09.0 Integer Programming.pdf

8/10/2019 09.0 Integer Programming.pdf

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Chapter 9

Integer ProgrammingCompanion slides of

Applied Mathematical Programming

by Bradley, Hax, and Magnanti(Addison-Wesley, 1977)

prepared by

Jos Fernando Oliveira

Maria Antnia Carravilla


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What are integer-programming

problems?

In many applications, integrality restrictions

reflect natural indivisibilities of the problem

under study. For example, when deciding how

many nuclear aircraft carriers to have in the

U.S. Navy, fractional solutions clearly are

meaningless. In these situations, the decision

variables are inherently integral by the natureof the decision-making problem.


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What are integer-programming

problems?

Linear programming problems in which

fractional solutions are not realistic.

Mixed integer programs when some, but not all,

variables are restricted to be integer.

Pure integer programs when all decision variables

must be integers.

Binary programs when all decision variables mustbe either 0 or 1.


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SOME INTEGER-PROGRAMMINGMODELS


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Capital Budgeting

In a typical capital-budgeting problem, decisions involve theselection of a number of potential investments. The investment decisions might be to choose among possible

plant locations, to select a configuration of capital equipment,or to settle upon a set of research-and-development projects.

Often it makes no sense to consider partial investments inthese activities, and so the problem becomes a gono-gointeger program, where the decision variables are taken tobexj= 0 or 1, indicating that thejth investment is rejectedor accepted.

Assume that: cjis the contribution resulting from thejth investment,

aijis the amount of resource iused on thejth investment.

bi is the limited availability of the ith resource.

The objective is to maximize total contribution from allinvestments.


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Model


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Note that

The coefficients aijrepresent the net cash flowfrom investmentjin period i. If the investment requires additional cash in period i,

then aij> 0,

If the investment generates cash in period i,then aij< 0.

The righthand-side coefficients birepresent theincremental exogenous cash flows.

If additional funds are made available in period i,then bi> 0,

If funds are withdrawn in period i,then bi< 0.


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The capital-budgeting model can be made

much richer by including logical

considerations The investment in a new product line is contingent

upon previous investment in a new plant

(contingency constraints)

Conflicting projects (multiple-choice constraints)


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The 0-1 knapsack problem

A capital budgeting problem with a single

resource:


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Warehouse Location

In modeling distribution systems, decisions must

be made about tradeoffs between transportation

costs and costs for operating distribution centers.

As an example, suppose that a manager mustdecide which of n warehouses to use for meeting

the demands ofm customers for a good.

The decisions to be made are which warehousesto operate and how much to ship from any

warehouse to any customer.


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Decision variables and relevant data

There are two types of constraints for the model: the demand djof each customer must be filled from the

warehouses,

goods can be shipped from a warehouse only if it is

opened.


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Model


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Scheduling

Consider the scheduling of airline flightpersonnel.

The airline has a number of routing legs to

be flown, such as 10 A.M. New York toChicago, or 6 P.M.Chicago to Los Angeles.

The airline must schedule its personnel crews

on routes to cover these flights. One crew, forexample, might be scheduled to fly a routecontaining the two legs just mentioned.


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Decision variables and relevant data

The coefficients aijdefine the acceptable combinations oflegs and routes, taking into account such characteristics assequencing of legs for making connections between flightsand for including in the routes ground time formaintenance.


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Model


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An alternative formulation

that permits a crew to ride as passengers

on a leg

Set-partitioning:

Set-covering


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Decision variables


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Model

Sub-tours are not prevented


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Sub-tours elimination

With n cities, (2n1)constraints of this nature must beadded True for this strategy and many other strategies used to

eliminate sub-tours in TSP models.

+


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FORMULATING INTEGERPROGRAMS


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Binary 0-1 variables

Suppose that we are to determine whether or not toengage in the following activities: (i) to build a new plant,

(ii) to undertake an advertising campaign, or

(iii) to develop a new product.

In each case, we must make a yesno or so-called gonogo decision. These choices are modeled easily by lettingxj= 1 if we engage in thejth activity andxj= 0otherwise.

Variables that are restricted to 0 or 1 in this way are termed

binary, bivalent, logical, or 01 variables. Binary variables are of great importance because they

occur regularly in many model formulations.

But can be also used as auxiliary variables


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Constraint Feasibility

Does a given choice of the decision variables

satisfies the constraint

?

Introduce a binary variable y with the

interpretation:

and write

B is chosen to be

large enough so

that the constraint

always is satisfied

if y = 1


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Alternative constraints

At least one, but not necessarily both, of these

constraints must be satisfied. This restriction can be modeled by combining the

technique just introduced with a multiple-choice

constraint as follows:


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We can save one integer variable in this

formulation:


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Conditional Constraints

These constraints have the form:

If we note that:

(pq) (pq)

we end up with the following alternativeconstraints:


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k-Fold Alternatives

We must satisfy at least k of the constraints:

Assuming that Bjforj = 1, 2, . . . , p, are chosen so

that the ignored constraints will not be binding, the

general problem can be formulated as follows:


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Compound Alternatives


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Representing Nonlinear Functions

Nonlinear functions can be represented by

integer-programming formulations


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Fixed Costs

Frequently, the objective function for a minimization problemcontains fixed costs (preliminary design costs, fixed

investment costs, fixed contracts, and so forth).


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Assume that the equipment has a capacity of B units.

Define y to be a binary variable that indicates when

the fixed cost is incurred, so that y = 1 whenx > 0

andy = 0 whenx = 0.

Then the contribution to cost due tox may bewritten as

with the constraints


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Piecewise Linear Representation


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To model the cost curve, we express any value

ofx as the sum of three variables 1, 2, 3,sothat the cost for each of these variables is

linear.

Hence,where

and the total variable cost is given by:


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Note that we have defined the variables so that:

1

corresponds to the amount by whichx exceeds 0,but is less than or equal to 4;

2is the amount by whichx exceeds 4, but is less thanor equal to 10; and

3is the amount by whichx exceeds 10, but is lessthan or equal to 15.

If this interpretation is to be valid, we must alsorequire that 1= 4 whenever 2> 0 and that 2=6 whenever

3> 0.

However, these restrictions on the variables aresimply conditional constraints and can bemodeled by introducing (more) binary variables:


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Diseconomies of Scale When marginal costs are increasing for a minimization

problem or marginal returns are decreasing for amaximization problem.


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SOME CHARACTERISTICS OFINTEGER PROGRAMS


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A sample problem

Determine:

subject to:


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Feasible region

Dots in the shaded region are feasible integer points.


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Somes notes If the integrality restrictions on variables are dropped,

the resulting problem is a linear program. We will call itthe associated linear program.

The optimal integer-programming solution is notobtained by rounding the linear-programmingsolution.

The closest point to the optimal linear-programsolution is not even feasible.

The nearest feasible integer point to the linear-program solution is far removed from the optimalinteger point.

It is not sufficient simply to round linear-programmingsolutions.

Systematic and explicit enumeration does not workeven for small problems.


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BRANCH-AND-BOUND


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Strategy of divide and conquer

An integer linear program is a linear program furtherconstrained by the integrality restrictions.

Thus, in a maximization problem, the value of theobjective function, at the linear-program optimum, will

always be an upper bound on the optimal integer-programming objective.

In addition, any integer feasible point is always a lowerbound on the optimal linear-program objective value.

The idea of branch-and-bound is to utilize theseobservations to systematically subdivide the linearprogramming feasible regionand make assessments ofthe integer-programming problem based upon thesesubdivisions.


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Further Considerations

There are three points that have yet to beconsidered with respect to the branch-and-boundprocedure:

i. Can the linear programs corresponding to thesubdivisions be solved efficiently?

ii. What is the best way to subdivide a given region,and which unanalyzed subdivision should beconsidered next?

iii. Can the upper bound (z = 41, in the example) on theoptimal value z of the integer program be improvedwhile the problem is being solved?

h l d


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Can the linear programs corresponding to

the subdivisions be solved efficiently?

Dual simplex method!

Adding the new constraint:

Results in:


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We must pivot to isolatex2in the second

constraint and re-express the system as:

and perform a dual simplex method iteration.

What is the best way to subdivide a given


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What is the best way to subdivide a given

region, and which unanalyzed subdivision

should be considered next? If we can make our choice of subdivisions in such a way

as to rapidly obtain a good (with luck, near-optimal)integer solution z, then we can eliminate manypotential subdivisionsimmediately

as if any region has its linear programming value z < z,then the objective value of no integer point in that regioncan exceed z and the region need not be subdivided.

There is no universal method for making the requiredchoice, although several heuristic procedures have

been suggested, such as selecting the subdivision: with the largest optimal linear-programming value.

on a last-generatedfirst-analyzed basis (depth-first).

Rules for determining which fractional variables to use

in constructing subdivisions are more subtle

Can the upper bound on the optimal value


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Can the upper bound on the optimal value

z*of the integer program be improved

while the problem is being solved?

Initial upper bound

= 41.25

41 if we recall thatthe objective

function coefficients

are also integer.

Current upperbound = 40 5/9

40, for the same

reasons.


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Summary

The essential idea of branch-and-bound is to subdividethe feasible region to develop boundszL< z* < z

Uonz*.

For a maximization problem, the lower boundzLis the

highest value of any feasible integer pointencountered.

The upper bound is given by the optimal value of theassociated linear program or by the largest value for

the objective function at any hanging box. After considering a subdivision, we must branch to

(move to) another subdivision and analyze it.


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Summary

If eitheri. the linear program over Ljis infeasible;

ii. the optimal linear-programming solution over Ljisinteger; or

iii. the value of the linear-programming solution zjoverLjsatisfiesz

j zL(if maximizing),

then Ljneed not be subdivided. In these cases,integer-programming terminology says that Lj has

beenfathomed. Case

(i) is termed fathoming by infeasibility,

(ii) fathoming by integrality, and

(iii) fathoming by bounds.


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BRANCH-AND-BOUNDFOR MIXED-INTEGER PROGRAMS

Branch and Bound


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Branch-and-Bound

for Mixed-Integer Programs

The branch-and-bound approach just

described is easily extended to solve problems

in which some, but not all, variables are

constrained to be integral.

Subdivisions then are generated solely by the

integral variables.

In every other way, the procedure is the sameas that specified above.


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Example


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Search Tree


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Simplex tableaus


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Simplex tableaus


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BINARY BRANCH-AND-BOUND

Implicit enumeration

A special branch and bound procedure for


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A special branch-and-bound procedure for

integer programs with only binary variables

The algorithm has the advantage that itrequires no linear-programming solutions.

In the ordinary branch-and-bound procedure,

subdivisions were analyzed by maintaining thelinear constraints and dropping the integralityrestrictions relax the variables

Here, we adopt the opposite tactic of alwaysmaintaining the 01 restrictions, but ignoringthe linear inequalities relax the constraints


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General idea

The idea is to utilize a branch-and-bound (orsubdivision) process to fix some of the variablesat 0 or 1.

The variables remaining to be specified are calledfree variables.

Note that, when the inequality constraints areignored, the objective function is maximized by

setting the free variables to zero, since theirobjective function coefficients are negative.

This is mandatory to apply this algorithm.


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Example

subject to:

all coefficients are negativemaximization problem

constraints are specified as

less than or equal to

inequalities


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We start with no fixed variables, and

consequently every variable is free and set tozero.

The solution does not satisfy the inequality

constraints, and we must subdivide to searchfor feasible solutions.

One subdivision choice might be:

For subdivision 1 :x1= 1, For subdivision 2 :x1= 0.


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Now variablex1is fixed in each subdivision. If theinequalities are ignored, the optimal solution over each

subdivision hasx2= x3= x4= x5= 0.

The resulting solution in subdivision 1 gives

z = 8(1) 2(0) 4(0) 5(0) + 10 = 2

and happens to satisfy the inequalities, so the optimalsolution to the original problem is at least 2, z* 2.

Also, subdivision 1 has been fathomed: The above solution is best among all 01 combinations

withx1= 1.

No other feasible 01 combination in subdivision 1 needsto be evaluated explicitly.

These combinations have been considered implicitly.


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The solution withx2= x3= x4= x5= 0 in

subdivision 2 is the same as the originalsolution with every variable at zero, and is

infeasible.

Consequently, the region must be subdividedfurther, say withx2= 1 orx2= 0, giving:

For subdivision 3 :x1= 0, x2= 1;

For subdivision 4 :x1= 0, x2= 0.


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Enumeration tree to this point

Completing it


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Complete

tree


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Note that

1. At, the solutionx1= 0, x2= x3= 1 , with freevariablesx4= x5= 0,is feasible, with z = 4, thusproviding an improved lower bound on z.

2. At, the solutionx1

= x3

= 0, x2

= x4

= 1, andfree variablex5= 0, hasz = 1 < 4, so that nosolution in that subdivision can be as good asthat generated at.

3. Atand, every free variable is fixed. In eachcase, the subdivisions contain only a singlepoint, which is infeasible, and furthersubdivision is not possible.


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Generalizing

Subdivisions are fathomed if any of three

conditions hold:

i. the integer program is known to be infeasible over

the subdivision, for example, by the previousinfeasibility test;

ii. the 01 solution obtained by setting free variables

to zero satisfies the linear inequalities; or

iii. the objective value obtained by setting freevariables to zero is no larger than the best feasible

01 solution previously generated.


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Problem transformation

Minimization problems are transformed tomaximization by multiplying cost coefficients by1.

Ifxjappears in the maximization form with a

positive coefficient, then the variable substitutionxj= 1xj everywhere in the model leaves thebinary variablexj with a negative objective-

function coefficient. Greater than or equal to constraints can be

multiplied by 1 to become less than or equalto constraints

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