Multiple objective programming with piecewise linear functions

JOURNAL OF MULTI-CRITERIA DECISION ANALYSIS

J. Multi-Crit. Decis. Anal. 8: 322–332 (1999)

Multiple Objective Programming with Piecewise Linear Functions

STEFAN NICKEL* and MARGARET M. WIECEK†

Fachbereich Mathematik, Universitat Kaiserslautern, Germany

ABSTRACT

An approach to generating all efficient solutions for multiple objective programs with piecewise linear objectivefunctions and linear constraints is presented. The approach is based on the decomposition of the feasible set intosubsets, referred to as cells, so that the original problem reduces to a series of single objective linear programs andfeasibility tests over the cells. The concepts of cell-efficiency and complex-efficiency are introduced and theirrelationship with efficiency is examined. A generic algorithm for finding efficient solutions for bi-objectivepiecewise linear programs is proposed. Applications in location theory as well as in worst case analysis arehighlighted. Copyright © 1999 John Wiley & Sons, Ltd.

KEY WORDS: multiple objective programming; piecewise linear functions; cell-efficiency; complex-efficiency

1. INTRODUCTION

The theory and methodology of multiple objectivelinear programming have already been intensivelystudied and applied to a variety of decision-making problems. In particular, the efficient set ofthose problems can be now easily computed bymeans of various algorithms. See Yu and Zeleny(1975), Gal (1977), Isermann (1977), Ecker et al.(1980), Steuer (1986), Armand (1993), Ehrgott etal. (1997) and Wiecek and Zhang (1997) for dif-ferent methodologies of finding the efficient set.Similarly, problems with convex objective func-tions over a convex feasible set have been treated,although their efficient set, in general, is not avail-able in such an elegant form as it is for linearproblems (see Haimes and Chankong, 1983). Mo-tivated by applications, in this paper we examinepossibly the simplest class of convex problems, i.e.multiple objective problems with piecewise linearobjective functions and linear constraints, forwhich we develop a description of the efficient setand propose an approach to finding this set.

Single or multiple objective piecewise linearprograms have also been studied by other au-thors. Fourer (1985, 1988, 1992) extended thesimplex method for linear programming to permit

the minimization of a convex piecewise linearfunction. Achary (1989) developed a simplex typemethod for a transportation problem with twoobjective functions: a piecewise linear convexfunction and a piecewise constant function repre-senting cost and time, respectively.

Multiple objective piecewise linear programs(MOPLPs) specifically arise in location theory,where—given a number of existing facilities—anew facility has to be located so that some objec-tives are optimized. The objectives are often in theform of piecewise linear functions owing to thefact that, typically, the sum (or the maximum) ofweighted distances from the existing facilities tothe new facility is chosen as a criterion. Moreover,these distances are often derived from norms witha polyhedral unit ball. See Durier and Michelot(1986), Pelegrin and Fernandez (1988), Puertoand Fernandez (1994) and Hamacher and Nickel(1996) for various MOPLP models in locationtheory.

Another area of application is multiple objec-tive worst case analysis. If, for example, the coef-ficients of a linear objective function shouldrepresent the cost of introducing a new producton the market, their exact numerical values areusually unknown, They may, however, be avail-able from market analysis in the form of a collec-tion of vectors. Under this uncertainty, the worstcase analysis dictates that the resulting objectivefunction be determined through the maximizationover the linear functions corresponding to thevectors in the collection and, hence, the objectivefunction becomes a piecewise linear function.

* Correspondence to: Fachbereich Mathematik, Uni-versitat Kaiserslautern, 67653 Kaiserslautern, Ger-many. E-mail: [email protected]† On sabbatical leave from the Department of Mathe-matical Sciences, Clemson University, South Carolina,USA.

Copyright © 1999 John Wiley & Sons, Ltd.Recei6ed 10 No6ember 1998

Accepted 6 May 2000

MULTIPLE OBJECTIVE PROGRAMMING 323

Furthermore, in order to obtain a more struc-tured description of the efficient set of convexmultiple objective programs, one may considerapproximating the convex objective functions bypiecewise linear functions. Such an approximationmethod including error bound analysis was pro-posed by Burkard et al. (1991).

We note that because a single objective piece-wise linear program can be transformed into alinear program (see Murty, 1983, p. 18), it is alsopossible to convert an MOPLP into a multipleobjective linear program (MOLP) for which solu-tion techniques are available, as is previouslymentioned. Such a transformation, however, sig-nificantly increases the number of variables andconstraints. The approach presented in this paperdoes not increase the size of the problem. Rather,it is based on the decomposition of the feasible setinto subsets, referred to as cells, so that the origi-nal problem reduces to a series of single objectivelinear programs and feasibility tests over the cells.The approach is an extension of the algorithmproposed by Nickel (1995) for multiple objectiveplanar location problems. While comparing thepresented approach with the classical one of con-verting an MOPLP to an MOLP, one may recog-nize a few advantages. An MOPLP solved as anMOLP features efficient solutions on theboundary of its feasible set that includes the feasi-

ble set of the original MOPLP as a subspace. Inparticular, converting a bi-objective piecewise lin-ear program to a bi-objective linear program usu-ally results in using the parametric cost simplexmethod to identify the efficient extreme solutions(see Geoffrion, 1967). The proposed approach,however, treats an MOPLP as a convex problemand finds its efficient solutions located in theinterior of its original feasible set. In this way, thisinterior-point-type approach preserves the origi-nal geometry of the problem. In addition, yieldingefficient solutions in the original space makesthem readily available for the decision stage ofmultiple criteria decision-making when the mostpreferred efficient solution is to be chosen by adecision-maker.

In the next section, the MOPLP is formulated

and basic concepts are presented. Decompositionof the feasible set is studied in Section 3. Theconcepts of cell-efficiency and complex-efficiencyare introduced in Section 4 and their relationshipwith efficiency is examined for the generalMOPLP. Section 5 includes a generic algorithmfor finding the efficient set of bi-objective piece-wise linear programs, and an illustrative exampleis contained in Section 6. Section 7 concludes thepaper.

2. DEFINITIONS AND BASIC CONCEPTS

Consider the following MOPLP:

minimize[ f1(x), . . . fm(x)] "MOPLP

s.t. x�Xwhere

fi(x)= max15k5Ki

{ fi1(x), . . . fiKi(x)} i=1, . . . m

fik(x)= (c ik)Tx+dik k=1, . . . Ki

cik is an n×1 finite vector, dik is a scalar, Ki is thenumber of affine functions fik defining the piece-wise linear function fi, X={x�Rn: Ax=b, x]0}, A is an l×n matrix of full rank, and b is anl×1 vector.

For every fi, i=1, . . . m, we can derive a sub-division of X by defining

We say that fij generates C0 ij. A non-empty set C0 i

j

is called a cell and is denoted by Cij. Then we have

X= .Ki

k=1

C0 ik= .

Qi

j=1

Cij i=1, . . . m

with dim(Cij)=n (because int(Ci

j)"¥) and 15Qi5Ki, where Qi denotes the number of cells withrespect to the objective function fi and we assumewithout loss of generality that C0 i

j"¥ for j=1, . . . Qi. We say that a function fik is active in acell Ci

j if

fi(x)= fik(x) for every x�Cij

This construction is illustrated in Figure 1,for fi(x)=max{ fi1(x), fi2(x), fi3(x), fi4(x), fi5(x)},where fi1(x)=x1+x2−20, fi2(x)=x1−x2,fi3(x)= −x1+x2, fi4(x)= −x1−x2+20 andfi5(x)=1

2 x1+10.

C0 ij�

!x�X : fij(x)] fik(x) for k=1, . . . Ki if ×x�X : fij(x)\ fik(x)

¥ otherwise for k=1, . . . Ki k" j

Copyright © 1999 John Wiley & Sons, Ltd. J. Multi-Crit. Decis. Anal. 8: 322–332 (1999)

S. NICKEL AND M.M. WIECEK324

Figure 1. A subdivision into cells. A cell is representedby the function which is active in it.

such that fi(x)5 fi(x°), i=1, . . . m, with at leastone strict inequality. Let XE denote the set ofefficient solutions of the MOPLP.

Geometrically related to the concept of effi-ciency is the concept of level sets and level curves.

Let y�Rm.

