Assignment 2 Answers Introduction to Management Science 2003

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1 Assignment Answers Introduction to Management Science 00. a. Top management will need to know how much to produce in each quarter. Thus, the decisions are the production levels in quarters,,, and. The objective is to maximize the net profit. b. Ending inventory(q) = Starting Inventory(Q) + Production(Q) Sales(Q) =,000 +,000,000 =,000 Ending inventory(q) = Starting Inventory(Q) + Production(Q) Sales(Q) =,000 +,000,000 =,000 Profit from sales(q) = Sales(Q) * ($0) = (,000)($0) = $0,000 Profit from sales(q) = Sales(Q) * ($0) = (,000)($0) = $0,000 Inventory Cost(Q) = Ending Inventory(Q) * ($) = (,000)($) = $,000 Inventory Cost(Q) = Ending Inventory(Q) * ($) = (,000)($) = $,000 c. Inventory Holding Cost Gross Profit from Sales Starting Maximum Demand/ Ending Inventory Gross Profit Inventory Production Production Sales Inventory Cost from Sales Quarter ² ³ Quarter ² ³ Quarter ² ³ Quarter ² ³ d. e. Net Profit A B C D E F G H I J K L M Inventory Holding Cost $ Gross Profit from Sales $0 Starting Maximum Demand/ Ending Inventory Gross Profit Inventory Production Production Sales Inventory Cost from Sales Quarter,000,000 ²,000,000 0 ³ 0 $0 $0,000 Quarter 0,000 ²,000,000 0 ³ 0 $0 $0,000 Totals $0 $,000 Net Profit $,000 A B C D E F G H I J K L M Inventory Holding Cost $ Gross Profit from Sales $0 Starting Maximum Demand/ Ending Inventory Gross Profit Inventory Production Production Sales Inventory Cost from Sales Quarter,000,000 ²,000,000,000 ³ 0 $,000 $0,000 Quarter,000,000 ²,000,000,000 ³ 0 $,000 $0,000 Quarter,000,000 ²,000,000,000 ³ 0 $,000 $,000 Quarter,000,000 ²,000,000 0 ³ 0 $0 $,000 Totals $0,000 $0,000 Net Profit $00,000

2 . a. Web Mercantile needs to know each month how many square feet to lease and for how long. The decisions therefore are for each month how many square feet to lease for one month, for two months, for three months, etc. The objective is to minimize the overall leasing cost. b. = (0,000 square feet)($ per square foot) + (0,000 square feet)($0 per square foot) = $. million. c. Month of Lease: Leased Required Length of Lease: (sq. ft.) (sq. ft.) Month >= Month >= Month >= Month >= Month >= d. e. Cost of Lease Lease (sq. ft.) A B C D E F G Month of Lease: Leased Required Length of Lease: (sq. ft.) (sq. ft.) Month 0,000 ³ 0,000 Month 0,000 ³ 0,000 Cost of Lease $ $0 $ Lease (sq. ft.),000 0,000 0 $,0,000 A B C D E F G H I J K L M N O P Q R S Month of Lease: Leased Required Length of Lease: (sq. ft.) (sq. ft.) Month 0,000 ³ 0,000 Month 0,000 ³ 0,000 Month 0,000 ³ 0,000 Month 0,000 ³,000 Month 0,000 ³ 0,000 Cost of Lease $ $0 $ $ $ $ $0 $ $ $ $0 $ $ $0 $ Lease (sq. ft.) , , ,000 $,0,000. a. Al will need to know how much to invest in each possible investment each year. Thus, the decisions are how much to invest in investment A in year,,, and ; how much to invest in B in year,, and ; how much to invest in C in year ; and how much to invest in D in year. The objective is to accumulate the maximum amount of money by the beginning of year.

3 b. Ending Cash (Y) = $0,000 (Starting Balance) $0,000 (A in Y) = $0,000 Ending Cash (Y) = $0,000 (Starting Balance) $0,000 (B in Y) $0,000 (C in Y) = $0 Ending Cash (Y) = $0 (Starting Balance) + $0,000(.) (for investment A) = $,000 Ending Cash (Y) = $,000 (Starting Balance) Ending Cash (Y) = $,000 (Starting Balance) + $0,000(.) (investment B) = $,000 Ending Cash (Y) = $,000 (Starting Balance) + $0,000(.) (investment C) = $0,000 c. Beginning Balance Minimum Balance Investment A A A A B B B C D Ending Minimum Year Balance Balance Year ³ Year ³ Year ³ Year ³ Year ³ Year ³ d. e. Dollars Invested A B C D E F G H I J K Beginning Balance $0,000 Minimum Balance $0 Investment A A A B B B C Ending Minimum Year Balance Balance Year - - $0 ³ $0 Year $0 ³ $0 Year. - - $,000 ³ $0 Dollars Invested $0,000 $0 $0 $0 $0 $0 $0 A B C D E F G H I J K L M Beginning Balance $0,000 Minimum Balance $0 Investment A A A A B B B C D Ending Minimum Year Balance Balance Year - - $0 ³ $0 Year $0 ³ $0 Year. - - $0 ³ $0 Year. -. $0 ³ $0 Year.. - $0 ³ $0 Year.... $,0 ³ $0 Dollars Invested $0,000 $0 $,000 $0 $0 $0 $0 $0 $,00

