Reference: <An Introduction to Management Science Quantitative Approaches to Decision Making, Revised 13th Edition>

A Make-or-Buy Decision: Company forecasters indicate that 3000 Financial Manager calculators and 2000 Technician calculators will be needed. However, manufacturing capacity is limited. The company has 200 hours of regular manufacturing time and 50 hours of overtime that can be scheduled for the calculators. Overtime involves a premium at the additional cost of $9 per hour. Table 4.6 shows manufacturing times (in minutes) for the components.

A1:Build model as below:

BM = number of bases manufactured

BP = number of bases purchased

FCM = number of Financial cartridges manufactured

FCP = number of Financial cartridges purchased

TCM = number of Technician cartridges manufactured

TCP = number of Technician cartridges purchased

FTM = number of Financial tops manufactured

FTP = number of Financial tops purchased

TTM = number of Technician tops manufactured

TTP = number of Technician tops purchased

Production Scheduling: Let us consider the case of the Bollinger Electronics Company, which produces two different electronic components for a major airplane engine manufacturer. The airplane engine manufacturer notifies the Bollinger sales office each quarter of its monthly requirements for components for each of the next three months. The monthly requirements for the components may vary considerably, depending on the type of engine the airplane engine manufacturer is producing. The order shown in Table 4.7 has just been received for the next three-month period.

After the order is processed, a demand statement is sent to the production control department. The production control department must then develop a three-month production plan for the components. In arriving at the desired schedule, the production manager will want to identify the following:

1. Total production cost

2. Inventory holding cost

3. Change-in-production-level costs

To develop the model, we let x_im denote the production volume in units for product i in month m. Here i =1, 2, and m =1, 2, 3; i =1 refers to component 322A, i =2 refers to component 802B, m 1 refers to April, m =2 refers to May, and m=3 refers to June. The purpose of the double subscript is to provide a more descriptive notation. We could simply use x₆ to represent the number of units of product 2 produced in month 3, but x₂₃ is more descriptive, identifying directly the product and month represented by the variable.

If component 322A costs $20 per unit produced and component 802B costs $10 per unit produced. Bollinger determined that on a monthly basis inventory holding costs are 1.5% of the cost of the product;Bollinger estimates that the cost associated with increasing the production level for any month is $0.50 per unit increase. A similar cost associated with decreasing the production level for any month is $0.20 per unit.

Suppose that the inventories at the beginning of the three-month scheduling period were 500 units for component 322A and 200 units for component 802B.

company specifies a minimum inventory level at the end of the three-month period of at least 400 units of component 322A and at least 200 units of component 802B.

Suppose that the production levels for March, the month before the start of the current production scheduling period, had been 1500 units of component 322A and 1000 units of component 802B

 Workforce Assignment:McCormick Manufacturing Company produces two products with contributions to profit per unit of $10 and $9, respectively. The labor requirements per unit produced and

the total hours of labor available from personnel assigned to each of four departments are shown in Table 4.11.

Suppose that McCormick has a cross-training program that enables some employees to be transferred between departments. By taking advantage of the cross-training skills, a limited number of employees and labor-hours may be transferred from one department to another. For example, suppose that the cross-training permits transfers as shown in Table 4.12. Row 1 of this table shows that some employees assigned to department 1 have cross-training skills that permit them to be transferred to department 2 or 3. The right-hand column shows that, for the current production planning period, a maximum of 400 hours can be transferred from department 1. Similar cross-training transfer capabilities and capacities are shown for departments 2, 3, and 4.

A:build model as below

P1 = units of product 1

P2 = units of product 2

b_i =the labor-hours allocated to department i for i = 1, 2, 3, and 4

t_ij =the labor-hours transferred from department i to department j

Blending Problems:Data available show that regular gasoline can be sold for $2.90 per gallon and premium gasoline for $3.00 per gallon. For the current production planning period, Grand Strand can obtain the three petroleum components at the cost per gallon and in the quantities shown in Table Product specifications for the regular and premium gasolines restrict the amounts of each component that can be used in each gasoline product. Table 4.14 lists the product specifications. Current commitments to distributors require Grand Strand to produce at least 10,000 gallons of regular gasoline.

The Grand Strand blending problem is to determine how many gallons of each component should be used in the regular gasoline blend and how many should be used in the premium gasoline blend. The optimal blending solution should maximize the firm’s profit,subject to the constraints on the available petroleum supplies shown in Table 4.13, the product specifications shown in Table 4.14, and the required 10,000 gallons of regular gasoline.

x_ij =gallons of component i used in gasoline j, where i = 1, 2, or 3 for components 1, 2, or 3, and j = 1 if regular or j = 2 if premium