RADA COLLEGE
OPERATION MANAGEMENT HOME TAKE EXAM
I-Write your answer for the following Questions
1) What are some important assumptions made in formulating a linear programming model?
2) What does it mean by Decision in organization business context ?
3) How are business decisions made in business organizations?
4) What are the possible preconditions to make best decisions?
5) How do intuition, judgment, and creativity affect decision making?
6) How can the decision-making process be managed?
Please first try to understand the following example and do your assignment as problem-I, Problem-II and Problem-III
Example:-MAXIMISATION CASE:
(Please refere the given e-book form Page 45-55 for your good understanding)
A factory manufactures two products A and B on three machines X, Y, and Z. Product A requires 10 hours of machine X and 5 hours of machine Y a one our of machine Z. The requirement of product B is 6 hours, 10 hours and 2 hours of machine X, Y and Z respectively. The profit contribution of products A and B are Birr 23/– per unit and Birr. 32 /– per unit respectively. In the coming planning period the available capacity of machines X, Y and Z are 2500 hours, 2000 hours and 500 hours respectively.
Required:-
1) Find the optimal product mix for maximizing the profit.
A) Using Graphic Method:
B) Using Simplex method
Step-I
You can summarize the above given information into the following data in table
Machines Product Capacity in Hours
A B
X 10 6 2500
Y 5 10 2000
Z 1 2 500
Profit per Unit in Birr 23 32
Step-II
Let the company manufactures “a” units of A and “b” units of B. Then the inequalities of the constraints (machine capacities) are:
Maximise Z = 23 a + 32 b S.T……………………………... OBJECTIVE FUNCTION
10a + 6b ≤ 2500
5a + 10b ≤ 2000 ……………………………………. STRUCTURAL CONSTRAINTS.
1a + 2b ≤ 500
And
both a and b are ≥ 0. ……………………………………….NON-NEGATIVITY CONSTRAINT.
Stpe-III (Now the above inequalities are to be converted into equations.)
Take machine X: One unit of product A requires 10 hours of machine X and one unit of product B require 6 units. But company is manufacturing a units of A and b units of B, hence both put together must be less than or equal to 2,500 hours. Suppose a = 10 and b = 10 then the total consumption is 10 × 10 + 6 × 10 = 160 hours. That is out of 2,500 hours, 160 hours are consumed, and 2,340 hours are still remaining idle. So if we want to convert it into an equation then 100 + 60 + 2,340 = 2,500. As we do not know the exact values of decision variables a and b how much to add to convert the inequality into an equation. For this we represent the idle capacity by means of a SLACK VARIABLE represented by S. Slack variable for first inequality is S1, that of second one is S2 and that of ‘n’th inequality is Sn.
Regarding the objective function, if we sell one unit of A it will fetch the company Birr 23.00 per unit and that of B is Birr. 32.00 per unit. If company does not manufacture A or B, all resources remain idle. Hence the profit will be Zero Birr. This clearly shows that the profit contribution of each hour of idle resource is zero. In Linear Programming language, we can say that the company has capacity of manufacturing 2,500 units of S1, i.e., S1 is an imaginary product, which require one hour of machine X alone. Similarly, S2 is an imaginary product requires one hour of machine Y alone and S3 is an imaginary product, which requires one hour of machine Z alone. In simplex language S1, S2 and S3 are Surplus resources or idle resources. The profit earned by keeping all the machines idle is Birr0.00 (zero). Hence the profit contributions of S1, S2 and S3 are Birr 0.00 per unit. By using this concept, the inequalities are converted into equations as shown below:
Maximize Z = 23 a + 32 b + 0S1 + 0S2 + 0S3
Subject to:
10a + 6 b + 1S1 = 2500
` 5a + 10 b + 1S2 = 2000
1a + 2 b + 1S3 = 500 and
a, b, S1, S2 and S3 all ≥ 0.
Step-IV
A) Solve the above using graphic methods.
In Graphical method, while finding the profit by Isoprofit line, we use to draw Isoprofit line at origin and use to move that line to reach the far off point from the origin. This is because starting from zero profit; we want to move towards the maximum profit. Here also, first we start with zero profit, i.e., considering the slack variables as the basis variables (problem variables) in the initial program and then improve the program step by step until we get the optimal profit. Let us start the first program or initial program by rewriting the entries as shown in the above simplex table.
B) Solve the problem using Simplex method (Maximization model).
1) Formulate the equation in simplex format
In Simplex version, all variables must be available in all equations.
Hence the Simplex format of the model is:
Maximise Z = 23 a + 32 b + 0S1 + 0S2 + 0S3
Subject to:
10 a + 6 b + 1S1 + 0S2 + 0S3 = 2500
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