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LESSON 14 INVENTORY MODELS (DETERMINISTIC) RESOURCE CONSTRAINED SYSTEMS Outline • Resource Constrained Multi-Product Inventory Systems – Fund constraint – Space constraint 1 Resource Constrained Multiple Product Inventory Systems • So far, we have discussed inventory control models assuming that there is only one product of interest. • Often, there can be multiple products that may compete for the same resource such as fund, space, etc. • In such a case, the EOQ solutions may be satisfactory if the fund/space required by the EOQ solutions is less than that available. • However, The EOQ solutions cannot be implemented if the fund/space required by the EOQ solutions is more than that available. 2 Resource Constrained Multiple Product Inventory Systems • An important observation on the resource constrained models is the following: If the following ratio Resource required per unit Holding cost per unit per year is the same for all products, an optimal order quantity of each product can be obtained by reducing its EOQ value by a constant multiplication factor. • We shall discuss two cases: – Fund constraint (satisfies the above condition) – Space constraint (may not satisfy the condition) 3 Fund Constraint Same Interest Rate For All Products • If the same interest rate is applied on all products, the ratio Cost per unit Holding cost per unit per year Cost per unit 1 (Cost per unit)(Interest Rate) Interest Rate is the same for all products. • Consequently, an optimal order quantity of each product can be obtained by reducing its EOQ value by a constant multiplication factor. 4 Fund Constraint Same Interest Rate For All Products Steps 1. Compute the EOQ values and the total investment required by the EOQ lot sizes. If the investment required does not exceed the budget constraint, stop. 2. Reduce the lot sizes proportionately. To do this, multiply each EOQ value by the constant multiplier, Budget m Total Investment Required by EOQ Lot Sizes 5 Example: Fund Constraint Same Interest Rate For All Products Example 4: A vegetable stand wants to limit the investment in inventory to a maximum of $300. The appropriate data are as follows: Tomatoes Lettuce Zucchini Annual demand 1000 1500 750 (in pounds) Cost/pound $0.29 $0.45 $0.25 The ordering cost is $5 in each case and the annual interest rate is 25%. What are the optimal quantities that should be purchased? 6 Computation for Tomatoes,i 1 h1 Ic1 2 K11 Q1 EOQ1 h1 Fund required c1Q1 Computation for Lettuce,i 2 h2 Ic2 2 K 2 2 Q2 EOQ2 h2 Fund required c2Q2 7 Computation for Zucchini, i 3 h3 Ic3 2 K 3 3 Q3 EOQ3 h3 Fund required c3Q3 Total fund required Fund available Since fund available is less than required, reduce each EOQ value by the constant multiplication factor, Fund available m Total fund required 8 Optimal order quantity for Tomatoes: Q1* mQ1 Optimal order quantity for Lettuce: Q2 mQ2 . * Optimal order quantity for Zucchini : Q3 mQ3 * 9 Example: Fund Constraint Same Interest Rate For All Products Tomatoes Lettuce Zucchini Index, i 1 2 3 Annual Demand, i 1000 1500 750 Ordering/Set-up Cost, Ki 5 5 5 Holding cost/unit/year, hi 0.0725 0.1125 0.0625 Cost/unit, wi 0.29 0.45 0.25 Lot sizes, Qi Fund required Total fund required Fund available 300 Constant multiplier, m Qi reduced proportionately, Q'i 10 Space Constraint • For each product, compute the following ratio Space required per unit Holding cost per unit per year • If the above ratio is the same for all products, a procedure similar to the one for the budget constraint may be applied. • Assume that the above ratios are different for different products (a reason may be that space requirement is not necessarily proportional to costs). 