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tings for A and B have some workstations in common (i.e., shared) but also contain unique (unshared) workstations. Because Youohimga does not always have suf?cient capacity to meet demand, especially during the peak demand period (i.e., the months near the start of the school year in September), in the past it has contracted out production of some of its products to vendors (i.e., the vendors manufacture devices that are shipped out under Youohimga’s label). This year, Youohimga has decided to

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nufacture devices that are shipped out under Youohimga’s label). This year, Youohimga has decided to use a systematic aggregate planning process to determine vendoring needs and a long-term production plan. (a) Using the following notation X it = units of family i (i = A, B) produced in month t (t = 1, . . . , 24) and available to meet demand in month t 598 Part III Principles in Practice Vit = units of family i purchased from vendor in month t and available to meet demand in month t Iit =

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it = units of family i purchased from vendor in month t and available to meet demand in month t Iit = ?nished goods inventory of family i at end of month t dit = units of family i demanded (and shipped) during month t c jt = hours available on work center j( j = 1, . . . , 10) in month t ai j = hours required at work center j per unit of family i v i = premium (i.e., extra cost) per unit of family i that is vendored instead of being produced in-house h i = holding cost to carry one unit of fami

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ly i that is vendored instead of being produced in-house h i = holding cost to carry one unit of family i in inventory from one month to the next formulate a linear program that minimizes the cost (holding plus vendoring premium) over a two-year (24-month) planning horizon of meeting monthly demand (i.e., no backorders are permitted). You may assume that vendor capacity for both families is unlimited and that there is no inventory of either family on hand at the beginning of the planning horizo

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mited and that there is no inventory of either family on hand at the beginning of the planning horizon. (b) Which of the following factors might make sense to examine in the aggregate planning model to help formulate a sensible vendoring strategy? r Altering machine capacities r Sequencing and scheduling r Varying size of workforce r Alternate shop ?oor control mechanisms r Vendoring individual operations rather than complete products r All the above (c) Suppose you run the model in part (a) an

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operations rather than complete products r All the above (c) Suppose you run the model in part (a) and it suggests vendoring 50 percent of the total demand for family A and 50 percent of the demand for B. Vendoring 100 percent of A and 0 percent of B is capacity-feasible, but results in a higher cost in the model. Could the 100–0 plan be preferable to the 50–50 plan in practice? If so, explain why. 6. Mr. B. O’Problem of Rancid Industries must decide on a production strategy for two top-secret

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hy. 6. Mr. B. O’Problem of Rancid Industries must decide on a production strategy for two top-secret products, which for security reasons we will call A and B. The questions concern (1) whether to produce these products at all and (2) how much of each to produce. Both products can be produced on a single machine, and there are three brands of machine that can be leased for this purpose. However, because of availability problems, Rancid can lease at most one of each brand of machine. Thus, O’Pro

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because of availability problems, Rancid can lease at most one of each brand of machine. Thus, O’Problem must also decide which, if any, of the machines to lease. The relevant machine and product data are given below: Machine Hours to Produce One Unit of A Hours to Produce One Unit of B Weekly Capacity (hours) Weekly Lease + Operating Cost ($) Brand 1 Brand 2 Brand 3 0.5 0.4 0.6 1.2 1.2 0.8 80 80 80 20,000 22,000 18,000 Product Maximum Demand (units/week) Net Unit Pro?t ($/unit)

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0.8 80 80 80 20,000 22,000 18,000 Product Maximum Demand (units/week) Net Unit Pro?t ($/unit) A B 200 100 150 225 Chapter 16 599 Aggregate and Workforce Planning (a) Letting X i j represent the number of units of product i produced per week on machine j (for example, X A1 is the number of units of A produced on the brand 1 machine), formulate an LP to maximize weekly pro?t (including leasing cost) subject to the capacity and demand constraints. (Hint: Observe that the leasing/oper

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ng leasing cost) subject to the capacity and demand constraints. (Hint: Observe that the leasing/operating cost for a particular machine is only incurred if that machine is used and that this cost is ?xed for any nonzero production level. Carefully de?ne 0–1 integer variables to represent the all-or-nothing aspects of this decision.) (b) Suppose that the suppliers of brand 1 machines and brand 2 machines are feuding and will not service the same company. Show how to modify your formulation to e

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machines are feuding and will not service the same company. Show how to modify your formulation to ensure that Rancid leases either brand 1 or brand 2 or neither, but not both. 7. All-Balsa, Inc., produces two models of bookcases, for which the relevant data are summarized as follows: Selling price Labor required Bottleneck machine time required Raw material required Bookcase 1 Bookcase 2 $15 0.75 hour/unit 1.5 hours/unit 2 bf/unit $8 0.5 hour/unit 0.8 hour/unit 1 bf/unit P1 = units of b

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15 0.75 hour/unit 1.5 hours/unit 2 bf/unit $8 0.5 hour/unit 0.8 hour/unit 1 bf/unit P1 = units of bookcase 1 produced per week P2 = units of bookcase 2 produced per week OT = hours of overtime used per week RM = board-feet of raw material purchased per week A1 = dollars per week spent on advertising bookcase 1 A2 = dollars per week spent on advertising bookcase 2 Each week, up to 400 board feet (bf ) of raw material is available at a cost of $1.50/bf. The company employs four workers, who wor

