"style":4642},[154],[109,5459,657],{"className":5460},[159],[106,5462,5463,5464],{},"exclusive set ",[109,5465,5467,5486],{"className":5466},[112],[109,5468,5470],{"className":5469},[116],[118,5471,5472],{"xmlns":120},[122,5473,5474,5483],{},[125,5475,5476,5478,5480],{},[131,5477,5291],{},[652,5479,1349],{},[131,5481,5482],{"mathvariant":1352},"E",[138,5484,5485],{"encoding":140},"e \\in \\mathcal{E}",[109,5487,5489,5507],{"className":5488,"ariaHidden":146},[145],[109,5490,5492,5495,5498,5501,5504],{"className":5491},[150],[109,5493],{"className":5494,"style":3662},[154],[109,5496,5291],{"className":5497},[159,163],[109,5499],{"className":5500,"style":1646},[691],[109,5502,1349],{"className":5503},[1682],[109,5505],{"className":5506,"style":1646},[691],[109,5508,5510,5513],{"className":5509},[150],[109,5511],{"className":5512,"style":3681},[154],[109,5514,5482],{"className":5515,"style":5516},[159,1686],"margin-right:0.0894em;",[106,5518,5519],{},"at-most-one row (set packing)",[106,5521,5522],{},"At most one offer of an exclusive set is used. For example, a product's prices for 1–9 packs and 10–99 packs cannot be mixed.",[85,5524,5525,5528,5860,5915,5918],{},[106,5526,5527],{},"(5)",[106,5529,5530],{},[109,5531,5533,5589],{"className":5532},[112],[109,5534,5536],{"className":5535},[116],[118,5537,5538],{"xmlns":120},[122,5539,5540,5586],{},[125,5541,5542,5559,5567,5569,5575,5577],{},[128,5543,5544,5546],{},[652,5545,1341],{},[125,5547,5548,5550,5552],{},[131,5549,377],{},[652,5551,1349],{},[128,5553,5554,5556],{},[131,5555,1413],{"mathvariant":1352},[131,5557,5558],{},"r",[267,5560,5561,5563,5565],{},[131,5562,3732],{},[131,5564,377],{},[275,5566,768],{},[275,5568,1385],{},[128,5570,5571,5573],{},[131,5572,578],{},[131,5574,377],{},[652,5576,3302],{},[267,5578,5579,5581,5583],{},[131,5580,3732],{},[131,5582,5558],{},[275,5584,5585],{},"stock",[138,5587,5588],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_r} Q^{\\text{pack}}_o\\, n_o \\le Q^{\\text{stock}}_r",[109,5590,5592,5797],{"className":5591,"ariaHidden":146},[145],[109,5593,5595,5599,5685,5688,5745,5748,5788,5791,5794],{"className":5594},[150],[109,5596],{"className":5597,"style":5598},[154],"height:1.2489em;vertical-align:-0.3998em;",[109,5600,5602,5605],{"className":5601},[1535],[109,5603,1341],{"className":5604,"style":3802},[1535,1699,3801],[109,5606,5608],{"className":5607},[168],[109,5609,5611,5677],{"className":5610},[172,173],[109,5612,5614,5674],{"className":5613},[177],[109,5615,5617],{"className":5616,"style":3815},[181],[109,5618,5619,5622],{"style":3818},[109,5620],{"className":5621,"style":190},[189],[109,5623,5625],{"className":5624},[194,195,196,197],[109,5626,5628,5631,5634],{"className":5627},[159,197],[109,5629,377],{"className":5630},[159,163,197],[109,5632,1349],{"className":5633},[1682,197],[109,5635,5637,5640],{"className":5636},[159,197],[109,5638,1413],{"className":5639,"style":1984},[159,1686,197],[109,5641,5643],{"className":5642},[168],[109,5644,5646,5666],{"className":5645},[172,173],[109,5647,5649,5663],{"className":5648},[177],[109,5650,5652],{"className":5651,"style":1997},[181],[109,5653,5654,5657],{"style":2000},[109,5655],{"className":5656,"style":1771},[189],[109,5658,5660],{"className":5659},[194,1775,1776,197],[109,5661,5558],{"className":5662,"style":1984},[159,163,197],[109,5664,206],{"className":5665},[205],[109,5667,5669],{"className":5668},[177],[109,5670,5672],{"className":5671,"style":1789},[181],[109,5673],{},[109,5675,206],{"className":5676},[205],[109,5678,5680],{"className":5679},[177],[109,5681,5683],{"className":5682,"style":3883},[181],[109,5684],{},[109,5686],{"className":5687,"style":692},[691],[109,5689,5691,5694],{"className":5690},[159],[109,5692,3732],{"className":5693},[159,163],[109,5695,5697],{"className":5696},[168],[109,5698,5700,5737],{"className":5699},[172,173],[109,5701,5703,5734],{"className":5702},[177],[109,5704,5706,5717],{"className":5705,"style":310},[181],[109,5707,5708,5711],{"style":3912},[109,5709],{"className":5710,"style":190},[189],[109,5712,5714],{"className":5713},[194,195,196,197],[109,5715,377],{"className":5716},[159,163,197],[109,5718,5719,5722],{"style":325},[109,5720],{"className":5721,"style":190},[189],[109,5723,5725],{"className":5724},[194,195,196,197],[109,5726,5728],{"className":5727},[159,197],[109,5729,5731],{"className":5730},[159,338,197],[109,5732,768],{"className":5733},[159,197],[109,5735,206],{"className":5736},[205],[109,5738,5740],{"className":5739},[177],[109,5741,5743],{"className":5742,"style":351},[181],[109,5744],{},[109,5746],{"className":5747,"style":692},[691],[109,5749,5751,5754],{"className":5750},[159],[109,5752,578],{"className":5753},[159,163],[109,5755,5757],{"className":5756},[168],[109,5758,5760,5780],{"className":5759},[172,173],[109,5761,5763,5777],{"className":5762},[177],[109,5764,5766],{"className":5765,"style":182},[181],[109,5767,5768,5771],{"style":410},[109,5769],{"className":5770,"style":190},[189],[109,5772,5774],{"className":5773},[194,195,196,197],[109,5775,377],{"className":5776},[159,163,197],[109,5778,206],{"className":5779},[205],[109,5781,5783],{"className":5782},[177],[109,5784,5786],{"className":5785,"style":213},[181],[109,5787],{},[109,5789],{"className":5790,"style":1646},[691],[109,5792,3302],{"className":5793},[1682],[109,5795],{"className":5796,"style":1646},[691],[109,5798,5800,5803],{"className":5799},[150],[109,5801],{"className":5802,"style":290},[154],[109,5804,5806,5809],{"className":5805},[159],[109,5807,3732],{"className":5808},[159,163],[109,5810,5812],{"className":5811},[168],[109,5813,5815,5852],{"className":5814},[172,173],[109,5816,5818,5849],{"className":5817},[177],[109,5819,5821,5832],{"className":5820,"style":310},[181],[109,5822,5823,5826],{"style":3912},[109,5824],{"className":5825,"style":190},[189],[109,5827,5829],{"className":5828},[194,195,196,197],[109,5830,5558],{"className":5831,"style":1984},[159,163,197],[109,5833,5834,5837],{"style":325},[109,5835],{"className":5836,"style":190},[189],[109,5838,5840],{"className":5839},[194,195,196,197],[109,5841,5843],{"className":5842},[159,197],[109,5844,5846],{"className":5845},[159,338,197],[109,5847,5585],{"className":5848},[159,197],[109,5850,206],{"className":5851},[205],[109,5853,5855],{"className":5854},[177],[109,5856,5858],{"className":5857,"style":351},[181],[109,5859],{},[106,5861,5862,5863],{},"stock set ",[109,5864,5866,5885],{"className":5865},[112],[109,5867,5869],{"className":5868},[116],[118,5870,5871],{"xmlns":120},[122,5872,5873,5882],{},[125,5874,5875,5877,5879],{},[131,5876,5558],{},[652,5878,1349],{},[131,5880,5881],{"mathvariant":1352},"R",[138,5883,5884],{"encoding":140},"r \\in \\mathcal{R}",[109,5886,5888,5906],{"className":5887,"ariaHidden":146},[145],[109,5889,5891,5894,5897,5900,5903],{"className":5890},[150],[109,5892],{"className":5893,"style":3662},[154],[109,5895,5558],{"className":5896,"style":1984},[159,163],[109,5898],{"className":5899,"style":1646},[691],[109,5901,1349],{"className":5902},[1682],[109,5904],{"className":5905,"style":1646},[691],[109,5907,5909,5912],{"className":5908},[150],[109,5910],{"className":5911,"style":3681},[154],[109,5913,5881],{"className":5914},[159,1686],[106,5916,5917],{},"capacity row (integer knapsack)",[106,5919,5920],{},"Do not order more units than the supplier has in stock.",[11,5922,5923],{},"For the supplier variables, we have two more:",[79,5925,5926,5940],{},[82,5927,5928],{},[85,5929,5930,5932,5934,5936,5938],{},[88,5931,3248],{},[88,5933],{},[88,5935,3253],{},[88,5937,3256],{},[88,5939,3259],{},[101,5941,5942,6790],{},[85,5943,5944,5947,6257,6311,6403],{},[106,5945,5946],{},"(6)",[106,5948,5949],{},[109,5950,5952,6008],{"className":5951},[112],[109,5953,5955],{"className":5954},[116],[118,5956,5957],{"xmlns":120},[122,5958,5959,6005],{},[125,5960,5961,5977,5983,5985,5989,5995,5997,5999],{},[128,5962,5963,5965],{},[652,5964,1341],{},[125,5966,5967,5969,5971],{},[131,5968,377],{},[652,5970,1349],{},[128,5972,5973,5975],{},[131,5974,1413],{"mathvariant":1352},[131,5976,862],{},[128,5978,5979,5981],{},[131,5980,374],{},[131,5982,377],{},[652,5984,3302],{},[652,5986,5988],{"stretchy":5987},"false","∣",[128,5990,5991,5993],{},[131,5992,1413],{"mathvariant":1352},[131,5994,862],{},[652,5996,5988],{"stretchy":5987},[275,5998,1385],{},[128,6000,6001,6003],{},[131,6002,859],{},[131,6004,862],{},[138,6006,6007],{"encoding":140},"\\sum_{o \\in \\mathcal{O}_k} x_o \\le \\lvert \\mathcal{O}_k \\rvert\\, u_k",[109,6009,6011,6160],{"className":6010,"ariaHidden":146},[145],[109,6012,6014,6018,6108,6111,6151,6154,6157],{"className":6013},[150],[109,6015],{"className":6016,"style":6017},[154],"height:1.1556em;vertical-align:-0.4056em;",[109,6019,6021,6024],{"className":6020},[1535],[109,6022,1341],{"className":6023,"style":3802},[1535,1699,3801],[109,6025,6027],{"className":6026},[168],[109,6028,6030,6099],{"className":6029},[172,173],[109,6031,6033,6096],{"className":6032},[177],[109,6034,6036],{"className":6035,"style":3815},[181],[109,6037,6038,6041],{"style":3818},[109,6039],{"className":6040,"style":190},[189],[109,6042,6044],{"className":6043},[194,195,196,197],[109,6045,6047,6050,6053],{"className":6046},[159,197],[109,6048,377],{"className":6049},[159,163,197],[109,6051,1349],{"className":6052},[1682,197],[109,6054,6056,6059],{"className":6055},[159,197],[109,6057,1413],{"className":6058,"style":1984},[159,1686,197],[109,6060,6062],{"className":6061},[168],[109,6063,6065,6087],{"className":6064},[172,173],[109,6066,6068,6084],{"className":6067},[177],[109,6069,6072],{"className":6070,"style":6071},[181],"height:0.3448em;",[109,6073,6075,6078],{"style":6074},"top:-2.3488em;margin-left:-0.0278em;margin-right:0.0714em;",[109,6076],{"className":6077,"style":1771},[189],[109,6079,6081],{"className":6080},[194,1775,1776,197],[109,6082,862],{"className":6083,"style":905},[159,163,197],[109,6085,206],{"className":6086},[205],[109,6088,6090],{"className":6089},[177],[109,6091,6094],{"className":6092,"style":6093},[181],"height:0.1512em;",[109,6095],{},[109,6097,206],{"className":6098},[205],[109,6100,6102],{"className":6101},[177],[109,6103,6106],{"className":6104,"style":6105},[181],"height:0.4056em;",[109,6107],{},[109,6109],{"className":6110,"style":692},[691],[109,6112,6114,6117],{"className":6113},[159],[109,6115,374],{"className":6116},[159,163],[109,6118,6120],{"className":6119},[168],[109,6121,6123,6143],{"className":6122},[172,173],[109,6124,6126,6140],{"className":6125},[177],[109,6127,6129],{"className":6128,"style":182},[181],[109,6130,6131,6134],{"style":410},[109,6132],{"className":6133,"style":190},[189],[109,6135,6137],{"className":6136},[194,195,196,197],[109,6138,377],{"className":6139},[159,163,197],[109,6141,206],{"className":6142},[205],[109,6144,6146],{"className":6145},[177],[109,6147,6149],{"className":6148,"style":213},[181],[109,6150],{},[109,6152],{"className":6153,"style":1646},[691],[109,6155,3302],{"className":6156},[1682],[109,6158],{"className":6159,"style":1646},[691],[109,6161,6163,6167,6170,6211,6214,6217],{"className":6162},[150],[109,6164],{"className":6165,"style":6166},[154],"height:1em;vertical-align:-0.25em;",[109,6168,5988],{"className":6169},[1817],[109,6171,6173,6176],{"className":6172},[159],[109,6174,1413],{"className":6175,"style":1984},[159,1686],[109,6177,6179],{"className":6178},[168],[109,6180,6182,6203],{"className":6181},[172,173],[109,6183,6185,6200],{"className":6184},[177],[109,6186,6188],{"className":6187,"style":893},[181],[109,6189,6191,6194],{"style":6190},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[109,6192],{"className":6193,"style":190},[189],[109,6195,6197],{"className":6196},[194,195,196,197],[109,6198,862],{"className":6199,"style":905},[159,163,197],[109,6201,206],{"className":6202},[205],[109,6204,6206],{"className":6205},[177],[109,6207,6209],{"className":6208,"style":213},[181],[109,6210],{},[109,6212,5988],{"className":6213},[2268],[109,6215],{"className":6216,"style":692},[691],[109,6218,6220,6223],{"className":6219},[159],[109,6221,859],{"className":6222},[159,163],[109,6224,6226],{"className":6225},[168],[109,6227,6229,6249],{"className":6228},[172,173],[109,6230,6232,6246],{"className":6231},[177],[109,6233,6235],{"className":6234,"style":893},[181],[109,6236,6237,6240],{"style":410},[109,6238],{"className":6239,"style":190},[189],[109,6241,6243],{"className":6242},[194,195,196,197],[109,6244,862],{"className":6245,"style":905},[159,163,197],[109,6247,206],{"className":6248},[205],[109,6250,6252],{"className":6251},[177],[109,6253,6255],{"className":6254,"style":213},[181],[109,6256],{},[106,6258,921,6259],{},[109,6260,6262,6280],{"className":6261},[112],[109,6263,6265],{"className":6264},[116],[118,6266,6267],{"xmlns":120},[122,6268,6269,6277],{},[125,6270,6271,6273,6275],{},[131,6272,862],{},[652,6274,1349],{},[131,6276,1476],{"mathvariant":1352},[138,6278,6279],{"encoding":140},"k \\in \\mathcal{K}",[109,6281,6283,6302],{"className":6282,"ariaHidden":146},[145],[109,6284,6286,6290,6293,6296,6299],{"className":6285},[150],[109,6287],{"className":6288,"style":6289},[154],"height:0.7335em;vertical-align:-0.0391em;",[109,6291,862],{"className":6292,"style":905},[159,163],[109,6294],{"className":6295,"style":1646},[691],[109,6297,1349],{"className":6298},[1682],[109,6300],{"className":6301,"style":1646},[691],[109,6303,6305,6308],{"className":6304},[150],[109,6306],{"className":6307,"style":3681},[154],[109,6309,1476],{"className":6310,"style":2333},[159,1686],[106,6312,6313,6314,6402],{},"linking row for the fixed charge ",[109,6315,6317,6336],{"className":6316},[112],[109,6318,6320],{"className":6319},[116],[118,6321,6322],{"xmlns":120},[122,6323,6324,6334],{},[125,6325,6326],{},[267,6327,6328,6330,6332],{},[131,6329,271],{},[131,6331,862],{},[275,6333,974],{},[138,6335,977],{"encoding":140},[109,6337,6339],{"className":6338,"ariaHidden":146},[145],[109,6340,6342,6345],{"className":6341},[150],[109,6343],{"className":6344,"style":987},[154],[109,6346,6348,6351],{"className":6347},[159],[109,6349,271],{"className":6350,"style":297},[159,163],[109,6352,6354],{"className":6353},[168],[109,6355,6357,6394],{"className":6356},[172,173],[109,6358,6360,6391],{"className":6359},[177],[109,6361,6363,6374],{"className":6362,"style":310},[181],[109,6364,6365,6368],{"style":1008},[109,6366],{"className":6367,"style":190},[189],[109,6369,6371],{"className":6370},[194,195,196,197],[109,6372,862],{"className":6373,"style":905},[159,163,197],[109,6375,6376,6379],{"style":325},[109,6377],{"className":6378,"style":190},[189],[109,6380,6382],{"className":6381},[194,195,196,197],[109,6383,6385],{"className":6384},[159,197],[109,6386,6388],{"className":6387},[159,338,197],[109,6389,974],{"className":6390},[159,197],[109,6392,206],{"className":6393},[205],[109,6395,6397],{"className":6396},[177],[109,6398,6400],{"className":6399,"style":1044},[181],[109,6401],{},", in summed big-M form",[106,6404,6405,6406,6434,6435,6504,6505,6593,6594,6697,6698,1518],{},"If any offer of supplier ",[109,6407,6409,6422],{"className":6408},[112],[109,6410,6412],{"className":6411},[116],[118,6413,6414],{"xmlns":120},[122,6415,6416,6420],{},[125,6417,6418],{},[131,6419,862],{},[138,6421,862],{"encoding":140},[109,6423,6425],{"className":6424,"ariaHidden":146},[145],[109,6426,6428,6431],{"className":6427},[150],[109,6429],{"className":6430,"style":947},[154],[109,6432,862],{"className":6433,"style":905},[159,163]," is used, 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must be 1, and the charge 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is paid. Here ",[109,6595,6597,6624],{"className":6596},[112],[109,6598,6600],{"className":6599},[116],[118,6601,6602],{"xmlns":120},[122,6603,6604,6621],{},[125,6605,6606,6609,6611,6613,6619],{},[131,6607,6608],{},"M",[652,6610,4554],{},[652,6612,5988],{"stretchy":5987},[128,6614,6615,6617],{},[131,6616,1413],{"mathvariant":1352},[131,6618,862],{},[652,6620,5988],{"stretchy":5987},[138,6622,6623],{"encoding":140},"M = \\lvert \\mathcal{O}_k \\rvert",[109,6625,6627,6645],{"className":6626,"ariaHidden":146},[145],[109,6628,6630,6633,6636,6639,6642],{"className":6629},[150],[109,6631],{"className":6632,"style":3681},[154],[109,6634,6608],{"className":6635,"style":3349},[159,163],[109,6637],{"className":6638,"style":1646},[691],[109,6640,4554],{"className":6641},[1682],[109,6643],{"className":6644,"style":1646},[691],[109,6646,6648,6651,6654,6694],{"className":6647},[150],[109,6649],{"className":6650,"style":6166},[154],[109,6652,5988],{"className":6653},[1817],[109,6655,6657,6660],{"className":6656},[159],[109,6658,1413],{"className":6659,"style":1984},[159,1686],[109,6661,6663],{"className":6662},[168],[109,6664,6666,6686],{"className":6665},[172,173],[109,6667,6669,6683],{"className":6668},[177],[109,6670,6672],{"className":6671,"style":893},[181],[109,6673,6674,6677],{"style":6190},[109,6675],{"className":6676,"style":190},[189],[109,6678,6680],{"className":6679},[194,195,196,197],[109,6681,862],{"className":6682,"style":905},[159,163,197],[109,6684,206],{"className":6685},[205],[109,6687,6689],{"className":6688},[177],[109,6690,6692],{"className":6691,"style":213},[181],[109,6693],{},[109,6695,5988],{"className":6696},[2268],", the number of its offers, the most the sum can be, so the row is harmless when ",[109,6699,6701,6723],{"className":6700},[112],[109,6702,6704],{"className":6703},[116],[118,6705,6706],{"xmlns":120},[122,6707,6708,6720],{},[125,6709,6710,6716,6718],{},[128,6711,6712,6714],{},[131,6713,859],{},[131,6715,862],{},[652,6717,4554],{},[648,6719,657],{},[138,6721,6722],{"encoding":140},"u_k = 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1",[109,7398,7400,7455],{"className":7399,"ariaHidden":146},[145],[109,7401,7403,7406,7446,7449,7452],{"className":7402},[150],[109,7404],{"className":7405,"style":1082},[154],[109,7407,7409,7412],{"className":7408},[159],[109,7410,1067],{"className":7411,"style":1089},[159,163],[109,7413,7415],{"className":7414},[168],[109,7416,7418,7438],{"className":7417},[172,173],[109,7419,7421,7435],{"className":7420},[177],[109,7422,7424],{"className":7423,"style":893},[181],[109,7425,7426,7429],{"style":1104},[109,7427],{"className":7428,"style":190},[189],[109,7430,7432],{"className":7431},[194,195,196,197],[109,7433,862],{"className":7434,"style":905},[159,163,197],[109,7436,206],{"className":7437},[205],[109,7439,7441],{"className":7440},[177],[109,7442,7444],{"className":7443,"style":213},[181],[109,7445],{},[109,7447],{"className":7448,"style":1646},[691],[109,7450,4554],{"className":7451},[1682],[109,7453],{"className":7454,"style":1646},[691],[109,7456,7458,7461],{"className":7457},[150],[109,7459],{"className":7460,"style":4642},[154],[109,7462,657],{"className":7463},[159],", the