MIS209 - Assignment Solutions 1

Transcription

1 MIS209 - Assignment s 1 Homework 23 1 Breakeven Point and Linear Programming Modeling Three types of manufacturing equipment for engine gaskets are under consideration. Their fixed costs and resulting variable costs per unit are shown in the table below. Fixed cost ($) Variable cost ($/unit) Equipment A 4, Equipment B 7, Equipment C 12, Table 1: Three types of manufacturing equipment and the cost structure. a. At what volume of production would Equipment A and Equipment B cost the same? b. Establish this breakeven point for equipments B and C. c. Suppose the volume anticipated was 8,000 units. Which equipment should be purchased? d. Suppose the volume anticipated was 12,900 units. Which equipment should then be purchased? FC i Fixed cost ($) of Equipment i. vc j Variable cost ($/unit) of Equipment j. Q Volume of production (unit). Table 2: Definitions. a. At what volume of production would Equipment A and Equipment B cost the same? FC A + (vc A Q) = FC B + (vc B Q) 4, (1.90Q) = 7, (1.40Q) Q = 6, 000 b. Establish this breakeven point for equipments B and C. FC B + (vc B Q) = FC C + (vc C Q) 7, (1.40Q) = 12, (1.00Q) Q = 12, 500 c. At Q = 8, 000, Equipment B should be purchased (see graph). d. At Q = 12, 900, Equipment C should be purchased (see graph).

2 mis209 - assignment solutions 2 $(x10 3 ) A B C Eq. A Eq. B Eq. C Unit(x10 3 ) Homework 47 Two brothers want to open a small neighborhood bakery, where they will specialize in loaves of bread. They have carefully estimated their fixed costs to be $110,000 per year. This includes a salary of $35,000 for each brother. The facility they will use is a former pizza kitchen. One problem is that there is limited capacity, so there will be an upper limit on the number of loaves of bread they can produce. Baking 6 days a week, the brothers believe they can produce 150 loaves per day. The cost to them for each loaf is $0.55. They believe they will be able to sell all the bread they can bake. They plan to bake 312 days during the year. a. Given these projections, how much will the brothers have to charge per loaf to stay in business? b. Suppose they are willing to take a much lower salary to get their business started. If they each agreed to take only $25,000 per year, how much would they have to charge for their bread? FC = f c 1 + f c 2 + f c 3 FC = 110, 000 f c 1 = 35, 000, f c 2 = 35, 000, f c 3 = 40, 000

3 mis209 - assignment solutions 3 FC Fixed cost ($) of the bakery. f c 1 Salary for Brother 1. f c 2 Salary for Brother 2. f c 3 Other fixed costs. VC Variable cost ($/unit) of each bread baked. Q Volume of production (unit). p Price for per loaf ($). Table 3: Definitions. Q = = 46, 800 a. Given these projections, how much will the brothers have to charge per loaf to stay in business? Total Income = Total Cost 46, 800 p 1 = 110, ( , 800) p 1 = $2.90 b. Suppose they are willing to take a much lower salary to get their business started. If they each agreed to take only $25,000 per year, how much would they have to charge for their bread? 46, 800 p 2 = 90, ( , 800) p 2 = $2.48 Homework 793 The ThinkBig Company currently manufactures a product which sells for $1.30. The fixed costs associated with this operation are $18,000; the variable costs are $0.65 per unit on a volume of 35,000 units per month. They are considering new equipment which will increase the fixed costs to $26,000 and the variable costs to $0.75, but the demand is expected to increase to 55,000 units, so this option may be attractive. Should ThinkBig make this investment? Why or Why not? The ThinkBig Company is now ready to purchase the new equipment. Two changes to the above plan include a price change to $1.45 per unit, and a revised demand forecast to 45,000 units. Given these circumstances, should the investment be made? Current situation; p = 1.30

