Q3.1 Linear Programming – Product OutputA company produces 3 different products, A, B, and C. All products require time in up to 5 different processing departments. The amount of time each product requires with each department is listed in the table below:

A company Product B

Product C Available

Process #1 2 hrs 8 hr 0 hrs 1,100 hrs Process #2 3 hrs 6 hrs 9 hrs 1,140 hrs Process #3 10 hrs 7 hrs 1 hr 1,120 hrs Process #4 1 hrs 5 hr 3 hrs 1,170 hrs Process #5 4 hr 3 hrs 2 hrs 1,150 hrs

Suppose product A sells for $125/unit, product B sells for $175/unit, and product C sells for $100/unit. And suppose that all products have a guaranteed buyer (i.e., no matter how many are produced it will be sold for the asking price), and that you must produce at least 10 of each product. 1. What is the formula for the objective function? (2 points) 2. Please list all constraints (2 points) 3. What is the maximum revenue to be generated for this configuration? (8 points) 4. How many each product should be produced that maximizes revenue? (8 points) SolutionLet X1 units of product A, X2 units of product B and X3 units of product C to produce.

1. The revenue from X1 units of product A, X2 units of product B and X3 units of product C is 125X1+175X2+100X3. Therefore, the objective function is, maximize Z = 125X1+175X2+100X3.

2. The other constraints are given as

2X1+8X2+0X3 ≤ 11003X1+6X2+9X3 ≤ 114010X1+7X2+X3 ≤ 1120X1+5X2+3X3 ≤ 11704X1+3X2+2X3 ≤ 1150X1 ≥ 10, X2 ≥ 10, X3 ≥ 10, Xi, i =1,2,3 integers.

3. This problem can be solved using excel solver and the output is given below

Integer Programming  

VariablesX1 14X2 134X3 32

Constraints

2X1+8X2+0X3 1100

3X1+6X2+9X3 1134

10X1+7X2+X3 1110

X1+5X2+3X3 780

4X1+3X2+2X3 522

Objective functions Z 28400

The maximum revenue that can be generated is Z= $28400

4. To get the above maximum revenue we must produce X1 = 14, X2 = 134, X3 = 32. That is produce 14 units of Product A, 134 units of product B and 32 units of product C.

Q3.2 Linear Programming – Product Output Suppose your company sells 4 different kinds of products, Axes, Barometers, Cherry Pickers, and Door Hinges, where each Ax sells for $15, each Barometer sells for $18, each Cherry Picker sells for $12, and each Door Hinge sells for $11. Further, suppose to make these products, each must be processed on 2 different machines in your factory and each takes a different amount of time on the two machines. Specifically, to make one Ax requires 3 hours on machine #1 and 4 hours on machine #2. To make one Barometer requires 6 hours on machine #1 and 5 hours on machine #2. To make one Cherry Picker requires 1 hour on machine #1 and 3 hours on machine #2. And to make one Door Hinge requires 1 hour on machine #1 and 1.5 hours on machine #2. Also, suppose there can be no more than a total of 525 hours of processing time on machine number 1 and no more than a total of 750 hours of processing time on machine number 2. Finally, one cannot make more than 100 of any one product. 1. What is the formula for the objective function (2 points) 2. Please list all 6 constraints (2 points) 3. What is the maximum revenue to be generated for this configuration? (8 points) 4. How many each product should be produced that maximizes revenue? (8 points)

SolutionLet X1 units of Axes, X2 units of Barometers and X3 units Cherry Pickers and X4 units of Door Hinge to produce.

1. X1 units of Axes, X2 units of Barometers and X3 units Cherry Pickers and X4 units of Door Hinge is 15X1+18X2+12X3+11X4. Therefore, the objective function is, maximize Z = 15X1+18X2+12X3+11X4.

2. The other constraints are given as

3X1+6X2+X3+X4 ≤ 5254X1+5X2+3X3+1.5X4 ≤ 750X1 ≤ 100, X2 ≤ 100, X3 ≤ 100, X4 ≤ 100X1 ≥ 0, X2 ≥ 0, X3 ≥ 0, X4 ≥ 0, integers

3. This problem can be solved using excel solver and the output is given below

Integer Programming  

Variables

X1 75X2 0X3 100X4 100

Constraints3X1+6X2+X3+X4 425

4X1+5X2+3X3+1.5X4 750 

Objective functions Z 3425

The Maximum Revenue can be obtained = $3425

4. The following are the optimal solution is given below

Number of Axes = 75Number of Barometers = 0Number of Cherry Pickers = 100Number of Door Hinge = 100