Sample Chapters

The chapter begins with an exploratory problem designed to introduce the concept of linear programming with an objective function to maximize profits by optimizing a company’s product mix. The problem context involves assembling two types of computers with different profit margins and labor requirements. Students are led through a graphical solution to a two decision variable problem involving two constraints. They then proceed to add a series of constraints and explore changes to the optimal solution. Lastly, the problem is expanded to include more decision variables with a focus on formulating this more complex problem. Once there are more than two decision variables, the problem cannot be solved graphically.

The second product mix example involves a detailed totally worked-out example involving the manufacture of skateboards. Students are shown step-by-step how to formulate and solve this two decision variable problem graphically. A third decision variable is then added to motivate the need for EXCEL to solve larger problems. Students are taught how to use SOLVER as standard add-in to EXCEL to solve linear programming problems. The largest example involves six decision variables and five constraints. This section also discusses how to use the linear programming output to perform sensitivity analysis. There is also an optional section that discusses the Simplex algorithm that is the basis for computational solution of LP problems.

The third example is a sports shoe company and focuses on interpretation of results. The text presents a fully formulated and solved problem involving six decision variables and six constraints. The emphasis is on interpreting the output from SOLVER and answering a variety of what-if questions.

The concept of a zero or one (binary) variable has special application to problems that involve matching up individuals or machines with tasks. This is called the assignment problem. The specific example involves picking a set of four swimmers to be in a medley relay. This class of problems has a unique mathematical structure that enables it to be solved with efficient specialty algorithms. In the original problem the goal is to minimize the total time by optimally matching the four swimmers to the four events. We then expand the problem such that there are more swimmers than events. We eventually grow the problem to a size that makes it cumbersome to use the Hungarian algorithm. We then show how to use EXCEL to solve this same problem.