Amr s Supply Chan Model by S. Ashtab a,, R.J. Caron b E. Selvarajah c a Department of Industral Manufacturng System Engneerng b Department of Mathematcs Statstcs c Odette School of Busness Unversty of Wndsor Wndsor, ON N9B P4 WMSR # 14-01
Amr s Supply Chan Model Sah Ashtab a,, Rchard J. Caron b, Esagnan Selvarajah c a Department of Industral Manufacturng Systems Engneerng, Unversty of Wndsor, Wndsor, ON, Canada, N9B P4 b Department of Mathematcs Statstcs, Unversty of Wndsor, Wndsor, ON, Canada, N9B P4 c Odette School of Busness, Unversty of Wndsor, Wndsor, ON, Canada, N9B P4 Abstract We show that Amr s model admts alternate optmal solutons that are nconsstent wth the model s phlosophy we provde a set of constrants that ensure consstency of all optmal solutons. Keywords: Supply chan network desgn, Faclty locaton, Mult-echelon, Mult-capactated. 1. Introducton Amr [1] presented a model Lagrangan-based soluton technque for mult-echelon, mult-capactated supply chan network desgn. We show that Amr s model admts alternate optmal solutons some of whch are nconsstent wth the model phlosophy. We provde a new set of constrants to ensure that the optmal soluton provded by the model properly lnks the decson to buld a warehouse to the decsons to allocate dem to that warehouse; that the fractonal varables are ndeed, fractons, as requred. 2. The Amr Model [1] The set of customer zones s ndexed by, the potental warehouse locatons by j, the potental plant locatons by k, the warehouse capacty levels by r the plant capacty levels by h. Let a be the dem from customer zone durng the plannng horzon, b r j be the capacty of a warehouse at locaton j bult at level r e h k be the capacty of a plant at locaton k bult at level h. Let Yjk r be the fracton of the total dem of a warehouse at locaton j wth capacty level r that s delvered from a plant at locaton k let X j be the fracton of the total dem of customer zone that s delvered from a warehouse at locaton j. A warehouse wth capacty level r s bult at locaton Correspondng Author. Tel: +1-519-992-6274. Emal address: ashtab@uwndsor.ca (Sah Ashtab) Preprnt submtted to European Journal of Operatonal Research August 18, 2014
j f only f Uj r = 1 Vk h = 1 f only f a plant wth capacty level h s bult at locaton k. The bnary varables U V determne number, locaton capacty of the facltes whle the real-valued varables X Y determne the flow of goods. The objectve functon s to mnmze the sum of the transportaton costs the fxed costs of establshng plants warehouses. Let C jk be the cost of shppng one unt of dem to a warehouse at locaton j from a plant at locaton k let C j be the cost of shppng one unt of dem to customer zone from a warehouse at locaton j. The n-bound out-bound transportaton costs are gven by T 1 (Y ) = r,j,k C jk b r j Y r jk T 2 (X) =,j C j a X j, respectvely. The fxed costs of establshng operatng the warehouses plants over the plannng horzon are F 1 (U) = j,r F r j U r j F 2 (V ) = k,h G h k V h k, respectvely, where Fj r s the fxed cost of openng operatng warehouse at locaton j wth capacty level r G h k s the fxed cost of openng operatng plant at locaton k wth capacty level h. The objectve functon s a summaton of the four cost elements s gven by The constrants Z 0 (Y, X, U, V ) = T 1 (Y ) + T 2 (X) + F 1 (U) + F 2 (V ). X j = 1,, (1) j ensure that the dem at each customer zone s covered by bult warehouses whle constrants a X j b r j Uj r, j, (2) r ensure that the capacty level at each warehouse s suffcent to meet out-bound shpments. That each warehouse each plant s assgned a sngle capacty level s ensured by 1, j, () r h U r j respectvely. The set of constrants a X j k,r V h k 1, k, (4) b r j Y jk r, j, (5) 2
ensure that the total out-bound shpment from a warehouse s not greater than the total n-bound shpment from plants to that warehouse. The nequaltes j,r b r j Y r jk h e h k V h k, k, (6) ensure that the total n-bound shpment from a bult plant to the warehouses s not greater than the chosen capacty level of that plant. The fnal sets of constrants are X j 0,, j, (7) Y r jk 0, k, j, r, (8) whch, together wth the unstated constrants that Uj r V k h are bnary, gve Amr s model whch s to AM0: mn U,Vbnary { Z 0(Y, X, U, V ) (1) (8) }. The fact that there are alternate solutons, becomes clear wth the change of varables W jk = r b r jy r jk. (9) We now replace T 1 (Y ) wth T 1 (W) = j,k C jk W jk gvng the new expresson for the objectve Z 1 (W, X, U, V ) = T 1 (W) + T 2 (X) + F 1 (U) + F 1 (V ). We next replace replace constrants (5) (6) wth a X j W jk, j, (10) k W jk j h respectvely. Model AM0 s equvalent to e h k V h k, k, (11) AM1: mn U,Vbnary { Z 1(W, X, U, V ) (1) (4), (7), (8), (10), (11) } Gven the W jk from a soluton to AM1, any set of non-negatve solutons Y r jk to (9) s a soluton to AM0.
Example. Consder a supply chan wth one plant havng a sngle avalable capacty level of e 1 1 = 6 000 one warehouse wth avalable capacty levels of b 1 1 = 1 000, b 2 1 = 000 b 1 = 5 000. The three customer zones have dems a 1 = 1 500, a 2 = 1 500 a = 1 000. To be feasble we must have V1 1 = 1, U1 1 = 0, U1 2 = 0, U1 = 1, X 11 = 1, X 21 = 1, X 1 = 1 4 000 W 11 6 000. An optmal soluton wll have W 11 = 4 000. Consequently, any non-negatve soluton to 4 000 = 1 000 Y11 1 + 000 Y11 2 + 5 000 Y11 wll be feasble optmal for AM0. So, whle Y11 1 = 4, Y11 2 = 0, Y11 = 0 s optmal for AM0, t has Y varables greater than one whch s nconsstent wth the model phlosophy that they are fractons. The soluton Y11 1 = 1, Y11 2 = 1, Y11 = 0 s optmal for AM0 but t s nconsstent wth U1 1 = 0 U1 2 = 0. The example shows that the model allows alternate optmal solutons that could lead to two dfferent undesrable stuatons; Y varables greater than one Y varables nconsstent wth the U varables. Both of these stuatons are corrected wth the addton to the model of the constrants Yjk r Uj r, j, r. (12) The modfed model, wthout the change of varables, s k AM2: mn{ Z 0 (Y, X, U, V ) (1) (8), (12) }. For our example, these new constrant force the unque, acceptable, optmal soluton wth Y11 1 = 0, Y11 2 = 0, Y11 = 1. To test the mpact of ths addtonal set of constrants we created, usng the method descrbed by Amr, sx nstances of an example wth 500 customer zones, 0 warehouses 20 plants (the largest nstance solved by Amr) solved the AM0 the AM2 models usng LINGO 14.0 x64 on a DELL server wth two 2500 MHz CPUs. The tmes taken to reach optmalty for the AM0 ranged from 56 to 185 seconds wth an average of 122. The tmes taken to reach optmalty for the AM2 ranged from 5 to 277 seconds wth an average of about 121. Ths lmted numercal evdence suggests that the ablty to solve the model, n reasonable tme, s not compromsed by the addtonal constrants. References [1] A. Amr, Desgnng a dstrbuton network n a supply chan system: Formulaton effcent soluton procedure, European Journal of Operatonal Research 171 (2006) 567 576. 4