0305-0483/78/0501-0145/102.00/0OMEGA. The Int JI of Mgmt Sci. Vol 6. No 2. pp. 145-152() Pergamon Press LId 1978. Printed in Great Britain

Parametric Branch and Bound

FRED GLOVERUniversity of Colorado

(Receit't'd Septenlbe, 1977)

The paper describes a procedure for mixed integer programming that allows branches to be imposed'by degrees', which can subsequently be revised or weeded out according to their relative influence.It is an adaptive approach in which the branch and bound tree can be maillipulated and rl'Structured.The approach also yields measures of the costs of imposing the branches that lead to integersolutions, thus providing a built-in form of sensitivity analysis for evaluating the effect of integer

restrictions.

1. INTRODUCTION

THIS paper introduces a parametric branchand bound (B&B) procedure that has greaterflexibility than ordinary B&B. Ol)ce a branchis taken in ordinary B&B, it is largely irrevo-cable-i.e. all descendant branches must inheritthe limitations imposed by their predecessors.In parametric B&B, a descendant branch maypartly or wholly undo an antecedent branch.Moreover, once a feasible mixed integer pro-gramming (MIP) solution is obtained, then the'actual' branches that achieved this solutioncan be identified by weeding out uninfluentialbranches created during the solution process.

The strategy underlying parametric B&Bis to incorporate variables and constraintsinto the Qbjective function in a manner resem-bling 'multi-objective' or 'goal programming'approaches [1, 2,9]. The process is especiallydirect for 0-1 MIP problems, which do notrequire the creation of additional variables.However, even in the general case, all calcula-tions can be carried out relative to a compactbasis that is the same size as for the original

problem.The use of weighted variables and con-

straints in the objective function also bears aconnection to 'Lagrangean' approaches [3,6,10]. However, in contrast with Lagrangeantechniques, the weights are not designed tosolve a dual problem, but rather entirely over-

shoot dual feasibility. The process may beviewed as that of constructing tentative duals(at least implicitly), guided by considerationsrelevant to the B&B setting. Also, in contrastto standard B&B and its exploitation of dua-lity, parametric branches are conveniently han-dled by postoptimizing with the primal simplexmethod, whereas ordinary branches are oftenpreferably handled by postoptimizing with thedual simplex method. Instead of imposing abranch either fully or not at all, parametricB&B allows one to impose branches 'bydegrees'. This is extremely important for en-abling branching alternatives to be carefullyanalyzed-and revised, if desirable-at laterstages of the tree.

By its nature, parametric B&B yields infor-mation about the cost of imposing branchesthat lead to integer solutions. This informationcan then be used in a sensitivity analysis forevaluating the significance of integer restric-tions. This type of analysis can also be coupledwith an approach that attaches penalties to de-viations from constraints. Such an approachyields a combined sensitivity analysis charac-terizing the influence of integer assignments onproblem constraints.

2. PARAMETRIC B&B FOR 0-1PROBLEMS

The basic ideas of parametric B&B are rela-tively straightforward for 0-1 MIP problems,

;lover-Parametric

Branch and Bound146

and we first examine them in this setting. For Drawing on the fact that the constraint Zo = 0our purposes, the 0-1 problem will be written is equivalent to the 'partial assignment' Xj = 0,

...jEJ 0 and x j = 1, jEJ 1 we may use the objectiveMinimize Xo = L cJx, (2-1) d . I h.. I . )IN J (2-1 ") as a eVlce to compe t 1S part1a asslgn-

L alJxJ .$; bl i£M (2-2) ment to hold. Clearly, if the parameters dj are..selected large enough, then Zo = 0 must result

XJ ?; 0 jEM (2-3)

1 ?; xJ ?; 0 jEJ (2-4)

xJ integer jEJ (2-5)

where J is the index set of the integer variablesand is a subset of N. In our discussion of 0-1problems, we will confine ourselves to elabor-ating the principal ideas of parametric B&B,together with illustrating the more rudimentarytypes of considerations. In fact, most of the im-plementation aspects of the 0-1 case requireno commentary other than to indicate the rele-vant decision alternatives. Subsequently, moreadvanced aspects of implementation will be in-troduced for the general MIP case, and thenlinked to the 'sensitivity analysis' framework.