spend is at least ",[109,7466,7468,7488],{"className":7467},[112],[109,7469,7471],{"className":7470},[116],[118,7472,7473],{"xmlns":120},[122,7474,7475,7485],{},[125,7476,7477],{},[267,7478,7479,7481,7483],{},[131,7480,6831],{},[131,7482,862],{},[275,7484,6870],{},[138,7486,7487],{"encoding":140},"S^{\\text{free}}_k",[109,7489,7491],{"className":7490,"ariaHidden":146},[145],[109,7492,7494,7497],{"className":7493},[150],[109,7495],{"className":7496,"style":987},[154],[109,7498,7500,7503],{"className":7499},[159],[109,7501,6831],{"className":7502,"style":6990},[159,163],[109,7504,7506],{"className":7505},[168],[109,7507,7509,7546],{"className":7508},[172,173],[109,7510,7512,7543],{"className":7511},[177],[109,7513,7515,7526],{"className":7514,"style":310},[181],[109,7516,7517,7520],{"style":7236},[109,7518],{"className":7519,"style":190},[189],[109,7521,7523],{"className":7522},[194,195,196,197],[109,7524,862],{"className":7525,"style":905},[159,163,197],[109,7527,7528,7531],{"style":325},[109,7529],{"className":7530,"style":190},[189],[109,7532,7534],{"className":7533},[194,195,196,197],[109,7535,7537],{"className":7536},[159,197],[109,7538,7540],{"className":7539},[159,338,197],[109,7541,6870],{"className":7542},[159,197],[109,7544,206],{"className":7545},[205],[109,7547,7549],{"className":7548},[177],[109,7550,7552],{"className":7551,"style":1044},[181],[109,7553],{},"\"",[106,7556,7557,7558,7649,7650,7678,7679,7767,7768,7860],{},"Shipping can be free, ",[109,7559,7561,7582],{"className":7560},[112],[109,7562,7564],{"className":7563},[116],[118,7565,7566],{"xmlns":120},[122,7567,7568,7580],{},[125,7569,7570,7576,7578],{},[128,7571,7572,7574],{},[131,7573,1067],{},[131,7575,862],{},[652,7577,4554],{},[648,7579,657],{},[138,7581,7396],{"encoding":140},[109,7583,7585,7640],{"className":7584,"ariaHidden":146},[145],[109,7586,7588,7591,7631,7634,7637],{"className":7587},[150],[109,7589],{"className":7590,"style":1082},[154],[109,7592,7594,7597],{"className":7593},[159],[109,7595,1067],{"className":7596,"style":1089},[159,163],[109,7598,7600],{"className":7599},[168],[109,7601,7603,7623],{"className":7602},[172,173],[109,7604,7606,7620],{"className":7605},[177],[109,7607,7609],{"className":7608,"style":893},[181],[109,7610,7611,7614],{"style":1104},[109,7612],{"className":7613,"style":190},[189],[109,7615,7617],{"className":7616},[194,195,196,197],[109,7618,862],{"className":7619,"style":905},[159,163,197],[109,7621,206],{"className":7622},[205],[109,7624,7626],{"className":7625},[177],[109,7627,7629],{"className":7628,"style":213},[181],[109,7630],{},[109,7632],{"className":7633,"style":1646},[691],[109,7635,4554],{"className":7636},[1682],[109,7638],{"className":7639,"style":1646},[691],[109,7641,7643,7646],{"className":7642},[150],[109,7644],{"className":7645,"style":4642},[154],[109,7647,657],{"className":7648},[159],", only if the spend with supplier ",[109,7651,7653,7666],{"className":7652},[112],[109,7654,7656],{"className":7655},[116],[118,7657,7658],{"xmlns":120},[122,7659,7660,7664],{},[125,7661,7662],{},[131,7663,862],{},[138,7665,862],{"encoding":140},[109,7667,7669],{"className":7668,"ariaHidden":146},[145],[109,7670,7672,7675],{"className":7671},[150],[109,7673],{"className":7674,"style":947},[154],[109,7676,862],{"className":7677,"style":905},[159,163]," reaches ",[109,7680,7682,7701],{"className":7681},[112],[109,7683,7685],{"className":7684},[116],[118,7686,7687],{"xmlns":120},[122,7688,7689,7699],{},[125,7690,7691],{},[267,7692,7693,7695,7697],{},[131,7694,6831],{},[131,7696,862],{},[275,7698,6870],{},[138,7700,7487],{"encoding":140},[109,7702,7704],{"className":7703,"ariaHidden":146},[145],[109,7705,7707,7710],{"className":7706},[150],[109,7708],{"className":7709,"style":987},[154],[109,7711,7713,7716],{"className":7712},[159],[109,7714,6831],{"className":7715,"style":6990},[159,163],[109,7717,7719],{"className":7718},[168],[109,7720,7722,7759],{"className":7721},[172,173],[109,7723,7725,7756],{"className":7724},[177],[109,7726,7728,7739],{"className":7727,"style":310},[181],[109,7729,7730,7733],{"style":7236},[109,7731],{"className":7732,"style":190},[189],[109,7734,7736],{"className":7735},[194,195,196,197],[109,7737,862],{"className":7738,"style":905},[159,163,197],[109,7740,7741,7744],{"style":325},[109,7742],{"className":7743,"style":190},[189],[109,7745,7747],{"className":7746},[194,195,196,197],[109,7748,7750],{"className":7749},[159,197],[109,7751,7753],{"className":7752},[159,338,197],[109,7754,6870],{"className":7755},[159,197],[109,7757,206],{"className":7758},[205],[109,7760,7762],{"className":7761},[177],[109,7763,7765],{"className":7764,"style":1044},[181],[109,7766],{},". With ",[109,7769,7771,7793],{"className":7770},[112],[109,7772,7774],{"className":7773},[116],[118,7775,7776],{"xmlns":120},[122,7777,7778,7790],{},[125,7779,7780,7786,7788],{},[128,7781,7782,7784],{},[131,7783,1067],{},[131,7785,862],{},[652,7787,4554],{},[648,7789,650],{},[138,7791,7792],{"encoding":140},"f_k = 0",[109,7794,7796,7851],{"className":7795,"ariaHidden":146},[145],[109,7797,7799,7802,7842,7845,7848],{"className":7798},[150],[109,7800],{"className":7801,"style":1082},[154],[109,7803,7805,7808],{"className":7804},[159],[109,7806,1067],{"className":7807,"style":1089},[159,163],[109,7809,7811],{"className":7810},[168],[109,7812,7814,7834],{"className":7813},[172,173],[109,7815,7817,7831],{"className":7816},[177],[109,7818,7820],{"className":7819,"style":893},[181],[109,7821,7822,7825],{"style":1104},[109,7823],{"className":7824,"style":190},[189],[109,7826,7828],{"className":7827},[194,195,196,197],[109,7829,862],{"className":7830,"style":905},[159,163,197],[109,7832,206],{"className":7833},[205],[109,7835,7837],{"className":7836},[177],[109,7838,7840],{"className":7839,"style":213},[181],[109,7841],{},[109,7843],{"className":7844,"style":1646},[691],[109,7846,4554],{"className":7847},[1682],[109,7849],{"className":7850,"style":1646},[691],[109,7852,7854,7857],{"className":7853},[150],[109,7855],{"className":7856,"style":4642},[154],[109,7858,650],{"className":7859},[159]," the row holds trivially.",[66,7862,7864],{"id":7863},"cp-sat","CP-SAT",[11,7866,7867,7868,7871,7872,7874],{},"Here you can load the MILP from its zipped ",[61,7869,7870],{},".mps"," file and solve it with CP-SAT. Note that CP-SAT's performance can be quite sensitive to the number of ",[61,7873,63],{}," as well as the time limit.",[7876,7877,7882],"pre",{"className":7878,"code":7879,"language":7880,"meta":7881,"style":7881},"language-python shiki shiki-themes github-dark","from ortools.linear_solver.python import model_builder as mb\nimport gzip\n\n# define the model and load the instance data (21538 variables, 35578 constraint rows)\nmodel = mb.Model()\nmodel.import_from_mps_string(gzip.open(f\"{DATA}\u002F{INSTANCE}.mps.gz\", \"rt\").read())\n\nsolver = mb.Solver(\"CP-SAT\")\nsolver.set_time_limit_in_seconds(60)\nWORKERS = 8 # number of CP-SAT \"workers\" (roughly equal to number of CPUs)\nsolver.set_solver_specific_parameters(f\"num_workers:{WORKERS}\")\nstatus = solver.solve(model)\n\nprint(f\"status    {status.name}\")\nprint(f\"cost      {solver.objective_value:,.0f}\")\nprint(f\"time      {solver.wall_time:.1f} s on {WORKERS} workers\")\n","python","",[61,7883,7884,7908,7916,7923,7930,7941,7973,7978,7995,8006,8020,8038,8049,8054,8080,8106],{"__ignoreMap":7881},[109,7885,7888,7892,7896,7899,7902,7905],{"class":7886,"line":7887},"line",1,[109,7889,7891],{"class":7890},"snl16","from",[109,7893,7895],{"class":7894},"s95oV"," ortools.linear_solver.python ",[109,7897,7898],{"class":7890},"import",[109,7900,7901],{"class":7894}," model_builder ",[109,7903,7904],{"class":7890},"as",[109,7906,7907],{"class":7894}," mb\n",[109,7909,7911,7913],{"class":7886,"line":7910},2,[109,7912,7898],{"class":7890},[109,7914,7915],{"class":7894}," gzip\n",[109,7917,7919],{"class":7886,"line":7918},3,[109,7920,7922],{"emptyLinePlaceholder":7921},true,"\n",[109,7924,7926],{"class":7886,"line":7925},4,[109,7927,7929],{"class":7928},"sAwPA","# define the model and load the instance data (21538 variables, 35578 constraint rows)\n",[109,7931,7933,7936,7938],{"class":7886,"line":7932},5,[109,7934,7935],{"class":7894},"model ",[109,7937,4554],{"class":7890},[109,7939,7940],{"class":7894}," mb.Model()\n",[109,7942,7944,7947,7949,7952,7956,7959,7962,7965,7967,7970],{"class":7886,"line":7943},6,[109,7945,7946],{"class":7894},"model.import_from_mps_string(gzip.open(",[109,7948,1067],{"class":7890},[109,7950,7554],{"class":7951},"sU2Wk",[109,7953,7955],{"class":7954},"sDLfK","{DATA}",[109,7957,7958],{"class":7951},"\u002F",[109,7960,7961],{"class":7954},"{INSTANCE}",[109,7963,7964],{"class":7951},".mps.gz\"",[109,7966,3167],{"class":7894},[109,7968,7969],{"class":7951},"\"rt\"",[109,7971,7972],{"class":7894},").read())\n",[109,7974,7976],{"class":7886,"line":7975},7,[109,7977,7922],{"emptyLinePlaceholder":7921},[109,7979,7981,7984,7986,7989,7992],{"class":7886,"line":7980},8,[109,7982,7983],{"class":7894},"solver ",[109,7985,4554],{"class":7890},[109,7987,7988],{"class":7894}," mb.Solver(",[109,7990,7991],{"class":7951},"\"CP-SAT\"",[109,7993,7994],{"class":7894},")\n",[109,7996,7998,8001,8004],{"class":7886,"line":7997},9,[109,7999,8000],{"class":7894},"solver.set_time_limit_in_seconds(",[109,8002,8003],{"class":7954},"60",[109,8005,7994],{"class":7894},[109,8007,8009,8011,8014,8017],{"class":7886,"line":8008},10,[109,8010,63],{"class":7954},[109,8012,8013],{"class":7890}," =",[109,8015,8016],{"class":7954}," 8",[109,8018,8019],{"class":7928}," # number of CP-SAT \"workers\" (roughly equal to number of CPUs)\n",[109,8021,8023,8026,8028,8031,8034,8036],{"class":7886,"line":8022},11,[109,8024,8025],{"class":7894},"solver.set_solver_specific_parameters(",[109,8027,1067],{"class":7890},[109,8029,8030],{"class":7951},"\"num_workers:",[109,8032,8033],{"class":7954},"{WORKERS}",[109,8035,7554],{"class":7951},[109,8037,7994],{"class":7894},[109,8039,8041,8044,8046],{"class":7886,"line":8040},12,[109,8042,8043],{"class":7894},"status ",[109,8045,4554],{"class":7890},[109,8047,8048],{"class":7894}," solver.solve(model)\n",[109,8050,8052],{"class":7886,"line":8051},13,[109,8053,7922],{"emptyLinePlaceholder":7921},[109,8055,8057,8060,8062,8064,8067,8070,8073,8076,8078],{"class":7886,"line":8056},14,[109,8058,8059],{"class":7954},"print",[109,8061,1374],{"class":7894},[109,8063,1067],{"class":7890},[109,8065,8066],{"class":7951},"\"status    ",[109,8068,8069],{"class":7954},"{",[109,8071,8072],{"class":7894},"status.name",[109,8074,8075],{"class":7954},"}",[109,8077,7554],{"class":7951},[109,8079,7994],{"class":7894},[109,8081,8083,8085,8087,8089,8092,8094,8097,8100,8102,8104],{"class":7886,"line":8082},15,[109,8084,8059],{"class":7954},[109,8086,1374],{"class":7894},[109,8088,1067],{"class":7890},[109,8090,8091],{"class":7951},"\"cost      ",[109,8093,8069],{"class":7954},[109,8095,8096],{"class":7894},"solver.objective_value",[109,8098,8099],{"class":7890},":,.0f",[109,8101,8075],{"class":7954},[109,8103,7554],{"class":7951},[109,8105,7994],{"class":7894},[109,8107,8109,8111,8113,8115,8118,8120,8123,8126,8128,8131,8133,8136],{"class":7886,"line":8108},16,[109,8110,8059],{"class":7954},[109,8112,1374],{"class":7894},[109,8114,1067],{"class":7890},[109,8116,8117],{"class":7951},"\"time      ",[109,8119,8069],{"class":7954},[109,8121,8122],{"class":7894},"solver.wall_time",[109,8124,8125],{"class":7890},":.1f",[109,8127,8075],{"class":7954},[109,8129,8130],{"class":7951}," s on ",[109,8132,8033],{"class":7954},[109,8134,8135],{"class":7951}," workers\"",[109,8137,7994],{"class":7894},[8139,8140],"cell-output",{":lines":8141,"label":8142},"[\"status    FEASIBLE\",\"cost      162,129,469,300\",\"time      60.1 s on 8 workers\"]","Output",[66,8144,8146],{"id":8145},"quicopt","Quicopt",[11,8148,8149],{},"Below we rebuild the model with Pyomo (to illustrate how it's done), and then send it to Quicopt's API service for solving.",[7876,8151,8153],{"className":7878,"code":8152,"language":7880,"meta":7881,"style":7881},"import pyomo.environ as pyo\nfrom quicopt import Client\nfrom quicopt.structured import active, block, semiinteger\n\n# define a Pyomo model\nmodel = pyo.ConcreteModel()\n",[61,8154,8155,8167,8179,8191,8195,8200],{"__ignoreMap":7881},[109,8156,8157,8159,8162,8164],{"class":7886,"line":7887},[109,8158,7898],{"class":7890},[109,8160,8161],{"class":7894}," pyomo.environ ",[109,8163,7904],{"class":7890},[109,8165,8166],{"class":7894}," pyo\n",[109,8168,8169,8171,8174,8176],{"class":7886,"line":7910},[109,8170,7891],{"class":7890},[109,8172,8173],{"class":7894}," quicopt ",[109,8175,7898],{"class":7890},[109,8177,8178],{"class":7894}," Client\n",[109,8180,8181,8183,8186,8188],{"class":7886,"line":7918},[109,8182,7891],{"class":7890},[109,8184,8185],{"class":7894}," quicopt.structured ",[109,8187,7898],{"class":7890},[109,8189,8190],{"class":7894}," active, block, semiinteger\n",[109,8192,8193],{"class":7886,"line":7925},[109,8194,7922],{"emptyLinePlaceholder":7921},[109,8196,8197],{"class":7886,"line":7932},[109,8198,8199],{"class":7928},"# define a Pyomo model\n",[109,8201,8202,8204,8206],{"class":7886,"line":7943},[109,8203,7935],{"class":7894},[109,8205,4554],{"class":7890},[109,8207,8208],{"class":7894}," pyo.ConcreteModel()\n",[11,8210,8211,8212,3203,8267,8301,8302,8305,8306,8340],{},"Next we introduce the decision variables ",[109,8213,8215,8233],{"className":8214},[112],[109,8216,8218],{"className":8217},[116],[118,8219,8220],{"xmlns":120},[122,8221,8222,8230],{},[125,8223,8224,8226,8228],{},[131,8225,133],{"mathvariant":1317},[652,8227,654],{"separator":146},[131,8229,578],{"mathvariant":1317},[138,8231,8232],{"encoding":140},"\\boldsymbol{z}, \\boldsymbol{n}",[109,8234,8236],{"className":8235,"ariaHidden":146},[145],[109,8237,8239,8243,8252,8255,8258],{"className":8238},[150],[109,8240],{"className":8241,"style":8242},[154],"height:0.6389em;vertical-align:-0.1944em;",[109,8244,8246],{"className":8245},[159],[109,8247,8249],{"className":8248},[159],[109,8250,133],{"className":8251,"style":1570},[159,1569],[109,8253,654],{"className":8254},[687],[109,8256],{"className":8257,"style":692},[691],[109,8259,8261],{"className":8260},[159],[109,8262,8264],{"className":8263},[159],[109,8265,578],{"className":8266},[159,1569],[109,8268,8270,8283],{"className":8269},[112],[109,8271,8273],{"className":8272},[116],[118,8274,8275],{"xmlns":120},[122,8276,8277,8281],{},[125,8278,8279],{},[131,8280,374],{"mathvariant":1317},[138,8282,3184],{"encoding":140},[109,8284,8286],{"className":8285,"ariaHidden":146},[145],[109,8287,8289,8292],{"className":8288},[150],[109,8290],{"className":8291,"style":3157},[154],[109,8293,8295],{"className":8294},[159],[109,8296,8298],{"className":8297},[159],[109,8299,374],{"className":8300},[159,1569]," that we already encountered above. Note that ",[56,8303,8304],{},"constraint type (1)"," is already covered in the Pyomo declaration of ",[109,8307,8309,8322],{"className":8308},[112],[109,8310,8312],{"className":8311},[116],[118,8313,8314],{"xmlns":120},[122,8315,8316,8320],{},[125,8317,8318],{},[131,8319,578],{"mathvariant":1317},[138,8321,3220],{"encoding":140},[109,8323,8325],{"className":8324,"ariaHidden":146},[145],[109,8326,8328,8331],{"className":8327},[150],[109,8329],{"className":8330,"style":3157},[154],[109,8332,8334],{"className":8333},[159],[109,8335,8337],{"className":8336},[159],[109,8338,578],{"className":8339},[159,1569],":",[7876,8342,8344],{"className":7878,"code":8343,"language":7880,"meta":7881,"style":7881},"# introduce the binary variables z\nmodel.z = pyo.Var(V, within=pyo.Binary)\n\n# integer numbers of packs ordered, constraint type (1) via bounds and semiinteger\nmodel.n = pyo.Var(O, within=pyo.NonNegativeIntegers, bounds=lambda model, o: (N_min[o], N_max[o]))\nsemiinteger(model.n)\n\n# shortcut aliases for n and z\nz, n = model.z, model.n\n\n# the binary variables x become indicators of the pack counts n\nx = {o: active(n[o]) for o in O}\n\n# tell the solver how the variables are related\nfor i in I:\n  block(model, f\"item_{i}\", [n[o] for v in V_i[i] for o in O_v[v]] + [z[v] for v in V_i[i]])\n",[61,8345,8346,8351,8370,8374,8379,8405,8410,8414,8419,8429,8433,8438,8460,8464,8469,8481],{"__ignoreMap":7881},[109,8347,8348],{"class":7886,"line":7887},[109,8349,8350],{"class":7928},"# introduce the binary variables z\n",[109,8352,8353,8356,8358,8361,8365,8367],{"class":7886,"line":7910},[109,8354,8355],{"class":7894},"model.z ",[109,8357,4554],{"class":7890},[109,8359,8360],{"class":7894}," pyo.Var(V, ",[109,8362,8364],{"class":8363},"s9osk","within",[109,8366,4554],{"class":7890},[109,8368,8369],{"class":7894},"pyo.Binary)\n",[109,8371,8372],{"class":7886,"line":7918},[109,8373,7922],{"emptyLinePlaceholder":7921},[109,8375,8376],{"class":7886,"line":7925},[109,8377,8378],{"class":7928},"# integer numbers of packs ordered, constraint type (1) via bounds and semiinteger\n",[109,8380,8381,8384,8386,8389,8391,8393,8396,8399,8402],{"class":7886,"line":7932},[109,8382,8383],{"class":7894},"model.n ",[109,8385,4554],{"class":7890},[109,8387,8388],{"class":7894}," pyo.Var(O, ",[109,8390,8364],{"class":8363},[109,8392,4554],{"class":7890},[109,8394,8395],{"class":7894},"pyo.NonNegativeIntegers, ",[109,8397,8398],{"class":8363},"bounds",[109,8400,8401],{"class":7890},"=lambda",[109,8403,8404],{"class":7894}," model, o: (N_min[o], N_max[o]))\n",[109,8406,8407],{"class":7886,"line":7943},[109,8408,8409],{"class":7894},"semiinteger(model.n)\n",[109,8411,8412],{"class":7886,"line":7975},[109,8413,7922],{"emptyLinePlaceholder":7921},[109,8415,8416],{"class":7886,"line":7980},[109,8417,8418],{"class":7928},"# shortcut aliases for n and z\n",[109,8420,8421,8424,8426],{"class":7886,"line":7997},[109,8422,8423],{"class":7894},"z, n ",[109,8425,4554],{"class":7890},[109,8427,8428],{"class":7894}," model.z, model.n\n",[109,8430,8431],{"class":7886,"line":8008},[109,8432,7922],{"emptyLinePlaceholder":7921},[109,8434,8435],{"class":7886,"line":8022},[109,8436,8437],{"class":7928},"# the binary variables x become indicators of the pack counts n\n",[109,8439,8440,8443,8445,8448,8451,8454,8457],{"class":7886,"line":8040},[109,8441,8442],{"class":7894},"x ",[109,8444,4554],{"class":7890},[109,8446,8447],{"class":7894}," {o: active(n[o]) ",[109,8449,8450],{"class":7890},"for",[109,8452,8453],{"class":7894}," o ",[109,8455,8456],{"class":7890},"in",[109,8458,8459],{"class":7894}," O}\n",[109,8461,8462],{"class":7886,"line":8051},[109,8463,7922],{"emptyLinePlaceholder":7921},[109,8465,8466],{"class":7886,"line":8056},[109,8467,8468],{"class":7928},"# tell the solver how the variables are related\n",[109,8470,8471,8473,8476,8478],{"class":7886,"line":8082},[109,8472,8450],{"class":7890},[109,8474,8475],{"class":7894}," i ",[109,8477,8456],{"class":7890},[109,8479,8480],{"class":7894}," I:\n",[109,8482,8483,8486,8488,8491,8493,8495,8497,8499,8502,8504,8507,8509,8512,8514,8516,8518,8521,8523,8526,8528,8530,8532],{"class":7886,"line":8108},[109,8484,8485],{"class":7894},"  block(model, ",[109,8487,1067],{"class":7890},[109,8489,8490],{"class":7951},"\"item_",[109,8492,8069],{"class":7954},[109,8494,1346],{"class":7894},[109,8496,8075],{"class":7954},[109,8498,7554],{"class":7951},[109,8500,8501],{"class":7894},", [n[o] ",[109,8503,8450],{"class":7890},[109,8505,8506],{"class":7894}," v ",[109,8508,8456],{"class":7890},[109,8510,8511],{"class":7894}," V_i[i] ",[109,8513,8450],{"class":7890},[109,8515,8453],{"class":7894},[109,8517,8456],{"class":7890},[109,8519,8520],{"class":7894}," O_v[v]] ",[109,8522,1396],{"class":7890},[109,8524,8525],{"class":7894}," [z[v] ",[109,8527,8450],{"class":7890},[109,8529,8506],{"class":7894},[109,8531,8456],{"class":7890},[109,8533,8534],{"class":7894}," V_i[i]])\n",[11,8536,8537],{},"Next, we define the remaining constraints, followed by the objective function:",[7876,8539,8541],{"className":7878,"code":8540,"language":7880,"meta":7881,"style":7881},"# constraint type (2)\nmodel.covered = pyo.Constraint(V, rule=lambda model, v: sum(Q_pack[o] * n[o] - Q_lost[o] * x[o] for o in O_v[v]) >= Q_need[v] * z[v])\n\n# constraint type (3)\nmodel.one_variant = pyo.Constraint(I, rule=lambda model, i: sum(z[v] for v in V_i[i]) == 1)\n\n# constraint type (4)\nmodel.exclusive = pyo.Constraint(E, rule=lambda model, e: sum(x[o] for o in O_e[e]) \u003C= 1)\n\n# constraint type (5)\nmodel.stock = pyo.Constraint(R, rule=lambda model, r: sum(Q_pack[o] * n[o] for o in O_r[r]) \u003C= Q_stock[r])\n\n# the objective function, including the final two constraint types\npyo_obj_expr = (sum(C_choice[v] * z[v] for v in V)\n              + sum(C_pack[o] * n[o] + C_use[o] * x[o] for o in O)\n              + sum(pyo.Expr_if(IF=sum(x[o] for o in O_k[k]) >= 1,\n                                THEN=C_fee[k],\n                                ELSE=0) for k in K)  # constraint (6)\n              + sum(pyo.Expr_if(IF=sum(S_pack[o] * n[o] + S_use[o] * x[o] for o in O_k[k]) >= S_free[k],\n                                THEN=-C_ship[k],\n                                ELSE=0) for k in K)) # constraint (7)\n\n# give the objective to Pyomo\nmodel.cost = pyo.Objective(sense=pyo.minimize, expr=pyo_obj_expr)\n",[61,8542,8543,8548,8609,8613,8618,8657,8661,8666,8704,8708,8713,8752,8756,8761,8790,8823,8857,8868,8894,8938,8949,8972,8977,8983],{"__ignoreMap":7881},[109,8544,8545],{"class":7886,"line":7887},[109,8546,8547],{"class":7928},"# constraint type (2)\n",[109,8549,8550,8553,8555,8558,8561,8563,8566,8569,8572,8575,8578,8581,8584,8586,8589,8591,8593,8595,8598,8601,8604,8606],{"class":7886,"line":7910},[109,8551,8552],{"class":7894},"model.covered ",[109,8554,4554],{"class":7890},[109,8556,8557],{"class":7894}," pyo.Constraint(V, ",[109,8559,8560],{"class":8363},"rule",[109,8562,8401],{"class":7890},[109,8564,8565],{"class":7894}," model, v: ",[109,8567,8568],{"class":7954},"sum",[109,8570,8571],{"class":7894},"(Q_pack[o] ",[109,8573,8574],{"class":7890},"*",[109,8576,8577],{"class":7894}," n[o] ",[109,8579,8580],{"class":7890},"-",[109,8582,8583],{"class":7894}," Q_lost[o] ",[109,8585,8574],{"class":7890},[109,8587,8588],{"class":7894}," x[o] ",[109,8590,8450],{"class":7890},[109,8592,8453],{"class":7894},[109,8594,8456],{"class":7890},[109,8596,8597],{"class":7894}," O_v[v]) ",[109,8599,8600],{"class":7890},">=",[109,8602,8603],{"class":7894}," Q_need[v] ",[109,8605,8574],{"class":7890},[109,8607,8608],{"class":7894}," z[v])\n",[109,8610,8611],{"class":7886,"line":7918},[109,8612,7922],{"emptyLinePlaceholder":7921},[109,8614,8615],{"class":7886,"line":7925},[109,8616,8617],{"class":7928},"# constraint type (3)\n",[109,8619,8620,8623,8625,8628,8630,8632,8635,8637,8640,8642,8644,8646,8649,8652,8655],{"class":7886,"line":7932},[109,8621,8622],{"class":7894},"model.one_variant ",[109,8624,4554],{"class":7890},[109,8626,8627],{"class":7894}," pyo.Constraint(I, ",[109,8629,8560],{"class":8363},[109,8631,8401],{"class":7890},[109,8633,8634],{"class":7894}," model, i: ",[109,8636,8568],{"class":7954},[109,8638,8639],{"class":7894},"(z[v] ",[109,8641,8450],{"class":7890},[109,8643,8506],{"class":7894},[109,8645,8456],{"class":7890},[109,8647,8648],{"class":7894}," V_i[i]) ",[109,8650,8651],{"class":7890},"==",[109,8653,8654],{"class":7954}," 1",[109,8656,7994],{"class":7894},[109,8658,8659],{"class":7886,"line":7943},[109,8660,7922],{"emptyLinePlaceholder":7921},[109,8662,8663],{"class":7886,"line":7975},[109,8664,8665],{"class":7928},"# constraint type (4)\n",[109,8667,8668,8671,8673,8676,8678,8680,8683,8685,8688,8690,8692,8694,8697,8700,8702],{"class":7886,"line":7980},[109,8669,8670],{"class":7894},"model.exclusive ",[109,8672,4554],{"class":7890},[109,8674,8675],{"class":7894}," pyo.Constraint(E, ",[109,8677,8560],{"class":8363},[109,8679,8401],{"class":7890},[109,8681,8682],{"class":7894}," model, e: ",[109,8684,8568],{"class":7954},[109,8686,8687],{"class":7894},"(x[o] ",[109,8689,8450],{"class":7890},[109,8691,8453],{"class":7894},[109,8693,8456],{"class":7890},[109,8695,8696],{"class":7894}," O_e[e]) ",[109,8698,8699],{"class":7890},"\u003C=",[109,8701,8654],{"class":7954},[109,8703,7994],{"class":7894},[109,8705,8706],{"class":7886,"line":7997},[109,8707,7922],{"emptyLinePlaceholder":7921},[109,8709,8710],{"class":7886,"line":8008},[109,8711,8712],{"class":7928},"# constraint type (5)\n",[109,8714,8715,8718,8720,8723,8725,8727,8730,8732,8734,8736,8738,8740,8742,8744,8747,8749],{"class":7886,"line":8022},[109,8716,8717],{"class":7894},"model.stock ",[109,8719,4554],{"class":7890},[109,8721,8722],{"class":7894}," pyo.Constraint(R, ",[109,8724,8560],{"class":8363},[109,8726,8401],{"class":7890},[109,8728,8729],{"class":7894}," model, r: ",[109,8731,8568],{"class":7954},[109,8733,8571],{"class":7894},[109,8735,8574],{"class":7890},[109,8737,8577],{"class":7894},[109,8739,8450],{"class":7890},[109,8741,8453],{"class":7894},[109,8743,8456],{"class":7890},[109,8745,8746],{"class":7894}," O_r[r]) ",[109,8748,8699],{"class":7890},[109,8750,8751],{"class":7894}," Q_stock[r])\n",[109,8753,8754],{"class":7886,"line":8040},[109,8755,7922],{"emptyLinePlaceholder":7921},[109,8757,8758],{"class":7886,"line":8051},[109,8759,8760],{"class":7928},"# the objective function, including the final two constraint types\n",[109,8762,8763,8766,8768,8771,8773,8776,8778,8781,8783,8785,8787],{"class":7886,"line":8056},[109,8764,8765],{"class":7894},"pyo_obj_expr ",[109,8767,4554],{"class":7890},[109,8769,8770],{"class":7894}," (",[109,8772,8568],{"class":7954},[109,8774,8775],{"class":7894},"(C_choice[v] ",[109,8777,8574],{"class":7890},[109,8779,8780],{"class":7894}," z[v] ",[109,8782,8450],{"class":7890},[109,8784,8506],{"class":7894},[109,8786,8456],{"class":7890},[109,8788,8789],{"class":7894}," V)\n",[109,8791,8792,8795,8798,8801,8803,8805,8807,8810,8812,8814,8816,8818,8820],{"class":7886,"line":8082},[109,8793,8794],{"class":7890},"              +",[109,8796,8797],{"class":7954}," sum",[109,8799,8800],{"class":7894},"(C_pack[o] ",[109,8802,8574],{"class":7890},[109,8804,8577],{"class":7894},[109,8806,1396],{"class":7890},[109,8808,8809],{"class":7894}," C_use[o] ",[109,8811,8574],{"class":7890},[109,8813,8588],{"class":7894},[109,8815,8450],{"class":7890},[109,8817,8453],{"class":7894},[109,8819,8456],{"class":7890},[109,8821,8822],{"class":7894}," O)\n",[109,8824,8825,8827,8829,8832,8835,8837,8839,8841,8843,8845,8847,8850,8852,8854],{"class":7886,"line":8108},[109,8826,8794],{"class":7890},[109,8828,8797],{"class":7954},[109,8830,8831],{"class":7894},"(pyo.Expr_if(",[109,8833,8834],{"class":8363},"IF",[109,8836,4554],{"class":7890},[109,8838,8568],{"class":7954},[109,8840,8687],{"class":7894},[109,8842,8450],{"class":7890},[109,8844,8453],{"class":7894},[109,8846,8456],{"class":7890},[109,8848,8849],{"class":7894}," O_k[k]) ",[109,8851,8600],{"class":7890},[109,8853,8654],{"class":7954},[109,8855,8856],{"class":7894},",\n",[109,8858,8860,8863,8865],{"class":7886,"line":8859},17,[109,8861,8862],{"class":8363},"                                THEN",[109,8864,4554],{"class":7890},[109,8866,8867],{"class":7894},"C_fee[k],\n",[109,8869,8871,8874,8876,8878,8881,8883,8886,8888,8891],{"class":7886,"line":8870},18,[109,8872,8873],{"class":8363},"                                ELSE",[109,8875,4554],{"class":7890},[109,8877,650],{"class":7954},[109,8879,8880],{"class":7894},") ",[109,8882,8450],{"class":7890},[109,8884,8885],{"class":7894}," k ",[109,8887,8456],{"class":7890},[109,8889,8890],{"class":7894}," K)  ",[109,8892,8893],{"class":7928},"# constraint (6)\n",[109,8895,8897,8899,8901,8903,8905,8907,8909,8912,8914,8916,8918,8921,8923,8925,8927,8929,8931,8933,8935],{"class":7886,"line":8896},19,[109,8898,8794],{"class":7890},[109,8900,8797],{"class":7954},[109,8902,8831],{"class":7894},[109,8904,8834],{"class":8363},[109,8906,4554],{"class":7890},[109,8908,8568],{"class":7954},[109,8910,8911],{"class":7894},"(S_pack[o] ",[109,8913,8574],{"class":7890},[109,8915,8577],{"class":7894},[109,8917,1396],{"class":7890},[109,8919,8920],{"class":7894}," S_use[o] ",[109,8922,8574],{"class":7890},[109,8924,8588],{"class":7894},[109,8926,8450],{"class":7890},[109,8928,8453],{"class":7894},[109,8930,8456],{"class":7890},[109,8932,8849],{"class":7894},[109,8934,8600],{"class":7890},[109,8936,8937],{"class":7894}," S_free[k],\n",[109,8939,8941,8943,8946],{"class":7886,"line":8940},20,[109,8942,8862],{"class":8363},[109,8944,8945],{"class":7890},"=-",[109,8947,8948],{"class":7894},"C_ship[k],\n",[109,8950,8952,8954,8956,8958,8960,8962,8964,8966,8969],{"class":7886,"line":8951},21,[109,8953,8873],{"class":8363},[109,8955,4554],{"class":7890},[109,8957,650],{"class":7954},[109,8959,8880],{"class":7894},[109,8961,8450],{"class":7890},[109,8963,8885],{"class":7894},[109,8965,8456],{"class":7890},[109,8967,8968],{"class":7894}," K)) ",[109,8970,8971],{"class":7928},"# constraint (7)\n",[109,8973,8975],{"class":7886,"line":8974},22,[109,8976,7922],{"emptyLinePlaceholder":7921},[109,8978,8980],{"class":7886,"line":8979},23,[109,8981,8982],{"class":7928},"# give the objective to Pyomo\n",[109,8984,8986,8989,8991,8994,8997,8999,9002,9005,9007],{"class":7886,"line":8985},24,[109,8987,8988],{"class":7894},"model.cost ",[109,8990,4554],{"class":7890},[109,8992,8993],{"class":7894}," pyo.Objective(",[109,8995,8996],{"class":8363},"sense",[109,8998,4554],{"class":7890},[109,9000,9001],{"class":7894},"pyo.minimize, ",[109,9003,9004],{"class":8363},"expr",[109,9006,4554],{"class":7890},[109,9008,9009],{"class":7894},"pyo_obj_expr)\n",[11,9011,9012],{},"And then we send it off:",[7876,9014,9016],{"className":7878,"code":9015,"language":7880,"meta":7881,"style":7881},"# solve with our API service\nresult = Client().solve(model, time_limit=2.0, gzip=True)\nprint(result.display)\n",[61,9017,9018,9023,9053],{"__ignoreMap":7881},[109,9019,9020],{"class":7886,"line":7887},[109,9021,9022],{"class":7928},"# solve with our API service\n",[109,9024,9025,9028,9030,9033,9036,9038,9041,9043,9046,9048,9051],{"class":7886,"line":7910},[109,9026,9027],{"class":7894},"result ",[109,9029,4554],{"class":7890},[109,9031,9032],{"class":7894}," Client().solve(model, ",[109,9034,9035],{"class":8363},"time_limit",[109,9037,4554],{"class":7890},[109,9039,9040],{"class":7954},"2.0",[109,9042,3167],{"class":7894},[109,9044,9045],{"class":8363},"gzip",[109,9047,4554],{"class":7890},[109,9049,9050],{"class":7954},"True",[109,9052,7994],{"class":7894},[109,9054,9055,9057],{"class":7886,"line":7918},[109,9056,8059],{"class":7954},[109,9058,9059],{"class":7894},"(result.display)\n",[8139,9061],{":lines":9062,"label":8142},"[\"▄▀▀▀▄        ▄                     █\",\"█   █ █   █ ▄▄  ▄▀▀▀▄ ▄▀▀▀▄ █▀▀▀▄ ▀█▀▀\",\"█ ▀▄▀ █   █  █  █   ▄ █   █ █▄▄▄▀  █  ▄\",\" ▀▀ ▀  ▀▀▀▀ ▀▀▀  ▀▀▀   ▀▀▀  █       ▀▀\",\"\",\"├── shots\",\"│   └── 1 · primal   162044219300   1.98s  ◀ best\",\"├── status:     heuristic\",\"├── feasible:   true\",\"├── objective:  1.620442193e11\",\"├── x:          x1=1, x10=0, x100=0, x1000=0, x10000=0, x10001=0, …  (10933 variables)\",\"├── time_limit: 2 s\",\"└── solve_time: 1.9852 s\",\"\",\"    Intel Xeon Gold 6338N @ 2.20GHz · 16 cores, 1 core per shot\"]",[11,9064,9065],{},"Let's inspect the solution quality:",[7876,9067,9069],{"className":7878,"code":9068,"language":7880,"meta":7881,"style":7881},"print(f\"CP-SAT's total cost                              {solver.objective_value:,.0f}\")\nprint(f\"Total cost from Quicopt's API service            {result.objective:,.0f}\")\n",[61,9070,9071,9094],{"__ignoreMap":7881},[109,9072,9073,9075,9077,9079,9082,9084,9086,9088,9090,9092],{"class":7886,"line":7887},[109,9074,8059],{"class":7954},[109,9076,1374],{"class":7894},[109,9078,1067],{"class":7890},[109,9080,9081],{"class":7951},"\"CP-SAT's total cost                              ",[109,9083,8069],{"class":7954},[109,9085,8096],{"class":7894},[109,9087,8099],{"class":7890},[109,9089,8075],{"class":7954},[109,9091,7554],{"class":7951},[109,9093,7994],{"class":7894},[109,9095,9096,9098,9100,9102,9105,9107,9110,9112,9114,9116],{"class":7886,"line":7910},[109,9097,8059],{"class":7954},[109,9099,1374],{"class":7894},[109,9101,1067],{"class":7890},[109,9103,9104],{"class":7951},"\"Total cost from Quicopt's API service            ",[109,9106,8069],{"class":7954},[109,9108,9109],{"class":7894},"result.objective",[109,9111,8099],{"class":7890},[109,9113,8075],{"class":7954},[109,9115,7554],{"class":7951},[109,9117,7994],{"class":7894},[8139,9119],{":lines":9120,"label":8142},"[\"CP-SAT's total cost                              162,129,469,300\",\"Total cost from Quicopt's API service            162,044,219,300\"]",[11,9122,9123,9124,9127,9128,9131,9132,9136],{},"Quicopt found a ",[56,9125,9126],{},"better solution"," in less than ",[56,9129,9130],{},"two seconds!"," CP-SAT invested 60s on eight CPUs (which actually sometimes suffices to find the same solution as Quicopt). The ",[24,9133,9135],{"href":40,"rel":9134},[28],"notebook"," on our GitHub has additional proofs that the solution of Quicopt's solver is correct in the original MILP formulation.",[9138,9139,9140],"style",{},"html pre.shiki code .snl16, html code.shiki .snl16{--shiki-default:#F97583}html pre.shiki code .s95oV, html code.shiki .s95oV{--shiki-default:#E1E4E8}html pre.shiki code .sAwPA, html code.shiki .sAwPA{--shiki-default:#6A737D}html pre.shiki code .sU2Wk, html code.shiki .sU2Wk{--shiki-default:#9ECBFF}html pre.shiki code .sDLfK, html code.shiki .sDLfK{--shiki-default:#79B8FF}html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html pre.shiki code .s9osk, html code.shiki .s9osk{--shiki-default:#FFAB70}",{"title":7881,"searchDepth":7910,"depth":7910,"links":9142},[9143,9148,9149],{"id":68,"depth":7910,"text":69,"children":9144},[9145,9146,9147],{"id":73,"depth":7918,"text":74},{"id":1264,"depth":7918,"text":1265},{"id":3126,"depth":7918,"text":3127},{"id":7863,"depth":7910,"text":7864},{"id":8145,"depth":7910,"text":8146},"Let us take a deep dive into the so-called \"internet-shopping optimization problem\", also known as \"supplier selection and order allocation\". This is a very common problem that occurs basically whenever a large number of items needs to be ordered from a large number of suppliers. As it involves integer quantities, it is usually formulated as a mixed-integer linear problem (MILP). Unfortunately, MILPs are known to be NP-hard - which more or less says that they can be very hard to solve.","md",null,{},"\u002Fuse-cases\u002Finternet-shopping",{"title":6,"description":9150},"use-cases\u002Finternet-shopping","7YrfJ8_CrBXiWQ9NQvFECLwflj6hqECfcF6FE3sLWZI",{"id":9159,"title":9160,"body":9161,"description":17794,"extension":9151,"faq":9152,"meta":17795,"navigation":7921,"path":17796,"seo":17797,"stem":17798,"__hash__":17799},"content\u002Fuse-cases\u002Finternet-shopping.de.md","Das Internet-Shopping-Optimierungsproblem",{"type":8,"value":9162,"toc":17785},[9163,9184,9200,9210,9214,9218,9221,10299,10303,10322,11551,12086,12090,12197,14817,14820,16731,16733,16742,16932,16935,16937,16940,16990,17120,17278,17281,17671,17674,17712,17714,17717,17765,17767,17783],[11,9164,9165,9166,9169,9170,9173,9174,9179,9180,9183],{},"Wir tauchen tief in das sogenannte „Internet-Shopping-Optimierungsproblem“ ein, auch bekannt als „Lieferantenauswahl und Bestellaufteilung“. Es tritt praktisch immer dann auf, wenn eine große Zahl von Artikeln bei einer großen Zahl von Lieferanten bestellt werden muss. Da es um ",[15,9167,9168],{},"ganzzahlige"," Mengen geht, wird es üblicherweise als ",[15,9171,9172],{},"gemischt-ganzzahliges lineares Problem"," (MILP) formuliert. Leider gelten MILPs als ",[24,9175,9178],{"href":9176,"rel":9177},"https:\u002F\u002Fde.wikipedia.org\u002Fwiki\u002FNP-Schwere",[28],"NP-schwer"," – das heißt mehr oder weniger, dass sie ",[15,9181,9182],{},"sehr"," schwer zu lösen sein können.",[11,9185,9186,9187,9191,9192,9195,9196,9199],{},"Sie können alle Ergebnisse selbst nachvollziehen, indem Sie ",[24,9188,9190],{"href":40,"rel":9189},[28],"dieses Notebook"," ausführen. Beachten Sie dabei: Der Open-Source-Solver CP-SAT läuft ",[15,9193,9194],{},"auf Ihrem Rechner",", Quicopt dagegen auf ",[15,9197,9198],{},"unserem API-Server",". Die Hardware ist also nicht identisch, wie sie es für einen streng sauberen Vergleich sein müsste.",[11,9201,9202,9203,9206,9207,9209],{},"Selbst wenn Sie CP-SAT auf schnelleren CPUs ausführen als denen unseres Testservers, werden Sie trotzdem feststellen, dass ",[56,9204,9205],{},"Quicopt deutlich schneller ist!"," Außerdem nutzt CP-SAT 8 Kerne, Quicopt dagegen nur einen! (Die Zahl der Kerne von CP-SAT lässt sich über ",[61,9208,63],{}," einstellen, siehe unten. Bedenken Sie schließlich, dass CP-SAT nicht deterministisch arbeitet: Die Laufzeiten schwanken also recht stark, ebenso die Qualität der