A hybrid approach to supply chain modeling and optimization

Transcription

1 Proceeings of the 2013 Feerate Conference on Computer Science an Information Systems pp A hybri approach to supply chain moeling an optimization Paweł Site Kielce University of Technology Al lecia PP 7, Kielce, Polan Institute of Management Control Systems site@tu.ielce.pl Jarosław Wiare Kielce University of Technology Al lecia PP 7, Kielce, Polan Institute of Management Control Systems j.wiare@tu.ielce.pl Abstract The paper presents the concept an an outline of the implementation of a hybri approach to supply chain moeling an optimization. Two environments mathematical programming (MP) an logic programming (P) were integrate. The strengths of integer programming (IP) an constraint logic programming (CP), in which constraints are treate in a ifferent way an ifferent methos are implemente, were combine to use the strengths of both. The propose approach is particularly important for the ecision moels with an objective function an many iscrete ecision variables ae up in multiple constraints. In orer to verify the propose approach, the optimization moels were presente an implemente in both traitional mathematical programming an the hybri environment. A I. ITRDUCTI SUPPY chain is referre to as an integrate systems which synchronizes a series of relate business processes in orer to acquire raw materials an parts, transform them into finishe proucts an istribute to customers an retailers. The supply chain plays an important role in the automotive, electronics an foo inustries. Huang et al. [1] stuie information sharing in supply chain management. 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. 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 focuses on type of representation, in this case, mathematical relationships, an the aspects to be consiere in the supply chain. The literature mostly escribes an iscusses mathematical programming-base moeling: linear programming, integer programming or mixe integer linear programming moels [2] [6]. Minimization of integrate costs is the main purpose of the moels presente in the literature [6] [10]. Maximization of revenues or sales is consiere to a lesser extent [4], [11]. Share information: Information is share between networ noes etermine by the moel. This enables prouction, istribution, inventory an transport planning, epening on the purpose. The information sharing process is a vital aspect in an effective supply chain. The following groups of parameters are taen into account: resources, inventory, prouction, transport, eman, time, etc. This paper focuses on the moeling approach to optimization problems in supply chain. A type of representation together with aspects to consier in the supply chain maes up a moeling approach. The vast majority of the wors reviewe have formulate their moels as linear programming (P), integer programming (IP) an mixe integer linear programming (MIP) problems an solve them using the perations Research methos. onlinear programming, multi-objective programming, fuzzy programming with stochastic programming are use much less frequently [12]. Problems relate to the esign, integration an management of the supply chain affect many aspects of prouction, istribution, warehouse management, supply chain structure, transport moes etc. Those problems are usually closely relate to each other, some may influence one another to a greater or lesser extent. Because of the interconnecteness an a very large number of ifferent constraints: resource, time, technological, an financial, the constraint base environments are suitable for proucing natural solutions for highly combinatorial problems. In the literature, references to moeling an optimizing supply chain problems using constraint base environments are relatively few in number [11], [12]. A. Constraint-base environments Constraint satisfaction problems (CSPs), constraint programming (CP) an constraint logic programming (CP) [13] [15] offer a very goo framewor for representing the nowlege an information neee to eal with supply chain problems. Constraint satisfaction problems (CSPs) are mathematical problems efine as a set of elements whose state must satisfy a number of constraints. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which are solve by constraint satisfaction methos. CSPs are the subject of intense stuy in both artificial intelligence an operations research, since the regularity in their formulation provies a common basis /$25.00 c 2013, I 1211

