Linear Solution

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CHAPTER 8

Linear Programming Applications

Teaching Suggestions

Teaching Suggestion 8.1: Importance of Formulating Large LP Problems.

Since computers are used to solve virtually all business LP problems, the most important thing a student can do is to get experience in formulating a wide variety of problems. This chapter provides such a variety.

Teaching Suggestion 8.2: Note on Production Scheduling Problems.

The Greenberg Motor example in this chapter is largest large problem in terms of the number of constraints, so it provides a good practice environment. An interesting feature to point out is that LP constraints are capable of tying one production period to the next.

Teaching Suggestion 8.3: Labor Planning Problem—Hong Kong Bank of Commerce.

This example is a good practice tool and lead-in for the Chase Manhattan Bank case at the end of the chapter. Without this example, the case would probably overpower most students.

Teaching Suggestion 8.4: Ingredient Blending Applications.

Three points can be made about the two blending examples in this chapter. First, both the diet and fuel blending problems presented here are tiny compared to huge real-world blending problems. But they do provide some sense of the issues to be faced.

Second, diet problems that are missing the constraints that force variety into the diet can be terribly embarrassing. It has been said that a hospital in New Orleans ended up with an LP solution to feed each patient only castor oil for dinner because analysts neglected to add constraints forcing a well-rounded diet.

Alternative Examples

Alternative Example 8.1:  Natural Furniture Company manufactures three outdoor products, chairs, benches, and tables. Each product must pass through the following departments before it is shipped: sawing, sanding, assembly, and painting. The time requirements (in hours) are summarized in the tables below.

The production time available in each department each week and the minimum weekly production requirement to fulfill contracts are as follows:

| | | |Minimum | | |Capacity | |Production | |Department |(In Hours) |Product |Level | |Sawing |450 |Chairs |100 | |Sanding |400 |Benches |50 | |Assembly |625 |Tables |50 | |Painting |550 | | |

| |Hours Required |Unit | |Product |Sawing |Sanding |Assembly |Painting |Profit | |Chairs |1.5 |1.0 |2.0 |1.5 |$15 | |Benches |1.5 |1.5 |2.0 |2.0 |$10 | |Tables |2.0 |2.0 |2.5 |2.0 |$20 |

The production manager has the responsibility of specifying production levels for each product for the coming week. Formulate as a linear programming problem to maximize profit.

Let

X1= Number of chairs produced

X2= Number of benches produced

X3= Number of tables produced

The objective function is

Maximize profit = 15X1 + 10X2 + 20X3

Constraints

1.5X1 + 1.5X2 + 2.0X3( 450 hours of sawing available

1.0X1 + 1.5X2 + 2.0X3( 400 hours of sanding available

2.0X1 + 2.0X2 + 2.5X3( 625 hours of assembly available

1.5X1 + 2.0X2 + 2.0X3( 550 hours of painting available

X1+ 2.0X2 + 2.0X3( 100 chairs

X2 + 2.0X3( 50 benches

X3( 50 tables

X1, X2, X3( 0

What mix of products would yield...
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