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In the sequel, we consider illustrative situations so as to explain the process of problem formulation in goal programming.

In the sequel, we consider illustrative situations so as to explain the process of problem formulation in goal programming.

Let Du denote the under-achievement of the goal.

Let Do denote the over-achievement of the goal.

Denote the under-achievement and the over-achievement of one goal by Du1 and Do1 respectively.

Denote the under-achievement and the over-achievement of another goal by Du2 and Do2 respectively.

Alpha company is known for the manufacture of tables and chairs. There is a profit of Rs. 200 per table and Rs. 80 per chair. Production of a table requires 5 hours of assembly and 3 hours in finishing. In order to produce a chair, the requirements are 3 hours of assembly and 2 hours of finishing. The company has 105 hours of assembly time and 65 hours of finishing. The company manager is interested to find out the optimal production of tables and chairs so as to have a maximum profit of Rs. 4000. Formulate a goal programming problem for this situation.

The manager is interested not only in the maximization of profit but he has also fixed a target of Rs. 4000 as profit. Thus, the problem involves a single goal of achieving the specified amount of profit.

Let

The objective in the given situation is to minimize under achievement. Let Z be the objective function. Then the problem is the minimization of

Let the number of tables to be produced be x and let the number of chairs to be produced be Y.

Profit from x tables = Rs. 200 x

Profit from y chairs = Rs. 80 y

The total profit = Profit from x tables and y chairs + under achievement of the profit target -over achievement of the profit target

To produce x tables, the requirement of assembly time = 5 x hours. To produce y

chairs, the requirement is 3 y hours. So, the total requirement is 5

must be fulfilled

To produce x tables, the requirement of finishing time = 3 x. To produce y chairs, the requirement is 2 y . So, the total requirement is 3 x+ 2 y. But the availability is 65 hours. Hence we have the restriction

3

The number of tables and chairs produced, the under achievement of the profit target and the over achievement cannot be negative. Thus we have the restrictions

Minimize

subject to the constraints

Sweet Bakery Ltd. produces two recipes A and B. Both recipes are made of two food stuffs I and II. Production of one Kg of A requires 7 units of food stuff I and 4 units of food stuff II whereas for producing one Kg of B, 4 units of food stuff I and 3 units of food stuff II are required. The company has 145 units of food stuff I and 90 units of food stuff II. The profit per Kg of A is Rs. 120 while that of B is Rs. 90. The manager wants to earn a maximum profit of Rs. 2700 and to fulfil the demand of 12 Kgs of A. Formulate a goal programming problem for this situation.

The management has two goals.

To reach a profit of Rs. 2700

Production of 12 Kgs of recipe A.

Let

Let

Let

Let

The objective in this problem is to minimize the under achievement of the profit target and to minimize the under achievement of the production target for recipe A.

Let Z be the objective function. Then the problem is the minimization of

Suppose the company has to produce x kgs of recipe A and y kgs of recipe B in order to achieve the two goals.

Profit from x kgs of A = 120 x

Profit from y kgs of B = 90 y

The total profit = Profit from x kgs of A + Profit from y kgs of B + under achievement of the profit target – over achievement of the profit target

= 120

Thus we have the restriction

120

Constraint for food stuff I:

7

Constraint for food stuff II:

4

Non-negativity restrictions

The target production of A = optimal production of A + under achievement in production target of A – over achievement of the production target of A.

Thus we have the condition

Minimize

subject to the constraints

Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory

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