2 1212 PRCDIGS F TH FDCSIS. KRAKÓW, 2013 to analyze an solve problems of many unrelate families [13]. Formally, a constraint satisfaction problem is efine as a triple (X,D,C), where X is a set of variables, D is a omain of values, an C is a set of constraints. very constraint is in turn a pair (t,r) (usually represente as a matrix), where t is an n-tuple of variables an R is an n-ary relation on D. An evaluation of the variables is a function from the set of variables to the omain of values, v:x D. An evaluation v satisfies constraint ((x 1,,x n ),R) if (v(x 1 ),..v(x n )) R. A solution is an evaluation that satisfies all constraints. Constraint satisfaction problems on finite omains are typically solve using a form of search. The most use techniques are variants of bactracing, constraint propagation, an local search. ur experience as well as that of other researchers, confirms that constraint propagation is central to the process of solving a constraint problem [13], [14], [16]. Constraint propagation embes any reasoning that consists in explicitly forbiing values or combinations of values for some ecision variables of a problem because a given subset of its constraints cannot be satisfie otherwise. CSPs are often use in constraint programming. Constraint programming is the use of constraints as a programming language to encoe an solve problems. Constraint logic programming is a form of constraint programming, in which logic programming is extene to inclue concepts from constraint satisfaction. A constraint logic program is a logic program that contains constraints in the boy of clauses. Constraints can also be present in the goal. These environments are eclarative. B. rganization an structure of the paper In this paper, we focus on the problem of hybri moeling an optimization of the supply chain problems in the hybri environment. We propose a novel approach to supply chain moeling an optimization by eveloping integrate moels an methos using the complementary strengths of MIP an CP/CP (II, III). In this approach, both the hybri moel (V) an the hybri framewor (IV) to its efficient solution were evelope. In orer to verify the propose approach, the optimization moel in mixe linear integer programming (MIP) was create an implemente in traitional (IP) an hybri approaches. Finally, the hybri moel was optimize in the hybri framewor (V). II. MTIVATI Base on [1], [2], [13], [15], [16] an our previous wor [14], [17], [18] we observe some avantages an isavantages of these environments. An integrate approach of constraint programming (CP) an mixe integer programming (MIP) can help to solve optimization problems that are intractable with either of the two methos alone [20] [22]. Although perations Research (R) an Constraint Programming (CP) have ifferent roots, the lins between the two environments have grown stronger in recent years. Both MIP/MIP/IP an finite omain CP/CP involve variables an constraints. However, the types of the variables an constraints that are use, an the way the constraints are solve, are ifferent in the two approaches [23], [24]. III. STAT F TH ART As mentione earlier, the vast majority of ecision-maing moels for the problems of prouction, logistics, supply chain are formulate in the form of mathematical programming (MIP, MIP, IP). Due to the structure of these moels (aing together iscrete ecision variables in the constraints an the objective function) an a large number of iscrete ecision variables (integer an binary), they can only be applie to small problems. Another weaness is that only linear constraints can be use. In practice, the issues relate to the prouction, istribution an supply chain constraints are often logical, nonlinear, etc. For these reasons the problem was formulate in a new way In our hybri approach to moeling an optimization supply chain problems, we propose the environment, where: nowlege relate to supply chain can be presente in a linear an logical constraints (implement all types of constraints of previous MIP/MIP moels [18], [19] an introuce new types of constraints (logical, nonlinear, symbolic etc.)); the optimization moel solve by using the framewor can be formulate as a pure MIP/MIP moel, a CP/ CP moel or as a hybri moel; the novel metho of constraint propagation is introuce (obtaine by transformation of the optimization moel to explore its structure (feasible routes, capacities, etc.)); constraine omains of ecision variables, new constraints an values for some variables are transferre from CP/CP to MIP/MIP; the efficiency of fining solutions to the problems of larger sizes is increase. As a result, we obtaine the hybri optimization environment that ensures a better an easier way of moeling an optimization, an more effective search solution for a certain class of optimization problems. This class inclues quantitative moels relate to costs, customer service an inventories. Moels of this class are characterize by aing up many iscrete ecision variables in both constraints an the objective function. IV. HYBRID PTIMIZATI VIRMT In orer to implement all the assumptions an requirements outline in the previous chapter, both constraint logic programming (CP) an integer programming (MIP/MIP) ha to be combine an line.

