Exam 6: Optimization Models With Integer Variables

The higher the upper bound produced in the branch and bound solution process, the faster the algorithm will be.

The branching part of the branch and bound algorithm that Solver uses to solve integer programming models means that the algorithm:

The default integer optimality setting in Solver is:

Solver may be unable to solve some integer programming problems, even when they have an optimal 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. Suppose PharmaCo wants to have a backup for each city. What is the minimum number of representatives that can cover this region in that case? What are their home cities?

An algorithm which uses implicit enumeration:

Complete enumeration of all possible solutions in many integer programming problems is impractical.

Problems with a fixed cost that is incurred only if an activity is undertaken at a positive level are inherently nonlinear.

In a set-covering model, each member of a given set (set 1) must be "covered" by all members of another set (set 2).

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. Suppose the shipping capacity for all routes is 3000 tons per month. What is Sinclair's optimal shipping plan in that case? What is the total cost in that case?

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. This problem is an example of what type of integer programming model?

Any integer program involving 0 − 1 variables with ____ constraint(s) is called a knapsack problem.

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. 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. Which items should she take with her to her new home? What is the value of these items?

In a linear model, the target cell and constraints are ultimately:

The binary variables in the fixed cost models correspond to:

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. Formulate an integer programming model to identify a feasible deployment plan to the sales representatives. What is the minimum number of representatives that can cover this region? What are their home cities?

Fixed costs imply that the divisibility assumption of linear models no longer holds.

Rounding the solution of a linear programming to the nearest integer values provides an:

Optimization software, including Solver, typically has more difficulty solving an integer programming problem, relative to a linear programming problem.