In the 0-1 setting, consider any two disjointsubsets J 0 and J 1 of J and the 'contrived'objective function

Minimize Zo = do + L dJxJ -L dJxJ ;2tl')joJo joJ,

where all of the d j are positive, and where

do = L dJ,joJ,

By the form of Zo, if there exists a feasible solu-tion to the LP problem (2-1H2-4) that satisfiesXj = 0 for jEJo and Xj = 1 for jEJ 1 then thiswill be an optimal solution to the LP problemin which (2-1 ') replaces (2-1). (In fact, thisassignment of values to the x j for jEJ oU J 1 isuniquely optimal when feasible, and hencemust occur at an LP extreme point. Note thatthis gives a rather simple proof of the fact thatfeasible integer assignments are unique extremepoints of the pure 0-1 problem and occur atone or more extreme points of the mixed 0-1

problem.)In addition, by the definition of do, it follows

that Zo ~ 0 for all feasible LP solutions, andZo = 0 only for a solution that yields theassignment x j = 0 for jEJ 0 and x j = I for jEJ ,.Consequently, the creation of a compositeobjective function

Minimize Uo = Xo + Zo (2-1")

assures by the non-negativity of Zo thatMin Uo ?; Min xo'

provided the corresponding partial assignmentis feasible.

Handling the composite objective function inparametric B&B

The first stage of the parametric B&Bapproach for 0-1 problems utilizes these ideasto influence the creation of partial assignmentsthrough manipulation of the parameters d j'However, instead of assigning these parameterspreemptively large values, more moderatevalues are assigned and then monitored inorder to determine interactive effects relevantto the B&B setting. By the familiar Lagrangeantype of argument, we may observe

Min {xo subject to Zo = 0]= {Min "0 subject to Zo = 0: ~ Min "0

and hence the composite objective function(2-1") always yields a lower bound on the ordi-nary B&B objective (in which Zo = 0 is expli-citly imposed). This type of bound informationcan be used for fathoming in a manner resem-bling that of the ordinary B&B approach. Wewill discuss the way to accommodate fathomedalternatives in the parametric setting after in-troducing the notion of a 'parametric branch'.

0-1 Parametric branchingThe branch step for the 0-1 parametric

approach is a 'tentative' operation that eitherbecomes consolidated or revised on the basisof information subsequently generated Quitesimply, if the branch corresponds to the assign-ment x, = 0 then the current objective Uo isupdated to become Uo + drxr and if the branchcorresponds to the assignment Xr = 1 then Uois updated to become Uo + dr(1 -xr) or equi-valently Uo + drx~ where x~ is the slack variablefor the inequality Xr ~ 1. The weight dr isselected so that the updated representation ofUo is dual infeasible, and therefore, allows re-optimization with the primal simplex method.

To illustrate, suppose the current LP repre-sentations of Uo and Xr are given by

"0 = 12* + 5! .~3 + Ii X4 + 4xs,

x, = i-2x3 + X4 + ixs.

Omega, Vol. 6. No.2 147

,

Then for the branch x,. = 0 any value of d,.satisfying d,. > 5t/2 will cause the new objec-tive Uo + d"x,. to be dual infeasible. (The ratio5t/2 is exactly the pivot ratio for the dual sim-plex method that would be identified if onewere to introduce the constraint x,. ~ 0.) Thus,for example, selecting d,. = 3, we obtain thenew Uo objective

Uo = 13 -1 X3 + 4* X4 + 6t Xs.

The negative coefficient for X3 of course signalsthat re-optimization may now be undertakenwith the primal simplex method.

Alternatively, for the branch x,. = 1 a valueof d,. satisfying d,. > It/! will succeed in estab-lishing dual infeasibility for the objectiveUo + d,.x~ where X~ = 1 -X,.. (Here 1 t/! is thepivot ratio for the dual simplex method relativeto the constraint X,. 2: 1.) Thus, for example,selecting d,. = 2 we obtain the new objective

Uo = I~i + 91 X3 -i X4 + 2; Xs.

Values of d,. such as those selected in theseexamples may not be sufficiently large to insurethat the branches for X,. = 0 and x,. = 1 willutlimately be enforced. However, as previouslynoted, the procedure does not seek preemptivevalues but rather seeks to analyze the conse-quences of more moderate values. (In this con-nection, it is easily established that if a dualpivot on the constraint x,. ~ 0 or x,. 2: 1 wouldachieve primal feasibility in one step, then anyvalue of d,. that exceeds the dual pivot ratiowill achieve exactly this same solution in oneprimal iteration.)

This extremely simple type of 'local' imple-mentation step for 0-1 parametric B&B hasa direct analog in a variety of standard 'para-metric' approaches for ordinary linear pro-gramming. The crucial aspect in the B&B set-ting is the way in which this step is used-i.e.the manner in which the parametric branchesare processed to yield information and branch-ing alternatives not available to ordinary B&B.Thus, in particular, parametric branches of theform just illustrated are not regarded as iron-clad impositions nor are they initially assigneda sequential ranking. Indeed, it may well occurat a subsequent step that an 'earlier' branchwill be discovered to be superfluous in termsof other more influential branches. We discussthis aspect of the approach next.

Revised and augmented parametric branches

Branches that are currently uninfluential caneasily be singled out by the parametric B&Bprocedure as follows. Suppose the updated LPrepresentation of Uo gives Xr or x; (dependingon which of the two variables was previouslyassigned a weight) a coefficient that equals orexceeds dr. (The implication, of course, is thatXr or x; is currently non basic.) Then, reducingthis coefficient by dr still leaves the objectivedual feasible. This means that the parametricbranch can be eliminated without changing thecurrent LP solution.

The step of weeding out uninfluentialbranches (i.e. those subject to elimination) canbe postponed until a feasible 0-1 solution isobtained. Then, all variables whose parametervalues can be reduced to zero may be excludedfrom the branching category. (An exceptionoccurs for branches that are antecedents of acompulsory branch, as described in the nextsection.) The remaining variables and para-metric branches may further be ranked, forexample, according to the magnitude of the drvalues that are required to make the currentsolution optimal. In this fashion, the 'actual'branches of the B&B process, and theirsequence, are decided upon at a stage in whichthe true significance of these branches canmore accurately be assessed.

Another means for evaluating the relative in-fluence or significance of a branch occurs whena branching variable 'resists' its weight andbecomes basic. The decision must then bewhether to increase or decrease dr. (The lattermay be a compounded step that both decreasesdr to 0 for the current parametric branch andthen assigns dr a positive value for the alterna-tive branch.) By means of such decisions,earlier parametric branches may either berevised or reinforced.

The technical aspects of implementing theseideas are relatively simple, and hence, we con-fine ourselves to the task of discussing onlythe more prominent strategic considerations,deferring the particulars of implementation tothe general MIP case, where they are morecritical.

Compulsory branches and sequencesThe minimization of Uo as noted earlier pro-

vides a lower bound on the optimum value ofXo subject to the partial assignment associated

148 Glover-Parametric Branch and Bound

with Zo = O. Whenever this bound equals orexceeds the value of Xo for the best MIP solu-tion currently known (which automaticallyoccurs when the current LP solution in factprovides this best MIP solution), then allfurther continuations of the partial assignmentare identified as unproductive in the usualB&B sense. This immediately provides theoption of 'back-tracking'-i.e. of decidingwhich parametric branch underlying the cur-rent LP solution should be considered the 'lastbranch'. The observation that the determina-tion of the last branch can be deferred to yielda more flexible variant of the LIFO procedurewas first made by Tuan [11]. In the currentapproach, this determination can be based, forexample, on the current dj values (after weed-ing out uninfluential branches). Thus, in thisfashion, a sequential ordering is gradually im-posed on the branches.

Having identified a last branch, the alterna-tive branch is now imposed as compulsory. Thismeans that the method does not allow thisbranch to be reversed or discarded (followingthe usual backtracking rules), until the currentpartial assignment is fathomed. In particular,as soon as a compulsory branch is identified,all other parametric branches that are cur-rently in force (i.e. all those for which dj iscurrently positive) must be considered as priorto the compulsory branch. This does not im-pose any particular sequence on these priorbranches (until backtracking again compelsone to be identified as a 'last member'), butdoes impose induced bounds on the parametersd j. That is, these positive d j coefficients cannotbe reduced below their current values as longas the compulsory branch remains in effect.

There are, however, three important excep-tions to the strict maintenance of inducedbounds. The first and obvious exception occursby deciding upon some partial sequence for theparametric branches and employing a treesearch rule that jumps back over some of the'later' branches of this sequence. Then, the in-duced bounds can be disregarded for thebranches thus bypassed. (The same remarkholds for branches that are released or reversedas a result of carrying out a backtracking stepwith the LIFO rule.) The second exceptionoccurs by redeciding the status of the branchthat has been designated compulsory. That is,a compulsory branch may be revised, if the

reconfigured objective Uo that results after aseries of additional iterations makes thisappear eminently desirable. (Such a processmust, of course, be sufficiently systematized toguard against circularity.) Finally, just as com-pulsory branches may be imposed as standardbranches rather than parametric branches (asby the use of pre-emptive weights), so may theidentified antecedents of the compulsorybranches be imposed in the standard fashion.This imposition, however, does not carry withit an implied sequence for the antecedentsthemselves. Further, greater flexibility isachieved by monitoring the values of par-ameters that would suffice to maintain the im-posed branches, thereby retaining the optionof amending the status of these branches atsubsequent stages.

3. PARAMETRIC B&B FORGENERAL MIP PROBLEMS

Many of the same strategic notions discussedin the preceding section carryover to the gen-eral MIP problem. However, there is a verysignificant difference between the general prob-lem and the 0-1 problem. Specifically, in thegeneral case, there may not exist weights thatwill cause all of the branching inequalities tobe satisfied. Moreover, if such weights exist,they may be very hard to find, or cause someinequalities to be unnecessarily over-satisfied,thus failing to identify integer solutions. Wenow examine the special techniques that permitthese difficulties to be overcome and allow thegeneral approach to be implemented success-fully.

Branching schemes for the general MIPproblem may be viewed as the successive impo-sition of upper and lower bounds on the prob-lem variables. Thus, at any particular stage, theinteger variables are governed by restrictionsof the form

Vi ~ Xi ~ Li jEJ (3-1)

where U j and Lj may represent originalbounds on x j or those currently inherited bybranching. The general parametric approachseeks to incorporate such bound restrictionsinto the objective function in a manner thatwill enable the same types of evaluations andmanipulations that are possible in the 0-1 set-

ting.

Omega. Va/. 6, No.2 149

This is accomplished by introducing non-negative variables Z j and the inequalities

i

uJ+zJ?JxJ~LJ-zJ j£J. (3-2)

Thereupon, the Z j variables are incorporatedinto the minimization of Xo to create the com-posite objective function

Minimize Uo = I CJXJ + I dJzJ. (3-3))EN )E}

It is clear that for the weights dj sufficientlylarge all of the Z j will be 0 and (3-2) will reduceto (3-1}-provided, of course, that (3-1) is con-sistent with the original problem constraints.Also, for d j > 0 we have

Min Uo ~ Min Xo

andMin (xo subject to (3-1):

= Min {uo subject to (3-1): ~ Min Uo.

Thus, the same bound relationships hold in thegeneral MIP settings as in the 0-1 contextupon introducing (3-2) and (3-3).

The problem at hand is how to manage thesystem (3-2) and (3-3) effectively. This is ex-tremely important because otherwise, the intro-duction of (3-2) and (3-3) doubles the numberof integer variables and adds a correspondingnumber of new constraints. The followingapproach for dealing with this problem consti-tutes an adaptation of procedures for 'weighteddeviation' problems introduced in [7].