4 mis209 - assignment solutions 4 Current situation p Price ($) FC Fixed cost ($) vc Variable cost ($) Q Volume (unit) Table 4: Definitions. FC = 18, 000 vc = 0.65 Q = 35, 000 With new equipment; p = 1.30 FC new = 26, 000 vc new = 0.75 Q new = 55, 000 Current profit; Pro f it = Income Cost Pro f it = (35, ) (18, (35, )) Pro f it = 4, 750 Before purchasing the new equipment; Pro f it investment = (55, ) (26, (55, )) Pro f it investment = 4, 250 The company shouldn t make an investment due to the profit decreasing. After purchasing the new equipment; Pro f it new = (45, ) (26, (45, )) Pro f it new = 5, 500 In this situation the company should purchase the new equipment. Problem from the Textbook George Johnson recently inherited a large sum of money; he wants to use a portion of this money to set up a trust fund for his two children. The trust fund has two investment options: (1) a bond fund and (2) a stock fund. The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund. Whatever portion of the inheritance he finally decides to commit to

5 mis209 - assignment solutions 5 the trust fund, he wants to invest at least 30% of that amount in the bond fund. In addition, he wants to select a mix that will enable him to obtain a total return of at least 7.5%. Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives. Parameters; 6% Return for bonds 10% Return for stocks 30% Minimum investment for bonds 7.5% Desired minimum return Decision variables; x 1 x 2 Bond fund invested Stock fund invested LP Model; Maximize Z = 0.06x x 2 Subject to: 0.06x x (x 1 + x 2 ) x (x 1 + x 2 ) x 1, x 2 0 p.s. Unbounded solution. Problem from the Textbook The Sea Wharf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising. Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount spent on radio advertising. A marketing consultant developed an index that measures audience exposure per dollar of advertising on a scale from 0 to 100, with higher values implying greater audience exposure. If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio advertising is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure? Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of total audience exposure.

6 mis209 - assignment solutions 6 Parameters; $1000 Monthly advertising budget 25% Minimum spending for each type of media 50 Value of the index for local newspaper advertising 80 Value of the index for spot radio advertising Decision variables; x 1 x 2 Newspaper advertising budget Radio advertising budget LP Model; Maximize Z = 50x x 2 Subject to: x 1 + x x x x 1, x 2 0 p.s. Optimum Z = 72, 500, x 1 = 250, x 2 = 750 Problem from the Textbook Tom s, Inc., produces various Mexican food products and sells them to Western Foods, a chain of grocery stores located in Texas and New Mexico. Tom s, Inc., makes two salsa products: Western Foods Salsa and Mexico City Salsa. Essentially, the two products have different blends of whole tomatoes, tomato sauce, and tomato paste. The Western Foods Salsa is a blend of 50% whole tomatoes, 30% tomato sauce, and 20% tomato paste. The Mexico City Salsa, which has a thicker and chunkier consistency, consists of 70% whole tomatoes, 10% tomato sauce, and 20% tomato paste. Each jar of salsa produced weighs 10 ounces. For the current production period, Tom s, Inc., can purchase up to 280 pounds of whole tomatoes, 130 pounds of tomato sauce, and 100 pounds of tomato paste; the price per pound for these ingredients is $0.96, $0.64, and $0.56, respectively. The cost of the spices and the other ingredients is approximately $0.10 per jar. Tom s, Inc., buys empty glass jars for $0.02 each, and labeling and filling costs are estimated to be $0.03 for each jar of salsa produced. Tom s contract with Western Foods results in sales revenue of $1.64 for each jar of Western Foods Salsa and $1.93 for each jar of Mexico City Salsa. Develop a linear programming model that will enable Tom s to determine the mix of salsa products that will maximize the total profit contribution.

7 mis209 - assignment solutions 7 Product Price ($) WT (oz.) TS (oz.) TP (oz.) Table 5: Definitions. Western Foods Salsa Mexico City Salsa Capacity Cost $0.06 $0.04 $0.035 WT: Whole tomatoes, TS: Tomato sauce, TP: Tomato paste Cost o f product per jar = ingredients + jar + WT + TS + TP Cost o f Western Foods Salsa per jar = = $0.62 Cost o f Mexico City Salsa per jar = = $0.66 Decision variables; x 1 x 2 Amount of Western Foods Salsa produced Amount of Mexico City Salsa produced Profit function; Pro f it = Income Cost Z = (1.64x x 2 ) (0.62x x 2 ) Z = 1.02x x 2 LP Model; Maximize Z = 1.02x x 2 Subject to: 5x 1 + 7x WT 3x 1 + x TS 2x 1 + 2x TP x 1, x 2 0 p.s. Optimum Z = $876, x 1 = 560, x 2 = 240