3 SITK PAWŁ, WIKARK JARSŁAW: A HYBRID APPRACH T SUPPY CHAI MDIG AD PTIMIZATI 1213 The hybri environment consists of MIP/CP/Hybri moels an hybri optimization framewor to solve them (Fig. 1). The concept of this framewor an its phases (P1.. P5, G1..G3) are presente in Fig. 2. Fig. 1 Scheme of the hybri optimization environment The etails of the hybri environment have been iscusse in [24]. The motivation was to offer the most effective tools for moel specific constraints an solution efficiency. Fig. 2 Scheme of the hybri optimization framewor The constraints propagation of the transforme moel (phase P3) largely affecte the efficiency of the solution. Therefore phase P2 was introuce. During this phase, the transformation was performe using the structure an properties of the moel. The etails of this transformation are escribe in the following chapter. From a variety of tools for the implementation of the CP/CP environment, CiPSe software [25] was selecte. CiPSe is an opensource software system for the cost effective evelopment an eployment of constraint programming applications. nvironment for the implementation of MIP/MIP was IG by ID Systems. IG ptimization Moeling Software is a powerful tool for builing an solving mathematical optimization moels [26]. V. XAMPS F SUPPY CHAI PTIMIZATI The propose HS environment was verifie an teste on two moels. First moel was formulate as a mixe linear integer programming (MIP) problem [18], [19] uner constraints (2).. (23) in orer to test the propose environment (Fig. 1) against the classical integer programming environment [26]. Then the hybri moel (1).. (25) was implemente an solve. Inices, parameters an ecision variables use in the moels together with their escriptions are summarize in Tab. 1. The simplifie structure of the supply chain networ for this moel, compose of proucers, istributors an customers is presente in Fig. 3. Fig. 3 The simplifie structure of the supply chain networ Both moels are the cost moels that tae 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 the transport moe. The main assumptions mae in the construction of these moels were as follows: the share information process in the supply chain consists of resources (capacity, versatility, costs), inventory (capacity, versatility, costs, time), prouction (capacity, versatility, costs), prouct (volume), transport (cost, moe, time), eman, etc; part of the supply chain has a structure as in Fig. 3.; 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 are taen into account; 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);

4 1214 PRCDIGS F TH FDCSIS. KRAKÓW, 2013 the moels have linear or linear an logical constraints. Symbol j i s M F s P V s W i, C i, R s, Tp s, Tc j, Z j, Zt Pt Tf i,s, K1 i,s,, R1 i,s, A i,s, Koa s,j, Tm s,j, K2 s,j,, R2 s,j, G s,j, Kog s,j, X i,s,, Xa i,s, Xb i,s, Y s,j,, Ya s,j, Yb s,j, Tc s CW TAB I IDICS, PARAMTRS AD DCISI VARIABS Description Inices prouct type (=1..) elivery point/customer/city (j=1..m) manufacturer/factory (i=1..) istributor /istribution center (s=1..) moe of transport (=1..) number of manufacturers/factories number of elivery points/customers number of istributors number of prouct types number of moe of transport Input parameters the fixe cost of istributor/istribution center s the area/volume occupie by prouct istributor s maximum capacity/volume prouction capacity at factory i for prouct the cost of prouct at factory i if istributor s can eliver prouct then R s,=1, otherwise R s,=0 the time neee for istributor s to prepare the shipment of prouct the cut-off time of elivery to the elivery point/customer j of prouct customer eman/orer j for prouct the number of transport units using moe of transport the capacity of transport unit using moe of transport the time of elivery from manufacturer i to istributor s using moe of transport the variable cost of elivery of prouct from manufacturer i to istributor s using moe of transport if manufacturer i can eliver to istributor s using moe of transport then R1 i,s,=1, otherwise R1 i,s,=0 the fixe