Linear and Integer Programming for Buff System

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Buff Systems is a small company that produces three products daily: (1) CD players, (2) cell phone, and (3) camcorders. The company has a limited amount of resources used in the production of these products. The limited resources are labor, machine hours, and inventory space allocated to all products. Buff has 1,560 labor hours per day, 1,900 machine hours per day, and 1,400 square feet of storage space capacity availability to store the three different products each day. In addition, the management policies of the company dictate the following:

. The minimum number of cell phones produced needs to be at least 20 percent of all units produced for all three products. . The minimum number of CD players produced need to be at least 40 percent of all units produced for all three products.
Given the above resource requirements, Buff has the following resource requirements for product and profit for each product being produced.
Table 5.1 Resource Requirements

Product
Labor Hours
Machine Hours
Storage Space (sq ft)
Profit per Unit ($)

CD players
1
1.25
.5
27

Cell phones
5
5.00
4.5
55

Camcorders
3
3.50
2.5
80

A. If Buff's goal is to maximize the profit, what would be the production quantity of each product and the total profit? Show LP formulation.

B. Instead of a single objective of maximizing profit, the company has listed the following objectives in order of importance:

1. The owner of Buff wants to be able to provide a stable environment for his employees. In order to accomplish his goal of having a stable workforce, he does not want to use fewer than 1,560 hours of labor per day. 2. Buff would like to achieve a satisfactory daily profit level of $32,000. 3. The owner also is concerned about machine utilization. He wants to be able to effectively utilize the machine hours. In order to reach his goal of high machine utilization, he would like a utilization of at least 1,900 machine hours. 4. Buff does not want to utilize more than 1,400 square feet of inventory space available. 5. Because Buff has to pay overtime, and due to high overhead costs when the plant is kept open past the normal operating hours, the owner wants to minimize the amount of overtime by its employees. 6. Due to warehouse space regulations, the company wants to limit the production of cell phones to 180 units per day.
7. Due to warehouse space regulations, the company wants to limit the production of camcorders to 220 units per day.
FORMULATE AND SOLVE THE GOAL PROGRAMMING PROBLEM DESCRIBED IN PART B. REMEMBER when formulating a goal programming problem, FORMULATE the goal programming problem as a linear programming model. Formulate the goals as goal constraints. Use preemptive penalties and solve using the linear/integer algorithm. Do not use goal programming algorithm. In your formulation be sure to show all constraints (e.g., content, goal, non-negative); show definitions of decision and deviation variables; show objective function; assign values to the penalties on the deviation variables and explain why you chose those values. Delineate your solution and explain the results. Remember the value of the objective function does not have a physical meaning, and everyone may not have the same value for the objective function.