2 2 02 34 thus we have c2 34 j g carlsson u of

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Unformatted text preview: 2 when the inventory reaches a zero level? This is either at the end of period 2, period 3, or period 4: If inventory reaches a zero level at the end of period 2, then we just pay a fixed cost for period 2 plus the remaining cost C3 , which would give 2 + C3 = 2 + 2.4 = 4.4 If inventory reaches a zero level at the end of period 3, then we need to buy enough inventory at period 2 to cover periods 2 and 3 That’s r2 + r3 = 5 units; we pay a one-time fixed cost, plus a holding cost for the 3 remaining units at the end of period 2, plus the remaining cost C4 , which is 2 + 3 × 0.2 + C4 = 4.6 If inventory reaches a zero level at the end of period 4, then we need to buy enough inventory at period 2 to cover periods 2, 3, and 4 That’s r2 + r3 + r4 = 7 units; we pay a one-time fixed cost, plus a holding cost for the 5 remaining units at the end of period 2, plus a holding cost for the 2 remaining units at the end of period 3, which is 2 + 5 × 0.2 + 2 × 0.2 = 3.4 Thus, we have C2 = 3.4 J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 53 / 55 Periodic review dynamic program To find C1 we need to consider four cases: when is the first time after period 1 when the inventory reaches a zero level? This is either at the end of period 1, period 2, period 3, or period 4: If inventory reaches a zero level at the end of period 1, then we just pay a fixed cost for period 1 plus the remaining cost C2 , which would give 2 + C2 = 2 + 3.4 = 5.4 If inventory reaches a zero level at the end of period 2, then we need to buy enough inventory at period 1 to cover periods 1 and 2 That’s r1 + r2 = 5 units; we pay a one-time fixed cost, plus a holding cost for the 2 remaining units at the end of period 1, plus the remaining cost C3 , which is 2 + 2 × 0.2 + C3 = 2.4 + 2.4 = 4.8 If inventory reaches a zero level at the end of period 3, then we need to buy enough inventory at period 1 to cover periods 1, 2, and 3 That’s r1 + r2 + r3 = 8 units; we pay a one-time fixed cost, plus a holding cost for the 5 remaining units at the end of period 1, plus a holding cost for the 3 remaining units at the end of period 2, plus the remaining cost C4 , which is 2 + 5 × 0.2 + 3 × 0.2 + C4 = 5.6 If inventory reaches a zero level at the end of period 4, then we need to buy enough inventory at period 1 to cover periods 1, 2, 3, and 4 That’s r1 + r2 + r3 + r4 = 10 units; we pay a one-time fixed cost, plus a holding cost for the 7 remaining units at the end of period 1, plus a holding cost for the 5 remaining units at the end of period 2, plus a holding cost for the 2 remaining units at the end of period 3, which is 2 + 7 × 0.2 + 5 × 0.2 + 2 × 0.2 = 4.8 Thus, we have C1 = 4.8, which we can get by either buying 10 units at the start, or by buying 5 units at the start and 5 units in period 3 The general formula: Ci = min j=i,i+1,...,n J. G. Carlsson, U of MN ISyE {Cj+1 + K + h[ri+1 + 2ri+2 + 3ri+3 + · · · + (j − i)rj ]} Lecture 10: Inventory Models November 7, 2013 54 / 55 References I [1] F.S. Hillier and G.J. Lieberman. Introduction to Operations Research. Introduction to Operations Research. McGraw-Hill Higher Education, 2010. J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 55 / 55...
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This document was uploaded on 02/23/2014 for the course MANAGMENT 2201 at University of Michigan.

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