11 Space Constraint Steps 1. Compute the EOQ values and the total space required by the EOQ lot sizes. If the space required does not exceed the space constraint, stop. 2. By trial and error, find a value of such that the space required by the following lot sizes equals the space available: 2 K i i Q * hi 2 wi i 12 Space Constraint Where, Qi Order quantity for product i K i Set - up cost for product i i Annual demand for product i hi Holding cost per unit per year for product i wi Space requiredper unit for product i 13 Example: Space Constraint Example 5: A vegetable stand has exactly 500 square feet of space. The appropriate data are as follows: Tomatoes Lettuce Zucchini Annual demand 1000 1500 750 (in pounds) Space required 0.5 0.4 1 (square feet/pound) Cost/pound $0.29 $0.45 $0.25 The ordering cost is $5 in each case and the annual interest rate is 25%. What are the optimal quantities that should be purchased? 14 Step 1: Check if EOQ values requiremore space Tomatoes,i 1, h1 Ic1 2 K11 Q1 EOQ1 h1 Lettuce,i 2, h2 Ic2 2 K 2 2 Q2 EOQ2 h2 Zucchini, i 3, h3 Ic3 2 K 3 3 Q3 EOQ3 h3 Total space required w1Q1 w2Q2 w3Q3 Since space requiredis more than available, 15 go to Step 2 An optional step between Steps 1 and 2 : Find the range of possible values Space available 500 m 0.7373 Space reuiredby the EOQ lot sizes 678.16 1 2Kii i hi 2 wi m EOQi 2 1 2 5 1000 1 0.0725 0.0609 2 0.50 0.7373 371.392 1 2 5 1500 2 0.1125 0.1181 2 0.40 0.7373 365.152 1 2 5 750 3 0.0625 0.0262 2 1.00 0.7373 346.412 16 An optional step between Steps 1 and 2 (continued) : Find the range of possible values A lower bound on min 1 , 2 , 3 min 0.0609,0.1181,0.0262 0.0262 An upper bound on max1 , 2 , 3 max0.0609,0.1181,0.0262 0.1181 Hence, it is sufficient to search between 0.0262 and 0.1181 17 Step 2 : Trial value of 2 K11 Q1 h1 2w1 Space required w1Q1 2 K 2 2 Q2 h2 2w2 Space required w2Q2 2 K 3 3 Q3 h3 2w3 Space required w3Q3 Total space required Question: Will you increase or decreasethe trial value?18 Step 2 : Trial value of 2 K11 Q1 h1 2w1 Space required w1Q1 2 K 2 2 Q2 h2 2w2 Space required w2Q2 2 K 3 3 Q3 h3 2w3 Space required w3Q3 Total space required Question: Will you increase or decreasethe trial value?19 Step 2 : Trial value of 2 K11 Q1 h1 2w1 Space required w1Q1 2 K 2 2 Q2 h2 2w2 Space required w2Q2 2 K 3 3 Q3 h3 2w3 Space required w3Q3 Total space required Question: When do you stop the trial and error process? 20 Example: Space Constraint Trial value of Tomatoes Lettuce Zucchini Index, i 1 2 3 Annual Demand, i 1000 1500 750 Ordering/Set-up Cost, Ki 5 5 5 Holding cost/unit/year, hi 0.0725 0.1125 0.0625 Space requirement/unit, wi 0.5 0.4 1 Lot sizes, Qi Space required Total space required Space available 500 Conclusion 21 Example: Space Constraint Trial value of 0.1 Tomatoes Lettuce Zucchini Index, i 1 2 3 Annual Demand, i 1000 1500 750 Ordering/Set-up Cost, Ki 5 5 5 Holding cost/unit/year, hi 0.0725 0.1125 0.0625 Space requirement/unit, wi 0.5 0.4 1 Lot sizes, Qi 240.77 279.15 169.03 Space required 120.39 111.66 169.03 Total space required 401.0748 Space available 500 Conclusion Decrease trial value (why?) 22 Example: Space Constraint Trial value of 0.02 Tomatoes Lettuce Zucchini Index, i 1 2 3 Annual Demand, i 1000 1500 750 Ordering/Set-up Cost, Ki 5 5 5 Holding cost/unit/year, hi 0.0725 0.1125 0.0625 Space requirement/unit, wi 0.5 0.4 1 Lot sizes, Qi 328.80 341.66 270.50 Space required 164.40 136.66 270.50 Total space required 571.5639 Space available 500 Conclusion Increase trial value (why?)23 Example: Space Constraint Trial value of 0.04287 Tomatoes Lettuce Zucchini Index, i 1 2 3 Annual Demand, i 1000 1500 750 Ordering/Set-up Cost, Ki 5 5 5 Holding cost/unit/year, hi 0.0725 0.1125 0.0625 Space requirement/unit, wi 0.5 0.4 1 Lot sizes, Qi 294.41 319.66 224.93 Space required 147.21 127.86 224.93 Total space required 499.9997 Space available 500 Conclusion ? 24 Example: Space Constraint Optimal order quantity for Tomatoes: Q1* Q1 Optimal order quantity for Lettuce: Q2 Q2 * Optimal order quantity for Zucchini: Q3 Q3 * 25 READING AND EXERCISES Lesson 14 Reading: Section 4.8 , pp. 221-225 (4th Ed.), pp. 212-215 Exercise: 26, 28, p. 225 (4th Ed.), p. 215 26