Cost optimization of supply chain with multimodal transport

Proceeings of the Feerate Conference on Computer Science an Information Systems pp. 8 ISB 978-83-6080-5-4 Cost optimization of supply chain with multimoal transport Paweł Sitek Kielce University of Technology Al. 000-lecia PP 7, 25-34 Kielce, Polan Institute of Management Control Systems e- mail:sitek@tu.kielce.pl Jarosław Wikarek Kielce University of Technology Al. 000-lecia PP 7, 25-34 Kielce, Polan Institute of Management Control Systems e- mail:j.wikarek@tu.kielce.pl Abstract The article presents the problem of supply chain optimization from the perspective of a multimoal logistics provier an inclues a mathematical moel of multilevel cost optimization in the form of MIP (Mixe Integer inear Programming). The costs of prouction, transport, istribution an environmental protection were aopte as optimization criteria. Timing, volume, capacity an moe of transport were also taken into account. The moel was implemente in the IG ver.2 package. The implementation etails, the basics of IG as well as the results of the numerical tests are presente an iscusse. The numerical experiments were carrie out using sample ata to show the possibilities of practical ecision support an optimization of the supply chain. T I. ITRDUCTI H issue of the supply chain is the area of science an practice that has been strongly eveloping since the 80 of the last century. umerous efinitions escribe the term, an a supply chain reference moel has been esigne [, 2]. The supply chain is commonly seen as a collection of various types of companies (raw materials, prouction, trae, logistics, etc.) working together to improve the flow of proucts, information an finance. As the wors in the term inicate, the supply chain is a combination of its iniviual links in the process of supplying proucts (material an services) to the market. Huang et al. [3] stuie the share information of supply chain prouction. They consiere an propose four classification criteria: supply chain structure, ecision level, moeling approach an share information. Supply chain structure: It efines the way various organizations within the supply chain are arrange an relate to each other. The supply chain structure falls into four main types [4]: Convergent: each noe in the chain has at least one successor an several preecessors. Divergent: each noe has at least one preecessor an several successors. Conjoine: which is a combination of each convergent chain an one ivergent chain. etwork: which cannot be classifie as convergent, ivergent or conjoine, an is more complex than the three previous types. Decision level: Three ecision levels may be istinguishe in terms of the ecision to be mae: strategic, tactical an operational, with their corresponing perio, i.e., long-term, mi-term an short-term. Supply chain analytical moeling approach: This approach consists in the type of representation, in this case, mathematical relationships, an the aspects to be consiere in the supply chain. Most literature escribes an iscusses the linear programming-base moeling approach, mixe integer linear programming moels in particular [5,6,7,8,9]. Share information: This consists in the information share between each network noe etermine by the moel, which enables prouction, istribution an transport planning epenent on the purpose. The share information process is vital for effective supply chain prouction, istribution an transport planning. In terms of centralize planning, the information flows from each noe of the network where the ecisions are mae. Share information inclues the following groups of parameters: resources, inventory, prouction, transport, eman, etc. Minimization of total costs is the main purpose of the moels presente in the literature [9,0,,2,3], while maximization of revenues or sales is consiere to a smaller scale [7,4]. This paper eals with a mathematical moel for supply chain costs optimization in the form of MIP (Mixe Integer inear Programming Problem) [5] from the perspective of logistic provier. In this moel, share information process inclues such parameters as resources, inventory, prouction, transport, eman etc. In previous works, the authors stuie moels an algorithms for combinatorial optimization of cost in a supply chain. This paper focuses on the multimoal transport in the supply chain an its implementation aspects. It shoul be emphasize that the presente moel can be the basis for the ecision support in the supply chain management. ptimization results of this moel relate to two types of ecision. These are short-term ecisions, about how to supply at minimum cost (operational level), an long-term ecisions on the capacity of iniviual istributors or prouction capacity of iniviual proucers (tactical an strategic level). The article also presents various moels of outsource logistics management. The rest of the paper is organize as follows: Section II escribes the problem of SCM (Supply Chain Management) from the logistic provier perspective. Section III analyses the state of the art in this omain. Section IV gives the problem 978-83-6080-5-4/$25.00 c 202 I