cost of elivery from manufacturer i to istributor s using moe of transport the total cost of elivery from istributor s to customer j using moe of transport the time of elivery from istributor s to customer j using moe of transport the variable cost of elivery of prouct from istributor s to customer j using moe of transport if istributor s can eliver to customer j using moe of transport then R2 s,j,=1, otherwise R2 s,j,=0 the fixe cost of elivery from istributor s to customer j using moe of transport the total cost of elivery from istributor s to customer j using moe of transport the environmental cost of using moe of transport Decision variables elivery quantity of prouct from manufacturer i to istributor s using moe of transport if elivery is from manufacturer i to istributor s using moe of transport then Xa i,s,=1, otherwise Xa i,s,=0 the number of courses from manufacturer i to istributor s using moe of transport elivery quantity of prouct from istributor s to customer j using moe of transport if elivery is from istributor s to customer j using moe of transport then Ya s,j, =1, otherwise Ya s,j, =0 the number of courses from istributor s to customer j using moe of transport if istributor s participates in eliveries, then Tc s=1, otherwise Tc s=0 Arbitrarily large constant A. bjective function The objective function (1) efines the aggregate costs of the entire chain an consists of five elements. The first element comprises the fixe costs associate with the operation of the istributor involve in the elivery (e.g. istribution centre, warehouse, etc.). The secon element correspons to environmental costs of using various means of transport. Those costs are epenent on the number of courses of the given means of transport, an on the other han, on the environmental levy, which in turn may epen on the use of fossil fuels an carbon-ioxie emissions. The thir component etermines the cost of the elivery from the manufacturer to the istributor. Another component is responsible for the costs of the elivery 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. Formulating the objective function in this manner allows comprehensive cost optimization of various aspects of supply chain management. ach subset of the objective function with the same constrains provies a subset of the optimization area an maes it much easier to search for a solution. F Tc ( Xb s Koa Kog i,s, 1 j 1 l B. Constraints s Yb i,s, 1 j 1 M s,j, M (C X i 1 1 j,s, ) The moel was base on constraints (2).. (24) Constraint (2) specifies that all eliveries of prouct prouce by the manufacturer i an elivere to all istributors s using moe of transport o not excee the manufacturer s prouction capacity. Constraint (3) covers all customer j emans for prouct (Z j, ) through the implementation of elivery by istributors s (the values of ecision variables Y i,s,, ). The flow balance of each istributor s correspons to constraint (4). The possibility of elivery is epenent on the istributor s technical capabilities - constraint (5). Time constraint (6) ensures the terms of elivery are met. Constraints (7a), (7b), (8) guarantee eliveries with available transport taen into account. Constraints (9), (10), (11) set values of ecision variables base on binary variables Tc s, Xa i,s,, Ya s,j,. Depenencies (12) an (13) represent the relationship base on which total costs are calculate. In general, these may be any linear functions. The remaining constraints (14)..(23) arise from the nature of the moel (MIP). Constraint (24) allows the istribution of exclusively one of the two selecte proucts in the istribution center s. Similarly, constraint (25) allows the prouction of exclusively one of the two selecte proucts in the factory i. Those constraints result from technological, mareting, sales or safety reasons. Therefore, some proucts cannot be istribute an/or prouce together. The constraint can be re-use for ifferent pairs of prouct an for some of or all i,s,, ) (1)