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t (bf ) of raw material is available at a cost of $1.50/bf. The company employs four workers, who work 40 hours per week for a total regular-time labor supply of 160 hours per week. They work regardless of production volumes, so their salaries are treated as a ?xed cost. Workers can be asked to work overtime and are paid $6 per hour for overtime work. There are 320 hours per week available on the bottleneck machine. In the absence of advertising, 50 units per week of bookcase 1 and 60 units per

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e bottleneck machine. In the absence of advertising, 50 units per week of bookcase 1 and 60 units per week of bookcase 2 will be demanded. Advertising can be used to stimulate demand for each product. Experience shows that each dollar spent on advertising bookcase 1 increases demand for bookcase 1 by 10 units, while each dollar spent on advertising bookcase 2 increases demand for bookcase 2 by 15 units. At most, $100 per week can be spent on advertising. An LP formulation and solution of the pr

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5 units. At most, $100 per week can be spent on advertising. An LP formulation and solution of the problem to determine how much of each product to produce each week, how much raw material to buy, how much overtime to use, and how much advertising to buy are given below. Answer the following on the basis of this output. MAX 15 P1 + 8 P2 - 6 OT - 1.5 RM - A1 - A2 SUBJECT TO 2) P1 - 10 A1 <= 50 3) P2 - 15 A2 <= 60 4) 0.75 P1 + 0.5 P2 - OT <= 160 5) 2 P1 + P2 - RM <= 0 6) RM <= 400 7) A1 + A2 <= 1

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P2 - 15 A2 <= 60 4) 0.75 P1 + 0.5 P2 - OT <= 160 5) 2 P1 + P2 - RM <= 0 6) RM <= 400 7) A1 + A2 <= 100 8) 1.5 P1 + 0.8 P2 <= 320 END 600 Part III Principles in Practice OBJECTIVE FUNCTION VALUE 1) 2427.66700 VARIABLE P1 P2 OT RM A1 A2 VALUE 160.000000 80.000000 .000000 400.000000 11.000000 1.333333 REDUCED COST .000000 .000000 2.133334 .000000 .000000 .000000 ROW 2) 3) 4) 5) 6) 7) 8) SLACK OR SURPLUS .000000 .000000 .000000 .000000 .000000 87.666660 16.000000 DUAL PRICES .100000 .0

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SLACK OR SURPLUS .000000 .000000 .000000 .000000 .000000 87.666660 16.000000 DUAL PRICES .100000 .066667 3.866666 6.000000 4.500000 .000000 .000000 NO. ITERATIONS = 5 RANGES IN WHICH THE BASIS IS UNCHANGED: VARIABLE P1 P2 OT RM A1 A2 ROW 2 3 4 5 6 7 8 CURRENT COEF 15.000000 8.000000 -6.000000 -1.500000 -1.000000 -1.000000 OBJ COEFFICIENT RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE .966667 .533333 .266667 .483333 2.133334 INFINITY INFINITY 4.500000 1.000000 5.333335 1.000000 7.249999

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667 .533333 .266667 .483333 2.133334 INFINITY INFINITY 4.500000 1.000000 5.333335 1.000000 7.249999 CURRENT RHS 50.000000 60.000000 160.000000 .000000 400.000000 100.000000 320.000000 RIGHT-HAND SIDE RANGES ALLOWABLE ALLOWABLE INCREASE DECREASE 110.000000 876.666600 20.000000 1315.000000 27.500000 2.500000 6.666667 55.000000 6.666667 55.000000 INFINITY 87.666660 INFINITY 16.000000 (a) If overtime costs only $4 per hour (and all other parameters remain unchanged), how much overtime should All

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time costs only $4 per hour (and all other parameters remain unchanged), how much overtime should All-Balsa use? (b) If each unit of bookcase 1 sold for $15.50 (and all other parameters are unchanged), what will the optimal pro?t per week be—or can you not tell without resolving the LP? (c) What is the most All-Balsa should be willing to pay for another unit of raw material? (d) If each worker were required (as part of the regular workweek) to work 45 hours per week (and all other parameters re

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ere required (as part of the regular workweek) to work 45 hours per week (and all other parameters remained unchanged), what would the company’s pro?t be? (e) If each unit of bookcase 2 sold for $10 (and all other parameters remained unchanged), what would be the optimal quantity of bookcase 2 to produce—or can you not tell without resolving the LP? (f) Reconsider the All-Balsa problem formulation and suppose that instead of having 400 bf of raw material available at $1.50/bf, All-Balsa faces a

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on and suppose that instead of having 400 bf of raw material available at $1.50/bf, All-Balsa faces a two-tier pricing scheme such that the ?rst 200 bf/week costs $2.00/bf, but any amount above 200 bf/week up to a Chapter 16 601 Aggregate and Workforce Planning limit of an additional 300 bf/week costs $ p/bf. (Note: p is a constant, not a variable, and we cannot purchase the $ p/bf raw material unless we ?rst purchase 200 bf of the $2.00 raw material.) To modify the LP to compute an “optim

