Exam 6: Optimization Models With Integer Variables

Exhibit 6-2Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which can produce up to 10,000 tons per month and the Lakeview plant which can produce up to 7,000 tons per month. Sinclair sells its polyethylene to three different auto manufacturing plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The variable unit cost of shipping a ton is $1 per mile. The number of miles between the respective plants is shown in the table below: Grand Rapids Blue Ridge Sunset Ohio Valley 50 40 100 Lakeview 60 50 75 In addition to the variable unit costs of shipping, Sinclair incurs a fixed handling cost (of pickup and delivery) for any shipping routes it uses. These costs are discounted for longer routes, as shown below: Fixed Cost Levels Distance (mi) Fixed Cost <=25 \ 100,000 <=50 \ 60,000 <=75 \ 20,000 <=100 \ 0 -Refer to Exhibit 6-2. What is Sinclair's optimal shipping plan? What is the total cost in that case?

If Solver fails to find an optimal solution to an integer programming problem, we might be able to find a near optimal solution by specifying a smaller number for the integer optimality setting.

During the solution of a problem with the branch and bound algorithm, an incumbent solution represents an upper bound on the solution.

Exhibit 6-3PharmaCo wants to determine how to deploy sales representatives across its Western U.S. region to support a new drug for obesity. Sales representatives will be located in a "home city", which they serve, in addition to cities with feasible commuting distance, with the objective that all cities must be served by at least one sales representative. The feasible connections between each city in the region are listed below (1 indicates a feasible connection): -Refer to Exhibit 6-3. This problem is an example of what type of integer programming model?

Exhibit 6-1A recent college graduate is planning to move cross country from her college town to the city where her first job is located. She has rented a truck that can haul up to 1100 cubic feet of furniture. The volume and value of each item she is considering moving on the truck are given below. Bedroom Set Dining Set Stereo Sofa TV Value \ 450 \ 500 \ 150 \ 200 \ 100 Volume 800 600 300 400 80 -Refer to Exhibit 6-1. Suppose she can save $400 by renting a smaller truck, which has only 800 cubic feed of capacity. Would she be better off with the smaller truck?

Many inherently nonlinear problems can be transformed into linear models with the use of:

Exhibit 6-2Sinclair Plastics operates two chemical plants which produce polyethylene; the Ohio Valley plant which can produce up to 10,000 tons per month and the Lakeview plant which can produce up to 7,000 tons per month. Sinclair sells its polyethylene to three different auto manufacturing plants, Grand Rapids (demand = 3000 tons per month), Blue Ridge (demand = 5000 tons per month), and Sunset (demand = 4000 tons per month). The variable unit cost of shipping a ton is $1 per mile. The number of miles between the respective plants is shown in the table below: Grand Rapids Blue Ridge Sunset Ohio Valley 50 40 100 Lakeview 60 50 75 In addition to the variable unit costs of shipping, Sinclair incurs a fixed handling cost (of pickup and delivery) for any shipping routes it uses. These costs are discounted for longer routes, as shown below: Fixed Cost Levels Distance (mi) Fixed Cost <=25 \ 100,000 <=50 \ 60,000 <=75 \ 20,000 <=100 \ 0 -Refer to Exhibit 6-2. What problem will occur if we use an IF statement to model the fixed handling costs?

In a model with 8 changing cells, all of which are constrained to be binary, the number of potentially feasible solutions is:

The LP relaxation of an integer programming (IP) problem is typically easy to solve and provides a bound for the IP model.

The branch and bound algorithm uses: