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ISYE 6210Theory of Production SchedulingConstraint Generation Methods

Thomas Sharkey

April 23, 2013

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Background

We have previously seen the core idea for constraint

generation methods: We have a large number of constraints in the problem. We solve a relaxation of the problem with a reduced number of

constraints. We then determine if the optimal solution to the relaxation

satisfies all the constraints in the original problem. If so, then the current solution is optimal to the original

problem. Otherwise, we add a subset of the violated constraints to our

relaxation.

Note that it actually may be beneficial to formulate the

problem with a large number of constraints. This is a similar idea to the set-partitioning formulation of the

GAP. We will see a very similar method as Dantzig-Wolfe

Decomposition.

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Benders Decomposition: Traditional Motivation

Suppose that we are running a scheduling system where we

have a set of customers j= 1, . . . , n and a set of facilitiesi= 1, . . . , m.

However, unlike previous models discussed in class, theprocessing times of the jobs are uncertain.

In certain situations, the assignment of the customers tofacilities (i.e., machines) must be done well before thecharacteristics of the jobs are known with certainty.

The actual scheduling of the jobs associated with thecustomers assigned to a facility, though, can be done after the

characteristics of the jobs are known.

In this situation, we wish to allocate the customers to thefacilities in such a way to minimize our expected costsover allpossible scenarios of the characteristics of the jobs.

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Benders Decomposition: Stochastic Scheduling

We will let xdenote the first stage decisions, i.e., the

assignment of customers to facilities. Note that the cost of the second stage decisions will be a

function of: (i) the first stage decisions xand (ii)characteristics of the job.

We will denote ykas the second stage scheduling decisions forscenario k (where k= 1, . . . , K).

We will have first stage constraints and second stageconstraints.

Our objective function will be equal to:

minimize cx+K

k=1

kfyk.

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Benders Decomposition Formulation

The formulation of our problem becomes:

minimize cx+K

k=1

kfyk.

subject to (P)

Ax = b

Bkx+Dyk = dk fork= 1, . . . , K.

x Xy Y.

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The Idea

We will try to remove the variables yk from the formulation.

This is accomplished by introducing constraints on the xvariables that bound the objective function (or some portionof the objective function) from the problem.

For now, lets assume that the yvariables are continuous andthat there always exists a feasible second-stage solution givenany first stage solution x. We will see how we can reformulate(P).

We will define the function zk(x) to be equal to the optimalobjective function value in scenario kgiven first stagedecisions x.

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First Reformulation

The problem (P) now becomes:

minimize cx+K

k=1

kzk(x)

subject to (P1)

Ax = b

x X.

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The Second-Stage Problem

minimize f

yk

subject to (SP(k))

Dyk = dk Bkx

yk 0.

Its dual problem is:

maximize pk(dk Bkx)

subject to (D(k))

pkD f.

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Getting to the Second Reformulation

By the definition ofzk(x) and linear programming duality, wecan express zk(x) as the optimal solution to (D(k)).

We define EPto be the extreme points of the feasible regionof (D(k)) and note that:

zk(x) = maxiEP

(pi)(dk Bkx).

Alternatively, zk(x) is the smallest value ofzk such that:

(pi)(dk Bkx) zk for all i EP.

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Second Reformulation

The problem (P1) now becomes:

minimize cx+

K

k=1

kzk

subject to (P2)

Ax = b

(pi)(dk Bkx) zk for all i EP.

x X.

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Constraint Generation Approach/Cutting Plane Algorithm

We will relax (P2), which refer to as (R-P2), by removing theextreme point constraints.

We will then solve (R-P2) over a relaxed set of constraintsand arrive at a solution (x, z).

We then need to either verify that this solution is optimal ordetermine which constraints (i.e., cuts) need to be added tothe problem.

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The Pricing Problem

We have the option of solving either the primal or dualproblem associated with zk(x

).

The dual problem will immediately give us the constraint tobe added to (R-P2), i.e., the one that is violated by (x, z).

The primal problem will give us an optimal solution yk. It is easy to verify if this solution has an objective function

greater than or equal to zk. We then would know that a constraint needs to be added to

(R-P2), but it is not immediately clear which one. Recall that there is a complementary dual solution to yk that

is optimal to the dual.

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The Relaxation at Any Point

minimize cx+K

k=1

kzk

subject to (R-P2)

Ax=b

Constraints on (portions of) the objective function

x X.

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A Generic Problem for Benders Decomposition

minimize f(x, y)

subject to (P)

C(x, y) are satisfied

x Dx, y Dy.

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Generic Relaxation

minimize z

subject to (R-P2)

z Bxh(x) for h= 1, . . . , H

x Dx,

where xh is a previous optimal solution to the relaxation and

Bxh(x) is a cut on the objective function.

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The Cuts are Important

The driving factor in the success of a Benders decompositionis the quality of the cuts.

Good cuts are readily available when the subproblems for a

fixed xare continuous problems. But, scheduling problems are often discrete (i.e., integer

variables) and/or combinatorial.

The question now becomes how can we develop quality cuts

for these types of problems.

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