L5i (yi)�{x�X : fi(x)5yi} i=1, . . . mL=

i (yi)�{x�X : fi(x)=yi} i=1, . . . m

The following theorem uses several results fromthe literature and summarizes properties of XE forMOPLPs (see Geoffrion (1967, 1968), Warburton(1983), Gal (1986), Hamacher and Nickel (1996)and Ehrgott et al. (1997):

Theorem 2.1

1. Let x°�X and yi°= fi(x°), i=1, . . . m.x°�XE iff

-m

i=1

L5i (yi°)= -m

i=1

L=i (yi°) (1)

(see Figure 2).2. The set XE is connected.3. Let m=2 and xi* be an optimal solution of

minimizing fi over X, i=1, 2. We have

XE¤L51 ( f1(x2*))S L52 ( f2(x1*))

Part 3 of this theorem provides a good bound forthe region where efficient solutions may occur. It

Because we are dealing with multiple objectiveproblems, we have to find a common subdivisionof X for all objective functions fi, i=1, . . . m.

Let pi�{1, . . . Qi}, i=1, . . . m. Then

Cq�.m

i=1

Cipi

is called a cell with respect to ( f1, . . . fm), ifint(Cq)"¥. Such a cell can be described by avector of objective functions that are active in it.Again we have

X= -Q

q=1

Cq

where Q is the total number of cells with respectto ( f1, . . . fm) and

fi(x)= fiqi(x) for every x�Cq

where qi�{1, . . . Ki}, i=1, . . . m. Two distinctcells Cq and Cr are said to be disjoint ifint(Cq)S int(Cr)=¥.

It should be noted that there are other possibil-ities of finding a decomposition of the feasible setX into cells. In order to establish the results ofthis paper, a subdivision of X into cells, represent-ing common domains of linearity for all objectivefunctions, is needed. In addition, we are interestedin a subdivision in which the single cells are aslarge as possible. For more details on how cellsubdivisions can be derived, the reader is referredto Rourke and Sanderson (1972).

A feasible point x°�X is said to be an efficientsolution of the MOPLP if there is no other x�X,

Figure 2. Illustration for Theorem 2.1. x° is efficientwith respect to L5

1 and L52 . If we add L5

3 , then x° isno longer efficient.



should be noted that this result is not useful if weconvert the MOPLP into an MOLP.

In the following sections, we will use the pre-sented concepts to gain more insight into thestructure of the feasible set as well as the effi-ciency with respect to the cells.

3. PROPERTIES OF THE DECOMPOSITION

In this section, we examine properties specifyingactive functions in the cells. Consider first a singleobjective case of the MOPLP for which a functionfik can be active in only one cell.

Theorem 3.1Let fi(x)�max{ fi1(x), . . . fiKi

(x)} be a piecewiselinear (convex) function, where

fik : Rn�R1, k=1, . . . Ki

fik(x)= (c ik)Tx+dik

c ik is an n×1 vector and dik is a scalar and x�X,and X is a polyhedral set defined as in Section 2.Let Ci

q be a cell in X so that a functionfip, p�{1, . . . Ki} is active in it. Then, the func-tion fip cannot be active in any other cell in Xdisjoint with the cell Ci

q.

ProofThis follows from the fact that the region inwhich function fip is active can be written as{x�Rn: fip(x)] fik(x), k=1, . . . Ki}, which is apolyhedral set.

Consider now a bi-objective case of the MOPLP:

minimize[ f1(x), f2(x)]BOPLP

s.t. x�X

where

f1(x)�max{ f11(x), . . . f1t(x), . . .

f1u(x), . . . f1K(x)}

f2(x)�max{ f21(x), . . . f26(x), . . .

f2w(x), . . . f2L(x)}

and X is defined as in Section 2. The property ofTheorem 3.1 cannot hold with respect to eachobjective function separately. That is, if f1t(x)=f2w(x) for every x�X, then the same function

might be active in two disjoint cells with respectto two (different) criterion indices. However, twofunctions active in a cell cannot be active in anyother disjoint cell.

Theorem 3.2Consider the BOPLP. Let f1t(x)= f2w(x) for everyx�X. Let f1t(x) be active in two disjoint cells, sayCq and Cr (i.e. q"r), with respect to the two(different) criterion indices, respectively. Then theother two functions that are active in Cq and Cr

with respect to the two criterion indices, respec-tively, cannot be the same.

ProofLet CqSCr=¥. Assume that f1t and f26 areactive in cell Cq, and that f1u and f2w are active incell Cr, where

f2w(x)= f1t(x) for every x�X (2)

Then we have

f1t(x)] f1ux Öx�Cq and

k=1, . . . t, . . . u, . . . K (3)

and

f1u(x)] f1k(x) Öx�Cr and

k=1, . . . t, . . . u, . . . K (4)

Similarly

f26(x)] f2k(x) Öx�Cq and

k=1, . . . 6, . . . w, . . . L (5)

and

f1t(x)] f2k(x) Öx�Cr and

k=1, . . . 6, . . . w, . . . L (6)

Assume now that

f26(x)= f1u(x) for every x�X (7)

and consider points in Cq and Cr. For any xq�Cq

and xr�Cr there must be either

f1t(xq)\ f1t(xr)= f2w(xr) (8)

or

f1t(xq)B f1t(xr)= f2w(xr) (9)

Assume the case given by (8).From (3), (7), (5) and (2) we get

"



f1t(x)] f1u(x)= f26(x)] f2w(x)

= f1t(x) for any x�Cq

Therefore

f1u(x)= f2w(x) for any x�Cq (10)

From (4), (6) and (7) we get

f1u(x)] f1t(x)] f26(x)= f1u(x) for any x�Cr

Therefore

f1t(x)= f26(x) for any x�Cr (11)

Equations (2), (7) and (10) imply

f1t(x)= f1u(x)= f26(x)= f2w(x) for any x�Cq

(12)

and (2), (7) and (11) imply

f1t(x)= f1u(x)= f26(x)= f2w(x) for any x�Cr

(13)

Let h(x) be a function defined by (12) and (13).Then h(x) is active in two disjoint cells, Cq andCr, with respect to each criterion index, which byTheorem 3.1 is a contradiction.

We now define other concepts needed to analysethe MOPLP. Let C denote the set of all cells Cq inX. In each cell Cq�C, ( f1, . . . fm) can be writtenas a vector f q= ( f1

q, . . . fmq ) of m linear functions.

Observe that there are no two disjoint cells in C,say Cq and Cr, in which f q and f r are active,respectively, and

f q(x)= f r(x) (14)

that is, the same vector of objective functionsactive in a cell cannot be active in another disjointcell. Therefore, the total number Q of cells in C is

15Q55m

i=1

Ki (15)

Let F denote a face of a cell. A face F¦Cq ismaximal if there is no other face F%¦Cq suchthat F¦F%. The set C(F)�{Cq�C: F¦Cq} isreferred to as a cell complex or a complex.

Let x be an extreme point of a cell. DenoteC(x)�{Cq�C : x�Cq}. Two disjoint cells Cq andCr are said to be adjacent if there exists a face Fsuch that F¤CqSCr and dim(F)]1.

4. EFFICIENCY WITH RESPECT TO CELLS

In this section, we continue to use the cell subdivi-sion and compare the efficiency in the feasible setwith the efficiency with respect to a cell or a cellcomplex. We first use the properties of level setsof the MOPLP and generalize part 1 of Theorem2.1. As the level sets are determined by affinefunctions, they can be viewed as polyhedral cones.We then introduce the definitions of cell-efficiencyand complex-efficiency and relate them to effi-ciency over the entire feasible set.

The set L5i (yi°) is a polyhedral set whose defin-ing halfspaces are given by

fik(x)5yi°, k=1, . . . Ki (16)

The set L=i (yi°) represents the set of the hyper-

planes associated with the defining halfspaces(16). Let x°�Li

=(yi°). There is a finite number ofthe defining hyperplanes passing through x°. LetK(x) denote a cone with origin at point x and let(K(x) denote its boundary. Define Ki(x°) to bethe cone generated by the finite number of thedefining halfspaces in (16) whose associated hy-perplanes pass through x°. Then

L5i (y°i )¤Ki(x°) for every i=1, . . . m

and

-m

i=1

L5i (yi°)¤-m

i=1

Ki(x°) (17)

Accordingly

L=i (y°i )¤(Ki(x°) for every i=1, . . . m

and

-m

i=1

L=i (yi°)¤-

m

i=1

(Ki(x°) (18)

Theorem 4.1Let x°�X and y°i = fi(x°), i=1, . . . m. x°�XE iff

-m

i=1

Ki(x°)= -m

i=1

(Ki(x°) (19)

Proof( [ ) If x°�XE then (1) holds. Using (17) and(18), we obtain (19).( < ) Assume that x° is not efficient. Then thereexists an x�X such that fi(x)5 fi(x°) for all i=1, . . . m, and for at least one index k, k�{1, . . . m}, fk(x)Bfk(x°). Therefore, x�L5i (yi°)



for all i=1, . . . m, where yi°= fi(x°) andxQL=

k (yi°) for at least one index k. Consequently,

x�-m

i=1

L5i (y°i )

and, using (17) we get

x�-m

i=1

Ki(x°) (20)

Because xQL=k (yi°), then

x�(Kk(x°) (21)

From (20) and (21), we conclude acontradiction.

Let x°�Cq. Because fi(x)=fipi(x), pi�{1, . . . Ki},

for every x�Cq, the set L5i (yi°) includes exactlyone defining halfspace

H5i, q

(x°)={x�X : fipi(x)5yi°} (22)

whose associated hyperplane H=i, q(x°) passes

through x° and the set CqSL=i (yi°).

Definition 4.1A point x°�Cq is said to be cell-efficient iff thereis no other x�Cq such that fipi

(x)5 fipi(x°), i=

1, . . . m, with at least one strict inequality.

Theorem 4.2A point x°�Cq is cell-efficient iff

-m

i=1

H5i, q(x°)= -m

i=1

H=i, q(x°) (23)

Proof:In accordance with part 1 of Theorem 2.1, a pointx°�Cq is cell-efficient iff condition (1) holds.Using (22), this condition reduces to (23).

In the next two results, we take a point from theinterior of the cell to be tested, because we need aneighbourhood around that point which is com-pletely contained in the cell.

Theorem 4.3Let x°� int(Cq). A point x°�XE iff x° is cell-efficient.

Proof( [ ) the proof is obvious.( < ) If x° is cell-efficient then (23) holds. Inaddition, Ki(x°)=H5i, q(x°) and (Ki(x°)=

H=i, q(x°). Therefore, using (1) and (19), x° is

efficient.

Corollary 4.1Let x°� int(Cq). If x° is cell-efficient, thenCq�XE.

Let C(F)={Cq, q=1, . . . r}.

Definition 4.2Let x°�rel int(F). Then x° is said to be complex-efficient if it is cell-efficient for every cellCq�C(F).

Theorem 4.4Let x°�rel int(F) and C(F) be a cell complexcontaining the face F. Then x°�XE if x° iscomplex-efficient.

Proof( [ ) the proof is obvious.( < ) If x° is complex-efficient, then it is cell-efficient for every cell Cq�C(F). That is

-m

i=1

H5i, q(x°)= -m

i=1

H=i, q(x°) for all q=1, . . . r

Therefore

-r

q=1

�-m

i=1

H5i, q(x°)�

= -r

q=1

�-m

i=1

H=i, q(x°)

�which implies

-m

i=1

� -r

q=1

H5i, q(x°)�

= -m

i=1

� -r

q=1

H=i, q(x°)

�(24)

Now observe that because x�Cq for all q=1, . . . r, the cone Ki(x°) is defined by exactly rhalfspaces H5i, q(x°). We get

Ki(x°)= -r

q=1

H5i, q(x°) (25)

and

(Ki(x°)= -r

q=1

H=i, q(x°) (26)

From (24), (25), (26), (19) and (1), we concludethat x°�XE.

Theorem 4.5Let x°�rel int(F). If x° is complex-efficient, thenthe face F�XE.

ProofLet x�F and x"x°. Assume that x is not effi-cient, so it is not complex-efficient either. There



exists at least one cell Ct�C(F), t�{1, . . . r},such that x is not cell-efficient. By Definition 4.1,

-m

i=1

H5i, t(x)"-m

i=1

Hi, t(x) (27)

Because x°�F, then x°�Ct and because x° isefficient

-m

i=1

H5i, t(x°)= -m

i=1

Hi, t(x°) (28)

Observe that

-m

i=1

H5i, t(x) and -m

i=1

H5i, t(x°)

determine the same cones with distinct origins,and the co-existence of (27) and (28) is acontradiction.

5. ALGORITHM

According to the results of the previous section,efficiency over the entire feasible set of MOPLPsis equivalent to efficiency over a subset of thefeasible set being a cell or a cell complex. If afeasible point is efficient with respect to a subset(i.e. it is cell-efficient or complex-efficient), thenthe subset or its part is also efficient (i.e. thewhole cell or the related face of the complex isalso efficient). These results indicate that one candevelop a top–down approach to generating theefficient solutions of MOPLPs in which higherdimensional subsets of the efficient set (cells andfaces) are identified first. Recall that traditionalapproaches to MOLPs start by identifying effi-cient extreme points and then move on to findingefficient faces.

We now present our approach for the BOPLP.We first find the ‘ends’ of the efficient set byminimizing each objective function individuallyand then work on finding possibly the highest-dimensional subsets composing this set.

First, observe that the lexicographic optimalsolutions Xi

lex, i=1, 2, with respect to the permu-tation (1, 2) and (2, 1) of the BOPLP are in XE,and because XE is connected, it must consist of achain of efficient cells and/or faces of cells‘spanned’ in the feasible set between the sets oflexicographic solutions.

Xilex, i=1, 2, can be found by solving the fol-

lowing single objective problems:

minimize fj(x)

s.t. x�Xi* i" j

where Xi* is the optimal solution set of theproblem

minimize fi(x)

s.t. x�X

Given an extreme point in Xilex, an efficient cell

(or face) adjacent to that point is identified. Thena new extreme point, which belongs to that effi-cient cell (or face) and at which the objectivefunction fj, j" i, assumes the smallest value overthe whole cell (or face), is found. At the extremepoints subsequently found, the efficiency test for anew cell (or face) is performed until a newly foundextreme point is in Xj

lex, j" i. Then the completeefficient set is generated.

Below, we present this approach in genericform.

Initialization:

1. For i=1, 2 solve:

minimize fi(x)

s.t. x�X

Let Xi* be the set of optimal solutions.2. For j=1, 2 solve

minimize fj(x)

s.t. x�Xi* i" j

Let Xilex be the set of optimal solutions.

3. Set

XE�.2

i=1

Xilex

4. Let x�X1lex. If x is an extreme point of a cell,

go to the main step. Otherwise x�rel int(F)and use a procedure for finding an extremepoint of X1

lex. Then go to the main step.

Main step:Let y�X1

lex be an efficient extreme point of a cellCq.while yQX2

lex do

� Find C(y).� If there is an efficient cell Cr¦C(y), set

XE�XE @ Cr, find



z=arg minx�C r

f2(x)

and set y�z.� Otherwise find a maximal efficient face F s.t.

y�F¦C(y), set XE�XE @ F, find

z=arg minx�F

f2(x)

and set y�z.

end

Output: XE

The approach includes four major tasks, namely

(i) find an extreme point in Xilex,

(ii) find a cell complex C(y),(iii) perform efficiency test for a cell, and(iv) perform efficiency test for a face,

which we will now discuss in more detail.

5.1. Procedures for finding an extreme point inXi

lex

5.1.1. Procedure A

1. After performing step 1 of the initialization,enumerate all extreme points in Xi*. LetEP(Xi*) be the set of all extreme points in Xi*.

2. Perform step 2 of the initialization as follows:

minimize fj(x)

s.t. x�EP(Xi*) i" j

Output: an extreme point in Xilex.

5.1.2. Procedure B (see Bazaraa et al., 1990)Let F be a face of a cell Cq and x�rel int(F).Let the system Hx=h represent the hyperplanesbinding at x. Note that rank(H)5n−1. Find asolution d"0 to the system Hd=0 and computeg1=max{g : x+gd�F}B�. Let y1=y+g1d.Hence at y1 there is at least one additional lin-early independent hyperplane binding. If the newbinding hyperplane(s) along with Hx=h producea system of rank n, then y1 is an extreme point ofthe cell Cq. Otherwise, repeat this step at y1 until,after at most n−rank(H) such steps, an extremepoint of Cq satisfying Hy1=g is obtained.

Output: an extreme point in Xilex.

5.2. Cell complex C(y)Finding C(y) just involves identifying the indicesof the functions active in all the cells adjacent tothe extreme point y.

5.3. Efficiency test for a cell Cq (based onTheorem 4.2 and Corollary 4.1)Let a cell Cq be given and let the functions f1t andf26 be active in Cq, where

f1t(x)= (c1t)Tx+d1t

f26(x)= (c26)Tx+d26

If the vectors c1t and c26 are a negative linearcombination of each other, then the whole cell Cq

is efficient. Note that c1t=0 or c26=0 means thatwe are in the set of optimal solutions for f1 or f2,respectively.

5.4. Efficiency test for a face F (based onTheorem 4.4)This test is performed if there is no efficient cell inC(y).

Let Gq denote the 2×n matrix whose rows arecomposed of the gradients of the objective func-tions active in the cell Cq. The matrix Gq will bereferred to as the gradient matrix of the cell Cq.

In order to check the efficiency of an n−1dimensional face in C(y), construct the linearsystem:!Gqd50

Grd50(29)

where Gq and Gr are the gradient matrices of twoadjacent cells Cq and Cr in C(y), respectively.Solve system (29) for d"0. If system (29) isinfeasible, then the face being the intersection ofCq and Cr is efficient. In fact, it is a maximalefficient face adjacent to y. Otherwise, that face isnot efficient and one should search for n−2dimensional efficient faces in C(y). In general, inorder to perform the efficiency test for an n−kdimensional face in C(y), construct the linearsystem:

G1d50�Gk+1d50

(30)

where G j, j=1, . . . k+1, are the gradient ma-trices of the adjacent cells C j in C(y), respectively.

If system (30) is infeasible, then the face beingthe intersection of C j, j=1, . . . k+1, is efficient.

ÍÁ

Ä



Otherwise proceed to examining faces of a lowerdimension.

Observe that for some k, k=1, . . . n−1, sys-tem (30) has to be infeasible, implying that thecorresponding n−k dimensional face is efficient.

In every main step of the algorithm, at least amaximal efficient face is identified so that oneproceeds quickly from an initial efficient extremepoint in X1

lex to a final efficient extreme point inX2

lex. Although a (single objective) linear programis solved in the main step, it is solved over a subsetof the original feasible set, i.e. either over a cell ora face. The efficiency tests require comparingvectors or solving homogeneous linear systems,which are considered to be efficiently manageablecomputational tasks.

Comparing the algorithm with the parametriccost simplex method, we observe that while theyboth move from one efficient extreme point toanother, each of them moves at a different pace.The former works in a lower dimensional spacethan the latter and in this way avoids visitingefficient extreme points generated by the artificialvariables. Furthermore, the former is designed toproceed quickly as it may identify even a completecell as efficient, while the latter only confirms theefficiency of the edge connecting two adjacentefficient extreme points and, given these points,builds up the efficient faces of higher dimensions.

In order to generalize the algorithm forMOPLPs, specific bookkeeping techniques willhave to be developed. This is because in this casethe efficient set does not have the structure of asingle chain of efficient cells and/or faces. Thisdifficulty can be alleviated by a branching typealgorithm, examining various efficient chains andperforming a merging step at the end. On the otherhand, most of the steps of the proposed algorithmcan be modified in a straightforward fashion tohandle problems with multiple objective functions.

