Operations Research Answers

Kyle, a runner, is interested in mixing two cereals to minimize caloric intake while maintaining at least 96 units of calcium and 24 units of iron in each serving. Cheerios has 6 units of calcium, 3 units of iron and 14 calories per ounce. Fruitloops has 12 units of calcium, 2 units of iron and 11 calories per ounce. How much of each cereal should Kyle consume to minimize calorie intake while maintaining the minimum nutrient levels

A trading company buys and sells 10,000 bottles of pain-balm every year. The 

company’s cost of placing an order of pain-balm is Rs. 100. The holding cost per 

bottle on inventory is Rs. 0.3. (5) 

 (i) Determine the optimum order quantity and inventory cycle time for the pain-balm, 

bottles. 

 (ii) How many orders should be placed each year?

Which of the following statements are true and which are false? Give a short proof or a 

counter example in support of your answer. (10) 

(a) The optimal solution for the following LPP is 30 :

*

Z =

Max 1 2 3 3 Z = x − x + x

 Subject to x1 + x2 + x3 ≤10

 x1

, x2

, x3 ≥ .0

 (b) The optimal solution of an ILLP can be obtained by rounding off the optimal solution 

of its LP relaxation. 

 (c) If the availabilities and requirements of a balanced transportation problem are 

integers, the optimal solution to the problem will have integer values. 

 (d) The following max /3/4 F / F problem can be reduced to a machine problem. 

Job 

Processing time on

M1 M2 M3

1 8 6 10

2 5 2 13

3 4 11 11

4 6 7 10

 (e) For the mixed generator rn+1 = 5( rn + )7 (mod ),8 if r0 = ,4 then 3

r is zero.

1. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients,

oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats

and rice it should include in each box of cereal to meet the minimum requirements of 48 mil-

ligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats

contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice

contributes 6 milligrams of A and 2 milligrams of B. An ounce of oats costs $0.05, and an ounce

of rice costs $0.03.

a. Formulate a linear programming model for this problem.

b. Solve this model by using graphical analysis.

2. What would be the effect on the optimal solution in (Question 1) if the cost of rice increased from

$0.03 per ounce to $0.06 per ounce?

when Tracy McCoy wakes up Saturday morning, she remembers

that she promised the PTA she would make some cakes and/or homemade bread for its bake sale

that afternoon. However, she does not have time to go to the store to get ingredients, and she has

only a short time to bake things in her oven. Because cakes and breads require different baking

temperatures, she cannot bake them simultaneously, and she has only 3 hours available to bake.

A cake requires 3 cups of flour, and a loaf of bread requires 8 cups; Tracy has 20 cups of flour.

A cake requires 45 minutes to bake, and a loaf of bread requires 30 minutes. The PTA will sell a

cake for $10 and a loaf of bread for $6. Tracy wants to decide how many cakes and loaves of

bread she should make.

a. Formulate a linear programming model for this problem.

b. Solve this model by using graphical analysis.

QUESTION 1

Food For U Sdn Bhd., a catering company is to make breakfast for a business meeting. It will serve chicken sandwiches, egg sandwiches, and vegetarian sandwiches. A chicken sandwich has 1 serving of vegetables, 4 slices of chicken, 2 slices of cheese, and 2 slices of bread. An egg sandwich has 2 serving of vegetables, 2 servings of egg, 2 slices of cheese and 2 slices of bread. A vegetarian sandwich has 3 servings of vegetables, 2 slices of cheese, and 2 slices of bread. A total of 5 bags of chicken are available, each of which has 40 slices; 19 loaves of bread are available, each with 14 slices; 200 servings of vegetables and eggs are available, and 15 bags of cheese, each with 60 slices, are available.

a) Formulate linear programming model based on the above problem by using Microsoft Excel. Show the answer and sensitivity report. (9 marks)

Food For U Sdn Bhd., a catering company is to make breakfast for a business meeting. It will serve chicken sandwiches, egg sandwiches, and vegetarian sandwiches. A chicken sandwich has 1 serving of vegetables, 4 slices of chicken, 2 slices of cheese, and 2 slices of bread. An egg sandwich has 2 serving of vegetables, 2 servings of egg, 2 slices of cheese and 2 slices of bread. A vegetarian sandwich has 3 servings of vegetables, 2 slices of cheese, and 2 slices of bread. A total of 5 bags of chicken are available, each of which has 40 slices; 19 loaves of bread are available, each with 14 slices; 200 servings of vegetables and eggs are available, and 15 bags of cheese, each with 60 slices, are available.

a) Formulate linear programming model based on the above problem by using Microsoft Excel. Show the answer and sensitivity report. (9 marks)

b) Given the resources, how many of each sandwich can be produced if the goal is to maximize the number of sandwiches? (3 marks)

10. The optimal solution of a maximization type LPP is given in the following table:

C sj

′ 6 4 0 0 0 

Solution

CB 

Basic Variables 

1

x 2

x 3

x 4

x 5

x

0 3

x 0 5/3 1 − 3/2 0 14 

0 5

x 0 − 3/1 0 1/3 1 5 

6 1

x 1 2/3 0 1/3 0 8 

Cj − Z j

0 0 0 − 2 0 Z = 48

 (i) Find the alternative optimal basic feasible solution. 

 (ii) Find an alternative optimal non-basic feasible solution.

9. Solve the ILLP given below by graphical method: (10)

Maximum 95 1 100 2 Z = x + x

Subject to the Constraints

5x1 + 2x2 ≤ 20

x1 ≥ 3

5 x2 ≤

1 2

x x are non-negative integers.

8(b) A trading company buys and sells 10,000 bottles of pain-balm every year. The

company’s cost of placing an order of pain-balm is Rs. 100. The holding cost per

bottle on inventory is Rs. 0.3. (5)

(i) Determine the optimum order quantity and inventory cycle time for the pain-balm,

bottles.

(ii) How many orders should be placed each year?