Question

Western Technical Services is a small engineering firm in Colorado that provides a variety of technical and drafting services. Western Technical Services employs three engineers, five drafters, and three secretaries. Western has developed a leading reputation in the design of low-capacity and medium-capacity heat exchangers for electrical equipment. A large backlog of jobs has accumulated, and the firm has decided it must complete the back orders. The firm wants to maximize the number of jobs finished each day to eliminate its backlog. The average time required to design a medium-capacity heat exchangers is 4 hours of engineering time, 3 hours of time by a drafter, and 3 hours of secretarial work. Designing a low-capacity heat exchanger requires an average of 1.5 hours of work from an engineer, 4 hours of work from a drafter, and 1.5 hours of secretarial effort.

Formulate a linear programming problem that Western could use to determine how best to employ its resources in order to maximize the total number of heat exchangers that can be completed each day subject to the firm’s labor constraints. I.e. the objective of the firm in this problem is to maximized total output (Q). (Hint: Q = 1M + 1L where M = the number of medium capacity heat exchangers completed and L = the number of low capacity heat exchangers completed each day) Assume that the engineers and drafters each work 10-hour days, and the secretaries work 8 hours per day.

Solve the linear programming problem and interpret the solution. How many heat exchangers will be completed each day? Your assignment is to maximize output (i.e. maximize the number total number of heat exchangers produced subject to the constraints provided.) Hint: The total number, i.e. quantity, of heat exchangers is Total output Q = 1M + 1L. You are trying to determine the maximum number the firm can produce given its production constraints.

Answer 1

Let M be the number of medium capacity heat exchangers completed each day and L be the number of low capacity heat exchangers completed per day

these are the decision variables

The total number of heat exhangers produced per day is M+L

The firm wants to maximize the total number of exchangers produced per day

The objective function is to maximize M+L

Following are the constraints

Engineering hours

• Each of the 3 engineers can work 10 hours per day, for a total 30 hours of Engineering time per day. The engineering time required to produce M and L quantities is ​ and this has to be less than or equal to 30
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Drafter hours

• Each of the 5 drafters can work 10 hours per day, for a total 50 hours of drafter time per day. The drafter time required to produce M and L quantities is ​ and this has to be less than or equal to 50
  • ​

Secretarial hours

• Each of the 3 secretaries can work 8 hours per day, for a total 24 hours of secretarial time per day. The secretarial time required to produce M and L quantities is ​ and this has to be less than or equal to 24
  • ​

The linear programming model is

maximize

s.t.

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• 
• 
• 

We prepare the following excel

get these values

set up the solver using data--->solver

get the following solution

To maximize the total number of exhanges produced per day, given the production constraints, the firm needs to produce 2.8 medium capacity and 10.4 low capacity heat exhangers. The fractional values will indicate the partially completed heat exhangers during the day.