5 SITK PAWŁ, WIKARK JARSŁAW: A HYBRID APPRACH T SUPPY CHAI MDIG AD PTIMIZATI 1215 istribution centers s an factories i. A logical constraint lie this cannot be easily implemente in a linear moel. nly eclarative application environments base on constraint satisfaction problem (CSP) mae it possible to implement constraints such as (24), (25). The aition of constraints of that type changes the moel class. It is a hybri moel. Xa X 1 i,s,, R s, W for i 1.., 1.. i, ( Y R ) Zj, 1 Xi 1 s,j,,,s,, s, M Y j 1 1 s,j,, for j 1..M, 1.. for s 1.., 1.. ( P X ) Tc V for s 1.. s s i,s, 1 Tf i,s, 1 Xa i,s, i,s,, Tp Ya Tm Tc for i 1.., s 1.., j 1..M, 1.., 1.. Xb Yb X X R1 i,s, Xb i,s, s, s,j, Pt X i,s,, P s,j, for i 1..,s 1.., 1.., 1.. R2 Yb Pt Y P s,j, s,j, for s 1.., j 1..M, 1.., 1.. Xb i i,s, s,j,,s, Xb 1 M Yb j 1 i,s, j,s, Zt s,j,, for 1.. CW Tc for s 1.. s j, (2) (3) (4) (5) (6) (7a) (7b) (8) (9) CW Xa i,s, for i 1..,s 1.., 1.. (10) CW Ya s,j, for s 1.., j 1..M, 1.. (11) Koa A Xb K1 X i,s, i,s, i,s, i,s,, i,s,, 1 (12) for i 1..,s 1.., 1.. Kog G Yb K2 Y s, j, s, j, j,s, s,j,, s, j,, 1 (13) for s 1.., j 1..M, 1.. i,s,, i,s,, Y s,j,, 0for i 1..,s 1.., 1..0, 1.. (14) Xb i,s, 0for i 1..,s 1.., 1.., (15) 0 for s 1.., j 2..M, 1.., (16) C for i 1..,s 1.., 1..0, 1.., (17) C for i 1..,s 1.., 1.. (18) C for s 1.., j 1..M, 1..0, 1.. (19) Yb s,j, Xb i,s, Yb s,j, C for s 1.., j 1..M, 1.., (20) Xa i,s, 0,1 for i 1..,s 1.., 1.., (21) Ya s,j, 0,1 for s 1.., j 2..M, 1.., (22) Tc s 0,1 for s 1.. (23) xclusiond(x i,s,,, X i,s,l,, s) for l, s=1..s (24) xclusionp(x i,s,,, X i,s,l,, i) for l, i=1.. (25) C. Moel transformation Due to the nature of the ecision problem (aing up ecision variables an constraints involving a lot of variables), the constraint propagation efficiency ecreases ramatically. Constraint propagation is one of the most important methos in CP affecting the efficiency an effectiveness of the CP an hybri optimization environment (Fig. 1). For that reason, research into more efficient an more effective methos of constraint propagation was conucte. The results inclue ifferent representation of the problem an the manner of its implementation. The classical problem moeling in the CP environment consists in builing a set of preicates with parameters. ach CP preicate has a corresponing multi-imensional vector representation. While moeling both problems, (1).. (23) an (1).. (25), quantities i, s,, an ecision variable X i,s,, were vector parameters. The process of fining the solution may consist in using the constraints propagation methos, labeling of variables an the bactracing mechanism. The quality of constraints propagation an the number of bactracings are affecte to a high extent by the number of parameters that must be specifie/labele in the given preicate/vector. In both moels presente above, the classical problem representation inclue five parameters: i, s,, an X i,s,,. Consiering the omain size of each parameter, the process is complex an time-consuming. ur iea was to transform the problem by changing its representation without changing the very problem. All permissible routes were first generate base on the fixe ata an a set of orers, then the specific values of parameters i, s,, were assigne to each of the routes. In this way, only ecision variables X i,s,, (eliveries) ha to be specifie. This transformation funamentally improve the efficiency of the constraint propagation an reuce the number of bactracs. A route moel is a name aopte for the moels that unerwent the transformation. D. Decision-maing support The propose moels can support ecision-maing in the following areas: the optimization of total cost of the supply chain (objective function, ecision variables-appenix A2); the selection of the transport fleet number, capacity an moes for specific total costs; the sizing of istributor warehouses an the stuy of their impact on the overall costs (Appenix A3, Fig. 4, Fig. 5, Tab. V); the selection of transport routes for optimal total cost. Detaile stuies of these topics are being conucte an will be escribe in our future articles. We use the hybri approach to both moeling an solving. VI. UMRICA XPRIMTS In orer to