STRATEGIC DESIGN FOR THE SUPPLY NETWORK OF MULTI-SITE ENTERPRISES: A MATHEMATICAL MODEL AND SOLVING APPROACHES Florene Pirard 1, Fouad Riane 2, Serguei Iassinovski 2, Valérie Botta-Genoulaz 1 1 Laboratoire PRISMa, Institut National des Sienes Appliquées de Lyon 7, avenue Jean Capelle, 69621 Villeurbanne edex, Frane florene.pirard@insa-lyon.fr, valerie.botta@insa-lyon.fr 2 CREGI, Faultés Universitaires Catholiques de Mons 151, haussée de Binhe, 7000 Mons, Belgique riane@fuam.a.be, iassinov@fuam.a.be Abstrat: We fous on the problem of strategi network design in the ontext of multi-site enterprises. To handle this problem, we first formulate a generi multi-ommodity, multi-ehelon and multi-period mixed-integer program whih aims at maximizing the after-tax profit. We then use the branh and bound tehnique to solve this program to optimality. As the resolution proess requires a onsiderable amount of omputational time for different instanes of the supply hain design problem, we develop a heuristi solving approah and ondut a series of experimental tests. The numerial results show that the heuristi approah an be a good alternative to the branh and bound algorithm. We devote the present paper to the presentation of the mathematial program. We desribe the solving approahes and omment on the numerial obtained results. Keywords: Supply hain management, strategi network design problem, multi-site enterprises, mixedinteger program, branh and bound, heuristi approah. 1 Introdution Following a series of mergers or aquisitions, many firms are onfronted nowadays with problems relating to the management of their multi-site logisti network. At the strategi level, these ompanies attempt to determine the design of their supply hain that will enable them to maximize their eonomi and servie performanes during an extended period of time. Solving the strategi network design problem makes it possible to take simultaneously three kinds of long-term deisions: on the loations of the prodution and distribution failities, on the material flows in and between the different sites of the network and on the investments in human and material resoures in eah site. In the literature, these three ategories of deision problems, respetively, refer to the loation, alloation and apaity problems (Jayaraman and Pirkul, 2001). This paper fouses on the presentation of the generi mathematial program that we formulated for the network design problem as well as the suggested solution approahes. It presents the developed heuristi approah and the numerial results obtained for a test problem based on industrial ases. The remainder of this paper is organized as follows. In the next setion, we propose a brief literature review. In setion 3, we desribe our mathematial model. Setion 4 presents the solution approahes and the numerial results. Setion 5 onludes the paper.
2 Literature review During the last deade several researhers have been interested in the modeling and solution of the strategi network design problem. The reader an find an extensive review in Thomas and Griffin (1996), Vidal and Goetshalkx (1997), Beamon (1998) and Min and Zhou (2002). These reviews fous on one or several aspets of the strategi supply hain management. For the problem of onern, the most developed models are analytial ones based on mixed-integer programs. These models are purely deterministi and are solved using branh and bound algorithms, Benders deomposition or heuristi methods. The maority of the supply hain models have been developed for the faility loation and alloation problems. Usually, they aim at minimizing the total ost of the supply hain. Some of them are foused on the optimization of the distribution network: Nozik and Turnquist (2001), Zhou et al. (2002) and Eskigun et al. (2005) study the faility loation problem while Geoffrion and Graves (1974); Jayaraman and Ross (2003) and Amiri (2006) deal with the loation and alloation problems. Jayaraman and Pirkul (2001), Yan et al. (2003), Jolayemi and Olorunniwo (2004) attempt to determine the state of prodution and distribution failities (opened or losed) as well as the material flows among the entities of the network. Paquet et al. (2003) onsider not only the problem of solving faility loation and alloation but also look for the determination of the investments in human and material resoures based on their respetive apabilities. This last model is a mono-period model. Arntzen et al. (1995), Vidal and Goetshalkx (2001), Fandel and Stammen (2004) and Wilhelm et al. (2005) take into aount in their modelling approah the international features relating to taxes and duties, transfer pries and exhange rate flutuation. Reently, Martel (2005) proposes a paper disussing various formulations of the elements (e.g. hoie of an obetive funtion, definition of planning horizon, finanial flows modelling) of prodution-distribution network design models. He presents an international multi-season model. When we examine the literature onerning the strategi network design problem, we note that even if the problem is treated, it is rarely globally takled. Many researhers limit their analysis on a subset of issues involved, or on problems inluding a subset of supply hain ators. Usually, the developed models aim at determining the loation of prodution and distribution failities and the material flows among these entities. The determination of investments or disinvestments in resoures is not frequent. Important harateristis of the global problem are often negleted; harateristis suh as the initial state of the supply network and the osts relating to the opening or the losing of failities are seldom taken into aount. Moreover, the maority of the models are single-period whih does not make it possible to evaluate the impat of making deisions in the long-term. Conerning the solution approahes, most of those developed in the literature are only validated on fititious problems of small size, whih does not guarantee that these methods find the optimal solution, or even a good quality feasible solution, for real problems in a reasonable omputational time. 3 The Mathematial model To model the multi-site strategi network design problem, we formulated a multi-period mixed-integer program. This model is generi; i.e., it is appliable to different supply networks with different strutures. This model is developed for a multi-site enterprise that owns several prodution and distribution entres positioned in various geographial areas (figure 1). At the beginning of the
