Lecture10(1)

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Unformatted text preview: OQ model is the same at cycle lengths √ √ T∗ / 2! Recall that T∗ = K/g: EOQ EOQ √ f( 2T∗ ) = EOQ = = √∗ 2TEOQ and √ K √ ∗ + g · 2 · T∗ EOQ 2TEOQ √√ K √√ + g · 2 · K/g 2 · K/g (√ )√ √ √√ √ 2gK + gK/ 2 = 2 + 1/ 2 gK and √ f(T∗ / 2) = EOQ √ √ + g · T∗ / 2 EOQ 2 √ √ K √ + g · K/g/ 2 =√ K/g/ 2 (√ )√ √ √√ √ = 2gK + gK/ 2 = 2 + 1/ 2 gK J. G. Carlsson, U of MN ISyE K T∗ / EOQ Lecture 10: Inventory Models November 7, 2013 42 / 55 Analysis of Po2 model We just saw that the cost of the EOQ model is the same at cycle lengths (√ )√ √∗ √ √ 2TEOQ and T∗ / 2, which turns out to be 2 + 1/ 2 gK ≈ 2.1213 EOQ √ The cost for the original EOQ model without power-of-2 constraints was 2 Kg √ √ Thus, the cost of the EOQ model at cycle lengths 2T∗ and T∗ / 2 is EOQ EOQ approximately √ √ 2 + 1/ 2 ≈ 1.06 2 times as bad as the optimal solution, i.e. costs are worse by only 6% ∗ We showed earlier that the optimal Po2 cycle length, 2i · TL , must lie √ √ betweenT∗ / 2 and 2T∗ EOQ EOQ ∗ Since the objective function f(T) is convex, we know that the costs f(2i · TL ) √ √ are at least as good as the costs at T∗ / 2 and 2T∗ EOQ EOQ Thus: the Po2 policy can only increase our costs by up to 6%! J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 43 / 55 Periodic review model In the preceding examples we assumed that demand was known and constant In practice this is rarely the case, and the EOQ formula is not as useful to us J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 44 / 55 Periodic review model In the deterministic periodic-review model, we need to plan for n time periods regarding how much to produce or order In the ith period we will have a demand of ri These demands must be met on time (no shortages allowed), and we have no stock on hand initially The costs of interest are the same as in the EOQ: K = setup cost for ordering one batch c = unit cost for producing or purchasing each unit h = holding cost per unit per unit of time in inventory at the end of a period Actually, we don’t even care about the unit costs! Over the n periods, we should of course only produce/purchase r1 + · · · + rn units (the total demand) J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 45 / 55 Periodic review example We have K = $2, setup cost h = $0.2, holding cost r1 = 3 r2 = 2 r3 r4 = = 3 2 Clearly we need to order 3 units in the first period to satisfy r1 , but should we order more? J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 46 / 55 Periodic review example The above shows inventory levels if we buy 3 airplanes in the first period, 6 in the second, and 1 in the fourth The total cost would be: Setup costs: 3 orders placed, so 3 × $2 = $6 Holding costs: none left at end of first period; 4 left at end of second; 1 left at end of third; none left at end of fourth, thus (4 + 1) × $0.2 = $1 thus, $7 in total J. G. Carlsson, U of MN ISyE Lecture 10: Inventory Models November 7, 2013 47 / 55 Periodic review integer program K= h r1 , r2 , r3 , r4 $2, setup cost = $0.2, holding cost = 3, 2, 3, 2 Let xi denote the amount we buy in period i What should the objective function be? Fixed costs are...
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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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