verify an evaluate the propose approach, many numerical experiments were performe. All the examples relate to the supply chain with two manufacturers (i=1..2), three istributors (s=1..3), five customers (j=1..5), three moes of transport (=1..3), an ten types of proucts (=1..10). xperiments began with three examples of P1, P2, P3 for the optimization MIP moel (1).. (23). The examples iffer in terms of capacity available to the istributors s (V s ), the number of transport units using the moe of transport (Zt ) an the number of orers (o). The first series of experiments was esigne to show the benefits an avantages of the hybri approach. For this purpose the

6 1216 PRCDIGS F TH FDCSIS. KRAKÓW, 2013 moel (1).. (23) was implemente in both the hybri an integer programming environments. In aition, hybri implementation of the transforme moel was performe with an without constraint propagation, (MIPT2) an (MIPT1), respectively. The experiments that follow were conucte to optimize examples P4, P5, which are implementations of the moel (1).. (25) for the hybri approach. xamples P4, P5 were obtaine from P1, P3 by the aition of logical constraints (24), (25). umeric ata of input parameters for examples P1, P2, P3, P4, P5 are shown in Appenix A1. The results in the form of the objective function an the computation time are shown in Table II. ther results incluing the ecision variables for the optimal value of the objective function are given in Appenix A2. TAB II TH RSUTS F UMRICA XAMPS FR BTH APPRACHS Hybri Integer P(o) Programming MIPT1 MIPT2 MIP Fc T Fc T Fc T P1(10) 22401* 600** * 600** P2(10) 21167* 600** * 600** P3(20) * 600** P(o) MH Fc T P4(10) P5(20) Fc the value of the objective function T time of fining solution (in secons) the optimal value of the objective function after the time T * the feasible value of the objective function after the time T ** calculation was stoppe after 600 s MIP MIP moel implementation in the IP environment. MIP moel after transformation - implementation in the MIPT1 hybri optimization framewor without phase P3 MIP moel after transformation-implementation in the MIPT2 hybri optimization framewor. MH Hybri moel after transformation-implementation in the hybri optimization framewor. For each example the solution for the MIPT2 implementation was foun faster than that for the MIP implementation. Moreover, for examples P1.. P3, the traitional approach base on integer programming gives only feasible solution (calculation was stoppe after 600 s) espite using highly efficient IG solvers. It is obvious that the solution of the hybri moel (MH) was, ue to its nature, only possible using the hybri environment. Also, the propose environment brought the expecte results. The results were obtaine in only a slightly longer perio of time than that necessary for examples P1 an P3. VII. CCUSI The efficiency of the propose approach is base on the reuction of the combinatorial problem. This means that using the hybri approach practically for all moels of this class, the same or better solutions are foun faster (the optimal instea of the feasible solutions). Another element contributing to the high efficiency of the metho is a possibility to etermine the values or ranges of values for some of the ecision variables (phase P3). All effective IG solvers can be use in the hybri metho. Therefore, the propose solution is highly recommene for all types of ecision problems in supply chain or for other problems with similar structure. This structure is characterize by the constraints of many iscrete ecision variables an their summation. Furthermore, this metho can moel an solve problems with logical constraints. Further wor will focus on running the optimization moels with non-linear an other logical constraints, multiobjective, uncertainty etc. in the hybri optimization framewor. APPDIX A1 TAB III DATA FR CMPUTATIA XAMPS P1, P2, P3,P4,P5 V j s P 1,P 4 P 2, P 3,P 5 F s P1 1 1 C P2 1 2 C P3 3 3 C P4 2 4 P5 3 5 Zt Pt s P6 1 s P 1,P 4 P 2 P 3,P 5 s P7 1 i S P8 3 F1 S P9 2 F2 S P10 3 i s K i,s, T i,s, i