2 PRCDIGS F TH FDCSIS. WRCŁAW, 202 statement an provies an optimization moel for the consiere supply chain with multimoal transport. The implementation aspects of the optimization moel are explaine briefly in Section V. Computational examples an tests of the implemente moel are presente in Section VI. The iscussion on possible extensions of the propose approach an conclusions is inclue in Section VII. II. SUPPY CHAI MAAGMT The aim of supply chain management (SCM) is to increase sales, reuce costs an take full avantage of business assets by improving interaction an communication between all the actors forming the supply chain. The supply chain management is a ecision process that not only integrates all of its participants but also helps to coorinate the basic flows: proucts/services, information an funs. Changes in the global economy an the increasing globalization lea to the wiesprea use of IT tools, which enables continuous, real-time communication between the supply chain links. ne of the objectives is to optimize logistics an entrust it to specialize companies. This irection contribute to the evelopment of logistics outsource operators known as 3P, 4P or 5P [6]. The term 3P (Thir Party ogistics) refers to the use of external companies an organizations to carry out logistic functions that can involve the entire logistics process or its selecte features. The company offers an provies 3P services using its own means of transport, warehouses, equipment an other necessary resources, an acts as a "thir party" between a proucer an a customer. The resulting moel with the supply chain logistics services outsource to specialize 3P companies is shown in Figure. This kin of cooperation is frequently referre to as the logistics alliance. 4P (Fourth-Party ogistics) is a certain evolution of the 3P concept to provie greater flexibility an aaptation to the nees of the client. 4P companies an organizations operate primarily by managing the information flow within the entire supply chain. Unlike the 3P, responsible for only a selecte segment, a 4P coorinates logistics processes along the whole length of the chain (from raw materials to en-buyers). The 4P moel enables the 3P operator to become a coorinator an integrator of the flows, not just an operator of physical isplacement of goos. Very often, its subcontractors are 3P or even 2P (Secon Party ogistics) operators, i.e., transport companies an warehouses. The company that uses the services of a 4P provier is in contact with only one operator who manages an integrates all types of resources an oversees the entire functionality across the supply chain. 4P proviers, having a complete picture of the supply chain an large IT capabilities may offer optimization an ecision support avisory services [7]. Further evelopment of logistics outsourcing resulte in the creation of a 5P moel (Fifth Party ogistics) - proviers of integrate logistics services that can esign an implement flexible an networke supply chains to cater to the nees of all participants (manufacturers, suppliers, carriers an en users). Fig. The chart of the supply chain with logistics services outsource to a logistic provier. III. STAT F ART AD MTIVATI Simultaneously consiering the supply chain prouction, istribution processes in istribution centers an transportplanning problems greatly avances the efficiency of all processes. The literature in the fiel is vast, so an extensive review of existing research on the topic is extremely helpful in moeling an research. Comprehensive surveys on these problems an their generalizations were publishe, for example in [3]. In our approach, we are consiering a case of the supply chain where: the share information process in the supply chain consists of resources (capacity, versatility, costs), inventory (capacity, versatility, costs, time), prouction

