Exam 6: Optimization Models With Integer Variables

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

Exhibit 6-1 A 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. -Refer to Exhibit 6-1.This problem is an example of what type of integer programming model?

The binary variables in the fixed cost models correspond to:

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

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

Exhibit 6-1 A 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. -Refer to Exhibit 6-1.Which items should she take with her to her new home? What is the value of these items?

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

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

Exhibit 6-3 PharmaCo 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?

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

Exhibit 6-2 Sinclair 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 costs of shipping between the respective plants are shown in the table below: 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: -Refer to Exhibit 6-2.This problem is an example of what type of integer programming model?

Exhibit 6-2 Sinclair 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 costs of shipping between the respective plants are shown in the table below: 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: -Refer to Exhibit 6-2.What is Sinclair's optimal shipping plan? What is the total cost in that case?

Exhibit 6-2 Sinclair 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 costs of shipping between the respective plants are shown in the table below: 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: -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?

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

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

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

If Solver fails to find an optimal solution to an integer programming problem,we might be able to find a near optimal solution by decreasing the tolerance setting.

Exhibit 6-3 PharmaCo 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 6-2 Sinclair 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 costs of shipping between the respective plants are shown in the table below: 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: -Refer to Exhibit 6-2.What problem will occur if we use an IF statement to model the fixed handling costs?

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