THE DETERMINATION OF PARADOXICAL PAIRS IN A LINEAR TRANSPORTATION PROBLEM

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1 Publshed by European Centre for Research Tranng and Development UK ( THE DETERMINATION OF PARADOXICAL PAIRS IN A LINEAR TRANSPORTATION PROBLEM Ekeze Dan Dan Department of Statstcs, Imo State Unversty PMB 2, Owerr Ngera GSM: Chukwud J. Ogbonna, Federal Unversty of Technology Owerr Ngera PMB 526, Owerr Ngera GSM: Opara Jude Department of Statstcs, Imo State Unversty PMB 2, Owerr Ngera GSM: ABSTRACT: The transportaton paradox s related to the classcal transportaton problem. For partcular reasons of ths problem, an ncrease n the quantty of goods to be transported may lead to a decrease n the optmal total transportaton cost. In ths paper, an effcent algorthm for solvng a lnear programmng problem was dscussed, and t was concluded that paradox exsts. The North-West Corner method was used to obtan the optmal soluton usng the TORA Statstcal Software Package. The method however gves a step by step development of the soluton procedure for fndng all the paradoxcal par. KEYWORDS: Transportaton Paradox; Transportaton Problem; Paradoxcal Range of Flow; Lnear Programmng INTRODUCTION The term Paradox arses when a transportaton problem admts of a total cost whch s lower than the optmum and s attanable by shppng larger quanttes of goods over the same routes that were prevously desgnated as optmal. Ths unusual phenomenon however, was noted by Szwarc (7). The classcal transportaton problem s the name of a mathematcal model whch has a specal mathematcal structure. The mathematcal formulaton of a large number of problems conforms to ths specal structure. Htchcock () orgnally developed the basc lnear transportaton problem. Charnes et al (53) developed the steppng stone methods whch provde an alternatve way of determnng the smplex method nformaton. Dantzg (63) used the smplex method to the transportaton problem as the prmal smplex transportaton method.

2 Publshed by European Centre for Research Tranng and Development UK ( Appa (73) also developed the soluton procedure for solvng the transportaton problem and ts varants. Klngman and Russel (74 and 75) ntroduced a specalzed method for solvng a transportaton problem wth several addtonal lnear constrants. Hadley (87) gave the detaled soluton procedure for solvng lnear transportaton problem. Tll date, several researchers studed extensvely to solve cost mnmzng transportaton problem n varous ways. In some stuatons, f we obtan more flow wth lesser cost than the flow correspondng to the optmum cost then we say paradox occurs. Charnes and Klngman (7), Szwarc (73), Adlakha and Kowalsk (8) and Storoy (27) consdered the paradoxcal transportaton problem. Gupta et al (3) consdered a paradox n lnear fractonal transportaton problem wth mxed constrants. Josh and Gupta (2) studed paradox n lnear plus fractonal transportaton problem. In the early day of lnear programmng problem some of the poneers observed paradox but by whom no one knows. In ths paper we present a method for solvng transportaton problem wth lnear constrants. Thereby, we state the suffcent condton of exstence of paradox, paradoxcal range of flow and paradoxcal flow for a specfed flow n such type of lnear transportaton problem. Havng known that paradox does not ext regularly n so many lnear transportaton problems, the ratonale behnd ths research work s to unvel a numercal practcal example that wll sut the algorthm dscussed n ths paper. We also ustfy the theory by llustratng a numercal example, and at the same tme revewng past researches. The purpose of ths research paper s to obtan the best paradoxcal par from the optmal basc feasble soluton of the transportaton problem, and at the same tme, to obtan the paradoxcal range of flow. DEFINITIONS OF TERMS. Paradox n a transportaton problem: n a transportaton problem f we can obtan more flow (F ) wth lesser cost (Z ) than the optmum flow (F ) correspondng to the optmum cost (Z ).e. F > F and Z < Z, then we say that a paradox occurs n a transportaton problem. 2. Cost-flow par: f the value of the obectve functon s Z and the flow s F correspondng to the feasble soluton X of a transportaton problem, then the par (Z, F ) s called the cost-flow par correspondng to the feasble soluton X. 3. Paradoxcal par: A cost-flow par, (Z, F) of an obectve functon s called paradoxcal par f Z < Z and F > F where Z s the optmum cost and F s the optmum flow of the transportaton problem. 4. Best paradoxcal par: The paradoxcal par (Z*, F*) s called the best paradoxcal par of a transportaton problem f for all paradoxcal par (Z, F), ether Z* < Z or Z* = Z but F* > F. 5. Paradoxcal range of flow: f F be the optmum flow and F* be the flow correspondng to the best paradoxcal par of a transportaton problem then [F, F*] s called paradoxcal range of flow.

