Operations Research Answers

John sells notebooks and pencils and notebooks. In a week he can sell between 400 and 500 pencils and between 150 and 200 notebooks but not more than 650 items altogether. Each pencil costs P18 and sells for P25 while each notebook costs P28 and sells for P45. How many of each type should he get to gain the maximum profit per week? 1. Formulate the problem above as a linear programming problem with objective function and constraints . Start by defining the decision variables and the profit function

Problem 5: To establish a driver educational school, organizers must decide how

many cars, instructors, and students to have. Costs are estimated as follows. Annual

fixed costs to operate the school are 30000 birr. The annual cost per car is 3000 birr.

The cost per instructor is 11000 birr and one instructor in needed for each car.

Tuition for each student is 350 birr. Let X be the number of cars and Y be the

number of students.

A. Write an expression for total cost.

B. Write an expression for total revenue.

C. Write an expression for total profit.

D. The school offers the course eight times each year. Each time the course

is offered, there are two sessions. If the decided to operate five cars, and if

four students can be assigned to each car, will they break-even?

Problem 4: John, president of Hardrock Concrete Company, has plants in three

locations and is currently working on three major construction projects, located at

different sites. The shipping cost per truckload of concrete, plant capacities, and

project requirements are provided in the following table.

A. Formulate an initial feasible solution to Hardrock’s transportation problem

using the northwest corner rule.

B. Find the optimal solution using stepping-stone method

Problem 2: A small business enterprise makes dresses and trousers. To make a

dress requires 2 hours of cutting and 2 hours of sewing. To make a trousers requires

1 hour of cutting and 3 hours of sewing. The profit on a dress is 40 birr and on a

trouser 50 birr. The business has a maximum of 32 hours of cutting time and 48

hours of sewing time to operate per week.

Required:

A. Define the decision variables

B. Write down the constraints in terms of the variables.

C. Write down the Objective Function in terms of the variables.

D. Write down the standard form of the problem

E. Determine how many dresses and trousers should be made to maximize profit

and what the maximum profit is. Using simplex method

Problem 1: A farmer plans to mix two types of food to make a mix of low cost feed

for the animals in his farm. A bag of food A costs $10 and contains 40 units of

proteins, 20 units of minerals and 10 units of vitamins. A bag of food B costs $12

and contains 30 units of proteins, 20 units of minerals and 30 units of vitamins. How

many bags of food A and B should the consumed by the animals each day in order to

meet the minimum daily requirements of 150 units of proteins, 90 units of minerals

and 60 units of vitamins at a minimum cost?

A. Formulate the problem in to linear programming problem model

B. Use graphical approach to find the solution

Establish the initial feasible solutions of the LP model using NCR (North west Corner Rule), MCM ( Minimum Cost Method or Greedy Method), and VAM ( Vogel’s Approximation Method)

 Minimize: C = 14X_1A + 25X_1B + 13X_1C + 18 X_1D.+ 10X_2A + 12X_2B + 13X_2C + 11X_2D+ 15X_3A + 20X_3B + 11X_3C+ 25X_3D

Subject to: X_1A + X_1B + X_1C + X_1D = 140  X_1A + X_2A + X_3A = 100

                X_2A + X_2B + X_2C + X_2D = 160     X_1B + X_2B + X_3B = 100

               X_3A + X_3B + X_3C + X_3D = 50      X_1C + X_2C + X_3C = 50

              X_1D + X_2D + X_3D = 100 X_ij    

1.)  Establish the initial feasible solutions of the LP model using NCR (North west Corner Rule), MCM ( Minimum Cost Method or Greedy Method), and VAM ( Vogel’s Approximation Method)

 Minimize: C = 14X_1A + 25X_1B + 13X_1C + 18 X_1D.+ 10X_2A + 12X_2B + 13X_2C + 11X_2D+ 15X_3A + 20X_3B + 11X_3C+ 25X_3D

 Subject to: X_1A + X_1B + X_1C + X_1D = 140     X_1A + X_2A + X_3A = 100

                     X_2A + X_2B + X_2C + X_2D = 160     X_1B + X_2B + X_3B = 100

                     X_3A + X_3B + X_3C + X_3D = 50       X_1C + X_2C + X_3C = 50

                     X_1D + X_2D + X_3D = 100 X_ij >= 0

1.)  Establish the initial feasible solutions of the LP model using NCR (North west Corner Rule), MCM ( Minimum Cost Method or Greedy Method), and VAM ( Vogel’s Approximation Method)

 Minimize: C = 14X_1A + 25X_1B + 13X_1C + 18 X_1D.+ 10X_2A + 12X_2B + 13X_2C + 11X_2D+ 15X_3A + 20X_3B + 11X_3C+ 25X_3D

Subject to: X_1A + X_1B + X_1C + X_1D = 140   X_1A + X_2A + X_3A = 100

                X_2A + X_2B + X_2C + X_2D = 160     X_1B + X_2B + X_3B = 100

               X_3A + X_3B + X_3C + X_3D = 50       X_1C + X_2C + X_3C = 50

              X_1D + X_2D + X_3D = 100 X_ij      

A contractor has to supply 10000 bearings per day to an automobile manufacturer. He finds that when he starts production run, he can produce 25000 bearings per day. The cost of holding a bearing in stock for one year is $2 and the set-up cost of a production run is $180. Find the economic order quantity(EOQ). How frequently should the production run be made?

Exercise Sheet

Q1. A juice company has its products viz. canned apple and bottled juice with profit margin Rs.4

and Rs.2 respectively per unit. The following table shows the labor, equipment, and ingredients

to produce each product per unit.

Canned Apple Bottled Juice Total

Labor 2 3 12

Equipment 3.2 1 8

Ingredients 2.4 2 9

Formulate the linear programming problem (model) specifying the product mix which will

maximize the profit without exceeding the levels of resourc