Lösung.)",[66,9211,9213],{"id":9212},"das-problem-als-milp","Das Problem als MILP",[71,9215,9217],{"id":9216},"variablen","Variablen",[11,9219,9220],{},"Die Variablen unseres Optimierungsproblems stehen in der folgenden Tabelle:",[79,9222,9223,9239],{},[82,9224,9225],{},[85,9226,9227,9230,9233,9236],{},[88,9228,9229],{},"Symbol",[88,9231,9232],{},"Werte",[88,9234,9235],{},"Bedeutung",[88,9237,9238],{},"Kosten, die sie 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Angebote)",[106,10008,10009],{},[109,10010,10012,10031],{"className":10011},[112],[109,10013,10015],{"className":10014},[116],[118,10016,10017],{"xmlns":120},[122,10018,10019,10029],{},[125,10020,10021],{},[267,10022,10023,10025,10027],{},[131,10024,271],{},[131,10026,862],{},[275,10028,974],{},[138,10030,977],{"encoding":140},[109,10032,10034],{"className":10033,"ariaHidden":146},[145],[109,10035,10037,10040],{"className":10036},[150],[109,10038],{"className":10039,"style":987},[154],[109,10041,10043,10046],{"className":10042},[159],[109,10044,271],{"className":10045,"style":297},[159,163],[109,10047,10049],{"className":10048},[168],[109,10050,10052,10089],{"className":10051},[172,173],[109,10053,10055,10086],{"className":10054},[177],[109,10056,10058,10069],{"className":10057,"style":310},[181],[109,10059,10060,10063],{"style":1008},[109,10061],{"className":10062,"style":190},[189],[109,10064,10066],{"className":10065},[194,195,196,197],[109,10067,862],{"className":10068,"style":905},[159,163,197],[109,10070,10071,10074],{"style":325},[109,10072],{"className":10073,"style":190},[189],[109,10075,10077],{"className":10076},[194,195,196,197],[109,10078,10080],{"className":10079},[159,197],[109,10081,10083],{"className":10082},[159,338,197],[109,10084,974],{"className":10085},[159,197],[109,10087,206],{"className":10088},[205],[109,10090,10092],{"className":10091},[177],[109,10093,10095],{"className":10094,"style":1044},[181],[109,10096],{},[85,10098,10099,10170,10172,10203],{},[106,10100,10101],{},[109,10102,10104,10121],{"className":10103},[112],[109,10105,10107],{"className":10106},[116],[118,10108,10109],{"xmlns":120},[122,10110,10111,10119],{},[125,10112,10113],{},[128,10114,10115,10117],{},[131,10116,1067],{},[131,10118,862],{},[138,10120,1072],{"encoding":140},[109,10122,10124],{"className":10123,"ariaHidden":146},[145],[109,10125,10127,10130],{"className":10126},[150],[109,10128],{"className":10129,"style":1082},[154],[109,10131,10133,10136],{"className":10132},[159],[109,10134,1067],{"className":10135,"style":1089},[159,163],[109,10137,10139],{"className":10138},[168],[109,10140,10142,10162],{"className":10141},[172,173],[109,10143,10145,10159],{"className":10144},[177],[109,10146,10148],{"className":10147,"style":893},[181],[109,10149,10150,10153],{"style":1104},[109,10151],{"className":10152,"style":190},[189],[109,10154,10156],{"className":10155},[194,195,196,197],[109,10157,862],{"className":10158,"style":905},[159,163,197],[109,10160,206],{"className":10161},[205],[109,10163,10165],{"className":10164},[177],[109,10166,10168],{"className":10167,"style":213},[181],[109,10169],{},[106,10171,9316],{},[106,10173,9977,10174,10202],{},[109,10175,10177,10190],{"className":10176},[112],[109,10178,10180],{"className":10179},[116],[118,10181,10182],{"xmlns":120},[122,10183,10184,10188],{},[125,10185,10186],{},[131,10187,862],{},[138,10189,862],{"encoding":140},[109,10191,10193],{"className":10192,"ariaHidden":146},[145],[109,10194,10196,10199],{"className":10195},[150],[109,10197],{"className":10198,"style":947},[154],[109,10200,862],{"className":10201,"style":905},[159,163]," liefert versandkostenfrei: nein\u002Fja",[106,10204,10205,10298],{},[109,10206,10208,10229],{"className":10207},[112],[109,10209,10211],{"className":10210},[116],[118,10212,10213],{"xmlns":120},[122,10214,10215,10227],{},[125,10216,10217,10219],{},[652,10218,1174],{},[267,10220,10221,10223,10225],{},[131,10222,271],{},[131,10224,862],{},[275,10226,1183],{},[138,10228,1186],{"encoding":140},[109,10230,10232],{"className":10231,"ariaHidden":146},[145],[109,10233,10235,10238,10241],{"className":10234},[150],[109,10236],{"className":10237,"style":1196},[154],[109,10239,1174],{"className":10240},[159],[109,10242,10244,10247],{"className":10243},[159],[109,10245,271],{"className":10246,"style":297},[159,163],[109,10248,10250],{"className":10249},[168],[109,10251,10253,10290],{"className":10252},[172,173],[109,10254,10256,10287],{"className":10255},[177],[109,10257,10259,10270],{"className":10258,"style":1218},[181],[109,10260,10261,10264],{"style":1221},[109,10262],{"className":10263,"style":190},[189],[109,10265,10267],{"className":10266},[194,195,196,197],[109,10268,862],{"className":10269,"style":905},[159,163,197],[109,10271,10272,10275],{"style":1233},[109,10273],{"className":10274,"style":190},[189],[109,10276,10278],{"className":10277},[194,195,196,197],[109,10279,10281],{"className":10280},[159,197],[109,10282,10284],{"className":10283},[159,338,197],[109,10285,1183],{"className":10286},[159,197],[109,10288,206],{"className":10289},[205],[109,10291,10293],{"className":10292},[177],[109,10294,10296],{"className":10295,"style":1258},[181],[109,10297],{}," (eine Ersparnis)",[71,10300,10302],{"id":10301},"zielfunktion","Zielfunktion",[11,10304,10305,10306,10309,10310,10313,10314,10317,10318,10321],{},"Das Problem hat vier „Ebenen“: die zu kaufenden ",[56,10307,10308],{},"Artikel",", die ",[56,10311,10312],{},"Varianten"," jedes Artikels, die ",[56,10315,10316],{},"Angebote"," für jede Variante und schließlich die ",[56,10319,10320],{},"Lieferanten",", die diese Angebote machen. Die Kostenfunktion besteht aus zwei 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bedeutet: Für jeden Artikel ",[109,11555,11557,11570],{"className":11556},[112],[109,11558,11560],{"className":11559},[116],[118,11561,11562],{"xmlns":120},[122,11563,11564,11568],{},[125,11565,11566],{},[131,11567,1346],{},[138,11569,1346],{"encoding":140},[109,11571,11573],{"className":11572,"ariaHidden":146},[145],[109,11574,11576,11579],{"className":11575},[150],[109,11577],{"className":11578,"style":2615},[154],[109,11580,1346],{"className":11581},[159,163]," und jede Variante ",[109,11584,11586,11599],{"className":11585},[112],[109,11587,11589],{"className":11588},[116],[118,11590,11591],{"xmlns":120},[122,11592,11593,11597],{},[125,11594,11595],{},[131,11596,136],{},[138,11598,136],{"encoding":140},[109,11600,11602],{"className":11601,"ariaHidden":146},[145],[109,11603,11605,11608],{"className":11604},[150],[109,11606],{"className":11607,"style":247},[154],[109,11609,136],{"className":11610,"style":201},[159,163]," zahlen wir 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wenn die Variante gewählt wird, und außerdem für jedes verwendete Angebot ",[109,11702,11704,11717],{"className":11703},[112],[109,11705,11707],{"className":11706},[116],[118,11708,11709],{"xmlns":120},[122,11710,11711,11715],{},[125,11712,11713],{},[131,11714,377],{},[138,11716,377],{"encoding":140},[109,11718,11720],{"className":11719,"ariaHidden":146},[145],[109,11721,11723,11726],{"className":11722},[150],[109,11724],{"className":11725,"style":247},[154],[109,11727,377],{"className":11728},[159,163]," dieser Variante 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pro gekaufter Packung sowie die Fixkosten ",[109,11820,11822,11841],{"className":11821},[112],[109,11823,11825],{"className":11824},[116],[118,11826,11827],{"xmlns":120},[122,11828,11829,11839],{},[125,11830,11831],{},[267,11832,11833,11835,11837],{},[131,11834,271],{},[131,11836,377],{},[275,11838,486],{},[138,11840,489],{"encoding":140},[109,11842,11844],{"className":11843,"ariaHidden":146},[145],[109,11845,11847,11850],{"className":11846},[150],[109,11848],{"className":11849,"style":499},[154],[109,11851,11853,11856],{"className":11852},[159],[109,11854,271],{"className":11855,"style":297},[159,163],[109,11857,11859],{"className":11858},[168],[109,11860,11862,11899],{"className":11861},[172,173],[109,11863,11865,11896],{"className":11864},[177],[109,11866,11868,11879],{"className":11867,"style":518},[181],[109,11869,11870,11873],{"style":313},[109,11871],{"className":11872,"style":190},[189],[109,11874,11876],{"className":11875},[194,195,196,197],[109,11877,377],{"className":11878},[159,163,197],[109,11880,11881,11884],{"style":325},[109,11882],{"className":11883,"style":190},[189],[109,11885,11887],{"className":11886},[194,195,196,197],[109,11888,11890],{"className":11889},[159,197],[109,11891,11893],{"className":11892},[159,338,197],[109,11894,486],{"className":11895},[159,197],[109,11897,206],{"className":11898},[205],[109,11900,11902],{"className":11901},[177],[109,11903,11905],{"className":11904,"style":351},[181],[109,11906],{},". Die zweite Summe läuft über die Lieferanten: die Gebühr ",[109,11909,11911,11930],{"className":11910},[112],[109,11912,11914],{"className":11913},[116],[118,11915,11916],{"xmlns":120},[122,11917,11918,11928],{},[125,11919,11920],{},[267,11921,11922,11924,11926],{},[131,11923,271],{},[131,11925,862],{},[275,11927,974],{},[138,11929,977],{"encoding":140},[109,11931,11933],{"className":11932,"ariaHidden":146},[145],[109,11934,11936,11939],{"className":11935},[150],[109,11937],{"className":11938,"style":987},[154],[109,11940,11942,11945],{"className":11941},[159],[109,11943,271],{"className":11944,"style":297},[159,163],[109,11946,11948],{"className":11947},[168],[109,11949,11951,11988],{"className":11950},[172,173],[109,11952,11954,11985],{"className":11953},[177],[109,11955,11957,11968],{"className":11956,"style":310},[181],[109,11958,11959,11962],{"style":1008},[109,11960],{"className":11961,"style":190},[189],[109,11963,11965],{"className":11964},[194,195,196,197],[109,11966,862],{"className":11967,"style":905},[159,163,197],[109,11969,11970,11973],{"style":325},[109,11971],{"className":11972,"style":190},[189],[109,11974,11976],{"className":11975},[194,195,196,197],[109,11977,11979],{"className":11978},[159,197],[109,11980,11982],{"className":11981},[159,338,197],[109,11983,974],{"className":11984},[159,197],[109,11986,206],{"className":11987},[205],[109,11989,11991],{"className":11990},[177],[109,11992,11994],{"className":11993,"style":1044},[181],[109,11995],{},", wenn der Lieferant verwendet wird, abzüglich der Versandkosten ",[109,11998,12000,12019],{"className":11999},[112],[109,12001,12003],{"className":12002},[116],[118,12004,12005],{"xmlns":120},[122,12006,12007,12017],{},[125,12008,12009],{},[267,12010,12011,12013,12015],{},[131,12012,271],{},[131,12014,862],{},[275,12016,1183],{},[138,12018,3056],{"encoding":140},[109,12020,12022],{"className":12021,"ariaHidden":146},[145],[109,12023,12025,12028],{"className":12024},[150],[109,12026],{"className":12027,"style":1196},[154],[109,12029,12031,12034],{"className":12030},[159],[109,12032,271],{"className":12033,"style":297},[159,163],[109,12035,12037],{"className":12036},[168],[109,12038,12040,12077],{"className":12039},[172,173],[109,12041,12043,12074],{"className":12042},[177],[109,12044,12046,12057],{"className":12045,"style":1218},[181],[109,12047,12048,12051],{"style":1221},[109,12049],{"className":12050,"style":190},[189],[109,12052,12054],{"className":12053},[194,195,196,197],[109,12055,862],{"className":12056,"style":905},[159,163,197],[109,12058,12059,12062],{"style":1233},[109,12060],{"className":12061,"style":190},[189],[109,12063,12065],{"className":12064},[194,195,196,197],[109,12066,12068],{"className":12067},[159,197],[109,12069,12071],{"className":12070},[159,338,197],[109,12072,1183],{"className":12073},[159,197],[109,12075,206],{"className":12076},[205],[109,12078,12080],{"className":12079},[177],[109,12081,12083],{"className":12082,"style":1258},[181],[109,12084],{},", wenn der Versand kostenfrei ist.",[71,12087,12089],{"id":12088},"nebenbedingungen","Nebenbedingungen",[11,12091,12092,12093,3167,12127,12161,12162,12196],{},"Dieses Problem hat viele Nebenbedingungen, die die Entscheidungsvariablen sinnvoll einschränken. Für ",[109,12094,12096,12109],{"className":12095},[112],[109,12097,12099],{"className":12098},[116],[118,12100,12101],{"xmlns":120},[122,12102,12103,12107],{},[125,12104,12105],{},[131,12106,133],{"mathvariant":1317},[138,12108,3147],{"encoding":140},[109,12110,12112],{"className":12111,"ariaHidden":146},[145],[109,12113,12115,12118],{"className":12114},[150],[109,12116],{"className":12117,"style":3157},[154],[109,12119,12121],{"className":12120},[159],[109,12122,12124],{"className":12123},[159],[109,12125,133],{"className":12126,"style":1570},[159,1569],[109,12128,12130,12143],{"className":12129},[112],[109,12131,12133],{"className":12132},[116],[118,12134,12135],{"xmlns":120},[122,12136,12137,12141],{},[125,12138,12139],{},[131,12140,374],{"mathvariant":1317},[138,12142,3184],{"encoding":140},[109,12144,12146],{"className":12145,"ariaHidden":146},[145],[109,12147,12149,12152],{"className":12148},[150],[109,12150],{"className":12151,"style":3157},[154],[109,12153,12155],{"className":12154},[159],[109,12156,12158],{"className":12157},[159],[109,12159,374],{"className":12160},[159,1569]," und ",[109,12163,12165,12178],{"className":12164},[112],[109,12166,12168],{"className":12167},[116],[118,12169,12170],{"xmlns":120},[122,12171,12172,12176],{},[125,12173,12174],{},[131,12175,578],{"mathvariant":1317},[138,12177,3220],{"encoding":140},[109,12179,12181],{"className":12180,"ariaHidden":146},[145],[109,12182,12184,12187],{"className":12183},[150],[109,12185],{"className":12186,"style":3157},[154],[109,12188,12190],{"className":12189},[159],[109,12191,12193],{"className":12192},[159],[109,12194,578],{"className":12195},[159,1569]," gelten die folgenden fünf:",[79,12198,12199,12217],{},[82,12200,12201],{},[85,12202,12203,12206,12208,12211,12214],{},[88,12204,12205],{},"Zeile",[88,12207],{},[88,12209,12210],{},"für jede\u002Fn",[88,12212,12213],{},"bekannt als",[88,12215,12216],{},"in Worten",[101,12218,12219,12634,13904,14165,14426],{},[85,12220,12221,12223,12576,12628,12631],{},[106,12222,3266],{},[106,12224,12225],{},[109,12226,12228,12285],{"className":12227},[112],[109,12229,12231],{"className":12230},[116],[118,12232,12233],{"xmlns":120},[122,12234,12235,12283],{},[125,12236,12237,12247,12249,12255,12257,12263,12265,12275,12277],{},[267,12238,12239,12241,12243,12245],{},[131,12240,3285],{},[131,12242,377],{},[131,12244,1309],{},[652,12246,1312],{},[275,12248,1385],{},[128,12250,12251,12253],{},[131,12252,374],{},[131,12254,377],{},[652,12256,3302],{},[128,12258,12259,12261],{},[131,12260,578],{},[131,12262,377],{},[652,12264,3302],{},[267,12266,12267,12269,12271,12273],{},[131,12268,3285],{},[131,12270,377],{},[131,12272,3319],{},[652,12274,1312],{},[275,12276,1385],{},[128,12278,12279,12281],{},[131,12280,374],{},[131,12282,377],{},[138,12284,3332],{"encoding":140},[109,12286,12288,12409,12464],{"className":12287,"ariaHidden":146},[145],[109,12289,12291,12294,12357,12360,12400,12403,12406],{"className":12290},[150],[109,12292],{"className":12293,"style":3342},[154],[109,12295,12297,12300],{"className":12296},[159],[109,12298,3285],{"className":12299,"style":3349},[159,163],[109,12301,12303],{"className":12302},[168],[109,12304,12306,12349],{"className":12305},[172,173],[109,12307,12309,12346],{"className":12308},[177],[109,12310,12312,12323],{"className":12311,"style":3362},[181],[109,12313,12314,12317],{"style":3365},[109,12315],{"className":12316,"style":190},[189],[109,12318,12320],{"className":12319},[194,195,196,197],[109,12321,377],{"className":12322},[159,163,197],[109,12324,12325,12328],{"style":325},[109,12326],{"className":12327,"style":190},[189],[109,12329,12331],{"className":12330},[194,195,196,197],[109,12332,12334],{"className":12333},[159,197],[109,12335,12337,12340,12343],{"className":12336},[1535,197],[109,12338,3392],{"className":12339},[197],[109,12341,1346],{"className":12342},[197],[109,12344,578],{"className":12345},[197],[109,12347,206],{"className":12348},[205],[109,12350,12352],{"className":12351},[177],[109,12353,12355],{"className":12354,"style":351},[181],[109,12356],{},[109,12358],{"className":12359,"style":692},[691],[109,12361,12363,12366],{"className":12362},[159],[109,12364,374],{"className":12365},[159,163],[109,12367,12369],{"className":12368},[168],[109,12370,12372,12392],{"className":12371},[172,173],[109,12373,12375,12389],{"className":12374},[177],[109,12376,12378],{"className":12377,"style":182},[181],[109,12379,12380,12383],{"style":410},[109,12381],{"className":12382,"style":190},[189],[109,12384,12386],{"className":12385},[194,195,196,197],[109,12387,377],{"className":12388},[159,163,197],[109,12390,206],{"className":12391},[205],[109,12393,12395],{"className":12394},[177],[109,12396,12398],{"className":12397,"style":213},[181],[109,12399],{},[109,12401],{"className":12402,"style":1646},[691],[109,12404,3302],{"className":12405},[1682],[109,12407],{"className":12408,"style":1646},[691],[109,12410,12412,12415,12455,12458,12461],{"className":12411},[150],[109,12413],{"className":12414,"style":3468},[154],[109,12416,12418,12421],{"className":12417},[159],[109,12419,578],{"className":12420},[159,163],[109,12422,12424],{"className":12423},[168],[109,12425,12427,12447],{"className":12426},[172,173],[109,12428,12430,12444],{"className":12429},[177],[109,12431,12433],{"className":12432,"style":182},[181],[109,12434,12435,12438],{"style":410},[109,12436],{"className":12437,"style":190},[189],[109,12439,12441],{"className":12440},[194,195,196,197],[109,12442,377],{"className":12443},[159,163,197],[109,12445,206],{"className":12446},[205],[109,12448,12450],{"className":12449},[177],[109,12451,12453],{"className":12452,"style":213},[181],[109,12454],{},[109,12456],{"className":12457,"style":1646},[691],[109,12459,3302],{"className":12460},[1682],[109,12462],{"className":12463,"style":1646},[691],[109,12465,12467,12470,12533,12536],{"className":12466},[150],[109,12468],{"className":12469,"style":499},[154],[109,12471,12473,12476],{"className":12472},[159],[109,12474,3285],{"className":12475,"style":3349},[159,163],[109,12477,12479],{"className":12478},[168],[109,12480,12482,12525],{"className":12481},[172,173],[109,12483,12485,12522],{"className":12484},[177],[109,12486,12488,12499],{"className":12487,"style":518},[181],[109,12489,12490,12493],{"style":3365},[109,12491],{"className":12492,"style":190},[189],[109,12494,12496],{"className":12495},[194,195,196,197],[109,12497,377],{"className":12498},[159,163,197],[109,12500,12501,12504],{"style":325},[109,12502],{"className":12503,"style":190},[189],[109,12505,12507],{"className":12506},[194,195,196,197],[109,12508,12510],{"className":12509},[159,197],[109,12511,12513,12516,12519],{"className":12512},[1535,197],[109,12514,3392],{"className":12515},[197],[109,12517,24],{"className":12518},[197],[109,12520,374],{"className":12521},[197],[109,12523,206],{"className":12524},[205],[109,12526,12528],{"classN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Variablen, Big-M-Zeilen",[106,12632,12633],{},"Packungen