Simulation-based Switching Algorithm for Inventory Management in a Multi-echelon Supply Chain

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1 Simulatio-based Switchig Algorithm for Ivetory Maagemet i a Multi-echelo Supply Chai GALINA MERKURYEVA (a) ad OLESYA VECHERINSKA (b) Departmet of Modellig ad Simulatio Riga Techical Uiversity 1 Kalku Street, LV-1658, Riga LATVIA Abstract: Cyclic or o-cyclic plaig policies are traditioally used to maage ivetories i a multiechelo supply chai. Evaluatio of the differece betwee performaces of cyclic ad o-cyclic plaig policies allows aalysig a efficiecy of a specific plaig policy at the product life cycle specific stage, ad provides a cotrol mechaism to switch from oe to aother policy. The paper is aimed is to preset simulatio-based switchig algorithm that provide estimatig the differece betwee the total costs of cyclic ad o-cyclic plaig policies with optimal parameters ad decisio makig about applicatio of a appropriate oe. Simulatio-based switchig algorithm cotais two phases: cost compariso aalysis i the first phase, ad advaced aalysis based o additioal costs of a cyclic solutio (ACCS) i the secod phase. A umerical example ad a prototype of a switchig module are described i the paper. Key-Words: o-cyclic ad cyclic plaig policies, multi-echelo supply chai, product life cycle, additioal costs of cyclic solutios, ad simulatio-based switchig. 1 Itroductio A set of differet repleishmet policies is adopted for utilizatio i plaig process i a supply chai withi sigle product life cycle. These policies are broadly classified as cyclic ad o-cyclic, ad the most suitable oe is used i a specific applicatio. I case of three-phase product life cycle, where phases are determied as itroductio, maturity ad ed-oflife phase, utilizatio of both cyclic ad o-cyclic policies makes it possible to improve the product life cycle maagemet. Determiig of ecessity of switchig betwee cyclic ad o-cyclic policies is the mai research obective withi itegratio of plaig policies ito the product life cycle. I the paper, these plaig policies are defied by two mai repleishmet strategies such as cotiuous review (ROP) ad repleishmet cycle (ROP) strategies. 2 Switchig Methodology To determie the ecessity of switchig from oe repleishmet system to aother oe simulatiobased two-phase switchig algorithm (Figure 1) is proposed. It allows estimatig the differece betwee the total costs of cyclic ad o-cyclic policies with optimal parameters ad makig a decisio about applicatio of a appropriate oe. Simulatio-based switchig algorithm cotais the followig phases: cost compariso of repleishmet alteratives based o testig statistical hypotheses i the first phase ad advaced ACCS aalysis based o a set of parameters or its half-width ratio value (HWR) i the secod phase. To test statistical hypotheses about the differece betwee total costs, the paired-t cofidece iterval method is used. It is supposed that observatios are idepedet, ormally distributed ad a umber of observatios received for two policies are equal. The choice of a aalysis i the secod phased depeds o the results of hypotheses testig Iput Data To determie parameters of repleishmet policies, aalytical calculus that take ito accouts the product fill rates costraits, is used. The followig parameters are defied: for ROP policy a reorder poit ad a order quatity per each echelo, ad for POR policy a repleishmet cycle ad order-up-to level per each echelo. For a cyclic policy the optimal parameters are itroduced. Algorithms of aalytical calculatios for a supply chai liear etwork are give i [2]. They assume that demad is dyamic ad stochastic ad defied by the ormal distributio; lead times are costat; demad at the last echelo is idepedet; safety stocks are allowed; capacities are ifiite; ivetory costs are liear; setup ad orderig cost are fixed; ad backloggig is ot allowed. ISSN: ISBN:

2 Fig.1: Two-phase switchig algorithm Performace measures of repleishmet policies, i.e. the total costs mea values ad correspodet ACCS values) are received from simulatio experimets with a supply chai model. To estimate performace measures expected values, multiple observatios are used, ad the steady-state behaviour of the model is aalysed. 2.2 Cost compariso aalysis Cost compariso for plaig alteratives is based o estimatio of the differece betwee two mea values of the total costs by usig the Paired-t cofidece iterval method [1]. It is aimed to discover if these two mea total costs values are sigificatly differet. I practice simulatio techiques i stochastic eviromets lead to variaces i output data. As the result, simulatio observatios are aalysed ot oly by ispectig poit estimates of the mea values, but also by usig their cofidece itervals. Utilizatio of the cofidece iterval makes simulated decisios more iformative. Two methods such as Welch ad Paired-t based o cofidece iterval approach are commoly used to compare two alteratives. Comparative aalysis of these methods leads to coclusio that Paired-t method is the most appropriate to the problem assumptio. At the same time, usig of commo radom umbers allows to reduce variaces of simulatio results. Two statistical hypotheses about the differece betwee the total costs of plaig alterative,, the ull hypothesis H 0 ad alterative hypothesis H 1 are formulated. H 0 : POR - ROP = 0 (or (POR-ROP) = 0 for paired-t otatio) (see, Fig.2a). H 1 : POR - ROP 0 (or (POR-ROP) 0 for paired-t otatio) (see, Fig.2b ad 2c). Fig.2: Possible positios of a cofidece iterval relative to zero ISSN: ISBN:

3 Here, POR- - average total costs of POR policy; ROP - average total costs of ROP policy; ad (POR- ROP) defies the differece betwee average total costs of policies POR ad ROP. Based o testig above formulated statistical hypothesis the followig coclusios C 0 ad correspodet decisios D 0 are made. C 0 : If the cofidece iterval icludes zero, hypothesis H 0 is failed to reect. There is o a sigificat differece betwee the mea costs for two policies. C 1 : If the cofidece iterval excludes zero, hypothesis H 0 is reected ad H 1 is assumed. There is a sigificat differece betwee the mea costs for two policies. C 0 meas that two mea costs values of repleishmet policies are equal, (POR-ROP) = 0. Hypothesis H 0 is true, ad the paired-t cofidece iterval icludes zero with a probability 1-, where is a sigificace level of te cofidece iterval. So, POR is ot sigificatly differet tha ROP with level of sigificace. This coclusio leads to a decisio D 0 : do t reect the POR strategy as the most suitable oe ad cotiue a further aalysis by ivestigatig the half-width ratio HWR of the cofidece iterval, compared to its critical value (Step 7-9).. If the cofidece iterval does ot iclude zero, the hypothesis H 0 is reected. So, POR is sigificatly differet tha the value of ROP at sigificace level. This coclusio leads to the followig decisios: D 11 : if POR - ROP < 0, i.e. the cofidece iterval is completely to the left of zero (Fig.2, positio (c)), the choose POR policy as the most suitable oe, or D 12 : if POR - ROP > 0, the cofidece iterval is located completely to the right of zero, (Fig.2, positio (b)), the cotiue ACCS aalysis compared to its critical value (Step 10-12). After hypotheses testig the decisio about switchig could be made i oe of three cases (see Figure 2, positio (c)).two others situatios eed more complete aalysis of performace measures for fial decisio makig. This aspect leads to the aalysis described below. 2.3 Switchig based o ACCS or HWR aalysis At the phase 2 the fial decisio is made, based o the aalysis of additioal costs of a cyclic solutio (ACCS) or half-width ratio value (HWR). The compariso of HWR value with its critical value, i.e. HWR cr or ACCS value with its critical value ACCS cr, is the basis for decisio makig by usig IF-THEN rules. Here, HWR cr defies the maximum allowed ratio of half-width value hw ad the differece betwee total costs meas, i.e. (POR-ROP). The half-width critical value is assiged by a applicatio expert ad is used as a threshold for makig a fial decisio. ACCS cr defies the maximum allowed ratio betwee two policies mea costs differece, i.e. POR - ROP, ad mea cost of o-cyclic policy. ACCS critical values are assiged by a applicatio expert. ACCS cr also could be refied withi simulatiobased aalysis ad used as a threshold for makig a fial decisio. 