W i, C i, F1 C1 S2 2 3 F1 P F1 C1 S3 4 4 F1 P F1 C2 S2 4 2 F1 P F1 C2 S3 8 3 F1 P F1 C3 S2 6 2 F1 P F1 C3 S3 8 3 F1 P F2 C1 S2 5 4 F2 P F2 C1 S3 7 4 F2 P F2 C2 S2 2 6 F2 P F2 C2 S3 4 7 F2 P F2 C3 S2 2 6 F2 P F2 C3 S3 3 6 F2 P s j K s,j, T s,j, s j K s,j, T s,j, C1 M1 S1 2 1 C2 M3 S2 6 2 C1 M1 S2 4 2 C2 M4 S1 3 1 C1 M2 S1 2 1 C2 M4 S2 6 2 C1 M2 S2 4 2 C2 M5 S1 3 1 C1 M3 S1 2 1 C2 M5 S2 6 2 C1 M3 S2 4 2 C3 M1 S1 4 1 C1 M4 S1 2 1 C3 M1 S2 8 2 C1 M4 S2 4 2 C3 M2 S1 4 1 C1 M5 S1 2 1 C3 M2 S2 8 2 C1 M5 S2 4 2 C3 M3 S1 4 1 C2 M1 S1 3 1 C3 M3 S2 8 2 C2 M1 S2 6 2 C3 M4 S1 4 1 C2 M2 S1 3 1 C3 M4 S2 8 2 C2 M2 S2 6 2 C3 M5 S1 4 1 C2 M3 S1 3 1 C3 M5 S2 8 2 V s i s P5 F1 P6 P5 C1 P6 P5 F2 P6 P5 C2 P6 P2 F1 P8 P5 C3 P6 P2 F2 P8 P2 C1 P8 P2 C2 P8 P2 C3 P8

7 SITK PAWŁ, WIKARK JARSŁAW: A HYBRID APPRACH T SUPPY CHAI MDIG AD PTIMIZATI 1217 s T s, s T s, s T s, C1 P1 2 C2 P1 1 C3 P1 3 C1 P2 2 C2 P2 1 C3 P2 3 C1 P3 2 C2 P3 1 C3 P3 3 C1 P4 2 C2 P4 1 C3 P4 3 C1 P5 2 C2 P5 1 C3 P5 3 C1 P6 2 C2 P6 1 C3 P6 3 C1 P7 2 C2 P7 1 C3 P7 3 C1 P8 2 C2 P8 1 C3 P8 3 C1 P9 2 C2 P9 1 C3 P9 3 C1 P10 2 C2 P10 1 C3 P10 3 ame j T j, Z j, ame j T j, Z j, Z_01 p1 m Z_11 p1 m Z_02 p2 m Z_12 p2 m Z_03 p3 m Z_13 p3 m Z_04 p4 m Z_14 p4 m Z_05 p5 m Z_15 p5 m Z_06 p6 m Z_16 p6 m Z_07 p7 m Z_17 p7 m Z_08 p8 m Z_18 p8 m Z_09 p9 m Z_19 p9 m Z_10 p10 m Z_20 p10 m APPDIX A2 TAB IV RSUTS F PTIMIZATI FR CMPUTATIA XAMPS P1, P2 (FU) AD P3,P4,P5 (PART) xample P1 Fc opt = ame i s j 1 2 X is1 Y sj2 Z_01 F1 P1 C1 M1 S3 S Z_01 F1 P1 C1 M1 S3 S Z_02 F1 P2 C1 M2 S3 S Z_03 F1 P3 C1 M3 S3 S Z_03 F1 P3 C1 M3 S3 S Z_04 F1 P4 C1 M4 S3 S Z_05 F2 P5 C2 M5 S3 S Z_06 F2 P6 C1 M1 S3 S Z_07 F2 P7 C1 M2 S3 S Z_07 F2 P7 C3 M2 S3 S Z_08 F2 P8 C1 M3 S3 S Z_08 F2 P8 C2 M3 S3 S Z_09 F2 P9 C2 M4 S2 S Z_10 F2 P10 C2 M5 S3 S Z_10 F2 P10 C3 M5 S3 S i s Xb i,s, i s Xb i,s, F1 C1 S F2 C2 S F2 C1 S F2 C3 S F2 C2 S s j Yb s,j, s j Yb s,j, C1 M1 S C2 M3 S C1 M1 S C2 M4 S C1 M2 S C2 M5 S C1 M3 S C3 M2 S C1 M3 S C3 M5 S C1 M4 S xample P2 Fc opt = 2142 ame i s j 1 2 X is1 Y sj2 Z_01 F1 P1 C1 M1 S2 S Z_01 F1 P1 C1 M1 S3 S Z_01 F1 P1 C1 M1 S3 S Z_02 F1 P2 C1 M2 S3 S Z_03 F1 P3 C1 M3 S3 S Z_04 F1 P4 C1 M4 S3 S Z_05 F1 P5 C1 M5 S2 S Z_05 F1 P5 C1 M5 S3 S Z_06 F1 P6 C1 M1 S3 S Z_05 F2 P5 C1 M5 S3 S Z_06 F2 P6 C1 M1 S3 S Z_07 F2 P7 C1 M2 S3 S Z_07 F2 P7 C2 M2 S3 S Z_08 F2 P8 C2 M3 S3 S Z_09 F2 P9 C1 M4 S3 S Z_10 F2 P10 C1 M5 S3 S Z_10 F2 P10 C2 M5 S3 S i s Xb i,s, i s Xb i,s, F1 C1 S F2 C1 S F1 C1 S F2 C2 S s j Yb s,j, s j Yb s,j, C1 M1 S C1 M5 S C1 M1 S C2 M2 S C1 M2 S C2 M3 S C1 M3 S C2 M5 S C1 M4 S xample P3 Fc opt = ame i s j 1 2 X is1 Y sj2 Z_01 F1 P1 C2 M1 S3 S Z_11 F1 P1 C2 M3 S3 S Z_20 F2 P10 C2 M2 S2 S Z_20 F2 P10 C2 M2 S3 S Z_10 F2 P10 C2 M5 S3 S i s Xb i,s, i s Xb i,s, F1 C2 S F2 C2 S F1 C2 S F2 C2 S s j Yb s,j, s j Yb s,j, C2 M1 S C2 M4 S C2 M2 S C2 M4 S C2 M2 S C2 M5 S C2 M3 S C2 M5 S C2 M3 S xample P4 Fc opt = ame i s j 1 2 X is1 Y sj2 Z_01 F1 P1 C1 M1 S3 S Z_02 F1 P2 C1 M2 S3 S Z_03 F1 P3 C1 M3 S3 S Z_03 F1 P3 C2 M3 S2 S Z_03 F1 P3 C2 M3 S2 S Z_10 F2 P10 C2 M5 S3 S Z_10 F2 P10 C3 M5 S3 S i s Xb i,s, i s Xb i,s, F1 C1 S F2 C2 S F1 C2 S F2 C2 S F2 C1 S F2 C3 S s j Yb s,j, s j Yb s,j, C1 M1 S C2 M3 S C1 M2 S C2 M4 S C1 M3 S C2 M5 S C1 M4 S C3 M2 S C1 M4 S C3 M5 S C2 M3 S xample P5 Fc opt = ame i s j 1 2 X is1 Y sj2 Z_01 F1 P1 C1 M1 S2 S Z_11 F1 P1 C1 M3 S3 S Z_01 F1 P2 C1 M2 S3 S Z_12 F1 P2 C1 M4 S3 S Z_20 F2 P10 C1 M2 S3 S

8 1218 PRCDIGS F TH FDCSIS. KRAKÓW, 2013 Z_10 F2 P10 C1 M5 S3 S Z_20 F2 P10 C2 M2 S2 S Z_10 F2 P10 C2 M5 S3 S i s Xb i,s, i s Xb i,s, F1 C1 S F2 C2 S F1 C1 S F2 C2 S F2 C1 S s j Yb s,j, s j Yb s,j, C1 M1 S C1 M5 S C1 M2 S C1 M5 S C1 M2 S C2 M1 S C1 M3 S C2 M2 S C1 M4 S C2 M3 S C1 M4 S C2 M5 S APPDIX A3 TAB V AAYSIS F TH IMPACT PARAMTR VS FR FC (XAMP P2) V=V 1= V 2= V 3 Fc opt Distributor capacity (Vs )utilization V 1 V 2 V Fig. 4 The value of the objective function epening on the parameter V (xample P2) Fig. 5 Capacity utilization (Vs) for istributor s=1, s=2,s=3 (xample P2) References [1] Huang, G.Q., au, J.S.K., Ma, K.., The impacts of sharing prouction