6. EXAMPLE

Consider the following BOPLP:

minimize

ÆÃÃÃÈ

maxi=1, . . .5

{ f1i(x)}max

i=1, . . .5{ f2i(x)}

ÇÃÃÃÉ

s.t. x�X={x�R2: x]0},

where

f11(x)=x1+x2−20 f21(x)=x1+x2−60f12(x)=x1−x2 f22(x)=x1−x2−20f13(x)= −x1+x2 f23(x)= −x1+x2+20f14(x)= −x1−x2+20 f24(x)= −x1−x2+60f15(x)=1

2 x1+10 f25(x)= −12 x1+35

Observe that

X1*=X1lex=

!� 010�"

and

X2*=X2lex=

!�5020�"

so we can move to the main step of the algorithm.By simply evaluating the linear functions at X1*,

we get the cell complex

C(X1*)={( f13, f24), ( f14, f24), ( f15, f24)}

Note that throughout this example we describe acell simply by the two linear objective functionsactive in it. Because neither (−1, 1), (−1, −1)nor (1

2, 0) (the gradients of f13, f14 and f15, respec-tively) is a negative multiple of (−1, 1) (the gradi-ent of f24), no complete cell in C(X1*) is efficient.

We next test the faces in C(X1*) as described in(29). As we first examine the face defined by( f13, f24)S ( f14, f24), we test the simultaneous solv-ability of the two systems�−1 1

−1 −1�

d50 and�−1 −1

−1 −1�

d50

It is easy to find a common solution, for exampled= (1, 1). We next examine the face defined by( f13, f24)S ( f15, f24). Therefore, we test the simulta-neous solvability of the two systems�−1 1

−1 −1�

d50 and� 1

2 0−1 −1

�d50

The only solution to these systems is d= (0, 0),which means that this face is efficient. Denote thisface by F. Now we solve the linear program

y=arg minx�F

f2(x)

and continue with the efficient extreme point y=(20

3 , 20). We get the following cell complex

C(y)= (( f15, f23), ( f15, f24), ( f13, f24), ( f13, f23))



Again, no complete cell is efficient. After testingthe different faces of this complex using the sameprocedure as described above, we find that theface defined by ( f15, f23)S ( f15, f24) is efficient.This leads us (solving again a linear program) tothe next efficient extreme point y= (10, 20) forwhich we get the cell complex

C(y)= (( f15, f23), ( f15, f24), ( f15, f25))

Here we see that (12, 0) (the gradient of f15) is a

negative of (−12, 0) (the gradient of f25). There-

fore, the cell C= ( f15, f25) is efficient and wecontinue with the efficient extreme point y=(40, 10) found as the solution of the linearprogram

y=arg minx�C

f2(x)

We continue the procedure until we reach X2*.Summing up, the efficient set of this example

problem is the union of four maximal efficientfaces and one efficient cell.

XE=F1 @ F2 @ C1 @ F3 @ F4

where

F1=conv�� 0

10�

,� 20

3

20��

F2=conv��6.6

20�

,�10

20��

C1=conv��10

20�

,� 15

22.5�

,�35

7.5�

,�40

10��

F3=conv��40

10�

,�130

3

10��

F4=conv��130

3

10�

,�50

20��

and conv(x1, . . . xn) denotes the convex hull of{x1, . . . xn}.

The feasible set, cells and efficient set of thisproblem are depicted in Figure 3.

7. CONCLUSIONS

In this paper, MOPLPs are studied and a genericalgorithm for generating the efficient set for bi-

Figure 3. Illustration for the example. The grey area isthe efficient set. The dashed and dotted lines representthe cell subdivision given by f1 and f2, respectively.

objective problems is proposed. The new conceptsof cell-efficiency and complex-efficiency help ex-amine the efficiency of the solutions of theMOPLP. The approach is based on a decomposi-tion of the feasible set into cells so that theefficient set is available as the union of efficientcells and/or maximal efficient faces. The proposedalgorithm directly examines the efficiency of theelements of this union starting with the elementsof highest dimension (cells) and then checkinglower dimensional elements (cells’ faces).

We expect that this algorithm is more attractivethan the classical parametric cost method appliedto the bi-objective linear program to which theoriginal bi-objective piecewise linear program canbe reformulated. This classical approach worksexactly in the opposite direction: it first identifiesefficient extreme points and then uses this infor-mation to examine the efficiency of objects ofhigher dimensions.

In the future, we plan to perform computa-tional studies to verify this hypothesis as well asto extend the algorithm to problems with multipleobjective functions.

ACKNOWLEDGEMENTS

Stefan Nickel was partially supported by grantERBCHRXCT930087 of the European HC&MProgramme.

The authors would like to express their thanksto Ansgar Weißler for preparing the numericalexample.



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