Test Bank for Chapter 3

Test Bank for Chapter 3

Problem 3-1:

The Weigelt Corporation has three branch plants with excess production capacity.Fortunately, the corporation has a new product ready to begin production, and all three plants have this capability, so some of the excess capacity can be used in this way. This product can be made in three sizes--large, medium, and small--that yield a net unit profit of $420, $360, and $300, respectively. Plants 1, 2, and 3 have the excess capacity to produce 750, 900, and 450 units per day of this product, respectively, regardless of the size or combination of sizes involved.The amount of available in-process storage space also imposes a limitation on the production rates of the new product. Plants 1, 2, and 3 have 13,000, 12,000, and 5,000 square feet, respectively, of in-process storage space available for a day's production of this product. Each unit of the large, medium, and small sizes produced per day requires 20, 15, and 12 square feet, respectively.Sales forecasts indicate that if available, 900, 1,200, and 750 units of the large, medium, and small sizes, respectively, would be sold per day.At each plant, some employees will need to be laid off unless most of the plant’s excess production capacity can be used to produce the new product. To avoid layoffs if possible, management has decided that the plants should use the same percentage of their excess capacity to produce the new product.Management wishes to know how much of each of the sizes should be produced by each of the plants to maximize profit.Formulate a linear programming model for this problem.

Solution for Problem 3.1:

The decision variables can be denoted and defined as follows:

xP1L = number of large units produced per day at Plant 1, xP1M = number of medium units produced per day at Plant 1, xP1S = number of small units produced per day at Plant 1, xP2L = number of large units produced per day at Plant 2, xP2M = number of medium units produced per day at Plant 2, xP2S = number of small units produced per day at Plant 2, xP3L = number of large units produced per day at Plant 3, xP3M = number of medium units produced per day at Plant 3, xP3S = number of small units produced per day at Plant 3.Also letting P (or Z) denote the total net profit per day, the linear programming model for this problem is Maximize P = 420 xP1L + 360 xP1M + 300 xP1S + 420 xP2L + 360 xP2M + 300 xP2S

• 420 xP3L + 360 xP3M + 300 xP3S,

(Introduction to Operations Research 10e Fred Hillier ) (Test Bank) (There is Test Bank for Chapter 1and 2) 1 / 4

subject to xP1L + xP1M + xP1S  750 xP2L + xP2M + xP2S  900 xP3L + xP3M + xP3S  450 20 xP1L + 15 xP1M + 12 xP1S  13000 20 xP2L + 15 xP2M + 12 xP2S  12000 20 xP3L + 15 xP3M + 12 xP3S  5000 xP1L + xP2L + xP3L  900 xP1M + xP2M + xP3M  1200 xP1S + xP2S + xP3S  750 750

• ( xP1L + xP1M + xP1S ) - 900
• ( xP2L + xP2M + xP2S ) = 0

750

• ( xP1L + xP1M + xP1S ) - 450
• ( xP3L + xP3M + xP3S ) = 0

and

xP1L  0, xP1M  0, xP1S  0, xP2L  0, xP2M  0, xP2S  0, xP3L  0, xP3M  0, xP3S  0.

The above set of equality constraints also can include the following constraint:

1 900 x P2L +x P2M +x

P2S()-

1 450 x P3L +x P3M +x

P3S() = 0.

However, any one of the three equality constraints is redundant, so any one (say, this one) can be deleted.

Problem 3-2:

Comfortable Hands is a company which features a product line of winter gloves for the entire family — men, women, and children. They are trying to decide what mix of these three types of gloves to produce.Comfortable Hands’ manufacturing labor force is unionized. Each full-time employee works a 40-hour week. In addition, by union contract, the number of full-time employees can never drop below 20. Nonunion, part-time workers can also be hired with the following union-imposed restrictions: (1) each part-time worker works 20 hours per week, and (2) there must be at least 2 full-time employees for each part-time employee.All three types of gloves are made out of the same 100% genuine cowhide leather.Comfortable Hands has a long term contract with a supplier of the leather, and receives a 5,000 square feet shipment of the material each week. The material requirements and labor requirements, along with the gross profit per glove sold (not considering labor costs) is given in the following table.

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Glove Material Required (square feet) Labor Required (minutes) Gross Profit (per pair) Men’s 2 30 $8 Women’s 1.5 45 $10 Children’s 1 40 $6

Each full-time employee earns $13 per hour, while each part-time employee earns $10 per hour. Management wishes to know what mix of each of the three types of gloves to produce per week, as well as how many full-time and how many part-time workers to employ. They would like to maximize their net profit — their gross profit from sales minus their labor costs.Formulate a linear programming model for this problem.

Solution for Problem 3-2:

The decision variables can be denoted and defined as follows:

M = number of men’s gloves to produce per week, W = number of women’s gloves to produce per week, C = number of children’s gloves to produce per week, F = number of full-time workers to employ, PT = number of part-time workers to employ.

(Alternative notation for the decision variables is xM, xW, xC, xF, and xPT, respectively.) Also letting P (or Z) denote the total net profit per week, the linear programming model for this problem is

Maximize P = 8 M + 10 W + 6 C – 13(40)F – 10(20) PT, subject to

2 M + 1.5 W + C  5000

30 M + 45 W + 40 C  40(60) F + 20(60) PT

F  20

F  2 PT

and

M  0, W  0, C  0, F  0, PT  0.

Problem 3-3:

Slim-Down Manufacturing makes a line of nutritionally complete, weight-reduction beverages. One of their products is a strawberry shake which is designed to be a complete 3 / 4

meal. The strawberry shake consists of several ingredients. Some information about each of these ingredients is given below.

Ingredient Calories from fat (per tbsp) Total Calories (per tbsp) Vitamin Content (mg/tbsp)

Thickeners (mg/tbsp)

Cost (¢/tbsp) Strawberry flavoring 1 50 20 3 10 Cream 75 100 0 8 8 Vitamin supplement 0 0 50 1 25 Artificial sweetener 0 120 0 2 15 Thickening agent 30 80 2 25 6

The nutritional requirements are as follows. The beverage must total between 380 and 420 calories (inclusive). No more than 20% of the total calories should come from fat. There must be at least 50 milligrams (mg) of vitamin content. For taste reasons, there must be at least two tablespoons (tbsp) of strawberry flavoring for each tbsp of artificial sweetener. Finally, to maintain proper thickness, there must be exactly 15 mg of thickeners in the beverage.Management would like to select the quantity of each ingredient for the beverage which would minimize cost while meeting the above requirements.Formulate a linear programming model for this problem.

Solution for Problem 3-3:

The decision variables can be denoted and defined as follows:

S = Tablespoons of strawberry flavoring, CR = Tablespoons of cream, V = Tablespoons of vitamin supplement, A = Tablespoons of artificial sweetener, T = Tablespoons of thickening agent.

(Alternative notation for the decision variables is xS, xC, xV, xA, and xT, respectively.) Also letting C (or Z) denote cost, the linear programming model for this problem is

Minimize C = 10 S + 8 CR + 25 V + 15 A + 6 T, subject to

50 S + 100 CR + 120 A + 80 T  380

50 S + 100 CR + 120 A + 80 T  420

S + 75 CR + 30 T  0.2 (50 S + 100 CR + 120 A + 80 T)

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