Operations Management - Game Theory, Goal Programming & Queuing Theory
Formulation of Goal Programming Problems
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In the sequel, we consider illustrative situations so as to explain the process of problem formulation in goal programming.
Formulation of Goal
Programming Problems
In the sequel, we consider illustrative situations so as to explain the process of problem formulation in goal programming.
If there is a single goal, we have the following notations:
Let Du denote the under-achievement of the goal.
Let Do denote the over-achievement of the goal.
If there are two goals, we have the following notations:
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.
Solution
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 Du denote the under achievement of the target profit and let Do be the over achievement.
The objective in the given situation is to minimize under achievement. Let Z be the objective function. Then the problem is the minimization of Z = Du .
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
So we have the relationship 200 x
+ 80 y + Du - Do = 4000.
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 x + 3 y hours. But the available time for assembly is 105 hours. Therefore constraint
5 x + 3 y
≤ 105
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 x + 2 y ≤ 65
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
x ≥ 0, y ≥ 0, Du ≥ 0, Do ≥ 0
Minimize Z = Du
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.
Solution
The management has two goals.
To reach a profit of Rs. 2700
Production of 12 Kgs of recipe A.
Let Dup denote the under achievement of the profit target.
Let Dop denote the over achievement of the profit target.
Let DuA denote the under achievement of the production target for recipe A.
Let DoA denote the over achievement of the production target for recipe A.
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
Z = Dup + DuA
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 x + 90 y + Dup − Dop
Thus we have the restriction
120 x + 90 y + Dup − Dop = 2700
Constraint for food stuff I:
7 x + 4 y ≤ 145
Constraint for food stuff II:
4 x + 3 y ≤ 90
Non-negativity restrictions
x , y , Dup , Dop , DuA , DoA ≥ 0
The target production of A = optimal production of A + under achievement in production target of A – over achievement of the production target of A.
x + DuA − DoA =12
Minimize Z = Dup + DuA
subject to the constraints
In the sequel, we consider illustrative situations so as to explain the process of problem formulation in goal programming.
Notations
If there is a single goal, we have the following notations:
Let Du denote the under-achievement of the goal.
Let Do denote the over-achievement of the goal.
If there are two goals, we have the following notations:
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.
Problem 1
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.
Solution
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 Du denote the under achievement of the target profit and let Do be the over achievement.
The objective in the given situation is to minimize under achievement. Let Z be the objective function. Then the problem is the minimization of Z = Du .
Formulation of the constraints
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
Assembly time
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 x + 3 y hours. But the available time for assembly is 105 hours. Therefore constraint
must be fulfilled
Finishing time
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 x + 2 y ≤ 65
Non-negativity restrictions
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
x ≥ 0, y ≥ 0, Du ≥ 0, Do ≥ 0
Statement of the problem
Minimize Z = Du
subject to the constraints
Problem 2
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.
Solution
The management has two goals.
To reach a profit of Rs. 2700
Production of 12 Kgs of recipe A.
Let Dup denote the under achievement of the profit target.
Let Dop denote the over achievement of the profit target.
Let DuA denote the under achievement of the production target for recipe A.
Let DoA denote the over achievement of the production target for recipe A.
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
Constraints
Suppose the company has to produce x kgs of recipe A and y kgs of recipe B in order to achieve the two goals.
Condition on profit
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 x + 90 y + Dup − Dop
Thus we have the restriction
120 x + 90 y + Dup − Dop = 2700
Constraint for food stuff I:
7 x + 4 y ≤ 145
Constraint for food stuff II:
4 x + 3 y ≤ 90
Non-negativity restrictions
x , y , Dup , Dop , DuA , DoA ≥ 0
Condition on recipe A
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
Statement of the problem
Minimize Z = Dup + DuA
subject to the constraints
Tags : Operations Management - Game Theory, Goal Programming & Queuing Theory
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