3 Publshed by European Centre for Research Tranng and Development UK ( LITERATURE REVIEW The transportaton paradox s, however, hardly mentoned at all n any of the great number of textbooks and teachng materals where the transportaton problem s treated. The smulaton research reported by Fnke (8) ndcates, however, that the paradox may occur qute frequently. Apparently, several researchers have dscovered the paradox ndependently from each other. But most papers on the subect refer to the paper by Charnes and Klngman (7) and Szware (73) as the ntal papers. Charnes and Klngman (7) name t the more-for-less paradox and they wrte: The paradox was frst observed n the early days of lnear programmng hstory (by whom no one knows) and has been part of the folklore known to some (e.g. A. Charnes and Cooper) but unknown to the great maorty of workers n the feld of lnear programmng. Accordng to Appa (73), the transportaton paradox s known as Dogs paradox at the London School of Economcs, named after Alson Dog who used t n exams etc around 5 (Dog dd not publsh any paper on t). Snce the transportaton paradox seems not to be known to the maorty of those who are workng wth (or teachng) the transportaton problem, one may be tempted to beleve that ths phenomenon s only an academc curosty whch wll most probably not occur n any practcal stuaton. But that seems not to be true. Experments done by Fnke (78), wth randomly generated nstances of the transportaton problem of sze and allowng addtonal shpments (post-optmal) show that the paradoxcal propertes. More precsely, the average cost reductons acheved are reported to be 8.6% wth total addtonal shpments of 2.5%. In a recent paper, Deneko et al (23) develop necessary and suffcent condtons for a cost matrx C to be mmune aganst the transportaton paradox. Arora and Ahua (2) carred out a research work n paradox on a fxed charge transportaton problem. In ther fndngs, a paradox arses when the fxed charge transportaton problem admts of a total cost whch s lower than the optmum cost, by transportng larger quanttes of goods over the same routes. A suffcent condton for the exstence of a paradox s establshed. Paradoxcal Range of flow s obtaned for any gven flow n whch the correspondng obectve functon value s less than the optmum value of the fxed charge transportaton problem. Manusr et al (22) n ther research paper The Algorthm of Fndng all Paradoxcal Pars n a Lnear Transportaton problem establshed a suffcent condton for the presence of paradox n a lnear programmng problem, obtaned the paradoxcal pars and fnally obtaned the paradoxcal range of flow. Vshwas and Nlama (2) n ther research ttled On a Paradox n Lnear Fractonal Transportaton Problem dscovered that a paradoxcal stuaton arses n a lnear plus lnear fractonal transportaton problem (LPLFTP), when value of the obectve functon falls below the optmal value and ths lower value s attanable by transportng larger amount of quantty. In

4 Publshed by European Centre for Research Tranng and Development UK ( ther research paper, a new heurstc s proposed for fndng ntal basc feasble soluton for LPLFTP and a suffcent condton for the exstence of a paradoxcal soluton s establshed n LPLFTP. PROBLEM FORMULATION In ths paper, we consder the followng transportaton problem: Let the transportaton problem conssts of m sources and n destnatons, where x = the amount of product transported from the th source to the th destnaton, c = the cost nvolved n transportng per unt product from the th source to the th destnaton, a = the number of unts avalable at the th source, b = the number of unts requred at the th destnaton. In ths paper, we consder the cost mnmzng lnear transportaton problem as: m n : MnZ = cx = = P subect to the constrants n x = a = m x = b = ; I = (, 2,, m) ; J = (, 2,, n) and x (, ) I J. x (, ) I J be a basc feasble soluton correspondng to the bass B of the Let X = { } problem P and the value of the obectve functon Z correspondng to the basc feasble soluton X s Z = m = n = c x (say) Let F be the correspondng flow. Then F = a = b I I Now we consder the dual varables u for I and I such that u + v = c correspondng to the bass B. Also (I, ) B, let c = (u + v ) c If < (, ) B then the soluton s optmum. c Theorem: the suffcent condton for the exstence of paradoxcal soluton of (P ) s that f at least one cell (r, s) B n the optmum table of (P ) where a r and b s are replaced by a r + l and b s + l respectvely (l > ) then (u r + v s ) <. 2

