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Integer Programming

Optimisation Methods

Lecture Outline

- 1 Introduction
- 2 Integer Programming
- 3 Modeling with 0-1 (Binary) Variables
- 4 Goal Programming
- 5 Nonlinear Programming

Introduction

- Integer programming is the extension of LP that

solves problems requiring integer solutions. - Goal programming is the extension of LP that

permits more than one objective to be stated. - Nonlinear programming is the case in which

objectives or constraints are nonlinear. - All three above mathematical programming models

are used when some of the basic assumptions of LP

are made more or less restrictive.

Summary Linear Programming Extensions

- Integer Programming
- Linear, integer solutions
- Goal Programming
- Linear, multiple objectives
- Nonlinear Programming
- Nonlinear objective and/or constraints

Integer Programming

- Solution values must be whole numbers in integer

programming . - There are three types of integer programs
- pure integer programming
- mixed-integer programming and
- 01 integer programming.

Integer Programming

- The Pure Integer Programming problems are cases

in which all variables are required to have

integer values. - The Mixed-Integer Programming problems are cases

in which some, but not all, of the decision

variables are required to have integer values. - The ZeroOne Integer Programming problems are

special cases in which all the decision variables

must have integer solution values of 0 or 1.

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.

- Suppose that in a given subproblem (call it old

subproblem), assumes a fractional value between

the integers i and i1. 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.

- 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

- 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.

- 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.

Harrison Electric Company

- The Company produces two products popular with

home renovators old-fashioned chandeliers and

ceiling fans. - Both the chandeliers and fans require a two-step

production process involving wiring and assembly. - It takes about 2 hours to wire each chandelier

and 3 hours to wire a ceiling fan. Final assembly

of the chandeliers and fans requires 6 and 5

hours, respectively. - The production capability is such that only 12

hours of wiring time and 30 hours of assembly

time are available.

Harrison Electric Company

If each chandelier produced nets the firm 7 and

each fan 6, Harrisons production mix decision

can be formulated using LP as follows

maximize profit 7X1 6X2

subject to 2X1 3X2 12 (wiring hours) 6X1

5X2 30 (assembly hours) X1, X2 0

(nonnegative) X1 number of chandeliers

produced X2 number of ceiling fans produced

Harrison Electric Company

With only two variables and two constraints, the

graphical LP approach to generate the optimal

solution is given below

6X1 5X2 30

Possible Integer Solution

Optimal LP Solution (X1 33/4, X2 11/2,

Profit 35.25

2X1 3X2 12

Integer Solution to Harrison Electric Co.

Optimal solution

Solution if rounding off

Integer Programming

- Rounding off is one way to reach integer solution

values, but it often does not yield the best

solution. - An important concept to understand is that an

integer programming solution can never be better

than the solution to the same LP problem. - The integer problem is usually worse in terms of

higher cost or lower profit.

Branch and Bound Method

- Branch and Bound breaks the feasible solution

region into sub-problems until an optimal

solution is found. - There are Six Steps in Solving Integer

Programming Maximization Problems by Branch and

Bound. - The steps are given over the next several slides.

Branch and Bound Method The Six Steps

- Solve the original problem using LP.
- If the answer satisfies the integer constraints,

it is done. - If not, this value provides an initial upper

bound. - Find any feasible solution that meets the integer

constraints for use as a lower bound. - Usually, rounding down each variable will

accomplish this.

Branch and Bound Method Steps

- Branch on one variable from Step 1 that does not

have an integer value. - Split the problem into two sub-problems based on

integer values that are immediately above and

below the non-integer value. - For example, if X2 3.75 was in the final LP

solution, introduce the constraint X2 4 in the

first sub-problem and X2 3 in the second

sub-problem. - Create nodes at the top of these new branches by

solving the new problems.

Branch and Bound Method Steps

- 5.
- If a branch yields a solution to the LP problem

that is not feasible, terminate the branch. - If a branch yields a solution to the LP problem

that is feasible, but not an integer solution, go

to step 6.

Branch and Bound Method Steps

- 5. (continued)
- If the branch yields a feasible integer solution,

examine the value of the objective function. - If this value equals the upper bound, an optimal

solution has been reached. - If it is not equal to the upper bound, but

exceeds the lower bound, set it as the new lower

bound and go to step 6. - Finally, if it is less than the lower bound,

terminate this branch.

Branch and Bound Method Steps

- Examine both branches again and set the upper

bound equal to the maximum value of the objective

function at all final nodes. - If the upper bound equals the lower bound, stop.
- If not, go back to step 3.

Minimization problems involve reversing the roles

of the upper and lower bounds.

Harrison Electric Co

Figure 11.1 shows graphically that the optimal,

non-integer solution is X1 3.75

chandeliers X2 1.5 ceiling fans

profit 35.25

- Since X1 and X2 are not integers, this solution

is not valid. - The profit value of 35.25 will serve as an

initial upper bound. - Note that rounding down gives X1 3, X2 1,

profit 27, which is feasible and can be used

as a lower bound.

Integer Solution Creating Sub-problems

- The problem is now divided into two sub-problems

A and B. - Consider branching on either variable that does

not have an integer solution pick X1 this time.

Optimal Solution for Sub-problems

Optimal solutions are Sub-problem A X1 4 X2

1.2, profit35.20 Sub-problem B X13, X22,

profit33.00 (see figure on next slide)

- Stop searching on the Subproblem B branch because

it has an all-integer feasible solution. - The 33 profit becomes the lower bound.
- Subproblem As branch is searched further since

it has a non-integer solution. - The second upper bound becomes 35.20, replacing

35.25 from the first node.

Optimal Solution for Sub-problem

Sub-problems C and D

Subproblem As branching yields Subproblems C and

D.

Sub-problems C and D (continued)

- Subproblem C has no feasible solution at all

because the first two constraints are violated if

the X1 4 and X2 2 constraints are observed. - Terminate this branch and do not consider its

solution. - Subproblem Ds optimal solution is
- X1 4 , X2 1, profit 35.16.
- This non-integer solution yields a new upper

bound of 35.16, replacing the original 35.20. - Subproblems C and D, as well as the final

branches for the problem, are shown in the figure

on the next slide.

Harrison Electrics Full Branch and Bound

Solution

Subproblems E and F

- Finally, create subproblems E and F and solve for

X1 and X2 with the added constraints X1 4 and

X1 5. The subproblems and their solutions are

Subproblems E and F (continued)

Goal Programming

- Firms usually have more than one goal. For

example, - maximizing total profit,
- maximizing market share,
- maintaining full employment,
- providing quality ecological management,
- minimizing noise level in the neighborhood, and
- meeting numerous other non-economic goals.
- It is not possible for LP to have multiple goals

unless they are all measured in the same units

(such as dollars), - a highly unusual situation.
- An important technique that has been developed to

supplement LP is called goal programming.

Goal Programming (continued)

- Goal programming satisfices,
- as opposed to LP, which tries to optimize.
- Satisfice means coming as close as possible to

reaching goals. - The objective function is the main difference

between goal programming and LP. - In goal programming, the purpose is to minimize

deviational variables, - which are the only terms in the objective

function.

Example of Goal Programming

Harrison Electric Revisited

- Goals Harrisons management wants to
- achieve, each equal in priority
- Goal 1 to produce as much profit above 30 as

possible during the production period. - Goal 2 to fully utilize the available wiring

department hours. - Goal 3 to avoid overtime in the assembly

department. - Goal 4 to meet a contract requirement to produce

at least seven ceiling fans.

Example of Goal Programming

Harrison Electric Revisited

Need a clear definition of deviational variables,

such as

d1 underachievement of the profit target d1

overachievement of the profit target d2 idle

time in the wiring dept. (underused) d2

overtime in the wiring dept. (overused) d3

idle time in the assembly dept. (underused) d3

overtime in the wiring dept. (overused) d4

underachievement of the ceiling fan goal d4

overachievement of the ceiling fan goal

Ranking Goals with Priority Levels

- A key idea in goal programming is that one

goal is more important than another. Priorities

are assigned to each deviational variable.

Priority 1 is infinitely more important than

Priority 2, which is infinitely more important

than the next goal, and so on.

Analysis of First Goal

Analysis of First and Second Goals

Analysis of All Four Priority Goals

Goal Programming Versus Linear Programming

- Multiple goals (instead of one goal)
- Deviational variables minimized (instead of

maximizing profit or minimizing cost of LP) - Satisficing (instead of optimizing)
- Deviational variables are real (and replace slack

variables)

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

- Stockco is considering four investments
- Each investment
- Yields a determined NPV
- Requires a 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

- Begin by defining a variable for each decision

that Stockco must make. - The NPV obtained by Stockco is Total NPV

obtained by Stocko 16x1 22x2 12x3 8x4 - Stockcos objective function is max z 16x1

22x2 12x3 8x4 - Stockco faces the constraint that at most 14,000

can be invested. - Stockcos 0-1 IP is

max z 16x1 22x2 12x3 8x4 s.t. 5x1

7x2 4x3 3x4 14 xj 0 or 1 (j 1,2,3,4)

Set Covering as an IP

- 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 2 that are

required to cover all the elements in set 1.

Piece-wise linear functions as IP

- 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 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 cannot be used to

solve the optimization problem.

Piece-wise linear functions as IP

- By using 0-1 variables, however, a piecewise

linear function can be represented in a linear

form. Suppose the piecewise linear function f (x)

has break points .

Piece-wise linear functions as IP

- 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

Piece-wise linear functions as IP

- 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.

9.4 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 non-integers. - 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

- 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 xi 0 and an xi 1 branch. - The LP relaxation may be solved by inspection.

max z c1x1 c2x2 cnxn s.t. a1x1

a2x2 anxn b x1 0 or 1 (i

1, 2, , n)

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?

- 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.

- 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.

- 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.

- 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

- 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.

minimize s.t.

- cij distance from city i to city j
- xij 1 if tour visits i then j, and 0 otherwise

(binary) - ti arbitrary real numbers we need to solve for

9.7 Implicit Enumeration

- The method of implicit enumeration is often used

to solve 0-1 IPs. Many IP problems can be

converted to 0-1 IP problems. - 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.

- 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 any 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 if 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.

- 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

- An alternative method to the branch-and-bound

method is the cutting plane algorithm. - Summary of the cutting plane algorithm
- Step 1 Find the optimal tableau for the IPs

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 variables

coefficient in the form x f, where 0 lt f lt 1.

9.8 Cutting Plane Algorithm

- 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 lt 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.

9.8 Cutting Plane Algorithm

- A cut generated by the method above has the

following properties - Any feasible point for the IP will satisfy the

cut - The current optimal solution to the LP relaxation

will not satisfy the cut