information on supply chain ynamics: a review of the literature. International Journal of Prouction Research 41, 2003, pp [2] Kanyalar, A.P., Ail, G.K., An integrate aggregate an etaile planning in a multi-site prouction environment using linear programming. International Journal of Prouction Research 43, 2005, pp [3] Perea-lopez,., Ystie, B.., Grossmann, I.., A moel preictive control strategy for supply chain optimization. Computers an Chemical ngineering 27, 2003, pp [4] Par, Y.B.,. An integrate approach for prouction an istribution planning in supply chain management. International Journal of Prouction Research 43, 2005, pp [5] Jung, H., Jeong, B., ee, C.G., An orer quantity negotiation moel for istributor-riven supply chains. International Journal of Prouction conomics 111, 2008, pp [6] Riz,., Martel, A., D amours, S., Multi-item ynamic prouction istribution planning in process inustries with ivergent finishing stages. Computers an perations Research 33, 2006, pp [7] Selim, H., Am, C., zarahan, I., Collaborative prouction istribution planning in supply chain: a fuzzy goal programming approach. Transportation Research Part -ogistics an Transportation Review 44, 2008, pp [8] ee, Y.H., Kim, S.H., ptimal prouction istribution planning in supply chain management using a hybri simulation-analytic approach. Proceeings of the 2000 Winter Simulation Conference 1 an 2, 2000, pp [9] Chern, C.C., Hsieh, J.S., A heuristic algorithm for master planning that satisfies multiple objectives. Computers an perations Research 34, 2007, pp [10] Jang, Y.J., Jang, S.Y., Chang, B.M., Par, J., A combine moel of networ esign an prouction/istribution planning for a supply networ. Computers an Inustrial ngineering 43, 2002, pp [11] Timpe, C.H., Kallrath, J., ptimal planning in large multi-site prouction networs. uropean Journal of perational Research 126, 2000, pp [12] Mula J., Peiro D., Diaz-Maronero M., Vicens., Mathematical programming moels for supply chain prouction an transport planning, uropean Journal of perational Research 204, 2010, pp [13] Apt K., Wallace M., Constraint ogic Programming using clipse, Cambrige University Press, [14] Site P., Wiare J., A Declarative Framewor for Constraine Search Problems. ew Frontiers in Applie Artificial Intelligence, ecture otes in Artificial Intelligence, guyen, T., et al. (s.), Vol. 5027, Springer-Verlag, Berlin-Heielberg, 2008, pp [15] Bocewicz G., Wójci R., Banasza Z., AGVs istribute control subject to imprecise operation time. In: Agent an Multi-Agent Systems: Technologies an Applications, ecture otes in Artificial Intelligence, AI, Springer-Verlag, Vol. 4953, 2008, pp [16] Rossi F., Van Bee P., Walsh T., Hanboo of Constraint Programming (Founations of Artificial Intelligence), lsevier Science Inc. ew Yor, Y, USA, [17] Site P., Wiare J., Cost optimization of supply chain with multimoal transport, Feerate Conference on Computer Science an Information Systems (FeCSIS), 2012, pp [18] Site P., Wiare J., Supply chain optimization base on a MIP moel from the perspective of a logistics provier, Management an Prouction ngineering Review, 2012 pp [19] Site P., Wiare J., The Declarative Framewor Approach to Decision Support for Constraine Search Problems, ITCH, 2011, pp [20] Jain V., Grossmann I.., Algorithms for hybri MIP/CP moels for a class of optimization problems, IFRMS Journal on Computing 13(4), 2001, pp [21] Milano M., Wallace M., Integrating perations Research in Constraint Programming, Annals of perations Research vol. 175 issue 1, 2010, pp [22] Achterberg T., Berthol T., Koch T., Wolter K., Constraint Integer Programming. A ew Approach to Integrate CP an MIP, ecture otes in Computer Science Volume 5015, 2008, pp [23] Bocmayr A., Kasper T., Branch-an-Infer, A Framewor for Combining CP an IP, Constraint an Integer Programming perations Research/Computer Science Interfaces Series, Volume 27, 2004, pp [24] Site P., Wiare J., A hybri metho for moeling an solving constraine search problems, Feerate Conference on Computer Science an Information Systems (FeCSIS), 2013.pp [25] [26]