stammen nur aus verwendeten Angeboten, und zwar zwischen der kleinsten und der größten Menge, die das Angebot zulässt.",[85,12635,12636,12638,13153,13205,13277],{},[106,12637,3695],{},[106,12639,12640],{},[109,12641,12643,12722],{"className":12642},[112],[109,12644,12646],{"className":12645},[116],[118,12647,12648],{"xmlns":120},[122,12649,12650,12720],{},[125,12651,12652,12668,12670,12678,12684,12686,12694,12700,12702,12704,12712,12714],{},[128,12653,12654,12656],{},[652,12655,1341],{},[125,12657,12658,12660,12662],{},[131,12659,377],{},[652,12661,1349],{},[128,12663,12664,12666],{},[131,12665,1413],{"mathvariant":1352},[131,12667,136],{},[652,12669,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,12671,12672,12674,12676],{},[131,12673,3732],{},[131,12675,377],{},[275,12677,768],{},[128,12679,12680,12682],{},[131,12681,578],{},[131,12683,377],{},[652,12685,1174],{},[267,12687,12688,12690,12692],{},[131,12689,3732],{},[131,12691,377],{},[275,12693,3753],{},[128,12695,12696,12698],{},[131,12697,374],{},[131,12699,377],{},[652,12701,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[652,12703,3764],{},[267,12705,12706,12708,12710],{},[131,12707,3732],{},[131,12709,136],{},[275,12711,3773],{},[275,12713,1385],{},[128,12715,12716,12718],{},[131,12717,133],{},[131,12719,136],{},[138,12721,3784],{"encoding":140},[109,12723,12725,12929,13047],{"className":12724,"ariaHidden":146},[145],[109,12726,12728,12731,12817,12823,12880,12920,12923,12926],{"className":12727},[150],[109,12729],{"className":12730,"style":3794},[154],[109,12732,12734,12737],{"className":12733},[1535],[109,12735,1341],{"className":12736,"style":3802},[1535,1699,3801],[109,12738,12740],{"className":12739},[168],[109,12741,12743,12809],{"className":12742},[172,173],[109,12744,12746,12806],{"className":12745},[177],[109,12747,12749],{"className":12748,"style":3815},[181],[109,12750,12751,12754],{"style":3818},[109,12752],{"className":12753,"style":190},[189],[109,12755,12757],{"className":12756},[194,195,196,197],[109,12758,12760,12763,12766],{"className":12759},[159,197],[109,12761,377],{"className":12762},[159,163,197],[109,12764,1349],{"className":12765},[1682,197],[109,12767,12769,12772],{"className":12768},[159,197],[109,12770,1413],{"className":12771,"style":1984},[159,1686,197],[109,12773,12775],{"className":12774},[168],[109,12776,12778,12798],{"className":12777},[172,173],[109,12779,12781,12795],{"className":12780},[177],[109,12782,12784],{"className":12783,"style":1997},[181],[109,12785,12786,12789],{"style":2000},[109,12787],{"className":12788,"style":1771},[189],[109,12790,12792],{"className":12791},[194,1775,1776,197],[109,12793,136],{"className":12794,"style":201},[159,163,197],[109,12796,206],{"className":12797},[205],[109,12799,12801],{"className":12800},[177],[109,12802,12804],{"className":12803,"style":1789},[181],[109,12805],{},[109,12807,206],{"className":12808},[205],[109,12810,12812],{"className":12811},[177],[109,12813,12815],{"className":12814,"style":3883},[181],[109,12816],{},[109,12818,12820],{"className":12819},[1817],[109,12821,1374],{"className":12822},[1821,1776],[109,12824,12826,12829],{"className":12825},[159],[109,12827,3732],{"className":12828},[159,163],[109,12830,12832],{"className":12831},[168],[109,12833,12835,12872],{"className":12834},[172,173],[109,12836,12838,12869],{"className":12837},[177],[109,12839,12841,12852],{"className":12840,"style":310},[181],[109,12842,12843,12846],{"style":3912},[109,12844],{"className":12845,"style":190},[189],[109,12847,12849],{"className":12848},[194,195,196,197],[109,12850,377],{"className":12851},[159,163,197],[109,12853,12854,12857],{"style":325},[109,12855],{"className":12856,"style":190},[189],[109,12858,12860],{"className":12859},[194,195,196,197],[109,12861,12863],{"className":12862},[159,197],[109,12864,12866],{"className":12865},[159,338,197],[109,12867,768],{"className":12868},[159,197],[109,12870,206],{"className":12871},[205],[109,12873,12875],{"className":12874},[177],[109,12876,12878],{"className":12877,"style":351},[181],[109,12879],{},[109,12881,12883,12886],{"className":12882},[159],[109,12884,578],{"className":12885},[159,163],[109,12887,12889],{"className":12888},[168],[109,12890,12892,12912],{"className":12891},[172,173],[109,12893,12895,12909],{"className":12894},[177],[109,12896,12898],{"className":12897,"style":182},[181],[109,12899,12900,12903],{"style":410},[109,12901],{"className":12902,"style":190},[189],[109,12904,12906],{"className":12905},[194,195,196,197],[109,12907,377],{"className":12908},[159,163,197],[109,12910,206],{"className":12911},[205],[109,12913,12915],{"className":12914},[177],[109,12916,12918],{"className":12917,"style":213},[181],[109,12919],{},[109,12921],{"className":12922,"style":1931},[691],[109,12924,1174],{"className":12925},[1935],[109,12927],{"className":12928,"style":1931},[691],[109,12930,12932,12935,12992,13032,13038,13041,13044],{"className":12931},[150],[109,12933],{"className":12934,"style":4005},[154],[109,12936,12938,12941],{"className":12937},[159],[109,12939,3732],{"className":12940},[1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(Überdeckung), eingeschaltet durch ",[109,13209,13211,13228],{"className":13210},[112],[109,13212,13214],{"className":13213},[116],[118,13215,13216],{"xmlns":120},[122,13217,13218,13226],{},[125,13219,13220],{},[128,13221,13222,13224],{},[131,13223,133],{},[131,13225,136],{},[138,13227,141],{"encoding":140},[109,13229,13231],{"className":13230,"ariaHidden":146},[145],[109,13232,13234,13237],{"className":13233},[150],[109,13235],{"className":13236,"style":155},[154],[109,13238,13240,13243],{"className":13239},[159],[109,13241,133],{"className":13242,"style":164},[159,163],[109,13244,13246],{"className":13245},[168],[109,13247,13249,13269],{"className":13248},[172,173],[109,13250,13252,13266],{"className":13251},[177],[109,13253,13255],{"className":13254,"style":182},[181],[109,13256,13257,13260],{"style":185},[109,13258],{"className":13259,"style":190},[189],[109,13261,13263],{"className":13262},[194,195,196,197],[109,13264,136],{"className":13265,"style":201},[159,163,197],[109,13267,206],{"className":13268},[205],[109,13270,13272],{"className":13271},[177],[109,13273,13275],{"className":13274,"style":213},[181],[109,13276],{},[106,13278,13279,13280,13368,13369,13457,13458,12161,13568,13678,13679,13838,13839,13903],{},"Die gewählte Variante muss gedeckt werden. Jedes Angebot liefert ",[109,13281,13283,13302],{"className":13282},[112],[109,13284,13286],{"className":13285},[116],[118,13287,13288],{"xmlns":120},[122,13289,13290,13300],{},[125,13291,13292],{},[267,13293,13294,13296,13298],{},[131,13295,3732],{},[131,13297,377],{},[275,13299,768],{},[138,13301,4374],{"encoding":140},[109,13303,13305],{"className":13304,"ariaHidden":146},[145],[109,13306,13308,13311],{"className":13307},[150],[109,13309],{"className":13310,"style":290},[154],[109,13312,13314,13317],{"className":13313},[159],[109,13315,3732],{"className":13316},[159,163],[109,13318,13320],{"className":13319},[168],[109,13321,13323,13360],{"className":13322},[172,173],[109,13324,13326,13357],{"className":13325},[177],[109,13327,13329,13340],{"className":13328,"style":310},[181],[109,13330,13331,13334],{"style":3912},[109,13332],{"className":13333,"style":190},[189],[109,13335,13337],{"className":13336},[194,195,196,197],[109,13338,377],{"className":13339},[159,163,197],[109,13341,13342,13345],{"style":325},[109,13343],{"className":13344,"style":190},[189],[109,13346,13348],{"className":13347},[194,195,196,197],[109,13349,13351],{"className":13350},[159,197],[109,13352,13354],{"className":13353},[159,338,197],[109,13355,768],{"className":13356},[159,197],[109,13358,206],{"className":13359},[205],[109,13361,13363],{"className":13362},[177],[109,13364,13366],{"className":13365,"style":351},[181],[109,13367],{}," Einheiten und verliert ",[109,13370,13372,13391],{"className":13371},[112],[109,13373,13375],{"className":13374},[116],[118,13376,13377],{"xmlns":120},[122,13378,13379,13389],{},[125,13380,13381],{},[267,13382,13383,13385,13387],{},[131,13384,3732],{},[131,13386,377],{},[275,13388,3753],{},[138,13390,4464],{"encoding":140},[109,13392,13394],{"className":13393,"ariaHidden":146},[145],[109,13395,13397,13400],{"className":13396},[150],[109,13398],{"className":13399,"style":290},[154],[109,13401,13403,13406],{"className":13402},[159],[109,13404,3732],{"className":13405},[159,163],[109,13407,13409],{"className":13408},[168],[109,13410,13412,13449],{"className":13411},[172,173],[109,13413,13415,13446],{"className":13414},[177],[109,13416,13418,13429],{"className":13417,"style":310},[181],[109,13419,13420,13423],{"style":3912},[109,13421],{"className":13422,"style":190},[189],[109,13424,13426],{"className":13425},[194,195,196,197],[109,13427,377],{"className":13428},[159,163,197],[109,13430,13431,13434],{"style":325},[109,13432],{"className":13433,"style":190},[189],[109,13435,13437],{"className":13436},[194,195,196,197],[109,13438,13440],{"className":13439},[159,197],[109,13441,13443],{"className":13442},[159,338,197],[109,13444,3753],{"className":13445},[159,197],[109,13447,206],{"className":13448},[205],[109,13450,13452],{"className":13451},[177],[109,13453,13455],{"className":13454,"style":351},[181],[109,13456],{},". Gilt zum Beispiel ",[109,13459,13461,13484],{"className":13460},[112],[109,13462,13464],{"className":13463},[116],[118,13465,13466],{"xmlns":120},[122,13467,13468,13482],{},[125,13469,13470,13478,13480],{},[267,13471,13472,13474,13476],{},[131,13473,3732],{},[131,13475,136],{},[275,13477,3773],{},[652,13479,4554],{},[648,13481,4557],{},[138,13483,4560],{"encoding":140},[109,13485,13487,13559],{"className":13486,"ariaHidden":146},[145],[109,13488,13490,13493,13550,13553,13556],{"className":13489},[150],[109,13491],{"className":13492,"style":290},[154],[109,13494,13496,13499],{"className":13495},[159],[109,13497,3732],{"className":13498},[159,163],[109,13500,13502],{"className":13501},[168],[109,13503,13505,13542],{"className":13504},[172,173],[109,13506,13508,13539],{"className":13507},[177],[109,13509,13511,13522],{"className":13510,"style":310},[181],[109,13512,13513,13516],{"style":3912},[109,13514],{"className":13515,"style":190},[189],[109,13517,13519],{"className":13518},[194,195,196,197],[109,13520,136],{"className":13521,"style":201},[159,163,197],[109,13523,13524,13527],{"style":325},[109,13525],{"className":13526,"style":190},[189],[109,13528,13530],{"className":13529},[194,195,196,197],[109,13531,13533],{"className":13532},[159,197],[109,13534,13536],{"className":13535},[159,338,197],[109,13537,3773],{"className":13538},[159,197],[109,13540,206],{"className":13541},[205],[109,13543,13545],{"className":13544},[177],[109,13546,13548],{"className":13547,"style":351},[181],[109,13549],{},[109,13551],{"className":13552,"style":1646},[691],[109,13554,4554],{"className":13555},[1682],[109,13557],{"className":13558,"style":1646},[691],[109,13560,13562,13565],{"className":13561},[150],[109,13563],{"className":13564,"style":4642},[154],[109,13566,4557],{"className":13567},[159],[109,13569,13571,13594],{"className":13570},[112],[109,13572,13574],{"className":13573},[116],[118,13575,13576],{"xmlns":120},[122,13577,13578,13592],{},[125,13579,13580,13588,13590],{},[267,13581,13582,13584,13586],{},[131,13583,3732],{},[131,13585,377],{},[275,13587,3753],{},[652,13589,4554],{},[648,13591,662],{},[138,13593,4673],{"encoding":140},[109,13595,13597,13669],{"className":13596,"ariaHidden":146},[145],[109,13598,13600,13603,13660,13663,13666],{"className":13599},[150],[109,13601],{"className":13602,"style":290},[154],[109,13604,13606,13609],{"className":13605},[159],[109,13607,3732],{"className":13608},[159,163],[109,13610,13612],{"className":13611},[168],[109,13613,13615,13652],{"className":13614},[172,173],[109,13616,13618,13649],{"className":13617},[177],[109,13619,13621,13632],{"className":13620,"style":310},[181],[109,13622,13623,13626],{"style":3912},[109,13624],{"className":13625,"style":190},[189],[109,13627,13629],{"className":13628},[194,195,196,197],[109,13630,377],{"className":13631},[159,163,197],[109,13633,13634,13637],{"style":325},[109,13635],{"className":13636,"style":190},[189],[109,13638,13640],{"className":13639},[194,195,196,197],[109,13641,13643],{"className":13642},[159,197],[109,13644,13646],{"className":13645},[159,338,197],[109,13647,3753],{"className":13648},[159,197],[109,13650,206],{"className":13651},[205],[109,13653,13655],{"className":13654},[177],[109,13656,13658],{"className":13657,"style":351},[181],[109,13659],{},[109,13661],{"className":13662,"style":1646},[691],[109,13664,4554],{"className":13665},[1682],[109,13667],{"className":13668,"style":1646},[691],[109,13670,13672,13675],{"className":13671},[150],[109,13673],{"className":13674,"style":4642},[154],[109,13676,662],{"className":13677},[159],", lautet die Zeile 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es werden also 6 Packungen benötigt. Ein zweites verwendetes Angebot würde ebenfalls sein ",[109,13840,13842,13859],{"className":13841},[112],[109,13843,13845],{"className":13844},[116],[118,13846,13847],{"xmlns":120},[122,13848,13849,13857],{},[125,13850,13851],{},[4935,13852,13853,13855],{},[131,13854,3732],{},[275,13856,3753],{},[138,13858,4943],{"encoding":140},[109,13860,13862],{"className":13861,"ariaHidden":146},[145],[109,13863,13865,13868],{"className":13864},[150],[109,13866],{"className":13867,"style":4953},[154],[109,13869,13871,13874],{"className":13870},[159],[109,13872,3732],{"className":13873},[159,163],[109,13875,13877],{"className":13876},[168],[109,13878,13880],{"className":13879},[172],[109,13881,13883],{"className":13882},[177],[109,13884,13886],{"className":13885,"style":310},[181],[109,13887,13888,13891],{"style":325},[109,13889],{"className":13890,"style":190},[189],[109,13892,13894],{"className":13893},[194,195,196,197],[109,13895,13897],{"className":13896},[159,197],[109,13898,13900],{"className":13899},[159,338,197],[109,13901,3753],{"className":13902},[159,197]," verlieren. Bei einer nicht gewählten Variante ist die rechte Seite 0, sodass es erlaubt ist, nichts zu kaufen.",[85,13905,13906,13908,14106,14159,14162],{},[106,13907,4994],{},[106,13909,13910],{},[109,13911,13913,13950],{"className":13912},[112],[109,13914,13916],{"className":13915},[116],[118,13917,13918],{"xmlns":120},[122,13919,13920,13948],{},[125,13921,13922,13938,13944,13946],{},[128,13923,13924,13926],{},[652,13925,1341],{},[125,13927,13928,13930,13932],{},[131,13929,136],{},[652,13931,1349],{},[128,13933,13934,13936],{},[131,13935,1368],{"mathvariant":1352},[131,13937,1346],{},[128,13939,13940,13942],{},[131,13941,133],{},[131,13943,136],{},[652,13945,4554],{},[648,13947,657],{},[138,13949,5037],{"encoding":140},[109,13951,13953,14097],{"className":13952,"ariaHidden":146},[145],[109,13954,13956,13959,14045,14048,14088,14091,14094],{"className":13955},[150],[109,13957],{"className":13958,"style":5047},[154],[109,13960,13962,13965],{"className":13961},[1535],[109,13963,1341],{"className":13964,"style":3802},[1535,1699,3801],[109,13966,13968],{"className":13967},[168],[109,13969,13971,14037],{"className":13970},[172,173],[109,13972,13974,14034],{"className":13973},[177],[109,13975,13977],{"className":13976,"style":3815},[181],[109,13978,13979,13982],{"style":3818},[109,13980],{"className":13981,"style":190},[189],[109,13983,13985],{"className":13984},[194,195,196,197],[109,13986,13988,13991,13994],{"className":13987},[159,197],[109,13989,136],{"className":13990,"style":201},[159,163,197],[109,13992,1349],{"className":13993},[1682,197],[109,13995,13997,14000],{"className":13996},[159,197],[109,13998,1368],{"className":13999,"style":1751},[159,1686,197],[109,14001,14003],{"className":14002},[168],[109,14004,14006,14026],{"className":14005},[172,173],[109,14007,14009,14023],{"className":14008},[177],[109,14010,14012],{"className":14011,"style":1764},[181],[109,14013,14014,14017],{"style":1767},[109,14015],{"className":14016,"style":1771},[189],[109,14018,14020],{"className":14019},[194,1775,1776,197],[109,14021,1346],{"className":14022},[159,163,197],[109,14024,206],{"className":14025},[205],[109,14027,14029],{"className":14028},[177],[109,14030,14032],{"className":14031,"style":1789},[181],[109,14033],{},[109,14035,206],{"className":14036},[205],[109,14038,14040],{"className":14039},[177],[109,14041,14043],{"className":14042,"style":3883},[181],[109,14044],{},[109,14046],{"className":14047,"style":692},[691],[109,14049,14051,14054],{"className":14050},[159],[109,14052,133],{"className":14053,"style":164},[159,163],[109,14055,14057],{"className":14056},[168],[109,14058,14060,14080],{"className":14059},[172,173],[109,14061,14063,14077],{"className":14062},[177],[109,14064,14066],{"className":14065,"style":182},[181],[109,14067,14068,14071],{"style":185},[109,14069],{"className":14070,"style":190},[189],[109,14072,14074],{"className":14073},[194,195,196,197],[109,14075,136],{"className":14076,"style":201},[159,163,197],[109,14078,206],{"className":14079},[205],[109,14081,14083],{"className":14082},[177],[109,14084,14086],{"className":14085,"style":213},[181],[109,14087],{},[109,14089],{"className":14090,"style":1646},[691],[109,14092,4554],{"className":14093},[1682],[109,14095],{"className":14096,"style":1646},[691],[109,14098,14100,14103],{"className":14099},[150],[109,14101],{"className":14102,"style":4642},[154],[109,14104,657],{"className":14105},[159],[106,14107,14108,14109],{},"Artikel 