Chapter 9 Integer Programming to accompany Operations Research: Applications and Algorithms 4th edition by Wayne L. Winston Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

9.1 Introduction to Integer Programming:

9.1 Introduction to Integer Programming An IP in which all variables are required to be integers is call a pure integer programming problem. An IP in which only some of the variables are required to be integers is called a mixed integer programming problem . An integer programming problem in which all the variables must be 0 or 1 is called a 0-1 IP. The LP obtained by omitting all integer or 0-1 constraints on variables is called LP relaxation of the IP.

9.2 Formulating Integer Programming Problems:

9.2 Formulating Integer Programming Problems Practical solutions can be formulated as IPs. The basics of formulating an IP model

Example 1: Capital Budgeting IP:

Example 1: Capital Budgeting IP Stockco is considering four investments Each investment Yields a determined NPV Requires at certain cash flow at the present time Currently Stockco has $14,000 available for investment. Formulate an IP whose solution will tell Stockco how to maximize the NPV obtained from the four investments.

Example 1: Solution:

Example 1: Solution Begin by defining a variable for each decision that Stockco must make. The NPV obtained by Stockco is Total NPV obtained by Stocko = 16 x 1 + 22 x 2 + 12 x 3 + 8 x 4 Stockco’s objective function is max z = 16 x 1 + 22 x 2 + 12 x 3 +8 x 4 Stockco faces the constraint that at most $14,000 can be invested. Stockco’s 0-1 IP is max z = 16 x 1 + 22 x 2 + 12 x 3 +8 x 4 s.t. 5 x 1 + 7 x 2 + 4 x 3 +3 x 4 ≤ 14 x j = 0 or 1 ( j = 1,2,3,4)

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Fixed-Charge Problems Suppose activity i incurs a fixed charge if undertaken at any positive level. Let = Level of activity i = 1 if activity i is undertaken at positive level = 0 if activity i is not undertaken at positive level Then a constraint of the form < must be added to the formulation. It must be large enough to ensure that will be less than or equal to .

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In a set-covering problem , each member of a given set must be “covered” by an acceptable member of some set. The objective of a set-covering problem is to minimize the number of elements in set 3 that are required to cover all the elements in set 1. Given two constraints ensure that at least one is satisfied by adding an either-or-constraint .

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M is a number chosen large enough to ensure that both constraints are satisfied for all values of that satisfy the other constraints in the problem. Suppose we want to ensure that > 0 implies . Then we include the following constraint in the formulation: Here, M is a large positive number, chosen large enough so that f < M and – g < M hold for all values of that satisfy the other constraints in the problem. This is called an if-then constraint.

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0-1 variables can be used to model optimization problems involving piecewise linear functions. A piecewise linear function consists of several straight line segments. The graph of the piecewise linear function is made of four straight-line segments. The points where the slope of the piecewise linear function changes are called the break points of the function. A piecewise linear function is not a linear function so linear programming can not be used to solve the optimization problem.

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By using 0-1 variables, however, a piecewise linear function can e represented in linear form. Suppose the piecewise linear function f ( x ) has break points . Step 1 Wherever f ( x ) occurs in the optimization problem, replace f ( x ) by . Step 2 Add the following constraints to the problem:

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If a piecewise linear function f(x) involved in a formulation has the property that the slope of the f(x) becomes less favorable to the decision maker as x increases, then the tedious IP formulation is unnecessary. LINDO can be used to solve pure and mixed IPs. In addition to the optimal solution, the LINDO output also includes shadow prices and reduced costs. LINGO and the Excel Solver can also be used to solve IPs.

9.3 The Branch-and-Bound Method for Solving Pure Integer Programming Problems:

9.3 The Branch-and-Bound Method for Solving Pure Integer Programming Problems In practice, most IPs are solved by some versions of the branch-and-bound procedure. Branch-and-bound methods implicitly enumerate all possible solutions to an IP. By solving a single subproblem, many possible solutions may be eliminated from consideration. Subproblems are generated by branching on an appropriately chosen fractional-valued variable.

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Suppose that in a given subproblem (call it old subproblem), assumes a fractional value between the integers i and i+ 1. Then the two newly generated subproblems are New Subproblem 1 Old subproblem + Constraint New Subproblem 2 Old subproblem + Constraint Key aspects of the branch-and-bound method for solving pure IPs If it is unnecessary to branch on a subproblem, we say that it is fathomed.

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These three situations (for a max problem) result in a subproblem being fathomed The subproblem is infeasible, thus it cannot yield the optimal solution to the IP. The subproblem yield an optimal solution in which all variables have integer values. If this optimal solution has a better z -value than any previously obtained solution that is feasible in the IP, than it becomes a candidate solution , and its z -value becomes the current lower bound (LB) on the optimal z -value for the IP. The optimal z -value for the subproblem does not exceed (in a max problem) the current LB, so it may be eliminated from consideration. A subproblem may be eliminated from consideration in these situations The subproblem is infeasible. The LB is at least as large as the z -value for the subproblem

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Two general approaches are used to determine which subproblem should be solved next. The most widely used is LIFO. LIFO leads us down one side of the branch-and-bound tree and quickly find a candidate solution and then we backtrack our way up to the top of the other side The LIFO approach is often called backtracking . The second commonly used approach is jumptracking . When branching on a node, the jumptracking method solves all the problems created by branching.

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When solving IP problems using Solver you can adjust a Solver tolerance setting. The setting is found under the Options. For example a tolerance value of .20 causes the Solver to stop when a feasible solution is found that has an objective function value within 20% of the optimal solution.

9.4 The Branch-and-Bound Method for Solving Mixed Integer Programming Problems:

9.4 The Branch-and-Bound Method for Solving Mixed Integer Programming Problems In mixed IP, some variables are required to be integers and others are allowed to be either integer or nonintegers. To solve a mixed IP by the branch-and-bound method, modify the method by branching only on variables that are required to be integers. For a solution to a subproblem to be a candidate solution, it need only assign integer values to those variables that are required to be integers

9.5 Solving Knapsack Problems by the Branch-and-Bound Method:

9.5 Solving Knapsack Problems by the Branch-and-Bound Method A knapsack problem is an IP with a single constraint. A knapsack problem in which each variable must be equal to 0 or 1 may be written as When knapsack problems are solved by the branch-and-bound method, two aspects of the method greatly simplify. Due to each variable equaling 0 or 1, branching on xi will yield in x i =0 and an x i =1 branch. The LP relaxation may be solved by inspection. max z = c 1 x 1 + c 2 x 2 + ∙∙∙ + c n x n s.t. a 1 x 1 + a 2 x 2 + ∙∙∙ + a n x n ≤ b x 1 = 0 or 1 ( i = 1, 2, …, n )

9.6 Solving Combinatorial Optimization Problems by the Branch-and-Bound Method:

9.6 Solving Combinatorial Optimization Problems by the Branch-and-Bound Method A combinatorial optimization problem is any optimization problem that has a finite number of feasible solutions. A branch-and-bound approach is often the most efficient way to solve them. Examples of combinatorial optimization problems Ten jobs must be processed on a single machine. It is known how long it takes to complete each job and the time at which each job must be completed. What ordering of the jobs minimizes the total delay of the 10 jobs?

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A salesperson must visit each of the 10 cities before returning to his home. What ordering of the cities minimizes the total distance the salesperson must travel before returning home? This problem is called the traveling sales person problem (TSP). In each of these problems, many possible solutions must be considered.

Example 11: Traveling Salesperson Problem:

Example 11: Traveling Salesperson Problem Joe State lives in Gary, Indiana and owns insurance agencies in Gary, Fort Wayne, Evansville, Terre Haute and South Bend. Each December he visits each of his insurance agencies. The distance between each agency is known. What order of visiting his agencies will minimize the total distance traveled?

Example 11: Solution:

Example 11: Solution To begin define Several branch-and-bound approaches have been developed for solving TSPs The approach we will use is the one in which the subproblems reduce to assignments problems.

Ex. 11 – Solution continued:

Ex. 11 – Solution continued First solve the assignment problem in subproblem 1. This solution contains two subtours and cannot be the optimal solution. Now branch on subproblem 1 in a way that will prevent one of subproblem 1’s subtours from recurring in solutions to subsequent subproblems. Now arbitrarily choose subproblem 2 to solve, applying the Hungarian method to the cost matrix shown. This solution can not be the optimal solution.

Ex. 11 – Solution continued:

Ex. 11 – Solution continued Now branch subproblem 2 in an effort to exclude the subtour 2-5-2. Thus we add two additional subproblems. Following the LIFO approach, next solve subproblem 4 or subproblem 5. By using the Hungarian method on subproblem 4, the optimal solution z =668 , x 15 = x 24 = x 31 = x 43 = x 52 =1. This is a candidate solution and any node that cannot yield a z -value < 668 may be eliminated from consideration. Following the LIFO rule, next solve subproblem 5. z =704 thus this subproblem is eliminated.

