Exam 11: Integer Linear Programming

The objective of the product design and market share optimization problem presented in the textbook is to choose the levels of each product attribute that will maximize the number of sampled customers preferring the brand in question.

If the acceptance of project A is conditional on the acceptance of project B,and vice versa,the appropriate constraint to use is a

Let x₁ and x₂ be 0 - 1 variables whose values indicate whether projects 1 and 2 are not done or are done.Which answer below indicates that project 2 can be done only if project 1 is done?

The constraint x₁  x₂ = 0 implies that if project 1 is selected,project 2 cannot be.

Given the following all-integer linear program: Max 3+2 s.t. 3+\leq9 +3\leq7 -+\leq1 ,\geq0 and integer a.Solve the problem as a linear program ignoring the integer constraints.Show that the optimal solution to the linear program gives fractional values for both x₁ and x₂. b.What is the solution obtained by rounding fractions greater than of equal to 1/2 to the next larger number? Show that this solution is not a feasible solution. c.What is the solution obtained by rounding down all fractions? Is it feasible? d.Enumerate all points in the linear programming feasible region in which both x₁ and x₂ are integers, and show that the feasible solution obtained in (c) is not optimal and that in fact the optimal integer is not obtained by any form of rounding.

Tower Engineering Corporation is considering undertaking several proposed projects for the next fiscal year.The projects,the number of engineers and the number of support personnel required for each project,and the expected profits for each project are summarized in the following table: $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text { Project }}$ Engineers Required Support Personnel Required Pro fit (\ 1,000,000 s) 1 20 15 1.0 2 55 45 1.8 3 47 50 2.0 4 38 40 1.5 5 90 70 3.6 6 63 70 2.2 Formulate an integer program that maximizes Tower's profit subject to the following management constraints: 1)Use no more than 175 engineers 2)Use no more than 150 support personnel 3)If either project 6 or project 4 is done,both must be done 4)Project 2 can be done only if project 1 is done 5)If project 5 is done,project 3 must not be done and vice versa 6)No more than three projects are to be done.

The solution to the LP Relaxation of a maximization integer linear program provides

Explain how integer and 0-1 variables can be used in a constraint to enable production.

Most practical applications of integer linear programming involve

The 0-1 variables in the fixed cost models correspond to

The solution to the LP Relaxation of a minimization problem will always be less than or equal to the value of the integer program minimization problem.

If x₁ + x₂  500y₁ and y₁ is 0 - 1,then if y₁ is 0,x₁ and x₂ will be 0.

The LP Relaxation contains the objective function and constraints of the IP problem,but drops all integer restrictions.

If the optimal solution to the LP relaxation problem is integer,it is the optimal solution to the integer linear program.

The Westfall Company has a contract to produce 10,000 garden hoses for a large discount chain.Westfall has four different machines that can produce this kind of hose.Because these machines are from different manufacturers and use differing technologies,their specifications are not the same. Machine Fixed Cost to Set Up Production Run Variable Cost Per Hose Capacity 1 750 1.25 6000 2 500 1.50 7500 3 1000 1.00 4000 4 300 2.00 5000 a.This problem requires two different kinds of decision variables. Clearly define each kind. b.The company wants to minimize total cost. Give the objective function. c.Give the constraints for the problem. d.Write a constraint to ensure that if machine 4 is used, machine 1 cannot be.

The use of integer variables creates additional restrictions but provides additional flexibility.Explain.

Consider a capital budgeting example with five projects from which to select.Let x_i = 1 if project i is selected,0 if not,for i = 1,...,5.Write the appropriate constraint(s)for each condition.Conditions are independent. a.Choose no fewer than three projects. b.If project 3 is chosen, project 4 must be chosen. c.If project 1 is chosen, project 5 must not be chosen. d.Projects cost 100, 200, 150, 75, and 300 respectively. The budget is 450. e.No more than two of projects 1, 2, and 3 can be chosen.

A business manager for a grain distributor is asked to decide how many containers of each of two grains to purchase to fill its 1,600 pound capacity warehouse.The table below summarizes the container size,availability,and expected profit per container upon distribution. Container Containers Container Grain Size Available Profit A 500 3 \ 1,200 B 600 2 \ 1,500 a. Formulate as a linear program with the decision variables representing the number of containers purchased of each grain. Solve for the optimal solution. b. What would be the optimal solution if you were not allowed to purchase fractional containers? c. There are three possible results from rounding an LP solution to obtain an integer solution:(1) the rounded optimal LP solution will be the optimal IP solution;(2) the rounded optimal LP solution gives a feasible, but not optimal IP solution;(3) the rounded optimal LP solution is an infeasible IP solution.For this problem (i) round down all fractions; (ii) round up all fractions; (iii) round off (to the nearest integer) all fractions (NOTE: Two of these are equivalent.) Which result above (1, 2, or 3) occurred under each rounding method?

Which of the following is the most useful contribution of integer programming?

In general,rounding large values of decision variables to the nearest integer value causes fewer problems than rounding small values.