PAW SITK: CST PTIMIZATI F SUPPY CHAI WITH MUTIMDA TRASPRT 3 (capacity, versatility, costs), prouct (volume), transport (cost, moe, time), eman, etc.(fig 2, Fig3); the transport is multimoal. (several moes of transport, a limite number of means of transport for each moe); the environmental aspects of use of transport moes; ifferent proucts are combine in one batch of transport; the cost of supplies is presente in the form of a function (in this approach linear function of fixe an variable costs); ifferent ecision levels are consiere simultaneously. Fig. 2 The part of the supply chain network with marke inices of iniviual participants (elements). Dashe line marks one of the possible routes of elivery. ecision variables introuce at the level of implementation, optimization moel in IG language, etc. The aim of this paper is to esign an implement the moel that can become the basis for making optimal ecisions at ifferent levels of supply chain management. The propose solution will also enable a comprehensive examination of the impact on cost an performance of various parameters of the share information. IV. PRBM STATMT A. BACKGRUD A key step in many ecision-making an esign processes is the optimization phase, which itself contains several stages. The purpose of the optimization process in our approach is to help etermine realistic an practical outcomes of management ecision-making an esign processes in the supply chain. There are two basic ways to optimize the problem, either the qualitative approach or the quantitative approach. Using only a qualitative approach, the problem optimization, when making a ecision, relies on personal jugment or experience acquire in ealings with similar problems in the past. In a few cases this approach may be aequate; however, there are many situations where a quantitative approach to the problem provies a betterstructure an logical path through the ecision-making process. We propose the quantitative approach for the cost optimization supply chain network moel (Fig. 2) esigne from a perspective of 3P/4P/5P proviers. Fig. 3 The selecte path of the supply chain along with the parameters that escribe the iniviual elements an its epenencies (share information). Decision levels in supply chains are mainly classifie by the extent or effect of the ecision to be mae in terms of time. For instance, at the strategic level, the ecisions mae in relation to selecting prouction, storage an istribution locations, etc shoul be ientifie. At the tactical level however, the aspects such as prouction an istribution planning, assigning prouction an transport capacities, inventories an managing safety inventories are ientifie. At the operational level, replenishment an elivery operations are classifie [3]. Most of the reviewe works focus on the tactical ecision level [6,7,8,0,,2,8,9]. nly few works eal with the problems taken together for the ifferent ecision levels [5,3]. Therefore, the motivation behin this work is to suggest an approach to multilevel supply chain cost optimization with multimoal transport from the perspective of a logistics provier, an to propose an optimization moel in the form of integer programming problem [20] which facilitates its solution using specialize software available on the market (IG, CPX). Many aspects of the propose moel implementation are feature here, incluing aitional B. PRBM FRMUATI The mathematical optimization moel was formulate as an integer linear programming problem [20] with the minimization of costs () uner constraints (2).. (23). Inices, parameters an ecision variables in the moel together with their escriptions are provie in Table. The propose optimization moel is a cost moel that takes into account three other types of parameters, i.e., the spatial parameters (area/volume occupie by the prouct, istributor capacity an capacity of transport unit), time (uration of elivery an service by istributor, etc.) an transport moe. The position of each parameter against the subsequent links of the supply chain is shown in Fig.3. Symbol Inices k j i s M Table I. Summary inices, parameters an ecision variables of the mathematical optimization moel Description prouct type (k=..) elivery point/customer/city (j=..m) manufacturer/factory (i=..) istributor /istribution center (s=..) moe of transport (=..) number of manufacturers/factories number of elivery points/customers number of istributors number of prouct types

