G carlsson u of mn isye 0 0 0 0 lecture 10 inventory

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Unformatted text preview: easy: introduce new variables yi = {0, 1} that indicate whether we place an order in period i: Fixed costs = K(y1 + y2 + y3 + y4 ) we need an appropriate constraint: Myi ≥ xi , where M is big J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 48 / 55 Periodic review integer program We have to pay a holding cost for the amount of inventory left over at the end of each period: left over at end of period 1 = x1 − r1 left over at end of period 2 = x1 − r1 + x2 − r2 left over at end of period 3 = x1 − r1 + x2 − r2 + x3 − r3 left over at end of period 4 = x1 − r1 + x2 − r2 + x3 − r3 + x4 − r4 Shortages aren’t allowed: x1 − r1 x1 − r1 + x2 − r2 x1 − r1 + x2 − r2 + x3 − r3 x1 − r1 + x2 − r2 + x3 − r3 + x4 − r4 J. G. Carlsson, U of MN ISyE ≥0 ≥0 ≥0 ≥0 Lecture 10: Inventory Models November 7, 2013 49 / 55 Periodic review integer program minimize K(y1 + y2 + y3 + y4 ) + h(x1 − r1 ) +h(x1 − r1 + x2 − r2 ) +h(x1 − r1 + x2 − r2 + x3 − r3 ) +h(x1 − r1 + x2 − r2 + x3 − r3 + x4 − r4 ) s . t. My1 ≥ x1 My2 ≥ x2 My3 My4 ≥ x3 ≥ x4 x 1 − r1 x1 − r1 + x2 − r2 x1 − r1 + x2 − r2 + x3 − r3 ≥0 ≥0 ≥0 x1 − r1 + x2 − r2 + x3 − r3 + x4 − r4 x1 , x2 , x3 , x4 y1 , y2 , y3 , y4 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models ≥0 ∈ integer, ∈ {0, 1} November 7, 2013 50 / 55 Periodic review integer program minimize 2(y1 + y2 + y3 + y4 ) + 0.2(x1 − 3) +0.2(x1 − 3 + x2 − 2) +0.2(x1 − 3 + x2 − 2 + x3 − 3) +0.2(x1 − 3 + x2 − 2 + x3 − 3 + x4 − 2) My1 My2 s.t. ≥ x1 ≥ x2 My3 My4 ≥ x3 ≥ x4 x1 − 3 x1 − 3 + x2 − 2 x1 − 3 + x2 − 2 + x3 − 3 ≥0 ≥0 ≥0 x1 − 3 + x2 − 2 + x3 − 3 + x4 − 2 x1 , x2 , x3 , x4 y1 , y2 , y3 , y4 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models ≥0 ∈ integer, ∈ {0, 1} November 7, 2013 51 / 55 Periodic review dynamic program Define Ci to be the total cost of an optimal policy for periods i, i + 1, . . . , n when period i starts with zero inventory (before producing) Thus we want to solve for C1 using dynamic programming Of course, C4 is easy because if we start period 4 with zero inventory, we need to purchase r4 = 2 units, and we just pay a fixed cost of 2 To find C3 we need to consider two cases: when is the first time after period 3 when the inventory reaches a zero level? This is either at the end of period 3, or the end of period 4: If inventory reaches a zero level at the end of period 3, then we just pay a fixed cost for period 3 and a fixed cost for period 4 and the cost is 2+2=4 If inventory reaches a zero level at the end of period 4, then we need to buy enough inventory at the beginning of period 3 to cover periods 3 and 4 That’s r3 + r4 = 5 units; we pay a one-time fixed cost, plus a holding cost for the 2 remaining units at the end of period 3 The cost is then 2 + 2 × 0.2 = 2.4 Thus we have C3 = 2.4 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 52 / 55 Periodic review dynamic program To find C2 we need to consider three cases: when is the first time after period...
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