3 Switchig Algorithm Let s defie: N i a umber of replicatios for i-th strategy ; xi - the average total costs from -th (=1,2,3,...,) replicatio of the simulatio model for the i-th (i=1,2) alterative desig; x ( 1 - the differece betwee the -th observatios from the two populatios; x( 1 - sample mea of the differece betwee two populatios that estimates true mea total costs ( 12 ) ; s (1-2) - sample stadard deviatio for total costs that estimates true stadard deviatio ( 1 ; t 1, a / 2 - factor, determied from the Studet s t table for α/2 value ad degree of freedom -1; hw - half-width of the cofidece iterval. 3.1 Preparatory steps 1. Set parameters of simulatio experimets (the warm-up period ad a umber of replicatios). 2. Ru the model usig commo radom umbers (i order to icrease the accuracy of simulatio results). 3. Fix the output statistics for two policies. 4. IF assumptios for the paired-t cofidece iterval method are valid THEN go to Phase 1, ELSE choose aother compariso method. 3.2 Phase 1 algorithm 1. Let defie N 1 = N 2 = ; ISSN: ISBN:

4 2. Defie a ew radom variable: x ( 1 2) = x 1 - x 2 3. Calculate the sample mea of the differece betwee two populatios: 1 x(1 x( Compute the sample stadard deviatio s for the differece betwee total costs meas: 1 2 s( 1 ( x(1 x(1 ) Assig value; 6. Determie factor t 1, a / 2 from the Studet s t table; 7. Determie the half-width of the cofidece ( t 1, a / 2) * s(1 iterval: hw 8. Defie the Paired-t cofidece iterval for a α level of sigificace: x( 1 2) hw (1 x(1 hw 9. Check the statistical hypothesis H IF H 0 is failed to reect THEN do Steps 13 16, ELSE go to Step IF H 0 is reected AND (POR-ROP) < 0 THEN choose POR strategy, ELSE go to Step IF H 0 is reected AND (POR-ROP) > 0 THEN do Steps Phase 2 algorithm 13. Set the HWR cr value. hw 14. Calculate HWR *100%. ( PORROP) 15. IF HWR HWR cr THEN choose POR strategy ELSE go to Step 16; 16. IF HWR > HWR cr THEN cotiue aalysis i order to decrease HWR value or choose ROP strategy. 17. Set ACCS cr value. POR ROP 18. Calculate ACCS *100%. ROP 19. IF ACCS ACCS cr THEN choose POR strategy ELSE Step 20; 20. IF ACCS > ACCS cr THEN choose ROP strategy. 4 Software Prototype ad Numerical Examples The switchig software prototype (see, Fig.3) is developed usig ProModel simulatio system at the preparatory steps, ad MS Excel ad VBA itegratio possibilities to execute simulatio-based switchig algorithm at Phases 1 ad 2. Fig.3: The mai widow of the prototype The prototype cotais the followig three modules: Modellig ad Simulatio module, that provides data iput to ProModel simulatio model, a model ru ad data export to other to prototype modules; Switchig module that executes the switchig algorithms; Advaced aalysis o a set of parameters to perform a sesitivity aalysis, what-if aalysis ad aalysis of gap evolutio i time. Examples of the prototype widows that describe coclusios ad decisios made at the phase 1 ad phase 2 are preseted i Fig. 4, 5. Fig.4: Example of the prototype widow at phase 1 Fig.5: Example of the prototype widow at phase 2 Below a umerical example is give for, 3-echelo liear supply chai. Its simulatio model has bee ISSN: ISBN:

5 created i ProModel software. Simulatio model s modules that model POR ad ROP repleishmet strategies are ru i parallel. The average costs statistics of both policies are collected ad aalysed based o the two phases of the algorithm. The mai steps of the calculatios are preseted below: 1. Replicatio umbers are equal: N 1 = N 2 = =15; 2. Sum of differece betwee two policies: x(1 2) ,843; 3. Sample mea of the differece betwee two policies: ,843 x( 1 x(1 2) 80543,86; Sample stadard deviatio s for the differece betwee total costs meas: 1 2 s( 1 ( x(1 2) x(1 2) ) , ; =0.1; 6. t 1, / 2 t14, ; 7. Half-width of the cofidece iterval: t1, / 2 t14,0.05 hw * s(1 2) * , Paired-t cofidece iterval for a 0,1 level of sigificace: ( Check the statistical hypothesis H 0 : ( PORROP) 0, i.e. the cofidece iterval is completely to the right of zero, the cotiue ACCS aalysis compared with ACCS critical value. 