A manufacturing company produces products 1, 2, and 3. The three products have t
ID: 3850241 • Letter: A
Question
A manufacturing company produces products 1, 2, and 3. The three products have the following resource requirements and produce the following profit: At present the firm has a daily labor capacity of 240 available hours and a daily supply of 400 pounds of material. Management has developed the following set of goals, arranged in order of their importance to the firm: Because of recent labor relations difficulties, management wants to avoid underutilization of normal production capacity. Management has established a satisfactory profit level of exist500 per day. Overtime is to be minimized as much as possible. Management wants to minimize the purchase of additional materials to avoid handling and storage problems. Formulate a goal programming model (multi-criteria model) to determine the number of each product to produce to best satisfy the goals.Explanation / Answer
there are 240 available hours and a daily supply of 400 pounds of materials
so maximize M = 3a+5b+2c
which is subject to
5a+6b+3c<=240
and 4a+6b+3c<=400
where a,b and c are >=0
the 2 goal model are as following: Labour Utilization, Profit Level and Purchase of materials
1. Labour Utilization
in the model, we will formulate the linear programming constraint which will avoid underutilization of labor and avoid overtime also (minimizing overtime).
the linear programming constraint is reformulated as:
5a+2b+4c+d1'-d1=240
where d1' and d1 are known as deviational variables. deviational variables represent the number of hours fewer than 240d1' i.e. underutilization and number of hours exceeding 240d1 (overtime).
Underutilization now can be specified as M= P1d1' this represents that the primary goal of organization is to minimize the underutilization of labor (d1')
where P1 is preemptive priority designation of underutilization d1'
Overtime now can be specified as M=P1d1'+P3d1 this represents a multidimensional function which is a combination of different priority aspects and correlated income unmeasurable objective principle.
where P3 is third party goal of overtime d1
2. Profit Level
the second target of management is to obtain the suitable target level of $500 per day. this objective can be formulated as
3a+5b+2c+d2'-d2=500
where d2' and d2 are profit target achievements
d2' is underachievement of benefit target
d2 is overachievement of benefit target
the profit target can be managed by reducing d2' at the second priority match.
therefore the minimization M=P1d1'+P2d2'+P3d1
3. Purchase of Materials
the last goal of management is to reduce the purchase of additional material to avert handling and storing issues.
this means that the management has to reduce the daily purchase of material in excess of 400 pounds
this objective can be formulated as below:
4a+6b+3c+d3'-d3=400
where d3' i s over usage of daily material requirements
and d3 is the purchase of additional material.
therefore the minimization at the fourth preference match is:
M=P1d1'+P2d2'+P3d1+P4d3 it shows that the management motive is to minimize the purchase of additional materials at a match of preference beneath those of other three targets.
finally, the goal of the programming model for the entire problem can be summarized as:
M=P1d1'+P2d2'+P3d1+P4d3
5a+2b+4c+d1'-d1=240
3a+5b+2c+d2'-d2=500
4a+6b+3c+d3'-d3=400
where
a,b,c,d1',d1,d2',d2,d3',d3 are >= 0
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