Operations Research Answers

Solve the ff linear programming graphically method maximize Z= 30×, + 100ײ 4×,+6ײ≤ 90 8×, +6×, ≤100 5×,+4×,≤ 80 ×, ×, ≥ 0

1.      Solve the following linear programming problem graphically:

Minimise Z = 200 x + 500 y

subject to the constraints:

x + 2y ≥ 10 ... (1)

3x + 4y ≤ 24 ... (2)

x ≥ 0, y ≥ 0 ... (3)

A company has found that its cost to purchase a component is ETB 50 per order and the carrying cost is 10% of the average inventory. The company currently purchases ETB 20,000 worth of components in a year. Assuming that same demand will be there in the next year,

a. Suggest a suitable policy of purchase in terms of no. of orders in a year and

b. Quantity to be ordered for each year. Assuming fractional order can be made.

Draw activity network of the project.

b. Crash the activities step by step until all paths are critical.

A firm produces three products A, B, and C, each of which passes through three departments: Fabrication, Finishing and Packaging. Each unit of product A requires 3, 4 and 2 hours; a unit of B requires 5, 4 and 4 hours while each unit of product C requires 2, 4, 5 hours respectively in the three departments. Every day, 60 hours are available in fabrication department, 72 hours in the finishing department and 100 hours in the packaging department. If the unit contribution of product A is ETB 5, of product B is ETB 10, and of product C is ETB 8, determine the number of units of each of the products, which should be made each day to maximize the total contribution. Also determine if any capacity would remain unutilised.

a. Write the formulation for this linear program.

b. Solve the Linear programming problem using simplex method.

Moore’s Meatpacking Company producesahot dog mixture in 1000- pound batches. The mixturecontains two ingredients- chicken and beef.The cost per pound of each of these ingredients is asfollows:

IngredientCost/lb.

Chicken 3

Beef 5

Each batch has the following recipe requirements:

1.At least 500 pounds of chicken

2.At least 200 pounds of beef

The ratio of chicken to beef must be at least 2 to 1.

1) The company wants to know the optimalmixture of ingredientsthatwillminimizecost.

2)Formulatea linear programming model for this problem.

A farmer is preparing to plant a crop in the spring and needs to fertilize a field. There are twobrands of fertilizer to choose from, Super-Gro and Crop Quick. Each brand yields a specific amountof Nitrogen and Phosphate per bag as follows.

Brand Chemical Contribution

Nitrogen (lb/bag) Phosphate (lb/bag)

Super-Gro 24

Crop Quick 43

The farmer’s field requires at least 16 pounds of nitrogen and at least 24 pounds of phosphate. Super- gro costs M6 per bag, and Crop Quick costs M3per bag.

1) The farmer wants to know how many bags of each brand to purchase in order to minimize thetotal cost of fertilizing.

2) Formulate a mathematical model of the problem.

A retail store stocks two types of shirts A and B. These are packed in attractive card boxes.During a week the store can sell a maximum of 400 shirts of type A and a maximum of 300 shirts oftype B. The storage capacity, however, is limited to a maximum of 600 of both types combined. TypeA fetches a profit of M2 per unit and Type B a profit of M5 per unit.

How many of each type thestore should stock per week to maximize the total profit? Formulate a mathematical model of theproblem.

Roma grower has a 50-acre farm on which to plant cabbages and tomatoes. The grower has available 300 hours of labourper week and 800 tons of fertilizer, and he has contracted shipping space for a maximum of 26 acre’s worth of cabbages and 37 acres’

worth of tomatoes. An acre of cabbages requires 10 hours of labour and 8 tons of fertilizer,whereas an acre of tomatoes requires 3 hours of labour and 20 tons of fertilizer.

The profit from an acre of cabbages is M400, and the profit from an acre of tomatoes is M300. Thefarmer wants to know the number of acres of cabbages and tomatoes to plant to maximize profit.

Formulate this as a linear programming

problem.

Paul Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, Phosphate,and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate,while a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounceof potassium. Ingredient 1 cost M3 per pound, and ingredient 2 costs M5 per pound. The companywants to know to know how many pounds of each chemical ingredient to put into a bag of fertilizerto meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and two ouncesof potassium while minimizing the cost.

•Formulate a linear programming model for this problem