Additional Homework Problems to accompany MODULE B: Linear Programming.
B.31 Andrew McCarroll is trying to determine how many units each of two ...
Additional Homework Problems to accompany MODULE B: Linear Programming
B.31 Andrew McCarroll is trying to determine how many units each of two cordless telephones to produce each day. One of these is the standard model; the other one is the deluxe model. The profit per unit on the standard model is $40, on the deluxe model $60. Each unit requires 30 minutes assembly time. The standard model requires 10 minutes of inspection time, the deluxe model 15 minutes. The company must fill an order for six standard phones. There are 450 minutes of assembly time and 180 minutes of inspection time available each day. How many units of each product should be manufactured to maximize profits? B.32 Solve the following linear programming problem graphically: Maximize
Z = 3X + 5Y
Subject to:
4X + 4Y ≤ 48 1X + 2Y ≤ 20 Y≥2 X, Y ≥ 0
B.33 The Outdoor Furniture Corporation manufactures two products, benches and picnic tables, for use in yards and parks. The firm has two main resources: its carpenters (labor force) and a supply of redwood for use in furniture. During the next production cycle, 1,200 hours of labor are available under a union agreement. The firm also has a stock of 3,500 board feet of quality redwood. Each bench that Outdoor Furniture produces requires 4 labor-hours and 10 board feet of redwood; each picnic table takes 6 laborhours and 35 board feet of redwood. Competed benches will yield a profit of $9 each, and tables will result in a profit of $20 each. How many benches and tables should Outdoor Furniture produce in order to obtain the largest possible profit? Use the graphical linear programming approach. B.34 Solve the following linear programming problem graphically: Maximize
Z = 4X 1 + 4X 2
Subject to:
3X 1 + 5X 2 ≤ 150 X 1 – 2X 2 ≤ 10 5X 1 + 3X 2 ≤ 150 X1, X2 ≥ 0
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B.35 Gupta Furniture manufactures two different types of china cabinets, a French provincial model and a Danish modern model. Each cabinet produced must go through three departments: carpentry, painting, and finishing. The accompanying table contains all relevant information concerning production times per cabinet produced and production capacities for each operation per day, along with net revenue per unit produced. The firm has a contract with a Miami distributor to produce a minimum of 300 of each cabinet per week (or 60 cabinets per day). Owner Sushil Gupta would like to determine a product mix to maximize his daily revenue.
Cabinet Style French provincial Danish modern Department capacity (hours)
Carpentry (hours per cabinet) 3 2 360
Painting (hours per cabinet) 1 1/2 1 200
Finishing (hours per cabinet) 3/4 3/4 125
Net Revenue per cabinet $28 $25
Formulate this as a linear programming problem and solve. B.36 This is the slack time of year at JES, Inc. The firm would actually like to shut down the plant, but if it laid off its core employees, they would probably go to work for a competitor. JES could keep its core (full-time, year-round) employees busy by making 10,000 round tables per month, or by making 20,000 square tables per month (or some ratio thereof). JES does, however, have a contract with a supplier to buy a minimum of 5,000 square tabletops per month. Handling and storage costs per round table will be $10; these costs would be $8 per square table. Draw a graph, algebraically describe the constraint inequalities and the objective function, identify the points bounding the feasible solution area, and find the cost at each point and the optimum solution. Let X 1 equal the thousands of round tables per month and X 2 equal the thousands of square tables per month. B.37 Solve the following linear programming problem graphically. Indicate the corner points on your graph. Maximize profit = $3X 1 + $5X 2 Subject to:
X2 ≤ 6 3X 1 + $2X 2 ≤ 18 X1, X2 ≥ 0
B.38 Greg Middleton, the advertising director for Diversey Paint and Supply, a chain of four retail stores on Chicago’s North Side, is considering two media possibilities. One plan is for a series of half-page ads in the Sunday Chicago Tribune newspaper, and the other is for advertising time on Chicago TV. The stores are expanding their line of do-it-yourself tools, and the advertising director is interested in an exposure level of at least 40% within the city’s neighborhood and 60% in northwest suburban areas. The TV viewing time under consideration has an exposure rating per spot of 5% in city homes and 3% in the northwest suburbs. The Sunday newspaper has corresponding exposure rates of 4% and 3% per ad. The cost of a half-page Tribune advertisement is $925; a television spot costs $2,000. Diversey Paint would like to select the least costly advertising strategy that would meet the desired exposure levels. Formulate and solve this LP problem.
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B.39 Andy’s Bicycle Company (ABC) has the hottest new product on the upscale toy market – boys’ and girls’ bikes in bright fashion colors, with oversize hubs and axles; shell design safety tires; strong padded frames; chrome-plated chains, brackets, and valves; and non-slip handlebars. Due to the seller’s market for high-quality toys for the newest baby boomers, ABC can sell all the bicycles it manufactures at the following prices: boys’ bikes, $220; girls’ bikes, $175. This is the price payable to ABC at its Orlando plant. The firm’s accountant, V. R. Dondeti, has determined that direct labor costs will be 45% of the price that ABC receives for the boys’ model and 40% of the price received for the girls’ model. Production costs, other than labor but excluding painting and packaging, and $44 per boys’ bicycle and $30 per girls’ bicycle. Painting and packaging are $20 per bike, regardless of model. The Orlando plant’s overall production capacity is 390 bicycles per day. Each boys’ bike requires 2.5 labor-hours, each girls’ model 2.4 hours. ABC currently employs 120 workers, each of whom puts in an 8-hour day. The firm has no desire to hire or fire to affect labor availability, for it believes its stable workforce is one of its biggest assets. Using a graphic approach, determine the best product mix for ABC. B.40 Mann Enterprises is a Houston manufacturer of tables and accessories for personal computers. The company is caught in a vicious cross fire between rapidly dropping market prices from competitors worldwide for its products and stable domestic costs for its materials. The 15 different products noted below must be scheduled to maximize profits or there will be no jobs and no firm in another 3 months. Your job as the new OM graduate is to address the issues raised this morning by Laura Mann, the president, in an emergency meeting in her office. Without being told so explicitly, you conclude that if you don’t get the schedule done accurately and in a timely manner, you will be history. To your relief, the industrial engineers and accountants have provided the data shown below. The issues, as recorded in your notes, are identified in (a) through (f)
Product A158 B179 C023 D045 E388 F422 G366 H600 I701 J802 K900 L901 M050 N150 P259 Availability per month
Steel Alloy Required (Lb.) 4 6 10 12 10 5 1 1 2 10 10
Plastic Required (Sq. Ft.) 0.4 0.5 0.4 1.2 1.4 1.4 1.0 0.4 0.3 0.2 1.8 2.7 1.1 -
Wood Required (Bd. Ft.) 0.7 1.8 1.5 2.0 1.2 1.5 7.0 5.0 1.5 5.0 5.8 6.2
Aluminum Required (Lb.) 5.8 10.3 1.1 8.1 7.1 6.2 7.3 10.0 11.0 12.5 13.1 15.0
Formica Required (Bd. Ft.) 10.9 2.0 2.3 4.9 10.0 11.1 12.4 5.2 6.1 7.7 5.0 2.1 1.0
Labor Required (Hr.) 3.1 1.0 1.2 4.8 5.5 0.8 9.1 4.8 1.9 1.4 1.0 5.1 3.1 7.7 6.6
980
400
600
2,500
1,800
1,000
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Minimum Monthly Demand (units) 20 10 10 20 10 20 50 20 20 10 20 10 10
Contribution to Profit $18.79 6.31 8.19 45.88 63.00 4.10 81.15 50.06 12.79 15.88 17.91 49.99 24.00 88.88 77.01
a) How many of each of the 15 products should be produced each month? b) Clearly explain the meaning of each shadow price. c) A number of workers interested in saving money for the holidays have offered to work overtime next month at a rate of $12.50 per hour. What should the response of management be? d) Two tons of steel alloy are available from an overstocked supplier at a total cost of $8,000. Should the steel be purchased? All or part of the supply? e) The accountants have just discovered that an error was made in the contribution to profit for product N150. The correct value is actually $8.88. What are the implications of this error? f) Management is considering the abandonment of five product lines (those beginning with the letters A through E). If no minimum monthly demand is established, what are the implications? Note that there already is no minimum for two of these products. Use the corrected value for N150.
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