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gewählt.",[85,14166,14167,14169,14367,14420,14423],{},[106,14168,5260],{},[106,14170,14171],{},[109,14172,14174,14211],{"className":14173},[112],[109,14175,14177],{"className":14176},[116],[118,14178,14179],{"xmlns":120},[122,14180,14181,14209],{},[125,14182,14183,14199,14205,14207],{},[128,14184,14185,14187],{},[652,14186,1341],{},[125,14188,14189,14191,14193],{},[131,14190,377],{},[652,14192,1349],{},[128,14194,14195,14197],{},[131,14196,1413],{"mathvariant":1352},[131,14198,5291],{},[128,14200,14201,14203],{},[131,14202,374],{},[131,14204,377],{},[652,14206,3302],{},[648,14208,657],{},[138,14210,5304],{"encoding":140},[109,14212,14214,14358],{"className":14213,"ariaHidden":146},[145],[109,14215,14217,14220,14306,14309,14349,14352,14355],{"className":14216},[150],[109,14218],{"className":14219,"style":5047},[154],[109,14221,14223,14226],{"className":14222},[1535],[109,14224,1341],{"className":14225,"style":3802},[1535,1699,3801],[109,14227,14229],{"className":14228},[168],[109,14230,14232,14298],{"className":14231},[172,173],[109,14233,14235,14295],{"className":14234},[177],[109,14236,14238],{"className":14237,"style":3815},[181],[109,14239,14240,14243],{"style":3818},[109,14241],{"className":14242,"style":190},[189],[109,14244,14246],{"className":14245},[194,195,196,197],[109,14247,14249,14252,14255],{"className":14248},[159,197],[109,14250,377],{"className":14251},[159,163,197],[109,14253,1349],{"className":14254},[1682,197],[109,14256,14258,14261],{"className":14257},[159,197],[109,14259,1413],{"className":14260,"style":1984},[159,1686,197],[109,14262,14264],{"className":14263},[168],[109,14265,14267,14287],{"className":14266},[172,173],[109,14268,14270,14284],{"className":14269},[177],[109,14271,14273],{"className":14272,"style":1997},[181],[109,14274,14275,14278],{"style":2000},[109,14276],{"className":14277,"style":1771},[189],[109,14279,14281],{"className":14280},[194,1775,1776,197],[109,14282,5291],{"className":14283},[159,163,197],[109,14285,206],{"className":14286},[205],[109,14288,14290],{"className":14289},[177],[109,14291,14293],{"className":14292,"style":1789},[181],[109,14294],{},[109,14296,206],{"className":14297},[205],[109,14299,14301],{"className":14300},[177],[109,14302,14304],{"className":14303,"style":3883},[181],[109,14305],{},[109,14307],{"className":14308,"style":692},[691],[109,14310,14312,14315],{"className":14311},[159],[109,14313,374],{"className":14314},[159,163],[109,14316,14318],{"className":14317},[168],[109,14319,14321,14341],{"className":14320},[172,173],[109,14322,14324,14338],{"className":14323},[177],[109,14325,14327],{"className":14326,"style":182},[181],[109,14328,14329,14332],{"style":410},[109,14330],{"className":14331,"style":190},[189],[109,14333,14335],{"className":14334},[194,195,196,197],[109,14336,377],{"className":14337},[159,163,197],[109,14339,206],{"className":14340},[205],[109,14342,14344],{"className":14343},[177],[109,14345,14347],{"className":14346,"style":213},[181],[109,14348],{},[109,14350],{"className":14351,"style":1646},[691],[109,14353,3302],{"className":14354},[1682],[109,14356],{"className":14357,"style":1646},[691],[109,14359,14361,14364],{"className":14360},[150],[109,14362],{"className":14363,"style":4642},[154],[109,14365,657],{"className":14366},[159],[106,14368,14369,14370],{},"exklusive Menge ",[109,14371,14373,14390],{"className":14372},[112],[109,14374,14376],{"className":14375},[116],[118,14377,14378],{"xmlns":120},[122,14379,14380,14388],{},[125,14381,14382,14384,14386],{},[131,14383,5291],{},[652,14385,1349],{},[131,14387,5482],{"mathvariant":1352},[138,14389,5485],{"encoding":140},[109,14391,14393,14411],{"className":14392,"ariaHidden":146},[145],[109,14394,14396,14399,14402,14405,14408],{"className":14395},[150],[109,14397],{"className":14398,"style":3662},[154],[109,14400,5291],{"className":14401},[159,163],[109,14403],{"className":14404,"style":1646},[691],[109,14406,1349],{"className":14407},[1682],[109,14409],{"className":14410,"style":1646},[691],[109,14412,14414,14417],{"className":14413},[150],[109,14415],{"className":14416,"style":3681},[154],[109,14418,5482],{"className":14419,"style":5516},[159,1686],[106,14421,14422],{},"Höchstens-eins-Zeile (Mengenpackung)",[106,14424,14425],{},"Von einer exklusiven Menge wird höchstens ein Angebot verwendet. Beispielsweise lassen sich die Preise eines Produkts für 1–9 Packungen und für 10–99 Packungen nicht mischen.",[85,14427,14428,14430,14758,14811,14814],{},[106,14429,5527],{},[106,14431,14432],{},[109,14433,14435,14488],{"className":14434},[112],[109,14436,14438],{"className":14437},[116],[118,14439,14440],{"xmlns":120},[122,14441,14442,14486],{},[125,14443,14444,14460,14468,14470,14476,14478],{},[128,14445,14446,14448],{},[652,14447,1341],{},[125,14449,14450,14452,14454],{},[131,14451,377],{},[652,14453,1349],{},[128,14455,14456,14458],{},[131,14457,1413],{"mathvariant":1352},[131,14459,5558],{},[267,14461,14462,14464,14466],{},[131,14463,3732],{},[131,14465,377],{},[275,14467,768],{},[275,14469,1385],{},[128,14471,14472,14474],{},[131,14473,578],{},[131,14475,377],{},[652,14477,3302],{},[267,14479,14480,14482,14484],{},[131,14481,3732],{},[131,14483,5558],{},[275,14485,5585],{},[138,14487,5588],{"encoding":140},[109,14489,14491,14695],{"className":14490,"ariaHidden":146},[145],[109,14492,14494,14497,14583,14586,14643,14646,14686,14689,14692],{"className":14493},[150],[109,14495],{"className":14496,"style":5598},[154],[109,14498,14500,14503],{"className":14499},[1535],[109,14501,1341],{"className":14502,"style":3802},[1535,1699,3801],[109,14504,14506],{"className":14505},[168],[109,14507,14509,14575],{"className":14508},[172,173],[109,14510,14512,14572],{"className":14511},[177],[109,14513,14515],{"className":14514,"style":3815},[181],[109,14516,14517,14520],{"style":3818},[109,14518],{"className":14519,"style":190},[189],[109,14521,14523],{"className":14522},[194,195,196,197],[109,14524,14526,14529,14532],{"className":14525},[159,197],[109,14527,377],{"className":14528},[159,163,197],[109,14530,1349],{"className":14531},[1682,197],[109,14533,14535,14538],{"className":14534},[159,197],[109,14536,1413],{"className":14537,"style":1984},[159,1686,197],[109,14539,14541],{"className":14540},[168],[109,14542,14544,14564],{"className":14543},[172,173],[109,14545,14547,14561],{"className":14546},[177],[109,14548,14550],{"className":14549,"style":1997},[181],[109,14551,14552,14555],{"style":2000},[109,14553],{"className":14554,"style":1771},[189],[109,14556,14558],{"className":14557},[194,1775,1776,197],[109,14559,5558],{"className":14560,"style":1984},[159,163,197],[109,14562,206],{"className":14563},[205],[109,14565,14567],{"className":14566},[177],[109,14568,14570],{"className":14569,"style":1789},[181],[109,14571],{},[109,14573,206],{"className":14574},[205],[109,14576,14578],{"className":14577},[177],[109,14579,14581],{"className":14580,"style":3883},[181],[109,14582],{},[109,14584],{"className":14585,"style":692},[691],[109,14587,14589,14592],{"className":14588},[159],[109,14590,3732],{"className":14591},[159,163],[109,14593,14595],{"className":14594},[168],[109,14596,14598,14635],{"className":14597},[172,173],[109,14599,14601,14632],{"className":14600},[177],[109,14602,14604,14615],{"className":14603,"style":310},[181],[109,14605,14606,14609],{"style":3912},[109,14607],{"className":14608,"style":190},[189],[109,14610,14612],{"className":14611},[194,195,196,197],[109,14613,377],{"className":14614},[159,163,197],[109,14616,14617,14620],{"style":325},[109,14618],{"className":14619,"style":190},[189],[109,14621,14623],{"className":14622},[194,195,196,197],[109,14624,14626],{"className":14625},[159,197],[109,14627,14629],{"className":14628},[159,338,197],[109,14630,768],{"className":14631},[159,197],[109,14633,206],{"className":14634},[205],[109,14636,14638],{"className":14637},[177],[109,14639,14641],{"className":14640,"style":351},[181],[109,14642],{},[109,14644],{"className":14645,"style":692},[691],[109,14647,14649,14652],{"className":14648},[159],[109,14650,578],{"className":14651},[159,163],[109,14653,14655],{"className":14654},[168],[109,14656,14658,14678],{"className":14657},[172,173],[109,14659,14661,14675],{"className":14660},[177],[109,14662,14664],{"className":14663,"style":182},[181],[109,14665,14666,14669],{"style":410},[109,14667],{"className":14668,"style":190},[189],[109,14670,14672],{"className":14671},[194,195,196,197],[109,14673,377],{"className":14674},[159,163,197],[109,14676,206],{"className":14677},[205],[109,14679,14681],{"className":14680},[177],[109,14682,14684],{"className":14683,"style":213},[181],[109,14685],{},[109,14687],{"className":14688,"style":1646},[691],[109,14690,3302],{"className":14691},[1682],[109,14693],{"className":14694,"style":1646},[691],[109,14696,14698,14701],{"className":14697},[150],[109,14699],{"className":14700,"style":290},[154],[109,14702,14704,14707],{"className":14703},[159],[109,14705,3732],{"className":14706},[159,163],[109,14708,14710],{"className":14709},[168],[109,14711,14713,14750],{"className":14712},[172,173],[109,14714,14716,14747],{"className":14715},[177],[109,14717,14719,14730],{"className":14718,"style":310},[181],[109,14720,14721,14724],{"style":3912},[109,14722],{"className":14723,"style":190},[189],[109,14725,14727],{"className":14726},[194,195,196,197],[109,14728,5558],{"className":14729,"style":1984},[159,163,197],[109,14731,14732,14735],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Lager hat.",[11,14818,14819],{},"Für die Lieferantenvariablen kommen zwei weitere hinzu:",[79,14821,14822,14836],{},[82,14823,14824],{},[85,14825,14826,14828,14830,14832,14834],{},[88,14827,12205],{},[88,14829],{},[88,14831,12210],{},[88,14833,12213],{},[88,14835,12216],{},[101,14837,14838,15671],{},[85,14839,14840,14842,15142,15194,15286],{},[106,14841,5946],{},[106,14843,14844],{},[109,14845,14847,14900],{"className":14846},[112],[109,14848,14850],{"className":14849},[116],[118,14851,14852],{"xmlns":120},[122,14853,14854,14898],{},[125,14855,14856,14872,14878,14880,14882,14888,14890,14892],{},[128,14857,14858,14860],{},[652,14859,1341],{},[125,14861,14862,14864,14866],{},[131,14863,377],{},[652,14865,1349],{},[128,14867,14868,14870],{},[131,14869,1413],{"mathvariant":1352},[131,14871,862],{},[128,14873,14874,14876],{},[131,14875,374],{},[131,14877,377],{},[652,14879,3302],{},[652,14881,5988],{"stretchy":5987},[128,14883,14884,14886],{},[131,14885,1413],{"mathvariant":1352},[131,14887,862],{},[652,14889,5988],{"stretchy":5987},[275,14891,1385],{},[128,14893,14894,14896],{},[131,14895,859],{},[131,14897,862],{},[138,14899,6007],{"encoding":140},[109,14901,14903,15047],{"className":14902,"ariaHidden":146},[145],[109,14904,14906,14909,14995,14998,15038,15041,15044],{"className":14905},[150],[109,14907],{"className":14908,"style":6017},[154],[109,14910,14912,14915],{"className":14911},[1535],[109,14913,1341],{"className":14914,"style":3802},[1535,1699,3801],[109,14916,14918],{"className":14917},[168],[109,14919,14921,14987],{"className":14920},[172,173],[109,14922,14924,14984],{"className":14923},[177],[109,14925,14927],{"className":14926,"style":3815},[181],[109,14928,14929,14932],{"style":3818},[109,14930],{"className":14931,"style":190},[189],[109,14933,14935],{"className":14934},[194,195,196,197],[109,14936,14938,14941,14944],{"className":14937},[159,197],[109,14939,377],{"className":14940},[159,163,197],[109,14942,1349],{"className":14943},[1682,197],[109,14945,14947,14950],{"className":14946},[159,197],[109,14948,1413],{"className":14949,"style":1984},[159,1686,197],[109,14951,14953],{"className":14952},[168],[109,14954,14956,14976],{"className":14955},[172,173],[109,14957,14959,14973],{"className":14958},[177],[109,14960,14962],{"className":14961,"style":6071},[181],[109,14963,14964,14967],{"style":6074},[109,14965],{"className":14966,"style":1771},[189],[109,14968,14970],{"className":14969},[194,1775,1776,197],[109,14971,862],{"className":14972,"style":905},[159,163,197],[109,14974,206],{"className":14975},[205],[109,14977,14979],{"className":14978},[177],[109,14980,14982],{"className":14981,"style":6093},[181],[109,14983],{},[109,14985,206],{"className":14986},[205],[109,14988,14990],{"className":14989},[177],[109,14991,14993],{"className":14992,"style":6105},[181],[109,14994],{},[109,14996],{"className":14997,"style":692},[691],[109,14999,15001,15004],{"className":15000},[159],[109,15002,374],{"className":15003},[159,163],[109,15005,15007],{"className":15006},[168],[109,15008,15010,15030],{"className":15009},[172,173],[109,15011,15013,15027],{"className":15012},[177],[109,15014,15016],{"className":15015,"style":182},[181],[109,15017,15018,15021],{"style":410},[109,15019],{"className":15020,"style":190},[189],[109,15022,15024],{"className":15023},[194,195,196,197],[109,15025,377],{"className":15026},[159,163,197],[109,15028,206],{"className":15029},[205],[109,15031,15033],{"className":15032},[177],[109,15034,15036],{"className":15035,"style":213},[181],[109,15037],{},[109,15039],{"className":15040,"style":1646},[691],[109,15042,3302],{"className":15043},[1682],[109,15045],{"className":15046,"style":1646},[691],[109,15048,15050,15053,15056,15096,15099,15102],{"className":15049},[150],[109,15051],{"className":15052,"style":6166},[154],[109,15054,5988],{"className":15055},[1817],[109,15057,15059,15062],{"className":15058},[159],[109,15060,1413],{"className":15061,"style":1984},[159,1686],[109,15063,15065],{"className":15064},[168],[109,15066,15068,15088],{"className":15067},[172,173],[109,15069,15071,15085],{"className":15070},[177],[109,15072,15074],{"className":15073,"style":893},[181],[109,15075,15076,15079],{"style":6190},[109,15077],{"className":15078,"style":190},[189],[109,15080,15082],{"className":15081},[194,195,196,197],[109,15083,862],{"className":15084,"style":905},[159,163,197],[109,15086,206],{"className":15087},[205],[109,15089,15091],{"className":15090},[177],[109,15092,15094],{"className":15093,"style":213},[181],[109,15095],{},[109,15097,5988],{"className":15098},[2268],[109,15100],{"className":15101,"style":692},[691],[109,15103,15105,15108],{"className":15104},[159],[109,15106,859],{"className":15107},[159,163],[109,15109,15111],{"className":15110},[168],[109,15112,15114,15134],{"className":15113},[172,173],[109,15115,15117,15131],{"className":15116},[177],[109,15118,15120],{"className":15119,"style":893},[181],[109,15121,15122,15125],{"style":410},[109,15123],{"className":15124,"style":190},[189],[109,15126,15128],{"className":15127},[194,195,196,197],[109,15129,862],{"className":15130,"style":905},[159,163,197],[109,15132,206],{"className":15133},[205],[109,15135,15137],{"className":15136},[177],[109,15138,15140],{"className":15139,"style":213},[181],[109,15141],{},[106,15143,9977,15144],{},[109,15145,15147,15164],{"className":15146},[112],[109,15148,15150],{"className":15149},[116],[118,15151,15152],{"xmlns":120},[122,15153,15154,15162],{},[125,15155,15156,15158,15160],{},[131,15157,862],{},[652,15159,1349],{},[131,15161,1476],{"mathvariant":1352},[138,15163,6279],{"encoding":140},[109,15165,15167,15185],{"className":15166,"ariaHidden":146},[145],[109,15168,15170,15173,15176,15179,15182],{"className":15169},[150],[109,15171],{"className":15172,"style":6289},[154],[109,15174,862],{"className":15175,"style":905},[159,163],[109,15177],{"className":15178,"style":1646},[691],[109,15180,1349],{"className":15181},[1682],[109,15183],{"className":15184,"style":1646},[691],[109,15186,15188,15191],{"className":15187},[150],[109,15189],{"className":15190,"style":3681},[154],[109,15192,1476],{"className":15193,"style":2333},[159,1686],[106,15195,15196,15197,15285],{},"Kopplungszeile für die Fixgebühr ",[109,15198,15200,15219],{"className":15199},[112],[109,15201,15203],{"className":15202},[116],[118,15204,15205],{"xmlns":120},[122,15206,15207,15217],{},[125,15208,15209],{},[267,15210,15211,15213,15215],{},[131,15212,271],{},[131,15214,862],{},[275,15216,974],{},[138,15218,977],{"encoding":140},[109,15220,15222],{"className":15221,"ariaHidden":146},[145],[109,15223,15225,15228],{"className":15224},[150],[109,15226],{"className":15227,"style":987},[154],[109,15229,15231,15234],{"className":15230},[159],[109,15232,271],{"className":15233,"style":297},[159,163],[109,15235,15237],{"className":15236},[168],[109,15238,15240,15277],{"className":15239},[172,173],[109,15241,15243,15274],{"className":15242},[177],[109,15244,15246,15257],{"className":15245,"style":310},[181],[109,15247,15248,15251],{"style":1008},[109,15249],{"className":15250,"style":190},[189],[109,15252,15254],{"className":15253},[194,195,196,197],[109,15255,862],{"className":15256,"style":905},[159,163,197],[109,15258,15259,15262],{"style":325},[109,15260],{"className":15261,"style":190},[189],[109,15263,15265],{"className":15264},[194,195,196,197],[109,15266,15268],{"className":15267},[159,197],[109,15269,15271],{"className":15270},[159,338,197],[109,15272,974],{"className":15273},[159,197],[109,15275,206],{"className":15276},[205],[109,15278,15280],{"className":15279},[177],[109,15281,15283],{"className":15282,"style":1044},[181],[109,15284],{},", in summierter Big-M-Form",[106,15287,15288,15289,15317,15318,15387,15388,15476,15477,15578,15579,15670],{},"Wird irgendein Angebot von Lieferant ",[109,15290,15292,15305],{"className":15291},[112],[109,15293,15295],{"className":15294},[116],[118,15296,15297],{"xmlns":120},[122,15298,15299,15303],{},[125,15300,15301],{},[131,15302,862],{},[138,15304,862],{"encoding":140},[109,15306,15308],{"className":15307,"ariaHidden":146},[145],[109,15309,15311,15314],{"className":15310},[150],[109,15312],{"className":15313,"style":947},[154],[109,15315,862],{"className":15316,"style":905},[159,163]," verwendet, muss ",[109,15319,15321,15338],{"className":15320},[112],[109,15322,15324],{"className":15323},[116],[118,15325,15326],{"xmlns":120},[122,15327,15328,15336],{},[125,15329,15330],{},[128,15331,15332,15334],{},[131,15333,859],{},[131,15335,862],{},[138,15337,865],{"encoding":140},[109,15339,15341],{"className":15340,"ariaHidden":146},[145],[109,15342,15344,15347],{"className":15343},[150],[109,15345],{"className":15346,"style":155},[154],[109,15348,15350,15353],{"className":15349},[159],[109,15351,859],{"className":15352},[159,163],[109,15354,15356],{"className":15355},[168],[109,15357,15359,15379],{"className":15358},[172,173],[109,15360,15362,15376],{"className":15361},[177],[109,15363,15365],{"className":15364,"style":893},[181],[109,15366,15367,15370],{"style":410},[109,15368],{"className":15369,"style":190},[189],[109,15371,15373],{"className":15372},[194,195,196,197],[109,15374,862],{"className":15375,"style":905},[159,163,197],[109,15377,206],{"className":15378},[205],[109,15380,15382],{"className":15381},[177],[109,15383,15385],{"className":15384,"style":213},[181],[109,15386],{}," gleich 1 sein, und die Gebühr ",[109,15389,15391,15410],{"className":15390},[112],[109,15392,15394],{"className":15393},[116],[118,15395,15396],{"xmlns":120},[122,15397,15398,15408],{},[125,15399,15400],{},[267,15401,15402,15404,15406],{},[131,15403,271],{},[131,15405,862],{},[275,15407,974],{},[138,15409,977],{"encoding":140},[109,15411,15413],{"className":15412,"ariaHidden":146},[145],[109,15414,15416,15419],{"className":15415},[150],[109,15417],{"className":15418,"style":987},[154],[109,15420,15422,15425],{"className":15421},[159],[109,15423,271],{"className":15424,"style":297},[159,163],[109,15426,15428],{"className":15427},[168],[109,15429,15431,15468],{"className":15430},[172,173],[109,15432,15434,15465],{"className":15433},[177],[109,15435,15437,15448],{"className":15436,"style":310},[181],[109,15438,15439,15442],{"style":1008},[109,15440],{"className":15441,"style":190},[189],[109,15443,15445],{"className":15444},[194,195,196,197],[109,15446,862],{"className":15447,"style":905},[159,163,197],[109,15449,15450,15453],{"style":325},[109,15451],{"className":15452,"style":190},[189],[109,15454,15456],{"className":15455},[194,195,196,197],[109,15457,15459],{"className":15458},[159,197],[109,15460,15462],{"className":15461},[159,338,197],[109,15463,974],{"className":15464},[159,197],[109,15466,206],{"className":15467},[205],[109,15469,15471],{"className":15470},[177],[109,15472,15474],{"className":15473,"style":1044},[181],[109,15475],{}," fällt an. Hier ist ",[109,15478,15480,15505],{"className":15479},[112],[109,15481,15483],{"className":15482},[116],[118,15484,15485],{"xmlns":120},[122,15486,15487,15503],{},[125,15488,15489,15491,15493,15495,15501],{},[131,15490,6608],{},[652,15492,4554],{},[652,15494,5988],{"stretchy":5987},[128,15496,15497,15499],{},[131,15498,1413],{"mathvariant":1352},[131,15500,862],{},[652,15502,5988],{"stretchy":5987},[138,15504,6623],{"encoding":140},[109,15506,15508,15526],{"className":15507,"ariaHidden":146},[145],[109,15509,15511,15514,15517,15520,15523],{"className":15510},[150],[109,15512],{"className":15513,"style":3681},[154],[109,15515,6608],{"className":15516,"style":3349},[159,163],[109,15518],{"className":15519,"style":1646},[691],[109,15521,4554],{"className":15522},[1682],[109,15524],{"className":15525,"style":1646},[691],[109,15527,15529,15532,15535,15575],{"className":15528},[150],[109,15530],{"className":15531,"style":6166},[154],[109,15533,5988],{"className":15534},[1817],[109,15536,15538,15541],{"className":15537},[159],[109,15539,1413],{"className":15540,"style":1984},[159,1686],[109,15542,15544],{"className":15543},[168],[109,15545,15547,15567],{"className":15546},[172,173],[109,15548,15550,15564],{"className":15549},[177],[109,15551,15553],{"className":15552,"style":893},[181],[109,15554,15555,15558],{"style":6190},[109,15556],{"className":15557,"style":190},[189],[109,15559,15561],{"className":15560},[194,195,196,197],[109,15562,862],{"className":15563,"style":905},[159,163,197],[109,15565,206],{"className":15566},[205],[109,15568,15570],{"className":15569},[177],[109,15571,15573],{"className":15572,"style":213},[181],[109,15574],{},[109,15576,5988],{"className":15577},[2268],", die Anzahl seiner Angebote und damit der größtmögliche Wert der Summe, sodass die Zeile für ",[109,15580,15582,15603],{"className":15581},[112],[109,15583,15585],{"className":15584},[116],[118,15586,15587],{"xmlns":120},[122,15588,15589,15601],{},[125,15590,15591,15597,15599],{},[128,15592,15593,15595],{},[131,15594,859],{},[131,15596,862],{},[652,15598,4554],{},[648,15600,657],{},[138,15602,6722],{"encoding":140},[109,15604,15606,15661],{"className":15605,"ariaHidden":146},[145],[109,15607,15609,15612,15652,15655,15658],{"className":15608},[150],[109,15610],{"className":15611,"style":155},[154],[109,15613,15615,15618],{"className":15614},[159],[109,15616,859],{"className":15617},[159,163],[109,15619,15621],{"className":15620},[168],[109,15622,15624,15644],{"className":15623},[172,173],[109,15625,15627,15641],{"className":15626},[177],[109,15628,15630],{"className":15629,"style":893},[181],[109,15631,15632,15635],{"style":410},[109,15633],{"className":15634,"style":190},[189],[109,15636,15638],{"className":15637},[194,195,196,197],[109,15639,862],{"className":15640,"style":905},[159,163,197],[109,15642,206],{"className":15643},[205],[109,15645,15647],{"className":15646},[177],[109,15648,15650],{"className":15649,"style":213},[181],[109,15651],{},[109,15653],{"className":15654,"style":1646},[691],[109,15656,4554],{"className":15657},[1682],[109,15659],{"className":15660,"style":1646},[691],[109,15662,15664,15667],{"className":15663},[150],[109,15665],{"className":15666,"style":4642},[154],[109,15668,657],{"className":15669},[159]," unschädlich ist.",[85,15672,15673,15675,16190,16242,16426],{},[106,15674,6794],{},[106,15676,15677],{},[109,15678,15680,15759],{"className":15679},[112],[109,15681,15683],{"className":15682},[116],[118,15684,15685],{"xmlns":120},[122,15686,15687,15757],{},[125,15688,15689,15705,15707,15715,15721,15723,15731,15737,15739,15741,15749,15751],{},[128,15690,15691,15693],{},[652,15692,1341],{},[125,15694,15695,15697,15699],{},[131,15696,377],{},[652,15698,1349],{},[128,15700,15701,15703],{},[131,15702,1413],{"mathvariant":1352},[131,15704,862],{},[652,15706,1374],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[267,15708,15709,15711,15713],{},[131,15710,6831],{},[131,15712,377],{},[275,15714,768],{},[128,15716,15717,15719],{},[131,15718,578],{},[131,15720,377],{},[652,15722,1396],{},[267,15724,15725,15727,15729],{},[131,15726,6831],{},[131,15728,377],{},[275,15730,486],{},[128,15732,15733,15735],{},[131,15734,374],{},[131,15736,377],{},[652,15738,1455],{"fence":146,"stretchy":146,"minsize":1418,"maxsize":1418},[652,15740,3764],{},[267,15742,15743,15745,15747],{},[131,15744,6831],{},[131,15746,862],{},[275,15748,6870],{},[275,15750,1385],{},[128,15752,15753,15755],{},[131,15754,1067],{},[131,15756,862],{},[138,15758,6881],{"encoding":140},[109,15760,15762,15966,16084],{"className":15761,"ariaHidden":146},[145],[109,15763,15765,15768,15854,15860,15917,15957,15960,15963],{"className":15764},[150],[109,15766],{"className":15767,"style":6891},[154],[109,15769,15771,15774],{"className":15770},[1535],[109,15772,1341],{"className":15773,"style":3802},[1535,1699,3801],[109,15775,15777],{"className":15776},[168],[109,15778,15780,15846],{"className":15779},[172,173],[109,15781,15783,15843],{"className":15782},[177],[109,15784,15786],{"className":15785,"style":3815},[181],[109,15787,15788,15791],{"style":3818},[109,15789],{"className":15790,"style":190},[189],[109,15792,15794],{"className":15793},[194,195,196,197],[109,15795,15797,15800,15803],{"className":15796},[159,197],[109,15798,377],{"className":15799},[159,163,197],[109,15801,1349],{"className":15802},[1682,197],[109,15804,15806,15809],{"className":15805},[159,197],[109,15807,1413],{"className":15808,"style":1984},[159,1686,197],[109,15810,15812],{"className":15811},[168],[109,15813,15815,15835],{"className":15814},[172,173],[109,15816,15818,15832],{"className":15817},[177],[109,15819,15821],{"className":15820,"style":6071},[181],[109,15822,15823,15826],{"style":6074},[109,15824],{"className":15825,"style":1771},[189],[109,15827,15829],{"className":15828},[194,1775,1776,197],[109,15830,862],{"className":15831,"style":905},[159,163,197],[109,15833,206],{"className":15834},[205],[109,15836,15838],{"className":15837},[177],[109,15839,15841],{"className":15840,"style":6093},[181],[109,15842],{},[109,15844,206],{"className":15845},[205],[109,15847,15849],{"className":15848},[177],[109,15850,15852],{"className":15851,"style":6105},[181],[109,15853],{},[109,15855,15857],{"className":15856},[1817],[109,15858,1374],{"className":15859},[1821,1776],[109,15861,15863,15866],{"className":15862},[159],[109,15864,6831],{"className":15865,"style":6990},[159,163],[109,15867,15869],{"className":15868},[168],[109,15870,15872,15909],{"className":15871},[172,173],[109,15873,15875,15906],{"className":15874},[177],[109,15876,15878,15889],{"className":15877,"style":310},[181],[109,15879,15880,15883],{"style":7005},[109,15881],{"className":15882,"style":190},[189],[109,15884,15886],{"className":15885},[194,195,196,197],[109,15887,377],{"className":15888},[159,163,197],[109,15890,15891,15894],{"style":325},[109,15892],{"className":15893,"style":190},[189],[109,15895,15897],{"className":15896},[194,195,196,197],[109,15898,15900],{"className":15899},[159,197],[109,15901,15903],{"className":15902},[159,338,197],[109,15904,768],{"className":15905},[159,197],[109,15907,206],{"className":15908},[205],[109,15910,15912],{"className":15911},[177],[109,15913,15915],{"className":15914,"style":351},[181],[109,15916],{},[109,15918,15920,15923],{"className":15919},[159],[109,15921,578],{"className":15922},[159,163],[109,15924,15926],{"className":15925},[168],[109,15927,15929,15949],{"className":15928},[172,173],[109,15930,15932,15946],{"className":15931},[177],[109,15933,15935],{"className":15934,"style":182},[181],[109,15936,15937,15940],{"style":410},[109,15938],{"className":15939,"style":190},[189],[109,15941,15943],{"className":15942},[194,195,196,197],[109,15944,377],{"className":15945},[159,163,197],[109,15947,206],{"className":15948},[205],[109,15950,15952],{"className":15951},[177],[109,15953,15955],{"className":15954,"style":213},[181],[109,15956],{},[109,15958],{"className":15959,"style":1931},[691],[109,15961,1396],{"className":15962},[1935],[109,15964],{"className":15965,"style":1931},[691],[109,15967,15969,15972,16029,16069,16075,16078,16081],{"className":15968},[150],[109,15970],{"className":15971,"style":4005},[154],[109,15973,15975,15978],{"className":15974},[159],[109,15976,6831],{"className":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die Big-M-Form von „wenn ",[109,16246,16248,16269],{"className":16247},[112],[109,16249,16251],{"className":16250},[116],[118,16252,16253],{"xmlns":120},[122,16254,16255,16267],{},[125,16256,16257,16263,16265],{},[128,16258,16259,16261],{},[131,16260,1067],{},[131,16262,862],{},[652,16264,4554],{},[648,16266,657],{},[138,16268,7396],{"encoding":140},[109,16270,16272,16327],{"className":16271,"ariaHidden":146},[145],[109,16273,16275,16278,16318,16321,16324],{"className":16274},[150],[109,16276],{"className":16277,"style":1082},[154],[109,16279,16281,16284],{"className":16280},[159],[109,16282,1067],{"className":16283,"style":1089},[159,163],[109,16285,16287],{"className":16286},[168],[109,16288,16290,16310],{"className":16289},[172,173],[109,16291,16293,16307],{"className":16292},[177],[109,16294,16296],{"className":16295,"style":893},[181],[109,16297,16298,16301],{"style":1104},[109,16299],{"className":16300,"style":190},[189],[109,16302,16304],{"className":16303},[194,195,196,197],[109,16305,862],{"className":16306,"style":905},[159,163,197],[109,16308,206],{"className":16309},[205],[109,16311,16313],{"className":16312},[177],[109,16314,16316],{"className":16315,"style":213},[181],[109,16317],{},[109,16319],{"className":16320,"style":1646},[691],[109,16322,4554],{"className":16323},[1682],[109,16325],{"className":16326,"style":1646},[691],[109,16328,16330,16333],{"className":16329},[150],[109,16331],{"className":16332,"style":4642},[154],[109,16334,657],{"className":16335},[159],", betragen die Ausgaben mindestens ",[109,16338,16340,16359],{"className":16339},[112],[109,16341,16343],{"className":16342},[116],[118,16344,16345],{"xmlns":120},[122,16346,16347,16357],{},[125,16348,16349],{},[267,16350,16351,16353,16355],{},[131,16352,6831],{},[131,16354,862],{},[275,16356,6870],{},[138,16358,7487],{"encoding":140},[109,16360,16362],{"className":16361,"ariaHidden":146},[145],[109,16363,16365,16368],{"className":16364},[150],[109,16366],{"className":16367,"style":987},[154],[109,16369,16371,16374],{"className":16370},[159],[109,16372,6831],{"className":16373,"style":6990},[159,163],[109,16375,16377],{"className":16376},[168],[109,16378,16380,16417],{"className":16379},[172,173],[109,16381,16383,16414],{"className":16382},[177],[109,16384,16386,16397],{"className":16385,"style":310},[181],[109,16387,16388,16391],{"style":7236},[109,16389],{"className":16390,"style":190},[189],[109,16392,16394],{"className":16393},[194,195,196,197],[109,16395,862],{"className":16396,"style":905},[159,163,197],[109,16398,16399,16402],{"style":325},[109,16400],{"className":16401,"style":190},[189],[109,16403,16405],{"className":16404},[194,195,196,197],[109,16406,16408],{"className":16407},[159,197],[109,16409,16411],{"className":16410},[159,338,197],[109,16412,6870],{"className":16413},[159,197],[109,16415,206],{"className":16416},[205],[109,16418,16420],{"className":16419},[177],[109,16421,16423],{"className":16422,"style":1044},[181],[109,16424],{},"“",[106,16427,16428,16429,16520,16521,16549,16550,16638,16639,16730],{},"Der Versand kann nur dann kostenfrei sein, ",[109,16430,16432,16453],{"className":16431},[112],[109,16433,16435],{"className":16434},[116],[118,16436,16437],{"xmlns":120},[122,16438,16439,16451],{},[125,16440,16441,16447,16449],{},[128,16442,16443,16445],{},[131,16444,1067],{},[131,16446,862],{},[652,16448,4554],{},[648,16450,657],{},[138,16452,7396],{"encoding":140},[109,16454,16456,16511],{"className":16455,"ariaHidden":146},[145],[109,16457,16459,16462,16502,16505,16508],{"className":16458},[150],[109,16460],{"className":16461,"style":1082},[154],[109,16463,16465,16468],{"className":16464},[159],[109,16466,1067],{"className":16467,"style":1089},[159,163],[109,16469,16471],{"className":16470},[168],[109,16472,16474,16494],{"className":16473},[172,173],[109,16475,16477,16491],{"className":16476},[177],[109,16478,16480],{"className":16479,"style":893},[181],[109,16481,16482,16485],{"style":1104},[109,16483],{"className":16484,"style":190},[189],[109,16486,16488],{"className":16487},[194,195,196,197],[109,16489,862],{"className":16490,"style":905},[159,163,197],[109,16492,206],{"className":16493},[205],[109,16495,16497],{"className":16496},[177],[109,16498,16500],{"className":16499,"style":213},[181],[109,16501],{},[109,16503],{"className":16504,"style":1646},[691],[109,16506,4554],{"className":16507},[1682],[109,16509],{"className":16510,"style":1646},[691],[109,16512,16514,16517],{"className":16513},[150],[109,16515],{"className":16516,"style":4642},[154],[109,16518,657],{"className":16519},[159],", wenn die Ausgaben bei Lieferant ",[109,16522,16524,16537],{"className":16523},[112],[109,16525,16527],{"className":16526},[116],[118,16528,16529],{"xmlns":120},[122,16530,16531,16535],{},[125,16532,16533],{},[131,16534,862],{},[138,16536,862],{"encoding":140},[109,16538,16540],{"className":16539,"ariaHidden":146},[145],[109,16541,16543,16546],{"className":16542},[150],[109,16544],{"className":16545,"style":947},[154],[109,16547,862],{"className":16548,"style":905},[159,163]," den Wert ",[109,16551,16553,16572],{"className":16552},[112],[109,16554,16556],{"className":16555},[116],[118,16557,16558],{"xmlns":120},[122,16559,16560,16570],{},[125,16561,16562],{},[267,16563,16564,16566,16568],{},[131,16565,6831],{},[131,16567,862],{},[275,16569,6870],{},[138,16571,7487],{"encoding":140},[109,16573,16575],{"className":16574,"ariaHidden":146},[145],[109,16576,16578,16581],{"className":16577},[150],[109,16579],{"className":16580,"style":987},[154],[109,16582,16584,16587],{"className":16583},[159],[109,16585,6831],{"className":16586,"style":6990},[159,163],[109,16588,16590],{"className":16589},[168],[109,16591,16593,16630],{"className":16592},[172,173],[109,16594,16596,16627],{"className":16595},[177],[109,16597,16599,16610],{"className":16598,"style":310},[181],[109,16600,16601,16604],{"style":7236},[109,16602],{"className":16603,"style":190},[189],[109,16605,16607],{"className":16606},[194,195,196,197],[109,16608,862],{"className":16609,"style":905},[159,163,197],[109,16611,16612,16615],{"style":325},[109,16613],{"className":16614,"style":190},[189],[109,16616,16618],{"className":16617},[194,195,196,197],[109,16619,16621],{"className":16620},[159,197],[109,16622,16624],{"className":16623},[159,338,197],[109,16625,6870],{"className":16626},[159,197],[109,16628,206],{"className":16629},[205],[109,16631,16633],{"className":16632},[177],[109,16634,16636],{"className":16635,"style":1044},[181],[109,16637],{}," erreichen. Für ",[109,16640,16642,16663],{"className":16641},[112],[109,16643,16645],{"className":16644},[116],[118,16646,16647],{"xmlns":120},[122,16648,16649,16661],{},[125,16650,16651,16657,16659],{},[128,16652,16653,16655],{},[131,16654,1067],{},[131,16656,862],{},[652,16658,4554],{},[648,16660,650],{},[138,16662,7792],{"encoding":140},[109,16664,16666,16721],{"className":16665,"ariaHidden":146},[145],[109,16667,16669,16672,16712,16715,16718],{"className":16668},[150],[109,16670],{"className":16671,"style":1082},[154],[109,16673,16675,16678],{"className":16674},[159],[109,16676,1067],{"className":16677,"style":1089},[159,163],[109,16679,16681],{"className":16680},[168],[109,16682,16684,16704],{"className":16683},[172,173],[109,16685,16687,16701],{"className":16686},[177],[109,16688,16690],{"className":16689,"style":893},[181],[109,16691,16692,16695],{"style":1104},[109,16693],{"className":16694,"style":190},[189],[109,16696,16698],{"className":16697},[194,195,196,197],[109,16699,862],{"className":16700,"style":905},[159,163,197],[109,16702,206],{"className":16703},[205],[109,16705,16707],{"className":16706},[177],[109,16708,16710],{"className":16709,"style":213},[181],[109,16711],{},[109,16713],{"className":16714,"style":1646},[691],[109,16716,4554],{"className":16717},[1682],[109,16719],{"className":16720,"style":1646},[691],[109,16722,16724,16727],{"className":16723},[150],[109,16725],{"className":16726,"style":4642},[154],[109,16728,650],{"className":16729},[159]," ist die Zeile trivial erfüllt.",[66,16732,7864],{"id":7863},[11,16734,16735,16736,16738,16739,16741],{},"Hier können Sie das MILP aus seiner gezippten ",[61,16737,7870],{},"-Datei laden und mit CP-SAT lösen. Beachten Sie, dass die Leistung von CP-SAT recht empfindlich auf die Zahl der ",[61,16740,63],{}," sowie auf das Zeitlimit reagieren