4 . a) A B C D E F G H TV Spots Magazine Ads Radio Ads SS Ads Exposures per Ad (thousands) Budget Budget Cost per Ad ($thousands) Spent Available Ad Budget ² 000 Planning Budget ² 00 Total Exposures TV Spots Magazine Ads Radio Ads SS Ads (thousands) Number of Ads 0,00 ² ² Max TV Spots Max Radio Spots Data cells: Changing cells: Target cell: B:E, B:E, H:H, B, and D B:E H F Budget Spent =SUMPRODUCT(B:E,$B$:$E$) =SUMPRODUCT(B:E,$B$:$E$) H Total Exposures (thousands) =SUMPRODUCT(B:E,B:E) b) This is a linear programming model because the decisions are represented by changing cells that can have any value that satisfy the constraints. Each constraint has an output cell on the left, a mathematical sign in the middle, and a data cell on the right. The overall level of performance is represented by the target cell and the objective is to maximize that cell. Also, the Excel equation for each output cell is expressed as a SUMPRODUCT function where each term in the sum is the product of a data cell and a changing cell. c) Let T = number of commercials on TV M = number of advertisements in magazines R = number of commercials on radio S = number of advertisements in Sunday supplements. Maximize Exposures (thousands) = T + 0M + 0R + 0S subject to 00T + M + 00R + 0S,000 ($thousands) 0T + 0M + 0R + 0S,000 ($thousands) T spots R spots and T 0, M 0, R 0, S 0.

5 . a & c) A B C D E F G H Activity Activity Activity Activity Contribution per unit $ $ $ $ Resource Usage Resource Resource per Unit of Activity Used Available Resource P - 00 ² 00 Resource Q - 00 ² 00 Resource R - 00 ² 00 Resource S - 00 ² 00 Activity Activity Activity Activity Total Contribution Level of Activity...0. $,. b) Below are five possible guesses (many answers are possible). (x, x, x, x ) Feasible? P (0,0,0,0) Yes $ (0,0,0,0) No (,,0,0) Yes $ (,,,0) Yes $ (,,,0) Yes $ Best. a) The activities are leasing space in each month for a number of months. The benefit is meeting the space requirements for each month. b) The decisions to be made are how much space to lease and for how many months. The constraints on these decisions are the minimum required space. The overall measure of performance is cost which is to be minimized. c) Month : (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) 0,000 square feet. Month : (M mo lease) + M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) 0,000 square feet. Month : (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) 0,000 square feet. Month : (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease),000 square feet. Month : (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) 0,000 square feet. Nonnegativity: (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M

6 mo lease) 0, (M mo lease) 0, (M mo lease) 0, (M mo lease) 0. d) Cost = ($0)[(M mo lease) + (M mo lease) + (M mo lease) + (M mo lease) + (M mo lease)] + ($,000)[(M mo lease) + (M mo lease) + (M mo lease) + (M mo lease)] + ($,0)[(M mo lease) + (M mo lease) + (M mo lease)] + ($,00)[(M mo lease) + (M mo lease)] + ($,00)[M mo lease] A B C D E F G H I J K L M N O P Q R S Month of Lease: Leased Required Length of Lease: (sq. ft.) (sq. ft.) Month 0,000 ³ 0,000 Month 0,000 ³ 0,000 Month 0,000 ³ 0,000 Month 0,000 ³,000 Month 0,000 ³ 0,000 Cost of Lease $ $0 $ $ $ $ $0 $ $ $ $0 $ $ $0 $ Lease (sq. ft.) , , ,000 $,0,000 Data cells: B:P, B:P, and S:S Changing cells: B:P Target cell: S Output cells: Q:Q Q Total Leased (sq. ft.) =SUMPRODUCT(B:P,$B$:$P$) =SUMPRODUCT(B:P,$B$:$P$) =SUMPRODUCT(B:P,$B$:$P$) =SUMPRODUCT(B:P,$B$:$P$) =SUMPRODUCT(B:P,$B$:$P$) S =SUMPRODUCT(B:P,B:P) e) Let x ij = square feet of space leased in month i for a period of j months. for i =,..., and j =,..., -i. Minimize C = $0(x + x + x + x + x ) + $,000(x + x + x + x ) +$,0(x + x + x ) + $,00(x + x ) + $,00x subject to x + x + x + x + x 0,000 square feet x + x + x + x + x + x + x + x 0,000 square feet x + x + x + x + x + x + x + x + x 0,000 sq. feet x + x + x + x + x + x + x + x,000 square feet x + x + x + x + x 0,000 square feet and x ij 0, for i =,..., and j =,..., -i.