Ex. 11 – Solution continued:

Ex. 11 – Solution continued Only subproblem 3 remains. The optimal solution is z =652. It is possible for this subproblem to yield a solution with no subtours that beats z= 668. Next branch on subproblem 3 creating subproblem 6 and 7. Both of these subproblems have a z -value that is larger than 668. Subproblem 4 thus yields the optimal solution.

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When using branch-and-bound methods to solve TSPs with many cities, large amounts of computer time is needed. Heuristic methods , or heuristics , can be used to quickly lead to a good solution. Heuristics is a method used to solve a problem by trial and error when an algorithm approach is impractical. Two types of heuristic methods can be used to solve TSP; nearest neighbor method and cheapest-insertion method.

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Nearest Neighbor Method Begin at any city and then “visit” the nearest city. Then go to the unvisited city closest to the city we have most recently visited. Continue in this fashion until a tour is obtained. After applying this procedure beginning at each city, take the best tour found. Cheapest Insertion Method (CIM) Begin at any city and find its closest neighbor. Then create a subtour joining those two cities. Next, replace an arc in the subtour (say, arc ( i, j ) by the combinations of two arcs---( i , k ) and ( k , j ), where k is not in the current subtour---that will increase the length of the subtour by the smallest (or cheapest) amount. Continue with this procedure until a tour is obtained. After applying this procedure beginning with each city, we take the best tour found.

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Three methods to evaluate heuristics Performance guarantees Gives a worse-case bound on how far away from optimality a tour constructed by the heuristic can be Probabilistic analysis A heuristic is evaluated by assuming that the location of cities follows some known probability distribution Empirical analysis Heuristics are compared to the optimal solution for a number of problems for which the optimal tour is known

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c ij = distance from city i to city j x ij = 1 if tour visits i then j, and 0 otherwise (binary) t i = arbitrary real numbers we need to solve for minimize: s.t. An IP formulation can be used to solve a TSP but can become unwieldy and inefficient for large TSPs. LINGO can be used to solve the IP of a TSP.

9.7 Implicit Enumeration:

9.7 Implicit Enumeration The method of implicit enumeration is often used to solve 0-1 IPs. Implicit enumeration uses the fact that each variable must be equal to 0 or 1 to simplify both the branching and bounding components of the branch-and-bound process and to determine efficiently when a node is infeasible. The tree used in the implicit enumeration method is similar to those used to solve 0-1 knapsack problems. Some nodes have variable that are specified called fixed variables .

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All variables whose values are unspecified at a node are called free variables . For any node, a specification of the values of all the free variables is called a completion of the node. Three main ideas used in implicit enumeration Suppose we are at ay node with fixed variables, is there an easy way to find a good completion of the node that is feasible in the original 0-1 TSP? Even is the best completion of a node is not feasible, the best completion gives us a bound on the best objective function value that can be obtained via feasible completion of the node. This bound can be used to eliminate a node from consideration.

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At any node, is there an easy way to determine if all completions of the node are infeasible? In general, check whether a node has a feasible completion by looking at each constraint and assigning each free variable the best value for satisfying the constraint.

9.8 Cutting Plane Algorithm:

9.8 Cutting Plane Algorithm An alternative method to the branch-and-bound method is the cutting plane algorithm . Summary of the cutting plane algorithm Step 1 Find the optional tableau for the IP’s programming relaxation. If all variables in the optimal solution assume integer values, we have found an optimal solution to the IP; otherwise, proceed to step2. Step 2 Pick a constraint in the LP relaxation optimal tableau whose right-hand side has the fractional part closest to 1/2. This constraint will be used to generate a cut. Step 2a For the constraint identified in step 2, write its right-hand side and each variable’s coefficient in the form [x] + f , where 0 <= f < 1.

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Step 2b Rewrite the constraint used to generate the cut as All terms with integer coefficients = all terms with fractional coefficients Then the cut is All terms with fractional coefficients <= 0 Step 3 Use the simplex to find the optimal solution to the LP relaxation, with the cut as an additional constraint . If all variables assume integer values in the optimal solution, we have found an optimal solution to the IP. Otherwise, pick the constraint with the most fractional right-hand side and use it to generate another cut, which is added to the tableau. We continue this process until we obtain a solution in which all variables are integers. This will be an optimal solution to the IP.

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