# Quantitative Methods

1. Solve the following linear programming model by using the computer:

Maximize Z = 5x1 + 8x2

Subject to

3x1 + 5x2 ≤ 50

2x1 + 4x2 ≤ 40

x1 ≤ 8

x2 ≤ 10

x1, x2 ≥ 0

2. Solve the following linear programming model by using the computer:

Minimize Z = 8x1 + 6x2

Subject to

4x1 + 2x2 ≥ 20

-6x1 + 4x2 ≤ 12

x1 + x2 ≥ 6

x1, x2 ≥ 0

3. Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday, and she must determine how much beer to stock. Betty stocks three brands of beer - Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is as follows:

Brand Cost/gallon

Yodel $1.50

Shotz 0.90

Rainwater 0.50

The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of Shotz, and 300 gallons of Rainwater. The tavern has a capacity to stock 1,000 gallons of beer; Betty wants to stock up completely. Betty wants to determine the number of gallons of each brand of beer to order so as to maximize profit.

a. Formulate a linear programming model for this problem (written in a format similar to the way Problems 1 and 2 were presented).

b. Solve this problem by using the computer.

4. A jeweler and her apprentice make silver pins and necklaces by hand. Each week they have 80 hours of labor and 36 ounces of silver available. It requires 8 hours of labor and 2 ounces of silver to make a pin, and 10 hours of labor and 6 ounces of silver to make a necklace. Each pin also contains a small gem of some kind. The demand for pins is no more than six per week. A pin earns the jeweler $400 in profit, and a necklace earns $100. The jeweler wants to know how many of each item to make each week to maximize profit.

a. Formulate an integer programming model for this problem (written in a format similar to the way Problems 1 and 2 were presented).

b. Solve this problem by using the computer (note: if using QM for Windows, be sure to use the Integer and Mixed Integer Programming Module).

5. A transportation problem involves the following costs, supply and demand.

To

From 1 2 3 4 Supply

1 $500 $750 $300 $450 12

2 650 800 400 600 17

3 400 700 500 550 11

Demand 10 10 10 10

a. Formulate a linear programming model for this problem (written in a format similar to the way Problems 1 and 2 were presented).

b. Solve this transportation problem by using the computer (note: if using QM for Windows, be sure to select the transportation module).

A shop has four machinists to be assigned to four machines. The hourly cost of having each machine operated by each machinist is as follows:

Machine

Machinist A B C D

1 $10 $2 $8 $6

2 9 5 11 9

3 12 7 14 14

4 3 1 4 2

a. Formulate a linear programming model for this problem.

b. Determine the optimal assignment and compute total minimum cost.

#### Solution Summary

Step by step solutions to all the problems are provided.