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rial unless we ?rst purchase 200 bf of the $2.00 raw material.) To modify the LP to compute an “optimal” production/advertising policy, we de?ne RM1 = bf of raw material purchased at $2.00/bf RM2 = bf of raw material purchased at $ p/bf To formulate an appropriate LP to represent this new pricing scheme, we ?rst replace 1.5RM in the objective function by 2RM1 + pRM2. i. If p > 2, what other changes in the previous LP make it properly re?ect the new pricing scheme? ii. If p < 2, what other chang

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ges in the previous LP make it properly re?ect the new pricing scheme? ii. If p < 2, what other changes in the previous LP make it properly re?ect the new pricing scheme? 8. Consider a production line with four workstations, labeled j = 1, 2, 3, and 4, in tandem (all products ?ow through all four machines in order). Three different products, labeled i = A, B, and C, are produced on the line. The hours required on each workstation for each product and the net pro?t per unit sold (ri ) are given

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hours required on each workstation for each product and the net pro?t per unit sold (ri ) are given as follows: j i 1 2 3 4 ri A B C 2.4 2.0 0.9 1.1 2.2 0.9 0.8 1.2 1.0 3.0 2.1 2.5 $50 $65 $70 The number of hours available (c jt ) and the upper and lower limits on demand (d¯it and d it ) for each product over the next four quarters are as follows: t 1 2 3 4 c1t c2t c3t c4t 640 640 1,920 1,280 640 640 1,920 1,280 1,280 640 1,920 1,280 1,280 640 1,920 2,560 d¯ At d At d¯

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40 640 1,920 1,280 640 640 1,920 1,280 1,280 640 1,920 1,280 1,280 640 1,920 2,560 d¯ At d At d¯ Bt d Bt d¯Ct d Ct 100 0 100 20 300 0 50 0 100 20 250 0 50 0 100 20 250 0 75 0 100 25 400 50 (a) Suppose we use a quarterly holding cost of $5 and a quarterly backorder cost of $10 per item on all products and allow backordering. Formulate an LP to maximize pro?t minus holding and backorder costs subject to the constraints on workstation capacity and minimum/maximum sales. (b) Using the LP s

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osts subject to the constraints on workstation capacity and minimum/maximum sales. (b) Using the LP solver of your choice, solve your formulation in part (a). Which constraints are binding in your solution? (c) Suppose that there is an inspect operation immediately after station 2 (which has plenty of capacity and therefore does not need to be modeled as an extra resource) and 20 602 Part III Principles in Practice percent of the parts (regardless of product type) are recycled back through

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Principles in Practice percent of the parts (regardless of product type) are recycled back through stations 1 and 2. Show how to modify your formulation in part a to model this. 9. A manufacturer of high-voltage switches projects demand (in units) for the upcoming year to be as follows. Jan Feb Mar Apr May Jun 1,000 1,000 1,000 2,000 2,400 2,500 Jul Aug Sep Oct Nov Dec 3,200 2,000 1,000 900 800 800 The plant runs 160 hours per month and produces at an average rate of 10 switches per hou

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00 800 800 The plant runs 160 hours per month and produces at an average rate of 10 switches per hour. Unit pro?t per switch sold is $50, and the estimated cost to hold a switch in inventory for 1 month is $5. There is no inventory at the start of the year. Overtime can be used at a cost of $300 per hour. (a) Compute the inventory-holding and overtime cost of a chase production strategy (i.e., producing the amount demanded in each month). (b) Compute the inventory holding and overtime cost of

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producing the amount demanded in each month). (b) Compute the inventory holding and overtime cost of a level production strategy (i.e., producing the same amount each month). If the monthly production quantity is set equal to average monthly demand, how much inventory will be left at the end of the year? (c) Compute a production strategy by solving a linear program to maximize pro?t (i.e., net sales revenue minus inventory carrying cost minus overtime cost). Is the amount of overtime in the pla

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ales revenue minus inventory carrying cost minus overtime cost). Is the amount of overtime in the plan reasonable? If not, what changes to the LP model could be made to generate a more reasonable solution? (d) How does the solution change if the inventory carrying cost is reduced to $3 per unit per month? If overtime costs are reduced to $200 per hour? Given that these costs are approximate, what do these results imply about the production plan? 10. Reconsider Problem 2 of Chapter 6 in which a

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t do these results imply about the production plan? 10. Reconsider Problem 2 of Chapter 6 in which a manufacturer produced three models of vacuum cleaner on a three-station production line. (a) Use linear programming to compute a monthly production plan that maximizes monthly pro?t, and compare it to the pro?t resulting from the current plan given in Chapter 6 and those suggested by the labor hours and ABA cost accounting calculations. (b) Could this LP solution have been arrived at by rank-ord

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urs and ABA cost accounting calculations. (b) Could this LP solution have been arrived at by rank-ordering the products according to pro?tability by a cost accounting scheme? What does this say about the effectiveness of using accounting methods to plan production schedules?