Let si and si denote slack variables for theinequalities of (3-2), yielding the equations

xJr-zJ+sj=Ujo (3-4)

-x. -z + s'~ = -L tJ J }

Adding these two ,equations, we obtain

-2z.+s'.+s J"=U.-L. (3-5)} J J r

Equation (3-5) identifies a 'primal relationship'between the variables Z j, si and si. On the'dual' side, let U j, vi and vi represent dual vari-ables associated with the non-negativity restric-tions on Zp si and s] respectively. (That is, thedefining equations for U j, vi and vi are respect-ively given by the Z p si and s] columns of theprimal.) Then, since the coefficients of zp si andsi are d p 0 and 0 in (3-3), it follows from (3-4)that the relationship between the columns forZj, si and si can be summarized by

UJ + vj + vi = dt (3-6)

This relationship, in conjunction with (3-5),provides the key to implementing the generalparamatric B&B approach without modifying

the size of the LP tableau. In particular, it isreadily established that at least one of the vari-ables Z p s} and sj must be basic and at leastone must be nonbasic.

Consequently, only one of the three primalvariables need be explicitly jncluded in the tab-leau at any given time, whereupon the formof the other variables is always known from(3-5) and (3-6). Similarly, Xj need not be in-cluded in the tableau since it is always capableof being recovered from an equation of (3-4).The fundamental relatiQnships of the approachas they apply in the present setting can be sum-marized as follows. (Formal proofs of theserelationships are omitted, since their derivationis a direct consequence of (3-5) and (3-6).)

(R.O}-Of the two variables (from the groupZ j' sj and sj) that are not explicitly includedin the LP tableau, one is currently basic andone is currently nonbasic.

(R.l}-If the explicit variable is nonbasic:

(a) The tableau row for the implicit basicvariable is precisely the primal equation (3-5).

(b) The tableau column for the implicit non-basic variable is the negative of the columnfor the explicit variable, except that the objec-tive function coefficient for the implicit variableis dj minus the objective function coefficientof the explicit variable, and the column coeffi-cient that corresponds to the implicit basicvariable is as given in (a).

(R.2}-If the explicit variable is basic:

(a) The tableau column for the implicit non-basic variable is precisely the dual equation(3-6). Thus, the objective function coefficient isdj and the column has unit coefficients in thepositions corresponding to its two companionbasic variables, with O's elsewhere.

(b) The tableau row for the implicit basicvariable is obtained by substituting the currentexpression for the explicit basic variable in(3-5) and giving the implicit nonbasic variablea unit coefficient [as in (a)].

We can now give the rules for implementingthe parametric B&B approach that result fromthese relationships. After stating these rules, wewill illustrate their use by numerical example.

1. The initial step solves the LP problem with-out the additional variables and constraintsof (3-4).

150 Glover-Parametric Branch and Bound

irrelevant unless the other slack becomesbasic-or until a new branch is introduced bytightening one of the U j or Lj values. Branch-ing can occur only if both of the slacks sj andsJ are basic and positive (since otherwise x jwould be assigned an integer value or lie out-side one of its provisional current bounds) andthus, in this case, one can simply jettison allthree of the Sft sJ and Z j variables [after reco-vering Xj from (3-4)] and start again with Step2.

In accordance with these observations, ifeither U j or Lj is a 'true' bound for x r thenthis bound can be rigidly enforced simply byrendering Z j an ineligible pivot column (e.g. byincreasing dj) whenever the slack variable as-sociated with the true bound is nonbasic. (IfZ j becomes basic when the alternative slack isnon basic, (3-5) implies that x j must satisfy itsappropriate bound, and condition 4(c) assuresthat the critical slack will not become nonbasicunless Z j is nonbasic.)

We now illustrate these considerations withthe following numerical example.

ExampleConsider the MIP problem represented by

the following 'condensed' tableau format,where Xl and X2 are the integer variables.

X2

2. The parametric branching step introducesthe restrictions (3-4) in a tableau for whichx j is basic and does not satisfy U j 2: x j 2: L j-

(a) Identify the equation of (3-4) which im-plies Z j > 0 for sj and sj nonnegative (i.e.the equation corresponding to the 'violatedbound');(b) Omit the slack variable sj or sj from theequation identified in (a), and use thisreduced equation to replace x j with Z j bydirect substitution. The omitted slack vari-able is the implicit non basic variable, andslack variable of the .other equation of (3-4)is the implicit basic variable. (These implicitvariables are irrelevant at this point). Thevariable Z j is now basic.