4 PRCDIGS F TH FDCSIS. WRCŁAW, 202 number of moe of transport Input parameters F s the fixe cost of istributor/istribution center s (s=..) P k the area/volume occupie by prouct k (k=..) V s istributor s maximum capacity/volume (s=..) W i,k prouction capacity at factory i for prouct k (i=..) (k=..) C i,k the cost of prouct k at factory i (i=..) (k=..) R s,k if istributor s (s=..) can eliver prouct k (k=..) then R sk=, otherwise R sk=0 Tp s,k the time neee for istributor s (s=..) to prepare the shipment of prouct k (k=..) Tc j,k the cut-off time of elivery to the elivery point/customer j (j=..m) of prouct k (k=..) Z j,k customer eman/orer j (j=..m) for prouct k (k=. ) Zt the number of transport units using moe of transport (=..) Pt the capacity of transport unit using moe of transport (=..) Tf i,s, the time of elivery from manufacturer i to istributor s using moe of transport (i=..) (s=..) (=..) the variable cost of elivery of prouct k from manufacturer i to K i,s,k, istributor s using moe of transport (=..) (i=..) (s=..) (k=..) if manufacturer i can eliver to istributor s using moe of R i,s, transport then R is=, otherwise R is=0 (=..) (s=..) (i=..) A i,s, the fixe cost of elivery from manufacturer i to istributor s using moe of transport (=..) (i=..) (s=..) Koa s,j, the total cost of elivery from istributor s to customer j using moe of transport (=..) (s=..) (j=..m) Tm s,j, the time of elivery from istributor s to customer j using moe of transport (=..) (s=..) (j=..m) the variable cost of elivery of prouct k from istributor s to K2 s,j,k, customer j using moe of transport (=..) (s=..) (k=..) (j=..m) R2 sj if istributor s can eliver to customer j using moe of transport then R2 sj=, otherwise R2 s,j,=0 (=..) (s=..) (j=..m) G s,j, the fixe cost of elivery from istributor s to customer j using moe of transport (s=..) (j=..m) (k=..) Kog s,j, the total cost of elivery from istributor s to customer j using moe of transport (=..) (s=..) (j=.m) (k=..) the environmental cost of using moe of transport (=..) Decision variables X i,s,k, elivery quantity of prouct k from manufacturer i to istributor s using moe of transport Xa i,s, if elivery is from manufacturer i to istributor s using moe of transport then Xa i,s,=, otherwise Xa i,s,=0 Xb i,s, the number of courses from manufacturer i to istributor s using moe of transport Y s,j,k, elivery quantity of prouct k from istributor s to customer j using moe of transport Ya s,j, if elivery is from istributor s to customer j using moe of transport then Ya s,j, =, otherwise Ya s,j, =0 Yb s,j, the number of courses from istributor s to customer j using moe of transport Tc s if istributor s participates in eliveries, then Tc s=, otherwise Tc s=0 CW Arbitrarily large constant C. PTIMIZATI CRITRIA The objective function () efines the aggregate costs of the entire chain an consists of five elements. The first is the fixe costs associate with the operation of the istributor involve in the elivery (e.g. istribution center, warehouse, etc.). The secon part sets out the environmental costs of using various means of transport. They are epenent on the number of runs of the means of transport, an on the environmental levy, which may epen on the use of fossil fuels an carbon-ioxie emissions. The thir component etermines the cost of supply from the manufacturer to the istributor. Another component is responsible for the costs of supply from the istributor to the en user (the store, the iniviual client, etc.). The last component of the objective function etermines the cost of manufacturing the prouct by the given manufacturer. F * Tc s s Koa ( Xb Kog i,s, i s s j l s Yb i,s, i s s j M s,j, M (C D. CSTRAITS j,s, ) * X ik i k s i,s,k, The moel was evelope subject to constraints (2).. (23). Constraint (2) specifies that all eliveries of prouct k prouce by the manufacturer i an elivere to all istributors s using moe of transport o not excee the manufacturer s prouction capacity. ( Xi,s,k,) Wi, k s for i..,k.. Constraint (3) covers all customer j emans for prouct k (Z j,k ) through the implementation of supply by istributors s (the values of ecision variables Y i,s,k, ). The constraint was esigne to take into account the specificities of the istributors resulting from environmental or technological constraints (i.e., whether the istributor s can eliver the prouct k or not ). ( Ys,j,k, * Rs,k ) Zj, k s for j..m,k.. The balance of each istributor s correspons to constraint (4). X i i,s,k, M Y s,j,k, j for s.., k.. The elivery, epenent on technical capabilities in the moel represente by istributor s volume/capacity- is efine by constraint (5). ( Pk * Xi,s,k, ) Tcs * Vs for s.. k i Constraint (6) ensures the fulfillment of the terms of elivery time. Xa i,s, *Tf i,s,a Xa i,s, *Tp Ya s,j, *Tm s, j, Tc for i.., s.., j..m, k..,.. s,k Constraints (7a), (7b), (8) guarantee eliveries with available transport taken into account. Ri,s, * Xbi,s, * Pt Xi,s,k, *Pk (7a) for i.., s.., k..,.. R2 s, j, * Yb s, j, * Pt Y s,j,k, * P for s.., j..m, k..,.. i s Xb M i,s, j s Yb j,s, Zt k for.. j, k ) () (2) (3) (4) (5) (6) (7b) Constraints (9), (0), () set values of ecision variables base on binary variables Tc s, Xa i,s,, Ya s,j, respectively. Xb i i,s, CW *Tc s for s.. (8) (9)