kann.",[7876,16743,16744],{"className":7878,"code":7879,"language":7880,"meta":7881,"style":7881},[61,16745,16746,16760,16766,16770,16774,16782,16804,16808,16820,16828,16838,16852,16860,16864,16884,16906],{"__ignoreMap":7881},[109,16747,16748,16750,16752,16754,16756,16758],{"class":7886,"line":7887},[109,16749,7891],{"class":7890},[109,16751,7895],{"class":7894},[109,16753,7898],{"class":7890},[109,16755,7901],{"class":7894},[109,16757,7904],{"class":7890},[109,16759,7907],{"class":7894},[109,16761,16762,16764],{"class":7886,"line":7910},[109,16763,7898],{"class":7890},[109,16765,7915],{"class":7894},[109,16767,16768],{"class":7886,"line":7918},[109,16769,7922],{"emptyLinePlaceholder":7921},[109,16771,16772],{"class":7886,"line":7925},[109,16773,7929],{"class":7928},[109,16775,16776,16778,16780],{"class":7886,"line":7932},[109,16777,7935],{"class":7894},[109,16779,4554],{"class":7890},[109,16781,7940],{"class":7894},[109,16783,16784,16786,16788,16790,16792,16794,16796,16798,16800,16802],{"class":7886,"line":7943},[109,16785,7946],{"class":7894},[109,16787,1067],{"class":7890},[109,16789,7554],{"class":7951},[109,16791,7955],{"class":7954},[109,16793,7958],{"class":7951},[109,16795,7961],{"class":7954},[109,16797,7964],{"class":7951},[109,16799,3167],{"class":7894},[109,16801,7969],{"class":7951},[109,16803,7972],{"class":7894},[109,16805,16806],{"class":7886,"line":7975},[109,16807,7922],{"emptyLinePlaceholder":7921},[109,16809,16810,16812,16814,16816,16818],{"class":7886,"line":7980},[109,16811,7983],{"class":7894},[109,16813,4554],{"class":7890},[109,16815,7988],{"class":7894},[109,16817,7991],{"class":7951},[109,16819,7994],{"class":7894},[109,16821,16822,16824,16826],{"class":7886,"line":7997},[109,16823,8000],{"class":7894},[109,16825,8003],{"class":7954},[109,16827,7994],{"class":7894},[109,16829,16830,16832,16834,16836],{"class":7886,"line":8008},[109,16831,63],{"class":7954},[109,16833,8013],{"class":7890},[109,16835,8016],{"class":7954},[109,16837,8019],{"class":7928},[109,16839,16840,16842,16844,16846,16848,16850],{"class":7886,"line":8022},[109,16841,8025],{"class":7894},[109,16843,1067],{"class":7890},[109,16845,8030],{"class":7951},[109,16847,8033],{"class":7954},[109,16849,7554],{"class":7951},[109,16851,7994],{"class":7894},[109,16853,16854,16856,16858],{"class":7886,"line":8040},[109,16855,8043],{"class":7894},[109,16857,4554],{"class":7890},[109,16859,8048],{"class":7894},[109,16861,16862],{"class":7886,"line":8051},[109,16863,7922],{"emptyLinePlaceholder":7921},[109,16865,16866,16868,16870,16872,16874,16876,16878,16880,16882],{"class":7886,"line":8056},[109,16867,8059],{"class":7954},[109,16869,1374],{"class":7894},[109,16871,1067],{"class":7890},[109,16873,8066],{"class":7951},[109,16875,8069],{"class":7954},[109,16877,8072],{"class":7894},[109,16879,8075],{"class":7954},[109,16881,7554],{"class":7951},[109,16883,7994],{"class":7894},[109,16885,16886,16888,16890,16892,16894,16896,16898,16900,16902,16904],{"class":7886,"line":8082},[109,16887,8059],{"class":7954},[109,16889,1374],{"class":7894},[109,16891,1067],{"class":7890},[109,16893,8091],{"class":7951},[109,16895,8069],{"class":7954},[109,16897,8096],{"class":7894},[109,16899,8099],{"class":7890},[109,16901,8075],{"class":7954},[109,16903,7554],{"class":7951},[109,16905,7994],{"class":7894},[109,16907,16908,16910,16912,16914,16916,16918,16920,16922,16924,16926,16928,16930],{"class":7886,"line":8108},[109,16909,8059],{"class":7954},[109,16911,1374],{"class":7894},[109,16913,1067],{"class":7890},[109,16915,8117],{"class":7951},[109,16917,8069],{"class":7954},[109,16919,8122],{"class":7894},[109,16921,8125],{"class":7890},[109,16923,8075],{"class":7954},[109,16925,8130],{"class":7951},[109,16927,8033],{"class":7954},[109,16929,8135],{"class":7951},[109,16931,7994],{"class":7894},[8139,16933],{":lines":8141,"label":16934},"Ausgabe",[66,16936,8146],{"id":8145},[11,16938,16939],{},"Im Folgenden bauen wir das Modell mit Pyomo nach (um zu zeigen, wie das geht) und schicken es dann zum Lösen an Quicopts API-Service.",[7876,16941,16942],{"className":7878,"code":8152,"language":7880,"meta":7881,"style":7881},[61,16943,16944,16954,16964,16974,16978,16982],{"__ignoreMap":7881},[109,16945,16946,16948,16950,16952],{"class":7886,"line":7887},[109,16947,7898],{"class":7890},[109,16949,8161],{"class":7894},[109,16951,7904],{"class":7890},[109,16953,8166],{"class":7894},[109,16955,16956,16958,16960,16962],{"class":7886,"line":7910},[109,16957,7891],{"class":7890},[109,16959,8173],{"class":7894},[109,16961,7898],{"class":7890},[109,16963,8178],{"class":7894},[109,16965,16966,16968,16970,16972],{"class":7886,"line":7918},[109,16967,7891],{"class":7890},[109,16969,8185],{"class":7894},[109,16971,7898],{"class":7890},[109,16973,8190],{"class":7894},[109,16975,16976],{"class":7886,"line":7925},[109,16977,7922],{"emptyLinePlaceholder":7921},[109,16979,16980],{"class":7886,"line":7932},[109,16981,8199],{"class":7928},[109,16983,16984,16986,16988],{"class":7886,"line":7943},[109,16985,7935],{"class":7894},[109,16987,4554],{"class":7890},[109,16989,8208],{"class":7894},[11,16991,16992,16993,12161,17046,17080,17081,17084,17085,17119],{},"Als Nächstes führen wir die Entscheidungsvariablen ",[109,16994,16996,17013],{"className":16995},[112],[109,16997,16999],{"className":16998},[116],[118,17000,17001],{"xmlns":120},[122,17002,17003,17011],{},[125,17004,17005,17007,17009],{},[131,17006,133],{"mathvariant":1317},[652,17008,654],{"separator":146},[131,17010,578],{"mathvariant":1317},[138,17012,8232],{"encoding":140},[109,17014,17016],{"className":17015,"ariaHidden":146},[145],[109,17017,17019,17022,17031,17034,17037],{"className":17018},[150],[109,17020],{"className":17021,"style":8242},[154],[109,17023,17025],{"className":17024},[159],[109,17026,17028],{"className":17027},[159],[109,17029,133],{"className":17030,"style":1570},[159,1569],[109,17032,654],{"className":17033},[687],[109,17035],{"className":17036,"style":692},[691],[109,17038,17040],{"className":17039},[159],[109,17041,17043],{"className":17042},[159],[109,17044,578],{"className":17045},[159,1569],[109,17047,17049,17062],{"className":17048},[112],[109,17050,17052],{"className":17051},[116],[118,17053,17054],{"xmlns":120},[122,17055,17056,17060],{},[125,17057,17058],{},[131,17059,374],{"mathvariant":1317},[138,17061,3184],{"encoding":140},[109,17063,17065],{"className":17064,"ariaHidden":146},[145],[109,17066,17068,17071],{"className":17067},[150],[109,17069],{"className":17070,"style":3157},[154],[109,17072,17074],{"className":17073},[159],[109,17075,17077],{"className":17076},[159],[109,17078,374],{"className":17079},[159,1569]," ein, denen wir oben bereits begegnet sind. Beachten Sie, dass ",[56,17082,17083],{},"Nebenbedingungstyp (1)"," bereits in der Pyomo-Deklaration von ",[109,17086,17088,17101],{"className":17087},[112],[109,17089,17091],{"className":17090},[116],[118,17092,17093],{"xmlns":120},[122,17094,17095,17099],{},[125,17096,17097],{},[131,17098,578],{"mathvariant":1317},[138,17100,3220],{"encoding":140},[109,17102,17104],{"className":17103,"ariaHidden":146},[145],[109,17105,17107,17110],{"className":17106},[150],[109,17108],{"className":17109,"style":3157},[154],[109,17111,17113],{"className":17112},[159],[109,17114,17116],{"className":17115},[159],[109,17117,578],{"className":17118},[159,1569]," enthalten ist:",[7876,17121,17122],{"className":7878,"code":8343,"language":7880,"meta":7881,"style":7881},[61,17123,17124,17128,17142,17146,17150,17170,17174,17178,17182,17190,17194,17198,17214,17218,17222,17232],{"__ignoreMap":7881},[109,17125,17126],{"class":7886,"line":7887},[109,17127,8350],{"class":7928},[109,17129,17130,17132,17134,17136,17138,17140],{"class":7886,"line":7910},[109,17131,8355],{"class":7894},[109,17133,4554],{"class":7890},[109,17135,8360],{"class":7894},[109,17137,8364],{"class":8363},[109,17139,4554],{"class":7890},[109,17141,8369],{"class":7894},[109,17143,17144],{"class":7886,"line":7918},[109,17145,7922],{"emptyLinePlaceholder":7921},[109,17147,17148],{"class":7886,"line":7925},[109,17149,8378],{"class":7928},[109,17151,17152,17154,17156,17158,17160,17162,17164,17166,17168],{"class":7886,"line":7932},[109,17153,8383],{"class":7894},[109,17155,4554],{"class":7890},[109,17157,8388],{"class":7894},[109,17159,8364],{"class":8363},[109,17161,4554],{"class":7890},[109,17163,8395],{"class":7894},[109,17165,8398],{"class":8363},[109,17167,8401],{"class":7890},[109,17169,8404],{"class":7894},[109,17171,17172],{"class":7886,"line":7943},[109,17173,8409],{"class":7894},[109,17175,17176],{"class":7886,"line":7975},[109,17177,7922],{"emptyLinePlaceholder":7921},[109,17179,17180],{"class":7886,"line":7980},[109,17181,8418],{"class":7928},[109,17183,17184,17186,17188],{"class":7886,"line":7997},[109,17185,8423],{"class":7894},[109,17187,4554],{"class":7890},[109,17189,8428],{"class":7894},[109,17191,17192],{"class":7886,"line":8008},[109,17193,7922],{"emptyLinePlaceholder":7921},[109,17195,17196],{"class":7886,"line":8022},[109,17197,8437],{"class":7928},[109,17199,17200,17202,17204,17206,17208,17210,17212],{"class":7886,"line":8040},[109,17201,8442],{"class":7894},[109,17203,4554],{"class":7890},[109,17205,8447],{"class":7894},[109,17207,8450],{"class":7890},[109,17209,8453],{"class":7894},[109,17211,8456],{"class":7890},[109,17213,8459],{"class":7894},[109,17215,17216],{"class":7886,"line":8051},[109,17217,7922],{"emptyLinePlaceholder":7921},[109,17219,17220],{"class":7886,"line":8056},[109,17221,8468],{"class":7928},[109,17223,17224,17226,17228,17230],{"class":7886,"line":8082},[109,17225,8450],{"class":7890},[109,17227,8475],{"class":7894},[109,17229,8456],{"class":7890},[109,17231,8480],{"class":7894},[109,17233,17234,17236,17238,17240,17242,17244,17246,17248,17250,17252,17254,17256,17258,17260,17262,17264,17266,17268,17270,17272,17274,17276],{"class":7886,"line":8108},[109,17235,8485],{"class":7894},[109,17237,1067],{"class":7890},[109,17239,8490],{"class":7951},[109,17241,8069],{"class":7954},[109,17243,1346],{"class":7894},[109,17245,8075],{"class":7954},[109,17247,7554],{"class":7951},[109,17249,8501],{"class":7894},[109,17251,8450],{"class":7890},[109,17253,8506],{"class":7894},[109,17255,8456],{"class":7890},[109,17257,8511],{"class":7894},[109,17259,8450],{"class":7890},[109,17261,8453],{"class":7894},[109,17263,8456],{"class":7890},[109,17265,8520],{"class":7894},[109,17267,1396],{"class":7890},[109,17269,8525],{"class":7894},[109,17271,8450],{"class":7890},[109,17273,8506],{"class":7894},[109,17275,8456],{"class":7890},[109,17277,8534],{"class":7894},[11,17279,17280],{},"Als Nächstes definieren wir die übrigen Nebenbedingungen, danach die Zielfunktion:",[7876,17282,17283],{"className":7878,"code":8540,"language":7880,"meta":7881,"style":7881},[61,17284,17285,17289,17335,17339,17343,17375,17379,17383,17415,17419,17423,17457,17461,17465,17489,17517,17547,17555,17575,17615,17623,17643,17647,17651],{"__ignoreMap":7881},[109,17286,17287],{"class":7886,"line":7887},[109,17288,8547],{"class":7928},[109,17290,17291,17293,17295,17297,17299,17301,17303,17305,17307,17309,17311,17313,17315,17317,17319,17321,17323,17325,17327,17329,17331,17333],{"class":7886,"line":7910},[109,17292,8552],{"class":7894},[109,17294,4554],{"class":7890},[109,17296,8557],{"class":7894},[109,17298,8560],{"class":8363},[109,17300,8401],{"class":7890},[109,17302,8565],{"class":7894},[109,17304,8568],{"class":7954},[109,17306,8571],{"class":7894},[109,17308,8574],{"class":7890},[109,17310,8577],{"class":7894},[109,17312,8580],{"class":7890},[109,17314,8583],{"class":7894},[109,17316,8574],{"class":7890},[109,17318,8588],{"class":7894},[109,17320,8450],{"class":7890},[109,17322,8453],{"class":7894},[109,17324,8456],{"class":7890},[109,17326,8597],{"class":7894},[109,17328,8600],{"class":7890},[109,17330,8603],{"class":7894},[109,17332,8574],{"class":7890},[109,17334,8608],{"class":7894},[109,17336,17337],{"class":7886,"line":7918},[109,17338,7922],{"emptyLinePlaceholder":7921},[109,17340,17341],{"class":7886,"line":7925},[109,17342,8617],{"class":7928},[109,17344,17345,17347,17349,17351,17353,17355,17357,17359,17361,17363,17365,17367,17369,17371,17373],{"class":7886,"line":7932},[109,17346,8622],{"class":7894},[109,17348,4554],{"class":7890},[109,17350,8627],{"class":7894},[109,17352,8560],{"class":8363},[109,17354,8401],{"class":7890},[109,17356,8634],{"class":7894},[109,17358,8568],{"class":7954},[109,17360,8639],{"class":7894},[109,17362,8450],{"class":7890},[109,17364,8506],{"class":7894},[109,17366,8456],{"class":7890},[109,17368,8648],{"class":7894},[109,17370,8651],{"class":7890},[109,17372,8654],{"class":7954},[109,17374,7994],{"class":7894},[109,17376,17377],{"class":7886,"line":7943},[109,17378,7922],{"emptyLinePlaceholder":7921},[109,17380,17381],{"class":7886,"line":7975},[109,17382,8665],{"class":7928},[109,17384,17385,17387,17389,17391,17393,17395,17397,17399,17401,17403,17405,17407,17409,17411,17413],{"class":7886,"line":7980},[109,17386,8670],{"class":7894},[109,17388,4554],{"class":7890},[109,17390,8675],{"class":7894},[109,17392,8560],{"class":8363},[109,17394,8401],{"class":7890},[109,17396,8682],{"class":7894},[109,17398,8568],{"class":7954},[109,17400,8687],{"class":7894},[109,17402,8450],{"class":7890},[109,17404,8453],{"class":7894},[109,17406,8456],{"class":7890},[109,17408,8696],{"class":7894},[109,17410,8699],{"class":7890},[109,17412,8654],{"class":7954},[109,17414,7994],{"class":7894},[109,17416,17417],{"class":7886,"line":7997},[109,17418,7922],{"emptyLinePlaceholder":7921},[109,17420,17421],{"class":7886,"line":8008},[109,17422,8712],{"class":7928},[109,17424,17425,17427,17429,17431,17433,17435,17437,17439,17441,17443,17445,17447,17449,17451,17453,17455],{"class":7886,"line":8022},[109,17426,8717],{"class":7894},[109,17428,4554],{"class":7890},[109,17430,8722],{"class":7894},[109,17432,8560],{"class":8363},[109,17434,8401],{"class":7890},[109,17436,8729],{"class":7894},[109,17438,8568],{"class":7954},[109,17440,8571],{"class":7894},[109,17442,8574],{"class":7890},[109,17444,8577],{"class":7894},[109,17446,8450],{"class":7890},[109,17448,8453],{"class":7894},[109,17450,8456],{"class":7890},[109,17452,8746],{"class":7894},[109,17454,8699],{"class":7890},[109,17456,8751],{"class":7894},[109,17458,17459],{"class":7886,"line":8040},[109,17460,7922],{"emptyLinePlaceholder":7921},[109,17462,17463],{"class":7886,"line":8051},[109,17464,8760],{"class":7928},[109,17466,17467,17469,17471,17473,17475,17477,17479,17481,17483,17485,17487],{"class":7886,"line":8056},[109,17468,8765],{"class":7894},[109,17470,4554],{"class":7890},[109,17472,8770],{"class":7894},[109,17474,8568],{"class":7954},[109,17476,8775],{"class":7894},[109,17478,8574],{"class":7890},[109,17480,8780],{"class":7894},[109,17482,8450],{"class":7890},[109,17484,8506],{"class":7894},[109,17486,8456],{"class":7890},[109,17488,8789],{"class":7894},[109,17490,17491,17493,17495,17497,17499,17501,17503,17505,17507,17509,17511,17513,17515],{"class":7886,"line":8082},[109,17492,8794],{"class":7890},[109,17494,8797],{"class":7954},[109,17496,8800],{"class":7894},[109,17498,8574],{"class":7890},[109,17500,8577],{"class":7894},[109,17502,1396],{"class":7890},[109,17504,8809],{"class":7894},[109,17506,8574],{"class":7890},[109,17508,8588],{"class":7894},[109,17510,8450],{"class":7890},[109,17512,8453],{"class":7894},[109,17514,8456],{"class":7890},[109,17516,8822],{"class":7894},[109,17518,17519,17521,17523,17525,17527,17529,17531,17533,17535,17537,17539,17541,17543,17545],{"class":7886,"line":8108},[109,17520,8794],{"class":7890},[109,17522,8797],{"class":7954},[109,17524,8831],{"class":7894},[109,17526,8834]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dann schicken wir es ab:",[7876,17675,17676],{"className":7878,"code":9015,"language":7880,"meta":7881,"style":7881},[61,17677,17678,17682,17706],{"__ignoreMap":7881},[109,17679,17680],{"class":7886,"line":7887},[109,17681,9022],{"class":7928},[109,17683,17684,17686,17688,17690,17692,17694,17696,17698,17700,17702,17704],{"class":7886,"line":7910},[109,17685,9027],{"class":7894},[109,17687,4554],{"class":7890},[109,17689,9032],{"class":7894},[109,17691,9035],{"class":8363},[109,17693,4554],{"class":7890},[109,17695,9040],{"class":7954},[109,17697,3167],{"class":7894},[109,17699,9045],{"class":8363},[109,17701,4554],{"class":7890},[109,17703,9050],{"class":7954},[109,17705,7994],{"class":7894},[109,17707,17708,17710],{"class":7886,"line":7918},[109,17709,8059],{"class":7954},[109,17711,9059],{"class":7894},[8139,17713],{":lines":9062,"label":16934},[11,17715,17716],{},"Sehen wir uns die Qualität der Lösung an:",[7876,17718,17719],{"className":7878,"code":9068,"language":7880,"meta":7881,"style":7881},[61,17720,17721,17743],{"__ignoreMap":7881},[109,17722,17723,17725,17727,17729,17731,17733,17735,17737,17739,17741],{"class":7886,"line":7887},[109,17724,8059],{"class":7954},[109,17726,1374],{"class":7894},[109,17728,1067],{"class":7890},[109,17730,9081],{"class":7951},[109,17732,8069],{"class":7954},[109,17734,8096],{"class":7894},[109,17736,8099],{"class":7890},[109,17738,8075],{"class":7954},[109,17740,7554],{"class":7951},[109,17742,7994],{"class":7894},[109,17744,17745,17747,17749,17751,17753,17755,17757,17759,17761,17763],{"class":7886,"line":7910},[109,17746,8059],{"class":7954},[109,17748,1374],{"class":7894},[109,17750,1067],{"class":7890},[109,17752,9104],{"class":7951},[109,17754,8069],{"class":7954},[109,17756,9109],{"class":7894},[109,17758,8099],{"class":7890},[109,17760,8075],{"class":7954},[109,17762,7554],{"class":7951},[109,17764,7994],{"class":7894},[8139,17766],{":lines":9120,"label":16934},[11,17768,17769,17770,17773,17774,17777,17778,17782],{},"Quicopt hat in weniger als ",[56,17771,17772],{},"zwei Sekunden"," eine ",[56,17775,17776],{},"bessere Lösung"," gefunden! CP-SAT hat 60 s auf acht CPUs investiert (womit es manchmal tatsächlich gelingt, dieselbe Lösung wie Quicopt zu finden). Das ",[24,17779,17781],{"href":40,"rel":17780},[28],"Notebook"," auf unserem GitHub enthält zusätzliche Nachweise, dass die Lösung des Quicopt-Solvers in der ursprünglichen MILP-Formulierung korrekt ist.",[9138,17784,9140],{},{"title":7881,"searchDepth":7910,"depth":7910,"links":17786},[17787,17792,17793],{"id":9212,"depth":7910,"text":9213,"children":17788},[17789,17790,17791],{"id":9216,"depth":7918,"text":9217},{"id":10301,"depth":7918,"text":10302},{"id":12088,"depth":7918,"text":12089},{"id":7863,"depth":7910,"text":7864},{"id":8145,"depth":7910,"text":8146},"Wir tauchen tief in das sogenannte „Internet-Shopping-Optimierungsproblem“ ein, auch bekannt als „Lieferantenauswahl und Bestellaufteilung“. Es tritt praktisch immer dann auf, wenn eine große Zahl von Artikeln bei einer großen Zahl von Lieferanten bestellt werden muss. Da es um ganzzahlige Mengen geht, wird es üblicherweise als gemischt-ganzzahliges lineares Problem (MILP) formuliert. Leider gelten MILPs als NP-schwer – das heißt mehr oder weniger, dass sie sehr schwer zu lösen sein können.",{},"\u002Fuse-cases\u002Finternet-shopping.de",{"title":9160,"description":17794},"use-cases\u002Finternet-shopping.de","r12peCNFMj90S6rC13aVD2hKgM7slIqATIaB-REFj_g",1792145071725]