Omega 36 (2008) 913–917www.elsevier.com/locate/omega

On step fixed-charge transportation problem�

Krzysztof Kowalskia, Benjamin Levb,∗aConnecticut Department of Transportation, 121 Cimarron Road, Middletown, CT 06457, USA

bSchool of Management, University of Michigan-Dearborn, Dearborn, MI 48126, USA

Received 14 November 2007; accepted 14 November 2007Available online 21 November 2007

Abstract

The fixed-charge problem is a nonlinear programming problem of practical interest in business and industry. One of itsvariations is the fixed-charge transportation problem (FCTP) where fixed cost is incurred for every route that is used in thesolution, along with the variable cost that is proportional to the amount shipped. That cost structure causes the value of theobjective function Z to also behave like a step function. Each time we open or close a route the objective function jumps a step.The step fixed-charge transportation problem (SFCTP) is a variation of the FCTP where the fixed cost is in the form of a stepfunction dependent on the load in a given route. While the value of the objective function Z in the FCTP is a step function,the introduction of the step fixed cost in the SFCTP results in the objective function Z being itself a step function with manymore steps. Fixed-charge problems are usually solved using sophisticated analytical or computer software. This paper discussesthe theory of SFCTP and presents a computationally simple heuristic algorithm for solving small SFCTPs.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Fixed charge; Transportation problem; Heuristic algorithm

1. Introduction

The fixed-charge transportation problem (FCTP) hasbeen a popular research topic in mathematical program-ming for quite some time. Its structure is almost identi-cal with that of a classical transportation problem (TP).An abundance of algorithms, heuristic and exact, areavailable in the literature to solve the FCTP. See, forexample, Adlakha and Kowalski [1,2], Adlakha et al.[3] for details and practical cases in Abd el-Wahed andLee [4], Liu [5], Kuo and Nicholls [6], Dorndorf et al.[7]. A new approach for formulating the FCTP was re-cently introduced by Kowalski and Lev [8] taking into

� This manuscript was processed by Area Editor B. Lev.∗ Corresponding author.

E-mail addresses: Krys.Kowalski@po.state.ct.us (K. Kowalski),blev@umich.edu (B. Lev).

0305-0483/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.omega.2007.11.001

account the distribution’s degeneracy. The step fixed-charge transportation problem (SFCTP) is a variation ofthe FCTP where the fixed cost is incurred by activatinga route and it includes a step function. The introductionof the step fixed costs in the SFCTP results in the ob-jective function Z being a “more frequent” step functioncompared to FCTP.

The SFCTP has received little attention in theavailable literature. It must be noted that there wasan effort by Sandrock [9] to introduce fixed charge,formulated as a step function of the load assignedto each source. The problem analyzed by Sandrockconstituted the so-called source induced fixed-chargetransportation problem, which has a completely dif-ferent formulation compared to the SFCTP. Sinceproblems with fixed charge are usually NP-hard (Non-deterministic Polynomial-time), the computational timeto obtain exact solutions increases in a polynomial

914 K. Kowalski, B. Lev / Omega 36 (2008) 913–917

fashion and very quickly becomes extremely difficultand long as the size of the problem increases. In thecase of the SFCTP due to the step function structureof the objective function Z, we are dealing with a“NP-super hard” problem with much “higher degree”of the polynomial complexity.

2. The SFCTP and its formulation

The SFCTP can be stated as a distribution problemin which there are m sources and n destinations. Sourcei has Si units of output and destination j demands Dj

units. Each of the m sources can ship to any of the ndestinations at a cost of cij per unit from source i todestination j, plus a fixed cost Kij assumed for openingthe route i, j. The objective is to determine which routesare to be opened and the amount of shipment on thoseroutes, so that the total cost is minimized while meetingdemand/source constraints.

SFCTP formulation

Minimize Z =m∑

i=1

n∑

j=1

cij xij +m∑

i=1

n∑

j=1

Kijyij (1)

Subject ton∑

j=1

xij = Si for i = 1, 2, . . . , m, (2)

m∑

i=1

xij = Dj for j = 1, 2, . . . , n, (3)

where

yij = 1 if xij > 0,

= 0 otherwise,

xij �0 for all (i, j) andm∑

i=1

Si =n∑

j=1

Dj .

Formulation of Kij : Assume that the fixed cost toopen a route i, j is kij,1 if the output is less than or equalto Aij . When the shipment exceeds Aij , an additionalfixed cost of kij,2 units is incurred.

Thus, Kij which is the total fixed charge associatedwith route i, j is

Kij = bij,1kij,1 + bij,2kij,2, (4)

where

bij,1 = 1 if xij > 0,

= 0 otherwise;

bij,2 = 1 if xij > Aij ,

= 0 otherwise;

and

kij,1, kij,2, Kij , Aij �0.

