Operations Research Tutorials Questions Linear Programming

Tutorial One

Question 1

Use the graphical method to solve the problem

0

44318

605210

.2

21

21

21

21

2

21

≥

≤+≤+

≤+≤

+=

XandXand

XXXX

XXXST

XXZMaximize

Question 2

The Good Quality Television Company has decided on the number of 27-and

20-inch sets to be produced at one of its factories. Market research

indicates that at most 40 of the 27-inch sets and 10 of the 20-inch sets can

be sold per month. The maximum number work-hours available are 500 per

month. A 27-inch set requires 20 work-hours and a 20-inch set requires 10

work-hours. Profits generated per unit sold from 27-inch and 20-inch are

$120 and $80 respectively. A wholesaler agreed to purchase all sets

provided they don’t exceed what has been indicated by the market research.

• Formulate a Linear-programming model for this problem.

• Use the graphical method to solve this model. What is the resulting

profit?

Question 3

The Magical Insurance Firm is introducing two new products lines: special

risk insurance and mortgages. The expected profit is $5 per unit and $2 per

unit of the special risk insurance and mortgages respectively. Management

wishes to establish sales quotas for the new product lines to maximize the

total expected profit. The work requirements are as follows

Work-Hours per Unit

Department Special Risk Mortgage

Work-Hours

Available

Underwriting 3 2 2400

Administration 0 1 800

Claims 2 0 1200

• Formulate a linear programming model for this problem.

• Use the graphical method to solve it.

• Verify the exact value of your optimal solution value from your

solution by solving algebraically for the simultaneous solution of the

relevant equations.

Question 4

Consider the following model

0&0202

123032

.5040

21

21

21

21

21

≥≥≥+≥+≥+

+=

XXXX

XXXX

STXXZMinimize

• Use the graphical method to solve this model.

• How does the optimal solution change if the objective function is

changed to 21 7040 XXZMinimize += ?

• How does the optimal solution change if the objective function is

changed to 152 21 ≥+ XX ?

Question 5

Happy Jack loves steaks and potatoes. Therefore, he has decided to go to a

steady diet of only these two foods (plus some liquids and vitamin

supplements) for all his meals. He realizes that this is not the healthiest

diet, so he wanted to make sure that he eats the right quantities of the two

foods to satisfy some key nutritional requirements. He has obtained the

following nutritional and cost information.

Grams of Ingredient

Per Serving

Ingredient Steak Potatoes

Daily Requirements (Grams)

Carbohydrates 5 15 ≥ 50

Protein 20 5 ≥ 40

Fat 15 2 ≤ 60

Cost per Serving $ 4 $ 2

Happy Jack wishes to determine the number of daily servings (may be

fractional) to steak and potatoes that will meet these requirements at a

minimum cost.

• Formulate a linear programming model for this problem.

• Use the graphical method to solve this model.

Question 6

The Capricorn metal company desires to blend a new alloy of 40 % tin, 35%

zinc, and 25% lead from several available alloys having the following

properties

Alloy

Property 1 2 3 4 5

% of Tin 60 25 45 20 50

% of Zinc 10 15 45 50 40

% of Lead 30 60 10 30 10

Cost

($/lb)

22 20 25 24 27

The objective is to determine the properties of these alloys that should be

blended to produce the new alloy at a minimum cost.

• Formulate a linear programming model for this problem.

More modeling LP Problems

Problem One

A company produces three electrical products-clocks, radios and toasters.

These products have the following resource requirements.

Resource Requirements

Cost-$/Unit Labor-hrs /Unit

Clock 7 2

Radio 10 3

Toaster 5 2

The manufacturer has a daily production budget of $2,000 and a maximum

of 600 hours of Labor. Maximum daily customer demand is for 200 clocks,

300 radios, and 150 toasters. Clocks sell for $15, radio for $20, and

toasters for $12. The company wants to know the optimal product mix that

will maximize profit.

Formulate a linear programming model for this problem.

Problem Two

A building supply company has received the following order for boards in

three lengths

Length Order (quantity)

7 ft 700 boards

9 ft 1,200 boards

10 ft 300 boards

The company has a 25-foot standard-length board in stock. Therefore, the

standard length boards must be cut into the lengths necessary to meet

order requirements. Naturally, the company wishes to minimize the number

of standard-length boards used. The company must therefore determine

how to cut up to 25 foot boards to meet order requirements and minimize

the number of standard-length boards used.

1 Formulate a linear programming model for this problem.

2 When a board is cut in a specific pattern, the amount of board left

over is known as “trim loss”. Reformulate the linear programming

for this problem, assuming that the objective is to minimize the

loss rather than to minimize the total of boards.