PAW SITK: CST PTIMIZATI F SUPPY CHAI WITH MUTIMDA TRASPRT 5 Xb i,s, CW *Xai,s, for i..,s..,.. (0) Ybs,j, CW*Yas,j, for s.., j..m,.. () Depenencies (2) an (3) represent the relationship by which total costs are calculate. In general, these may be any linear functions. Koa Kog i,s, s,j, Ai,s, * Xbi,s, K i,s,k, *Xi,s,k, k (2) for i..,s..,.. Gs,j, * Ybj,s, K2s,j,k, * Ys,j,k, k (3) for s.., j..m,.. The remaining constraints (4)..(23) arise from the nature of the moel. X i,s,k, 0 for i..,s..,k..0,.. (4) Xb i,s, 0 for i.., s..,.., (5) Yb s,j, 0 for s.., j 2..M,.., (6) X i,s,k, C for i..,s.., k..0,.., (7) Xb i,s, C for i.., s..,.. (8) Y s,j,k, C for s.., j..m, k..0,.. (9) Yb s, j, C for s.., j..m,.., (20) Xa s,j, 0, for i.., s..,.., (2) Ya s,j, 0, for s.., j..m,.., (22) Tc s 0, for s.. (23) V. MTHD DVPD The moel was implemente in "IG" by ID Systems [2]. "IG" ptimization Moeling Software is a powerful tool for builing an solving mathematical optimization moels. "IG" package provies the language to buil optimization moels an the eitor program incluing all the necessary features an built-in "solvers" in a single integrate environment. "IG" is esigne to moel an solve linear, nonlinear, quaratic, integer an stochastic optimization problems. Moel implementation is possible in two basic ways. The first way is to enter the moel into the "IG" eitor in the explicit form, that is, a full function of the objective with all the constraints, parameters, etc. Although it is an intuitive approach consistent with the stanar form of linear programming [20], it is not very useful in practice. This is ue to the size of moels implemente in practice. For the example presente in chapter Computational examples, the number of ecision variables an constraints was 45 an 862, respectively. The other way is to use the "IG" language of mathematical moeling, an integral part of the "IG" package, whose basic syntax elements are shown in Tab.. For the real examples with sizes exceeing several ecision variables, the construction an implementation of the moel is only possible using the moeling language (Tab.2, Fig. 7). The basic elements of mathematical moeling language syntax of "IG" are presente in Tab. 2. Table II. The basic syntax of "IG" mathematical moeling language Mathematical nomenclature IG syntax Minimum MI = Z jkt @sum(rdr (j,k,t)) j=..m for each customer (j) in the @FR(CUSTMRS (j)) set of customers * = = X integer @gin(x) X {0,} @bin(x) oa input parameters p from the p=@file(ane.lt) file ane.lt The moel can be save in a text file using any text eitor an with a stanar extension *. lng an *. lt ata file. The structure of the moel is compose of sections. The main section is the MD section, which begins with the wor MD: an ens with the wor D. ther sections may be integrate in this section. The most important sections, highlighte by the relevant keywors are: section STS (ST: DSTS) an DAT (DAT: enate). In the STS section one can efine types of simple or complex objects, an their mutual relationships. In the implemente moel, simple objects are exemplifie by types such as proucts, factories, etc.; complex objects: prouction, istribution, etc. In this section, the parameters an variables of the moel are assigne to particular types. DATA section allows initiating or assigning values to iniviual parameters of the moel. There are two methos to o it in the "IG" package. ither place the numerical ata irectly in the section or make references to the place where those ata files are inclue. This metho of moel construction ensures the separation of ata from the relevant moel, which is very important because the change in ata values or even their size oes not require any changes in the objective function or constraints. nly the moel implemente in the implicit form has such a feature. VI. CMPUTATIA XAMPS The cost optimization moel ()..