5 Publshed by European Centre for Research Tranng and Development UK ( Proof: Let Z be the value of the obectve functon and F be the optmum flow correspondng to the optmum soluton (X ) of problem (P ). The dual varables u and v are gven by u + v = c, (, ) B Then Z = c x = ( u + v )x = x u + = au + b v and F = a = b x v Now, let at least one cell (r, s) B, where a r are replaced by a r + l and b s + l, respectvely (l > ) n such a way that the optmum bass remans same, then the value of the obectve functon Ẑ s gven by ˆ Z = a ( ) ( ) [ ( )] u + b v + ur ar + l + vs bs + l = Z + l ur + vs r s The new flow Fˆ s gven by Fˆ = a + l = b + l = F + l ˆ F F = l > Therefore for the exstence of paradox we must have Ẑ Z <. Hence the suffcent condton for the exstence of paradox s that at least one cell (r, s) B n the optmum table of such that f a r and b s are replaced by a r + l and b s + l (l > ) then l(u r + u s ) <,.e. (u r + v s ) <. Now we state the followng algorthm to fnd all the paradoxcal par of the problem (P ). Algorthm: Step : Fnd the cost-flow par (Z, F ) for the optmum soluton X. Step 2: = Step 3: Fnd all cells (r, s) B such that (u r + v s ) < f t exsts otherwse go to step 8. Step 4: Among all cells (r, s) B satsfyng step 3 fnd mn flow for l = whch enter nto the exstng bass whose correspondng cost s mnmum. Let (Z, F ) be the new cost flow par correspondng to the optmum soluton X. Step 5: Wrte (Z, F ). Step 6: = +. Step 7: go to step 3 Step 8: We wrte the best paradoxcal par (Z*, F*) = (Z, F ) for the optmum soluton X* = X. Step : End. 3

6 Publshed by European Centre for Research Tranng and Development UK ( Ths algorthm gves all the paradoxcal pars. From these pars we can fnd the paradoxcal par for a specfed flow ( F ) also. DATA ANALYSIS The data used for ths research was extracted from Opara J. (2), Introducton to Operaton Research, Exercses 3 page 28. The estmated supply capactes of the fve warehouses, the demand requrements at the fve markets and the transportaton cost of each product are gven n Table I below: Table I (, ) M M 2 M 3 M 4 M 5 S W W W W W d Solvng the above problem usng the Least-Cost method through TORA Statstcal Software Package, the optmal transportaton table s presented n Table II. Table II (, ) M M 2 M 3 M 4 M 5 S V W W W d V

7 Publshed by European Centre for Research Tranng and Development UK ( The total cost s 456. We then check the sgn of (U r + V s ), where (r, s) B n Table II, we observe that U 4 + V 2 = - <, U 5 + V 2 = - 2 <, and U 5 + V 3 = - <. So Paradoxcal par of the problem (P ) applyng the algorthm dscussed n ths paper. Applyng Step : The cost-flow par s (Z o, F o ) = (456, 36) correspondng to the optmum soluton X = { x =, x 2 = 35, x 5 =, x 2 =, x 23 = 25, x 34 = 2, x 35 = 2, x 4 =, x 5 = }. Applyng step 2: set = Applyng step 3: Now we check the sgn of (U r + V s ) and we obtan for the non-basc cells (4,2), (5,2) and (5,3), the sgn that s negatve. Applyng step 4: For l = For the cell (4, 2) Table III (, ) M M 2 M 3 M 4 M 5 S V W W W d V The total cost of ths transportaton problem s 455. For cell (5, 2), the transportaton table s presented n Table IV 5

8 Publshed by European Centre for Research Tranng and Development UK ( Table IV (, ) M M 2 M 3 M 4 M 5 S V W W W d V The total cost of ths transportaton problem s 454. For the cell (5, 3), the transportaton table s presented n Table V. Table V (, ) M M 2 M 3 M 4 M 5 S V W W W d V The total cost s 455. The mn cost = mn{456, 454, 455} = 454. Hence l = enters n the optmum bass from the cell (5, 2) and correspondng table s Table IV, the correspondng paradoxcal par (Z, F ) = (454, 37). Employng steps 6 and 7. Then repeatng ths process, the next table s 6

