Chapter 7
10) For the linear program
find the optimal solution using the graphical solution procedure. What is the value of the objective function at the optimal solution?
13) Consider the following linear program
a. Show the feasible region. b. What are the extreme points of the feasible region? c. Find the optimal solution using the graphical procedure.
14) Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor believes a market exists for both a medium-priced golf bag, referred to as a standard model, and a high-priced golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can make the bags at a competitive price, the distributor will purchase all the bags that Par can manufacture over the next three months. A careful analysis of the manufacturing requirements resulted in the following table, which shows the production time requirements for the four required manufacturing operations and the accounting department’s estimate of the profit contribution per bag:
Production Time (hours)
Product
Cutting and Dyeing
Sewing Finishing
Inspection and Packaging
Profit per Bag Standard 7/10 1/2 1 1/10 $10 Deluxe 1 5/6 2/3 1/4 $9
The director of manufacturing estimates that 630 hours of cutting and dyeing time, 600 hours of sewing time, 708 hours of finishing time, and 135 hours of inspection and packaging time will be available for the production of golf bags during the next three months:
a. If the company wants to maximize total profit contribution, how many bags of each model should it manufacture? b. What profit contribution can Par earn on those production quantities? c. How many hours of production time will be scheduled for each operation? d. What is the slack time in each operation?
24) Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: a regular model and a catcher’s model. The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department. The production time requirements and the profit contribution per glove are given in the following table:
Production Time (hours)
Product
Cutting and Sewing
Finishing
Packaging and Shipping
Profit per Bag
Regular model 1 1/2 1/8 $5 Catcher's model 3/2 1/3 1/3 $8
Assuming that the company is interested in maximizing the total profit contribution, answer the following:
a. What is the linear programming model for this problem? b. Find the optimal solution using the graphical solution procedure. How many gloves of each model should Kelson manufacture? c. What is the total profit contribution Kelson can earn with the given production quantities? d. How many hours of production time will be scheduled in each department? e. What is the slack time in each department?
36) As part of a quality improvement initiative, Consolidated Electronics employees complete a three-day training program on team building and a two-day training program on problem solving. The manager of quality improvement has requested that at least 8 training programs on team building and at least 10 training programs on problem solving be offered during the next three1 months. In addition, senior-level management has specified that at least 25 training programs must be offered during this period. Consolidated Electronics uses a consultant to teach the training programs. During the next quarter, the consultant has 84 days of training time available. Each training program on team building costs $10,000 and each training program on problem solving costs $8000. 1 This is a correction from the textbook which originally read “during the next six months.” To be consistent with consulting time available in the “next quarter”, the question was changed to read three months instead of six.
a. Formulate a linear programming model that can be used to determine the number training programs on team building and the number of training programs on problem solving that should be offered in order to minimize total cost. b. Graph the feasible region. c. Determine the coordinates of each extreme point. d. Solve for the minimum-cost solution.
41) Southern Oil Company produces two grades of gasoline: regular and premium. The profit contributions are $0.30 per gallon for regular gasoline and $0.50 per gallon for premium gasoline. Each gallon of regular gasoline contains 0.3 gallons of grade A crude oil and each gallon of premium gasoline contains 0.6 gallons of grade A crude oil. For the next production period, Southern has 18,000 gallons of grade A crude oil available. The refinery used to produce the gasolines has a production capacity of 50,000 gallons for the next production period. Southern Oil’s distributors have indicated that demand for the premium gasoline for the next production period will be at most 20,000 gallons.
a. Formulate a linear programming model that can be used to determine the number of gallons of regular gasoline and the number of gallons of premium gasoline that should be produced in order to maximize total profit contribution. b. What is the optimal solution? c. What are the values and interpretations of th