(23) was implemente in the IG environment. Fig. 7 shows the implicit moel. ptimization was performe for six examples: P,P2,P3,P4,P5 an P6. All the cases relate to the supply chain with two manufacturers (i=..2), three istributors (s=..3), four recipients (j=..4), four moe of transport (=..4) an five types of proucts (k=..5). The examples iffer in capacity available to the istributors (V s ), number of transport units using moe of transport (Zt ) an the environmental cost of using moe of transport ( ). The numeric ata for all the moel parameters from Tab.I are presente in Appenix A. In the examples, istributors capacities are: V =V 2 =V 3 =050 (P); V =V 2 =V 3 =200 (P2),. V =V 2 =V 3 =2000 (P3), V =V 2 =V 3 =500 (P4,P5,P6). Parameters Zt are: Zt = Zt 2 = Zt 3 = 0 for all examples. The environmental cost of using moe of transport are: =0, 2 =30, 3 =400 (P,P2,P3), =0, 2 =30, 3 =00 (P4), =0, 2 =00, 3 =500 (P5), =0,

6 PRCDIGS F TH FDCSIS. WRCŁAW, 202 2 =60, 3 =800. ther etails are the same for all six examples. ptimization follows the implementation of the moel in the IG mathematical moeling language (Fig. 4). Moel: Sets: proucts /..@file(size.lt)/:p; factories /..@file(size.lt)/; customers /..@file(size.lt)/; istributors /..@file(size.lt)/:f,v,vx,t; moe /..@file(size.lt)/:pt,zt,o,x; orers (customers,proucts):z,tc; prouction (factories,proucts):c,w,wx; locations (istributors,proucts):r,tp; route_ (factories,istributors,moe):a,r,tf,xb,xa,ko_; route_2(istributors,customers,moe):g,r2,tm,yb,ya,ko_2; elivery_(factories, istributors,proucts,moe):x,k; elivery_2(istributors,customers,proucts,moe): Y,k2; elivery_3(factories, istributors, proucts); elivery_4(istributors, customers, proucts); nsets Data: p =@file(ane.lt);f =@file(ane.lt); r =@file(ane.lt); v =@file(ane.lt);pt =@file(ane.lt); tp =@file(ane.lt); zt =@file(ane.lt);z =@file(ane.lt); a =@file(ane.lt); tc =@file(ane.lt);c =@file(ane.lt); w =@file(ane.lt); r =@file(ane.lt);... ndata! bjective function; Min= @sum(punkty(s):f(s)*t(s))+ @sum(ostawa_(i,s,k,):ko_(i,s,))+ @sum(ostawa_2(s,j,k,):ko_2(s,j,))+ @sum(wytwarzanie(i,k):c(i,k)*(@sum(punkty(s): @sum(sroki():x(i,s,k,)))))+ @sum(sroki():o()*(@sum(fabryki(i): @sum(punkty(s):xb(i,s,)))+@sum(punkty(s): @sum(miasta(j):yb(s,j,))))); @for(trasy_(i,s,): ko_(i,s,)=a(i,s,)*xb(i,s,)+ @sum(proukty(k):k(i,s,k,)*x(i,s,k,))); @for(trasy_2(s,j,): ko_2(s,j,)=g(s,j,)*yb(s,j,)+ @sum(proukty(k):k2(s,j,k,)*y(s,j,k,)));! Constraint(); @for(prouction(i,k):@sum(istributors(s): @sum(moe(): X(i,s,k,))) <=w(i,k) ;! calculation of the auxiliary variable Wx; @sum(istributors (s):@sum(moe():x(i,s,k,)))=wx(i,k););! Constraint (2); @for(orers(j,k):@sum(istributors (s):@sum(moe():r(s,k)*y(s,j,k,)))>=z(j,k));...! binary Ts; @for(istributors (s):bin(t(s))); n Fig. 4 Part of the file scm.lng (the supply chain cost optimization moel in IG). ptimization results for all ecision variables are shown in Appenix B (Tab. 4) for P, P2, P3 an Fig. (only for P2) with the parameters of the process of searching for the optimal solution: the number of iterations, the optimization algorithm use (Branch-an-Boun) [20], the number of ecision variables in the integer constraints, etc. The optimization process involves fining the global solution for the specific ata Appenix A (Tab.III), which in this case means the lowest cost of satisfying customer nees through the supply chain an amounts to Fc opt =39445 for P, Fc opt =37825 for P2, Fc opt =37795 for P3, Fc opt =3775 for P4,. Fc opt =38585 for P5 an Fc opt =38505 for P6. Transportation networks iagrams showing the number of hauls (no number means one) corresponing to the optimal solutions for P, P2, P3, P4, P5, P6 are shown sequentially in Fig. 5, Fig. 6, Fig. 7, Fig. 8, Fig. 9, Fig. 0. At the same time, the specific values of ecision variables that minimize the cost are etermine (Tab.4 only for P,P2,P3). These values represent, among other things, the volume of supplies from the manufacturer to the istributor of selecte proucts using moe of transport (X isk ) an the supply of proucts from specific istributors to selecte customers/recipients (Y sjk ). Fig. 5 Transport network of multimoal optimal solution (Fc opt =39445) for P Fig. 7 Transport network of multimoal optimal solution (Fc opt =37795) for P3 Fig. 9 Transport network of multimoal optimal solution (Fc opt =38585) for P5 Fig. 6 Transport network of multimoal optimal solution (Fc opt =37825) for P2 Fig. 8 Transport network of multimoal optimal solution (Fc opt =3775) for P4 Fig. 0 Transport network of multimoal optimal solution (Fc opt =38505) for P6 Base on these variables, one can make a ecision at the current operating level. The values of ecision variables Yb s,j,, Xb i,s, etermine the number of runs using transport moe. Base on these variables one can make a ecision from the tactical level, which inclues the moe of transport an the nee for ifferent means of transport. Another way to use the implemente moel is to etermine the effect of the change in the moel parameters on the cost. ne can analyze in etail the sensitivity of solutions epening on the parameters Ko, A, G, C, T, V, Zt etc. The article focuse on the effect of parameter V,Zt,.