9 Publshed by European Centre for Research Tranng and Development UK ( Table VI (, ) M M 2 M 3 M 4 M 5 S V W W W d V Now repeatng ths process, the fnal table s presented n Table VII Table VII (, ) M M 2 M 3 M 4 M 5 S V W W W d V Hence, from the above table, the correspondng paradoxcal par (Z, F ) = (436, 6). Applyng step 8: The best paradoxcal par s (Z*, F*) = (436, 6) correspondng to the optmum soluton s X* = {x = ; x 2 = 45, x 5 =, x 2 =, x 23 = 25, x 34 = 2, x 35 = 2, x 4 =, x 5 = 2} and the paradoxcal range of flow s [F, F*] = [36, 6]. Thus, all the paradoxcal par are {(454, 37), (452, 38), (45, 3), (448, ), (446, ), (444, 2), (442, 3), (44, 4), (438, 5) and (436, 6). Ths research paper has really unveled the applcaton of the algorthms of paradoxcal pars n a lnear transportaton problem. Ths paper wll however go a long way to assst researchers who may wsh to embark on a smlar research topc. 7

10 Publshed by European Centre for Research Tranng and Development UK ( CONCLUSION In ths paper, we have been able to dscuss an effcent statstcal algorthm for computng paradox n a lnear transportaton problem f paradox does exst. The algorthm gves step by step development of the soluton procedure for fndng all the paradoxcal par, well understandng. The TORA statstcal software package was used to obtan the optmal soluton before mplementng the algorthm of paradoxcal pars. REFERENCES Adlakha V. and Kowalsk, K. (8): A quck suffcent soluton to the more-for-less paradox n a transportaton problem, Omega 26(4): Appa G.M. (73): The Transportaton problem and ts varants, Oper. Res. Q. 24:7-. Arora S.R., and Ahua A. (2): A paradox n a fxed charge transportaton problem. Indan J. pure appl. Math., 3(7): 8-822, July 2 prnted n Inda. Berge C. (62): Theory of Graphs and ts Applcatons (translated by Alson Dog), Methuen, London. Charnes A.; Cooper W.W. and Henderson (53): An Introducton to Lnear programmng (Wley, New Work). Charnes A. and Klngman D. (7): The more-for-less paradox n the dstrbuton model, Cachers du Centre Etudes de Recherche Operaonelle 3;-22. Dantzg, G.D. (63): Lnear Programmng and Extensve (Prnceton Unversty Press, NJ). Dantzg G.B. (5): Applcaton of the smplex method to a transportaton problem, n Actvty Analyss of Producton and Allocaton (T.C. Koopmans, ed.) Wley, New York, pp Deneko V.G., Klnz B. and Woegnger G.J. (23): Whch matrces re mmune aganst the transportaton paradox?, Dscrete Appled Mathematcs, Vol. 3, 23, pp Gupta, A. Khanna S and Pur, M.C. (3): A paradox n lnear fractonal transportaton problems wth mxed constrants, Optmzaton 27: Hadley G. (87): Lnear Programmng (Narosa Publshng House, New Delh). Htcock, F.L. (): The dstrbuton of a product from several resources to numerous localtes, J. Math. Phys. 2: Josh, V. D. and Gupta, N. (2): On a paradox n lnear plus fractonal transportaton problem, Mathematka 26(2): Jude, O. (2): Manual on Introducton to Operaton Research, Unpublshed. Klngman, D. and Russel, R. (75): Solvng constraned transportaton problems, Oper. Res. 23():-5. Manusr B., Debprasad A., and Atanu D. (22): The Algorthm of fndng all Paradoxcal Pars n a Lnear Transportaton Problem. Dscrete Math. Algorthm. Appl Downloaded from Storoy, S. (27): The transportaton paradox revsted, N-52 Bergen, Norway Szwarc W. (7): Naval Res. Logstcs Qly. 8(7) No.2, Szwarc W. (73): The transportaton paradox, Nav. Res. Logst. Q.8:

11 Publshed by European Centre for Research Tranng and Development UK ( Vshwas D.J., and Nlama G. (2): On a Paradox n Lnear Plus Lnear Plus Lnear Fractonal Transportaton Problem. MATHEMATIKA, 2, volume 26, Number 2, Department of Mathematcs, UTM.