Exam 8: Linear Programming Applications

The constraints in a transportation problem deal with requirements at each origin and capacities at each destination.

Table 8-4 The following is a linear programming formulation of a labor planning problem. There are four overlapping shifts, and management must decide how many employees to schedule to start work on each shift. The objective is to minimize the total number of employees required while the constraints stipulate how many employees are required at each time of day. The variables X₁ - X₄ represent the number of employees starting work on each shift (shift 1 through shift 4). -According to Table 8-4, which describes a labor planning problem and its solution, how many workers would be assigned to shift 2?

Determining the mixture of ingredients for a most economical feed or diet combination would be described as a production mix type of linear program.

A mail order firm, AmazingCo, can use one of 3 shipping couriers. Ajax Shipping charges $6 per pound and delivers in 2 days. Bilco Lanes charges $9 per pound but guarantees next day delivery. The final courier, Hobo Ltd., charges only $3 per pound but takes 4 days to deliver. AmazingCo has a quarterly budget of $280,000 and a reputation for timely delivery. Formulate the linear program so that the firm delivers as fast as possible within budget for 70,000 pounds of shipments per quarter.

The following table provides shipping costs from each of two regional warehouses to each of three destinations. The supplies available and the demands are also given in the table. Formulate this as a linear programming problem. Carefully define all decision variables.

The linear programming truck loading model always results in a practical solution.

Table 8-2 A small furniture manufacturer produces tables and chairs. Each product must go through three stages of the manufacturing process: assembly, finishing, and inspection. Each table requires 4 hours of assembly, 3 hours of finishing, and 1 hour of inspection. Each chair requires 3 hours of assembly, 2 hours of finishing, and 2 hours of inspection. The selling price per table is $140 while the selling price per chair is $90. Currently, each week there are 220 hours of assembly time available, 160 hours of finishing time, and 45 hours of inspection time. Assume that one hour of assembly time costs $5.00; one hour of finishing time costs $6.00; one hour of inspection time costs $4.50; and that whatever labor hours are not required for the table and chairs can be applied to another product. Linear programming is to be used to develop a production schedule. Define the variables as follows: -According to Table 8-2, which describes a production problem, suppose it was decided that all the labor hour costs have to be covered through the sale of the tables and chairs, regardless of whether or not all the labor hours are actually used. How would the objective function be written?

Linear programming is usually used by managers involved in portfolio selection to

Dr. Malcomb Heizer wishes to invest his retirement fund of $2,000,000 so that his return on investment is maximized, but he also wishes to keep the risk level relatively low. He has decided to invest his money in any of three possible ways: CDs that pay a guaranteed 4 percent; stocks that have an expected return of 14 percent; and a money market mutual fund that is expected to return 18 percent. He has decided that the total $2,000,000 will be invested, but any part (or all) of it may be put in any of the three alternatives. Thus, he may have some money invested in all three alternatives. He has also decided to invest, at most, 30 percent of this in stocks and at least 20 percent of this in money market funds. Formulate this as a linear programming problem and carefully define all the decision variables.

The linear programming model of the production scheduling process is usually used when we have to schedule the production of multiple products, each of which requires a set of resources not required by the other products, over time.

Describe the production scheduling linear programming application.

A computer start-up named Pear is considering entering the U.S. market with what they believe to be a smaller and faster computer than some of the existing products on the market. They want to get a feel for whether or not customers would be willing to switch from some of the existing bigger brands to consider their product. They want to collect a reasonable sample from across the U.S. representative of all age brackets. They have split the U.S. into 2 regions: East and West. They want to at least 65% of their sample to cover the East and no fewer than 25% of the West. They also have divided the age groups into 3 categories: 18-35, 36-69, and 70 and up. They want at least 50% of their sample to be between 18-35 and at least 40% to be between 36-69. The costs per person surveyed is given in the table below: Assume that exactly 1,000 people are to be surveyed. The problem is for Pear Company to decide how many people to survey from each age bracket within each region in order to minimize costs while meeting requirements. Formulate this problem as a linear program.

First Securities, Inc., an investment firm, has $380,000 on account. The chief investment officer would like to reinvest the $380,000 in a portfolio that would maximize return on investment while at the same time maintaining a relatively conservative mix of stocks and bonds. The following table shows the investment opportunities and rates of return. The Board of Directors has mandated that at least 60 percent of the investment consist of a combination of municipal and federal bonds, 25 percent Blue Chip Stock, and no more than 15 percent High Tech Stock. Formulate this portfolio selection problem using linear programming.

What are the decision variables in the diet problem?

Cedar Point amusement park management is preparing the park's annual promotional plan for the coming season. Several advertising alternatives exist: newspaper, television, radio, and displays at recreational shows. The information below shows the characteristics associated with each of the advertising alternatives, as well as the maximum number of placements available in each medium. Given an advertising budget of $250,000, how many placements should be made in each medium to maximize total audience exposure? Formulate this as a linear programming problem.

Table 8-1 A small furniture manufacturer produces tables and chairs. Each product must go through three stages of the manufacturing process: assembly, finishing, and inspection. Each table requires 3 hours of assembly, 2 hours of finishing, and 1 hour of inspection. Each chair requires 2 hours of assembly, 2 hours of finishing, and 1 hour of inspection. The profit per table is $120 while the profit per chair is $80. Currently, each week there are 200 hours of assembly time available, 180 hours of finishing time, and 40 hours of inspection time. Linear programming is to be used to develop a production schedule. Define the variables as follows: -According to Table 8-1, which describes a production problem, which of the following would be a necessary constraint in the problem?

Friendly Manufacturing has three factories (1, 2, and 3) and three warehouses (A, B, and C). The table below shows the shipping costs between each factory (in dollars) and warehouse, the factory manufacturing capabilities (in 1000s), and the warehouse capacities (in 1000s). Write the objective function and the constraint inequalities. Let X_1A = 1000s of units shipped from factory 1 to warehouse A, etc.

Since the production mix linear program applications are a special situation, the number of decision variables is limited to two.

Describe the truck loading linear programming application.

In the linear programming transportation model, the coefficients of the objective function can represent either the cost or the profit from shipping goods along a particular route.