3. The next component step of parametricbranching is to create (or increase) theweight d j for a selected variable Z j that iscurrently basic, To do this, add dj times thecurrent Z j equation to the objective functionequation, where dj denotes the increment indp selected large enough to require re-opti-mization with the primal method. (The effectof this step on any currently implicit non-basic variables is immediately given in rela-tion (R.l)(b), since none of the implicit non-basic variables governed by (R.2)(a) areaffected except for the companion of Z jwhose coefficient is always dj-)

4. An arbitrary iteration of the primal methodcan be implemented by the following obser-vations:

Xo = -9 -4

X4IXs =(a) An implicit nonbasic variable provides

an eligible pivot column if and only if its ...explicit companion is non basic and has a The problem constraInt equatIons are read dl-

coefficient exceeding d . (as It f I -rectly from the tableau (e.g. X4 = 7 -5Xl -

.J a resu 0 re a 2 3 ) P" h .I d Itions (R.2)(a) and (R l)(b» X2 + X3. Ivotmg on t e clrc e e ement

(b) If a n . m I.. t . b'. . bl .with the primal simplex method yields the opti-

I p ICl non aslc varIa e ISI LP blI t d l" h . I h .ma ta eause ec e lor t e PIvot co umn, t en It can

replace the previous explicit variable, whichnow becomes implicit.(c) No implicit basic variable need be con-sidered in determining the pivot row unlessboth slack variables sj and sj are basic. Ifan implicit basic variable provides the pivotrow, it replaces the currently explicit basic Suppose now that we wish to branch parame-variable as the new explicit variable. trically on the inequality X2 ~ 3 by Step 2 of

the preceding rules, thus obtaining U 2 = 3 andCondition 4(c) is a direct consequence of L2 = 0 (where L2 = 0 represents a true lower

(3-5) and implies that once a slack variable of bound). The first equation of (3-4) implies(3-4) first becomes implicitly basic, it remains z 2 > 0 and yields z 2 = Xx -U 2 upon omitting

Xl X4 X3

Xo = 2

~~~

XlX, =

Omega, Vol. 6, No.2 151

the slack variable 52. Thus, the tableau equa-tion for z 2 is given by

By Step 3, we again seek to add a sufficientmultiple of z 1 to the current objective functionto produce a dual infeasibility. Since the impli-cit non basic variable 82 has a coefficient whichis the negative of the explicit Z2 coefficient (i.e.-t), we must take the effect of 82 into accountin the manner noted parenthetically in Step 3.By relation (R.IXb), the current objective func-tion coefficient of z 2 is 1 -! = !. This yieldsthe 'expanded' current Uo and z 1 rows as fol-lows

XI X. X3

Z2 =X2 -~ =uI~~_-!J

Next, by Step 3, we add a sufficient multipleof this equation to the objective equation tocreate dual infeasibility. Any value exceedingthe 'dual pivot ratio' 1/(5/2) will do, and weselect a multiple of 1 to give Uo = Xo + lz2or

Z2 ,X4 X3 51

-131 !! J i cX4XI X3-Uo =

uo=l-lil-i r'~1-"~I-z___~ ZI

'Thus, putting Steps 2 and 3 together (replacingthe X2 equation by the Z2 equation and replac- From this, we see that ai. multiple of 2 sufficesing Xo by uo) we obtain the tableau to produce dual infeasibility thereby giving the

Xl -~4 X3 new objective function representation

-,

-ii-25

1~

Uo =Ii X4 X3

iZ2 =

Xs =

Uo =1

Z2 52

"0 =E~~E~~ 13

Note that the 'expansion' used in this exampleThe primal simplex method now pivots on the was not really necessary since the fact that Z2circled element to obtain has a positive coefficient immediately implies

that we consider the implicit variable insteadZ2 x4 X3 of-rather than in addition to--the explicit

variable.Because 52 gives the eligible pivot column,

we replace Z2 by 52 (so that Z2 now becomesimplicit) to yield the tableau

!1f

XI =Xs =

52 X4 X3----j ¥ t-+--

-t ! -!I

1/ -! -¥I

=-l2!1

43"

if

-j -t

-J?- ---¥

Since optimality is re-established in a singlepivot, this is the same tableau that would havebeen obtained by a 'dual' pivot on the con- Uo =

straint X2 S 3 (except for the Z2 column). With ZI =Z2 and 52 both non basic (the first explicitly, Xs =

the second implicitly), we may infer that thecurrent value of X2 is 3 (as seen by (3-4) with Re-optimization with the primal method givesU 2 = 3). Consequently, at the present stage theparametric branch has achieved the same effectas an ordinary branch, although we do notassign the branch a sequential 'slot' and leaveit open to subsequent revision.

Suppose that Xl ~ I is selected to providethe next parametric branch. By Step 2, the rele-vant expression for Zl is Zl = Ll -Xl =1 -Xl giving the tableau representation

z.

X4 X3.'---t 2.-J.--~

--J

"0 =

52 =Xs = 1O,

-,

The variables of (3-4) associated with Xl andX2 have integer values in this tableau, andhence so do Xl and X2' Specifically, direct ap-plication of (3-4) yields Xl = 1 and X2 = 1

(noting that s~ and Z2 are implicitly nonbasic).Also, since Zl and Z2 are both in the current

Zz x. X3

Zl =111 t!=:il

152 Glover--Parametric Branch and Bound

basic solution, it follows that Uo and Xo havethe same value.

We now analyze the parametric branchesthat provided this current integer solution. Byrelation (R.2)(a), the objective function coeffi-cient of Z2 is d2. Thus, reducing the coefficientof Z2 by d2 (which effectively removes Z2 asa weighted variable in the objective function)still yields the same LP optimum. By ourearlier observations for the 0-1 case, thismeans that the branching restriction that gaverise to Z2 is currently 'uninfluential', and thuswe can shrink the B&B tree to the single re-striction x I 2. t.

At this stage, since the current solution islocally optimal, it is unproductive to proceedwith the present line of branching. Thus Xl 2. 1is replaced by the compulsory branch X I ~ 0,and the solution process continues.

4. PARAMETRIC B&B AND MIPSENSITIVITY ANALYSIS

Two types of sensitivity analysis are avail-able with parametric B&B as illustrated in thepreceding section. The first involves identifyingthe 'cost' of branches that lead to particularinteger solutions. For example, in the earliernumerical example, it was noted that essen-tially no cost attached to branching on Xl toobtain the first integer solution. (Consequently,the integer restriction for Xl was judged 'condi-tionally superfluous' and the branch was dis-carded in this instance.) Further, the tableauthat yields the integer solution assigns z 1 anobjective function coefficient of l, which meansthat dl can be reduced from its present valueof 2 to 2 -l = I without altering the LP opti-mum. This places the cost of the X 1 branchunderlying this solution at 1, which providesa measure of the relative significance of thisbranch and the integer restriction on X 1 in thecurrent solution. Specifically, we can concludethat relaxing this branch will result in at least1 unit of improvement in the objective functionfor each unit of change in the value of Xl. Cor-responding interpretations apply to situationsin which multiple branches underlie an integersolution.

The second type of sensitivity analysis in-volves the explicit assignment of penalties todeviating from particular problem constraints.

Coupling this penalty procedure with para-metric 8&8, the interactive effect of constraintdeviations and parametric branches becomessusceptible to evaluation. For example, theupdated cost of a variable representing a con-straint violation identifies the net effective costof imposing the constraint relative to a giveninteger solution. Thus, if the constraint is infact violated at this solution, the d j values forthe parametric branches not only reflectbranching costs relative to the 'original' objec-tive function but also relative to the penaltyincurred from the violated constraint. Thisenables a more realistic form of problem solv-ing in situations where constraint deviationsmay be tolerable at a cost. The entire analysisand solution process can be conveniently car-ried out by merging the compact basis pro-cedures illustrated for the MIP problem withthe more elaborate procedures of [7] to main-tain the working tableau the same size as theoriginal.

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9. JOHNSEN E (1968) Studies in multiobjective decisionmodels. Suden Litteratur.

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ADDRESS FOR CORRESPONDENCE: Professor Fred Glover,College of Business Administration, University of Color-ado, Boulder, CO 80309, USA.