PAW SITK: CST PTIMIZATI F SUPPY CHAI WITH MUTIMDA TRASPRT 7 umerous analyses of that kin can be conucte. For these stuies an especially long-term ecision support, the optimization moel was extene at the implementation stage. Auxiliary variables were introuce at implementation stage Vx s (the value correspons to the istributor s uptake capacity), Wx ik (prouction capacity utilization rates for manufacturer i of prouct k) an Dx (the cumulative number of courses given moe). The analysis of the ecision variables values Vx s,wx ik an Dx Appenix B (Tab.5) has an impact on strategic ecision making level of prouction capacity or istributor location, capacity etc. To estimate the influence of parameters (V,Zt, ) on the solution, aitional experiments were carrie out. The effect of selecte parameters on the solution is presente on charts (Fig.2, Fig.3, Fig.4). Fig. IG results winow, for P2. Fig. 3 The impact of parameter for the solution. VII. CCUSI Fig. 2 The impact of parameter V for the solution Fig. 4 The impact of parameter Zt for the solution. The paper presents a moel of optimizing supply chain costs. Creating the moel in the form of a MIP problem unoubtely facilitates its solution using mathematical programming tools available in "IG" package [2] or "CPX" [22] an others. f course, the moel shoul be implemente in one, selecte environment package. Implementation of the moel in the "IG" package an the computational experiments were presente. The approach from the perspective of an optimizing logistics provier that has access to all ata an all participants in the ownstream chain is very interesting. After the implementation of the language from the mathematical moeling package "IG", a number of computational experiments were conucte. Six of them in the form of examples P.. P6 were escribe in the article. Base on the experimental results, analysis an previous experience, the authors can state that the propose moel an its implementation ensure a very large range of applications. First, they allow fining the istribution flows (ecision variables) for the moele supply chain, which minimize the global cost satisfying the customer emans. Secon, they offer numerous possibilities for ecision support in supply chain management through the solutions sensitivity analysis, etermination of the range an quality of the impact of various parameters on the cost an even on the structure of the supply chain. The analysis presente in the article, only in terms of the capacity available to istributors, the number of transport units an environmental costs, fully confirms this statement. RFRCS [] Simchi-evi, D., Kaminsky, P., Simchi-evi. Designing an Managing the Supply Chain: Concepts, Strategies, an Case Stuies. McGraw-Hill, ew York 2003. [2] Shapiro, J.F., Moeling the Supply Chain, Duxbury Press 200. [3] Huang, G.Q., au, J.S.K., Mak, K.., 2003. The impacts of sharing prouction information on supply chain ynamics: a review of the literature. International Journal of Prouction Research 4, 483 57. [4] Beamon, B.M., Chen, V.C.P., 200. Performance analysis of conjoine supply chains. International Journal of Prouction Research 39, 395 328. [5] Kanyalkar, A.P., Ail, G.K., 2005. An integrate aggregate an etaile planning in a multi-site prouction environment using linear programming. International Journal of Prouction Research 43, 443 4454. [6] Perea-lopez,., Ystie, B.., Grossmann, I.., 2003. A moel preictive control strategy for supply chain optimization. Computers an Chemical ngineering 27, 20 28. [7] Park, Y.B., 2005. An integrate approach for prouction an istribution planning in supply chain management. International Journal of Prouction Research 43, 205 224. [8] Jung, H., Jeong, B., ee, C.G., 2008. An orer quantity negotiation moel for istributor-riven supply chains. International Journal of Prouction conomics, 47 58. [9] Rizk,., Martel, A., D amours, S., 2006. Multi-item ynamic prouction istribution planning in process inustries with ivergent finishing stages. Computers an perations Research 33, 3600 3623. [0] Selim, H., Am, C., zkarahan, I., 2008. Collaborative