If Aij = 0 and the load is xij = 0 on the same route i, j

then there is no cost kij,1. But if the load is xij > 0 thenthe cost is kij,1 + kij,2 for opening that route, which issimply a single step function. If all Aij ’s are equal to 0,then the problem becomes an FCTP with a single fixedcost of kij,1+kij,2. If, however, Aij > Min (Si, Dj ) in agiven route then we never impose the penalty of kij,2 forthat route and the cost for opening that route has singlestep of kij,1. If all Aij > Min (Si, Dj ), then the prob-lem becomes an FCTP with a single fixed cost of kij,1.

Note that Kij is a step function. In this example ithas two steps. It could have multiple steps, dependingon the problem structure.

Number of positive variables in an optimal solu-tion: Kowalski [10] shows that the maximum numberof variables in the optimal solution for FCTP is equalto n + m − 1. But as it is shown next, the situation iscompletely different for the SFCTP. To demonstrate thiswe analyze the following example: Let us formulate asquare problem of 3×3 with S1 =S2 =S3 =D1 =D2 =D3 = 3; all cij = 2; all Aij = 1, all kij,1 = 1.5 and allkij,2 = 500. The optimal distribution is all xij = 1, withZ = 9 × 1 × 2 + 9 × 1.5 = 31.5. Any alteration of theloads will activate the large value of kij,2 and any costsavings will be offset by an increase of 500. Therefore,the proposed distribution is optimal and it includes allnine variables; which is more than m + n − 1 = 5. Onecan generate an example where the number of positivevariables in a 3 × 3 is three positive variables (alongthe diagonal), thus in an n × n problem the number ofpositive variables can vary from n to n2.

3. Balinski solution for FCTP

Balinski [11] provides a heuristic solution for FCTP.Assume the fixed cost is kij,1 then the Balinski matrixis obtained by formulating a linear version of FCTP byrelaxing the integer restriction on yij as follows:

yij = xij /Mij , (5)

where

Mij = min(Si, Dj ). (6)

So, the linear version of FCTP thus formulated wouldsimply be a classical TP with transportation costs as

Cij = kij,1/Mij + cij , (7)

where kij,1 is a fixed cost.

K. Kowalski, B. Lev / Omega 36 (2008) 913–917 915

Cost

Mij

kij,1

cij * Mij

Xij

Relaxed integer restriction

Fixed charge stepfunction

Quantity shipped

Fig. 1. Shipping cost as a function of quantity shipped along route(i, j) for FCTP.

The optimal solution {xij } to this TP can be easilymodified into a feasible solution of FCTP as follows:

yij = 0 if xij = 0

and

yij = 1 if xij > 0.

Balinski [11] shows that the optimal value of the linearversion TP provides a lower bound on the optimal valueZ of FCTP (ZFCTP)

m∑

i=1

n∑

j=1

Cijxij �ZFCTP. (8)

For every loaded location i, j the cost of the fixed-charge step function formulation is greater than the cor-responding cost of the relaxed integer restriction func-tion as shown in Fig. 1.

4. The SFCTP algorithm

Since there is no algorithm for SFCTP then anyheuristic method which provides a good solution shouldbe considered useful. Our approach is similar to othermethods used for solving FCTP. At first we obtain inPart 1 a “good initial solution” and then in Part 2 weconverge on a better solution. As shown next, some ofthe algorithms used for solving FCTP can also be ap-plicable for SFCTP. One of those methods is Balinski[11] method, useful for finding a good initial solutionin FCTP.

Part 1: We propose two heuristic algorithms, whichare similar to Balinski [11] mentioned in Section 3. Theonly difference is in Eq. (7) above.

Our first proposal is Relaxed Integer Restriction #1where we convert the SFCTP to a classical TP by re-placing Eq. (7) with Eq. (7.1) where

Cij = (kij,1 + kij,2)/Mij + cij . (7.1)

M A

kij,1

kij,2

c*(M-A)

c*A

Relaxed IntegerRestriction #2

Fixed chargedstep function

Cost

Quantity shipped

Relaxed IntegerRestriction #1

Fig. 2. Shipping cost as a function of quantity shipped along route(i, j) for SFCTP.

The second proposal is Relaxed Integer Restriction #2where we replace Eq. (7) with Eq. (7.2) and the trans-portation costs are

Cij = kij,2/(Mij − Aij ) + cij . (7.2)

It can be noticed that during this process we are “losing”the kij,1 factor, which means that this approach is moreeffective for problems with large kij,2 coefficients com-pared to kij,1 (Fig. 2).

The “good initial solution” is obtained by solving theTPs with coefficients similar to those in Balinski Matrix,for either of the two formulations. There can be severalother strategies including a “mixed” approach whereeach Cij is created separately and differently for each(i, j), based on the magnitude of kij,1 and kij,2 values.It could be Eqs. (7), (7.1), (7.2) or a mixture of all. Oureffort will be limited to the above two approaches.

