Linear Programming Model Assignment Questions
The homework problems for Module 4 are: 2-24, 34, 36, 38 (you only have to do part A for these problems. You do not have to do the part B graphical solutions) and 3-10, 12 (parts B and C for Problem 12), 28, 30. Please use Excel solver function.
I posted an annotated solution to Problem 2-5. This problem is an examplar for both the chapter 2 and 3 problems. In this example I show how I typically set up a problem. First I set up the linear programming model and then develop a parallel set-up to use as input to the Solver add-in.
I use this strategy because I first like to set up the problem solution before I worry about setting up the parameters for using Solver. 24. Universal Claims Processors processes insurance claims for large national insurance companies. Most claim processing is done by a large pool of computer operators, some of whom are perma-nent and some of whom are temporary.
A permanent operator can process 16 claims per day, whereas a temporary operator can process 12 per day, and on average the company processes at least 450 claims each day.
The company has 40 computer workstations. A permanent operator generates about 0.5 claim with errors each day, whereas a temporary operator averages about 1.4 defective claims per day. The company wants to limit claims with errors to 25 per day. A per-manent operator is paid $ 64 per day, and a temporary operator is paid $ 42 per day. The company wants to determine the number of permanent and temporary operators to hire in order to mini-mize costs. a. Formulate a linear programming model for this problem.
34. Gillian’s Restaurant has an ice- cream counter where it sells two main products, ice cream and frozen yogurt, each in a variety of flavors. The restaurant makes one order for ice cream and yogurt each week, and the store has enough freezer space for 115 gallons total of both products. A gallon of frozen yogurt costs $ 0.75 and a gallon of ice cream costs $ 0.93, and the restaurant budgets $ 90 each week for these products. The manager estimates that each week the restaurant sells at least twice as much ice cream as frozen yogurt. Profit per gallon of ice cream is $ 4.15, and profit per gal-lon of yogurt is $ 3.60. a. Formulate a linear programming model for this problem.
36. Copperfield Mining Company owns two mines, each of which produces three grades of ore— high, medium, and low. The company has a contract to supply a smelting company with at least 12 tons of high- grade ore, 8 tons of medium- grade ore, and 24 tons of low- grade ore. Each mine produces a certain amount of each type of ore during each hour that it operates. Mine 1 produces 6 tons of high- grade ore, 2 tons of medium- grade ore, and 4 tons of low- grade ore per hour.
Mine 2 pro-duces 2, 2, and 12 tons, respectively, of high-, medium-, and low- grade ore per hour. It costs Copperfield $ 200 per hour to mine each ton of ore from mine 1, and it costs $ 160 per hour to mine each ton of ore from mine 2. The company wants to determine the number of hours it needs to operate each mine so that its contractual obligations can be met at the lowest cost. a. Formulate a linear programming model for this problem.
38. A manufacturing firm produces two products. Each product must undergo an assembly process and a finishing process. It is then transferred to the warehouse, which has space for only a limited number of items. The firm has 80 hours available for assembly and 112 hours for finishing, and it can store a maximum of 10 units in the warehouse.
Each unit of product 1 has a profit of $ 30 and requires 4 hours to assemble and 14 hours to finish. Each unit of product 2 has a profit of $ 70 and requires 10 hours to assemble and 8 hours to finish. The firm wants to determine the quantity of each product to produce in order to maximize profit. a. Formulate a linear programming model for this problem.
10. A company produces two products, A and B, which have profits of $ 9 and $ 7, respectively. Each unit of product must be processed on two assembly lines, where the required production times are as follows. ————————————————-
Hours/ Unit————————————————-Product Line 1 Line 2————————————————-A 12 4————————————————-B 4 8————————————————-Total hours 60 40
a. Formulate a linear programming model to determine the optimal product mix that will maximize profit. 12. For the linear programming model formulated in Problem 10 and solved graphically in Problem 11: a. Determine the sensitivity ranges for the objective function coefficients, using graphical analysis. b. Verify the sensitivity ranges determined in ( a) by using the computer. c. Using the computer, determine the shadow prices for additional hours of production time on line 1 and line 2 and indicate whether the company would prefer additional line 1 or line 2 hours. 28. The Bluegrass Distillery produces custom- blended whiskey. A particular blend consists of rye and bourbon whiskey.
The company has received an order for a minimum of 400 gallons of the custom blend. The customer specified that the order must contain at least 40% rye and not more than 250 gallons of bourbon. The customer also specified that the blend should be mixed in the ratio of two parts rye to one part bourbon. The distillery can produce 500 gallons per week, regardless of the blend. The production manager wants to complete the order in 1 week. The blend is sold for $ 5 per gallon. The distillery company’s cost per gallon is $ 2 for rye and $ 1 for bourbon. The company wants to determine the blend mix that will meet customer requirements and maximize profits.
Formulate a linear programming model for this problem. 30. Solve the linear programming model formulated in Problem 28 for the Bluegrass Distillery by using the computer. a. Identify the sensitivity ranges for the objective function coefficients and explain what the upper and lower limits are. b. How much would it be worth to the distillery to obtain additional production capacity? c. If the customer decided to change the blend requirement for its custom- made whiskey to a mix of three parts rye to one part bourbon, how would this change the optimal solution?