prouction istribution planning in supply chain: a fuzzy goal programming approach. Transportation Research Part -ogistics an Transportation Review 44, 396 49. [] ee, Y.H., Kim, S.H., 2000. ptimal prouction istribution planning in supply chain management using a hybri simulation-analytic approach. Proceeings of the 2000 Winter Simulation Conference an 2, 252 259. [2] Chern, C.C., Hsieh, J.S., 2007. A heuristic algorithm for master planning that satisfies multiple objectives. Computers an perations Research 34, 349 353. [3] Jang, Y.J., Jang, S.Y., Chang, B.M., Park, J., 2002. A combine moel of network esign an prouction/istribution planning for a supply network. Computers an Inustrial ngineering 43, 263 28. [4] Timpe, C.H., Kallrath, J., 2000. ptimal planning in large multi-site prouction networks. uropean Journal of perational Research 26, 422 435. [5] Schrijver, A., Theory of inear an Integer Programming. ISB 0-47-98232-6, John Wiley & sons. 998. [6] Ho, H., & im, C. 200. The logistics players From P to 5P. Morgan Stanley: China ogistics, 8 9. [7] Jianming, Yao, 200. Decision optimization analysis on supply chain resource integration in fourth party logistics. Journal of Manufacturing Systems 29, 2-29. [8] Chern, C.C., Hsieh, J.S., 2007. A heuristic algorithm for master planning that satisfies multiple objectives. Computers an perations Research 34, 349 353. [9] Torabi, S.A., Hassini,., 2008. An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets an Systems 59,93 24. [20] Schrijver, A., Theory of inear an Integer Programming, John Wiley & sons. 998. [2] www.lino.com. 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8 PRCDIGS F TH FDCSIS. WRCŁAW, 202 [23] Hokey Min, Gengui Zhou, 2002. Supply chain moeling: past, present an future. Computers an Inustrial ngineering 43, 23-249. APPDIX A Data for computational examples P, P2, P3 Table III The set of parts of ata tables for examples P,P2 an P3 s F s P - V s P2 - V s P3 - V s P4, P5, P6 - V s 200 050 200 2 000 500 2 500 050 200 2 000 500 3 000 050 200 2 000 500 Pt Zt P, P2, P3 - P4- P5- P6-60 0 0 0 0 20 2 80 0 30 30 00 60 3 600 0 400 00 500 800 j k Z jk Tc jk j k Z jk Tc jk i k C ik W ik 0 0 2 0 0 00 00 2 0 0 2 2 20 0 2 200 00 3 5 0 2 3 0 0 3 200 00 4 0 0 2 4 20 0 4 300 00 5 5 20 2 5 0 20 5 300 00 3 0 0 4 0 0 2 50 00 3 2 0 0 4 2 0 0 2 2 20 00 3 3 0 0 4 3 0 0 2 3 50 00 3 4 0 0 4 4 0 0 2 4 250 00 3 5 5 20 4 5 5 20 2 5 350 00 k P k s k R sk Tp sk s k R sk Tp sk 0 2 2 2 5 2 2 2 2 3 5 3 2 2 3 0 4 0 4 2 2 4 5 20 5 0 2 2 5 3 3 3 3 3 3 2 3 3 4 0 3 3 5 3 i s A is R is Tf is i s A is R is Tf is 0 2 2 5 4 2 20 3 2 2 0 6 3 40 4 2 3 20 0 7 2 2 2 2 0 4 2 2 24 2 2 2 2 20 6 2 3 42 3 2 2 3 40 7 3 5 2 5 4 3 2 0 2 2 2 25 6 3 3 25 3 2 3 35 0 7 s j G is R2 is Tm sj s j G is R2 is Tm sj 2 2 4 2 4 2 2 8 3 0 0 2 2 3 6 2 2 2 2 2 3 2 2 5 2 2 2 6 2 3 2 2 2 2 3 5 2 3 4 2 3 5 3 2 2 2 3 2 0 3 3 20 2 2 3 3 5 0 2 4 5 2 4 2 4 2 3 2 4 2 4 4 3 30 2 2 4 3 0 0 2 3 2 3 3 6 3 2 4 3 3 2 0 3 3 0 2 3 3 3 20 2 3 2 3 3 4 4 3 2 2 6 3 4 2 8 3 2 3 4 2 3 4 3 20 2 APPDIX B Results of optimization for computational examples P, P2, P3) Table IV The set of parts of tables with results for examples P, P2,P3 xample P Fc opt = 39445 i s k X isk Xb is i s k X isk Xb is 2 2.00 2 3.00 2 2 2.00 2 3 2 24.00 2 3 3 28.00 2 4 2 36.00 3 2 3 8.00 2 2 4 4.00 3 5 3 25.00 s j k Y isk Yb is s j k Y isk Yb is 2 2 2.00 3 2 0.00 3 2 5.00 3 2 2 8.00 4 2 0.00 3 5 2 5.00 2 2 0.00 3 2 5 5.00 2 2 3 2 0.00 3 3 2 0.00 2 4 2 0.00 3 3 2 2 0.00 3 2 2 0.00 3 4 5 6.00 2 3 4 2 6.00 3 4 5 2 9.00 4 2 2.00 3 4 2 8.00 4 3 2 0.00 2 3 4 4.00 xample P2 Fc opt = 37825 i s k X isk Xb is i s k X isk Xb is 3 3 40.00 2 4 4.00 3 2 3 20.00 2 3 2 24.00 3 5 3 25.00 2 4 2 36.00 2 2 3.00 2 2 2 0.00 s j k Y isk Yb is s j k Y isk Yb is 2 2 2.00 3 2 0.00 3 2 5.00 3 2 2 8.00 4 2 0.00 3 5 2 5.00 2 3 2 0.00 3 2 2 0.00 2 4 2 0.00 3 2 5 2 5.00 3 4 2.00 3 3 2 0.00 3 4 2 8.00 3 3 2 2 2.00 2 3 2 2 8.00 3 4 5 6.00 2 4 3 2 0.00 3 4 2 0.00 xample P3 Fc opt = 37795 i s k X isk Xb is i s k X isk Xb is 3 3 40.00 2 4 4.00 3 2 3 22.00 2 3 2 24.00 3 5 3 25.00 2 2 2 8.00 2 2 3.00 2 4 2 36.00 s j k Y isk Yb is s j k Y isk Yb is 3 2 5.00 3 2 0.00 4 2 0.00 3 2 2 0.00 2 3 2 0.00 3 5 2 5.00 2 4 2 0.00 3 2 2 0.00 3 4 2.00 3 2 5 2 5.00 3 2 2 8.00 3 3 2 0.00 3 4 2 8.00 3 3 2 2 2.00 4 3 2 0.00 3 4 2 0.00 3 4 5 6.00 2 3 4 5 2 9.00 Table V The set of parts of tables with results for examples P, P2, P3 ecision variables Vx s, Dx P P2 P3 P P2 P3 s Vx s s Vx s s Vx s Dx Dx Dx 035 925 895 7 4 4 2 40 2 0 2 0 2 0 2 0 2 0 3 050 3 200 3 230 3 3 3