10. ACCS cr =10%; 11. POR ROP ACCS * 100% 29,12% ROP 12. ACCS > ACCS cr (29,12% > 10%)THEN ROP strategy is preferable. Estimatio of the ACCS parameter based o the three-echelo liear model provide more useful for aalysis values. Coefficiet of demad variatio (CODVAR) ifluece to ACCS value is show i Fig.6. ACCS parameter is defied based o the average maximum values of total costs. The average value of ACCS varies betwee ad The cofidece itervals have also big dispersio. But i geeral, ACCS parameter icreases if the CODVAR value icreases. Fig.6: ACCS as a fuctio of CODVAR Two variats of coclusio based o the hypotheses testig are possible aalyzig the ifluece of CODVAR that is from 0.1 till 1.0; i.e. o sigificat differece betwee two plaig policies ad average ROP policy s costs are smaller the average costs of POR policy. That is why HWR ad ACCS aalysis eed to be doe, accordigly. Obviously, big dispersio of the results eeds to refie simulatios experimets. 5 Sesitivity Aalysis Sesitivity aalysis for two factors [3], i.e. demad variatio ad lead time variatio, is performed. Sesitivity aalysis procedure itegrates differet techiques, e.g. oe-at-time experimetal desig ad estimatio of factors ifluece usig theory of probability. The omial values of both factors are defied by logormally distributed demad (10000; 2000) i items ad ormally distributed lead time (4; 0,5) i days. The utilizatio of oe-at-time (OAT) desig allows to calculate sesitivity idexes (SI) of both parameters with the aim to compare the ifluece of them. SI Dmax Dmi, Dmax where D max u D mi defie performace measure s (i.e. costs) values for maximal ad miimal factor values. The ifluece of both factors is ot sigificatly differet, i.e SI D oat = 0,48, ad SI LT oat = 0,66. Based o the SI values iteractive ifluece of these factors is aalysed with the help of Lati Hypercube Samplig (LHS) method. The values of factors are chaged uiformly i the rage of (0, 10) for demad variatio ad (0, 2) for lead time variatio with the step 0,5 ad 0,1, correspodigly. Samples of factors pairs are determied (see, Fig.7) ad the performace measures values are received ad aalysed after 100 rus. ISSN: ISBN:

6 Here, the probability of decisio chage icreases with the icrease of the demad variatio ad decrease of the lead time variatio. Fig.7: Geerated pairs of factors values The ifluece of factors iteractio is ivestigated while chagig the decisio about the repleishmet alterative. The positive cofidece iterval (301958; ) is received for factors omial values. The chage of the iterval positio aroud zero poit leads to possibility of factors ifluece o the decisio makig ad o the system i geeral. The decisio was chaged 27 times from 100 experimets. The probability aalysis of the ifluece of factors o the decisio chage is show graphically o the Figures 8 ad 9. Fig.8: Decisio chagig probability based o the demad variatio Fig.9: Decisio chagig probability based o lead time variatio 6 Coclusio This paper presets simulatio-based switchig algorithm that is based o evaluatio of the differece betwee performaces of cyclic ad ocyclic plaig policies ad provides a cotrol mechaism to switch from oe to aother policy. Simulatio-based switchig algorithm icludes the followig steps ad phases: Preparatory steps - to iput data, set parameters of simulatio experimets, ru a model; Phase 1. Cost compariso to test statistical hypotheses o the differece betwee the total costs meas of a cyclic ad o-cyclic policy, the paired-t cofidece iterval method is used. Phase 2. Advaced aalysis based o ACCS or HWR (half-width ratio) estimatios depedet o the results of hypotheses testig. A software prototype of a switchig algorithm is developed ad tested. Sesitivity aalysis of the demad variatio ad lead time variatio is performed shows the tedecy of icreasig of the probability of decisio chage whe the demad variatio icreases ad lead time variatio decreases. Preseted algorithm could be also applied for compariso of alterative decisio i stochastic eviromet. Refereces: [1] Harrell, C., Ghosh, B.K. ad Bowde, R.O., Simulatio Usig Promodel, 2d ed, McGraw- Hill Higher Educatio, [2] Merkuryeva, G. ad Vecheriska, O., Simulatio-Based Approach for Compariso of (s, Q) ad (R, S) Repleishmet Policies Utilizatio Efficiecy i Multi-echelo Supply Chais, Proceedigs of UKSIM Teth Iteratioal Coferece o Computer Modellig ad Simulatio, Cambridge, UK, April 2008, pp [3] Merkuryeva, G., Timmermas, S. ad Vecheriska, O., Evaluatig the optimality gap betwee cyclic ad o-cyclic plaig policies i supply chais, Proceedigs of Iteratioal Coferece o Productio Egieerig , Wroclaw, Polad, December 2006, pp ISSN: ISBN: