Buster Brown - Texas Tech University



Linear Programming

Capital Investment. Energy Management Corporation (EMC) must decide its level of capital investment in the six energy ventures described below. EMC wishes to maximize its total expected return on a maximum total investment of $10,000,000. At least half of this must be in the United States, including Alaska. No more than 20% can be in sour crude and coal investments.

|Venture (Location) |Expected Return |Minimum Investment |Primary Product |

|Wyoming Coal |75% |$1,000,000 |Coal |

|Colorado Shale |62% |$400,000 |Sour crude |

|Prudhoe Bay Alaska |125% |$1,000,000 |Sweet crude |

|Mexico |135% |None |Sweet crude |

|Alberta Tar Sands |80% |None |Sour crude |

|Virginia Coal |85% |$300,000 |Coal |

Formulate the LP model. Use fractions rather than percentages in your formulation of the objective function. Solve the model using Excel Solver. Interpret the results. What aspects of the computer-generated strategy are you in agreement with (based on a consideration of the sensitivity data)? What aspects of it would you disagree with? Why?

Today, we are covering linear programming. We want to learn two things—how to formulate linear programming models and how to interpret the computer generated output. In doing so, we shall construct models for the following two problems—Buster Brown, and Blending. We will consider first, Buster Brown.

Buster Brown

The Buster Brown Shoe Company manufactures two types of shoes—a dress shoe and a work shoe. It makes $16 profit on the sale of each pair of dress shoes and $12 on the sale of each pair of work shoes. Due to the additional leather work required, dress shoes require twice as much labor as work shoes. Within any given production period there is enough labor to produce 1,000 pairs of work shoes if no dress shoes were produced. In addition, there is enough leather to produce 1,000 shoes of either type since both require the same amount. The dress shoe requires a fancy heel of which there are 500 available each period, whereas the work shoe requires a different heel of which only 600 are available each period. Buster Brown wants to know how many shoes of each type to produce each period so that profit will be maximized. It can sell everything it produces.

The first thing we do in constructing linear programming models is determine and define the decision variables. The decision variables are what we want to find as a manager. We write out explicit definitions for each, including the units to be used. Next, we construct an objective function, using the decision variables defined. The objective function is a mathematization of the goal or objective that the manager has in mind. Then we determine the constraints. Let’s do this for the Buster Brown problem described above. There are just two decision variables. They are:

x1 = # of ordinary work shoes to produce in any given production period

x2 = # of dress (fancy) shoes to produce in any given production period

The objective function is

Max x0 = 12x1 + 16x2

It is customary to denote the objective function by x0. Notice that this objective function adds the profit contributions of both work shoes (12x1) and dress shoes (16x2). Doing so, gives us a function for total profit of the Buster Brown Shoe Company.

Next, we determine the constraints. The problem statement says that there is enough labor to complete 1,000 work shoes, but that dress shoes require twice as much labor as work shoes. We write this constraint as x1 + 2x2 ................
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