Part 2: The second part of the proposed method isbased on local or global concavity of the objective func-tion in the location obtained in Part 1. Depending onthe case a local or global minimum will be obtained. Itmust be noted that in the fixed-charge problems everysolution in an extreme point (or peak) is a local min-imum. Our analysis however is limited to the domainincluding the extreme points only and therefore ignoresthe “in between peaks” values. Graphically the situationis similar to a seesaw, where we are investigating onlythe “bottom of the teeth” points ignoring the “top of theteeth” and in between values. During our analysis weinvestigate the adjacent peaks to the load distributionobtained in Part 1. This is achieved through perturbingthe loads using the approach introduced by Adlakha andKowalski [1]. Since we are dealing with a transportation

916 K. Kowalski, B. Lev / Omega 36 (2008) 913–917

tableau, the search simply translates into perturbingeach load using single stepping-stone moves. There aremany such stepping-stone moves. Worse, since the num-ber of basic variables can be more than m + n − 1, thenumber of stepping-stone moves is very large. To alle-viate that difficulty we split the process into two phasesas described below.

Since the fixed charge is a step function we aredealing with a two-tier problem. This means that wemust treat separately the distributions “above” and“below” Aij .

Note 1: The initial distribution (computed in Part 1)is obtained from relaxed integer restriction problems(formulated as classical TPs); it always contains n +m − 1 or less loads. Any optimal solution requiringmore loads is obtained in Part 2 of the algorithm. Butthe n+m− 1 principle always governs the “upper” tier(“above” Aij ) distribution, which is governed by theTP/FCTP constrained problem principles.

It must be noted that the single stepping-stone movesin SFCTP are much more complex than in the case of anFCTP. They must take into consideration the intervalsof the step function. Thus, at first we try to reduce thenumber of the upper tier fixed charges by perturbingthe upper tier loads and “pushing” them “down” to thelower tier. That is when more than n+m−1 locations inthe transportation matrix can be populated. Followingthe decrease or Status Quo of the objective function weextend the analysis to all loaded locations. Unlike theFCTP, the analysis for SFCTP is much more complex.A possibility of a presence of many local minimumscan obscure the search for a global minimum. Also, thenumber of local minimums grows with the number ofsteps in the fixed-charge function.

5. A numerical example

Let Si , Dj and variable cost cij be as follows:

D1 = 10 D2 = 30 D3 = 10

S1 = 15 1 3 1S2 = 20 2 = cij 2 3S3 = 15 2 1 2

The fixed costs are as follows: the first number is kij,1for xij > 0 and the second number is kij,2 for xij > Aij

where Aij = 5.

10; 20 10; 10 10; 3010; 30 10; 20 10; 2010; 20 10; 30 10; 10

Relaxed Integer Restriction #1: Relaxed fixed-chargecoefficient (kij,1+kij,2)/Mij where Mij =min(Si, Dj ).

3 1.33 44 1.5 33 2.66 2

Combined variable and relaxed fixed-charge (Balinski)coefficients Cij = (kij,1 + kij,2)/Mij + cij .

4 4.33 56 3.5 65 3.66 4

Optimal distribution for Balinski Matrix.

10 520

5 10

The MODI (shadow price matrix) solution confirms op-timality to Balinski Matrix (cij − ui − vj for non-basicvariables and cij for basic variable—BV).

4 BV 4.33 BV .33 v1 = 02.83 3.5 BV 2.16 v2 = −.831.67 3.66 BV 4 BV v3 = −.67u1 = 4 u2 = 4.33 u3 = 4.67

Relaxed Integer Restriction #2: Slope/tangent coeffi-cients for relaxed fixed charges kij,2/(Mij − Aij ).

4 1 66 1.33 44 3 2

Combined variable and relaxed fixed-charge (Balinski)coefficients Cij = kij,2/(Mij − Aij ) + cij .

D1 = 10 D2 = 30 D3 = 10

S1 = 15 5 4 7S2 = 20 8 3.33 7S3 = 15 6 4 4

Optimal distribution for Balinski Matrix.

10 520

5 10

This is the same solution as earlier.

K. Kowalski, B. Lev / Omega 36 (2008) 913–917 917

In both cases Z=10+20+10+10+20+10+10+10+10×1+5×3+20×2+5×1+10×2=190. As can beseen the result of the two different heuristic approachesfor finding an initial solution yields the same solution.Now that we have a starting solution we apply Part 2to improve it. There are many possible stepping-stonemoves. The one we selected is a single stepping-stonemovement of 5 units of load from location 1,1 to 3,1and matching movement of 5 units of load from location3,3 to location 1,3. Those loads constitute the “above”Aij (equal to 5) portion of the original load of 10 andare being moved to “below” Aij “level” as shown below.This yields an improved solution:

5 5 520

5 5 5

With Z = 10 + 10 + 10 + 10 + 20 + 10 + 10 + 10 + 5 ×1+5×3+5×1+20×2+5×2+5×1+5×2=180.

6. Conclusion

We have developed a simple efficient heuristic proce-dure for solving two SFCTPs. We also introduced sev-eral new aspects of the problem showing the differencesand the similarities to the fixed-charge problem. Whilethe current research is limited to FCTP only, a lot ofreal life problems can only be described through a stepfixed-charge formulation. An example can be gradualtaxes activated by a higher turnover, or user fees start-ing after achieving some usage level.

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