Multi-site enterprise Suppliers Prodution entres Distribution entres Customers Potential loations Components Finished produts Figure 1: Example of supply hain struture planning horizon, some of these failities are opened. The losed failities orrespond to potential loations of ativities. In addition of its initial state, eah faility is haraterized by an initial available apaity (i.e., a maximum number of produts units to be treated in the site) and by a list of omponents and finished produts that an be manufatured, assembled or pakaged at the onsidered sites. As time passes by, the available apaity in eah site an be modified and thus varies in a given interval. We do not make assumptions on the struture of the network and we permit the transfer of produts between sites of the same level. In order to evaluate the profit of the multi-site enterprise, we develop a osting model that assoiates a set of reeipts and ost elements to eah entity and eah link between entities within the network (figure 2). The ustomers demands are known in advane and are deterministi. The suppliers are also predetermined. They deliver the omponents to the failities of the enterprise aording to its needs. We assume that the quantities of the omponents available from eah of the suppliers are limited. The supply hain is also haraterized by the omponents and the finished produts whih irulate between the various entities of the network. Bills of materials and routings are assoiated with these produts. The solution of the mathematial program enables deision-makers to determine, simultaneously and for eah onsidered period of time, the state of the total system as manifested in the opening and losing of failities (these deisions are modelled by binary variables), the available apaity in eah site, the investment and disinvestment in apaity in eah faility (modelled by integer variables), the material flow between the suppliers and the prodution sites, the material flow between sites, and between the sites and the ustomers, as well as a prodution plan (ontinuous variables). All these deisions are made in order to maximize the after-tax profit of the multi-site enterprise. We define this profit as the differene between the total reeipts and the sum of the osts inurred aording to our osting model. We multiply the resultant gross profit by a fator relating to taxation. The onstraints of our program are the following: the satisfation of ustomer s demands, the respet of bills of material in manufaturing, assembling and pakaging produts, the onservation of material flows, the respet of raw material inventory replenishment, prodution and distribution apaities and a set of logial onstraints relating to the state of the failities (e.g., onstraints to detet the openings and losings of the failities, onstraints to forbid the openings and losings of the failities during the same time period, and onstraints to forbid inoming, outoming material flows and prodution ativities when a faility is losed). Additional onstraints are also onsidered in our program in order to prohibit the opening and losing of a faility several times during the planning horizon. The mathematial model formulation is available in Appendix A.
Opening Purhasing ost Entity Cost Ownership Closing Prodution ost Reeipt Capaity ost Distribution Cost Investment Ownership Disinvestment Figure 2: The osting model 4 Solution approahes 4.1 Exat approah The most frequent approah to solve a mixed-integer programming model onsists in using the branh and bound algorithm. This kind of algorithm is implemented in various ommerial solvers. We used the algorithm available in LINGO 7 software (LINDO, 2002). 4.1.1 Numerial results The exat approah has been applied to a test problem based on two real life appliations enountered in a pharmaeutial and in a paper industry. The supply hain of the two multi-site enterprises was rather similar in terms of their struture (number of sites, number of ustomers; et.). We reated a fititious network similar to the two real ones. Conerning the data, we used the available information olleted on the real sites. For the unknown or unavailable data, we generated them or dedued them from offiial and publi douments suh as the balane sheets. The result was a multi-site enterprise that onsists of 16 failities spread over different geographial areas in Europe. Among theses sites, 6 are manufaturing plants and 10 are distribution entres. These latter sites deliver 20 families of finished produts to 20 ustomer zones. These produts are manufatured starting from 45 families of omponents. The enterprise is supplied upstream by 10 suppliers. We onsider 5 periods of time in the strategi planning. The assoiated mixed-integer programming requires 151681 variables, inluding 480 binary and integer variables and 11146 onstraints. For this problem, we generated several instanes based on different data sets orresponding to several ustomers' demand senarios. These data sets have the same total demand and are distinguished by the demand distribution along the 5 periods of time. Table 1 ontains the results obtained that orrespond to 10 data sets. For eah set, we indiated the number of iterations needed to find the optimal solution, the number of branhes that form the branh and bound tree used to solve the program and the required omputational time 1. The analysis of this table shows that for some data sets the algorithm finds the optimal solution in a reasonable omputational time. For others, this time is high. For other data sets not inluded in the table, we had to stop the solver after several tens or 1 The problems were solved on a Pentium IV, 2.4GHz proessor with 512Mo of RAM.
even hundreds of hours of alulation without reahing the optimum. These variations of the omputational time are due to the fat that the searh spae is not idential for the various data sets. This spae is defined with regard to the onstraints and with regard the fat that the deision variables are mixed integers. The unreasonably high omputational times of the exat method led us to develop a heuristi approah. In our numerial undertakings we have also ompared the single- and multi-period approahes. More preisely, we have examined the results obtained by establishing strategi plans over 5 periods of time and those obtained by elaborating 5 plannings, eah for one period of time. The solutions provided by these two approahes were slightly different. For example, for the deisions onerning the resoure investments, we noted that the multi-period program antiipates the future needs (i.e. no disinvestment in apaity in a site at a given period if this same site needs this apaity at the next periods). We should also note that the onatenation of single-period program solutions may not to respet the onstraint on number of sites state hanging. This has onsequenes on the value of the obetive funtion. Iterations Branhes Computational time Data 1 739375 9564 40 min 05 se Data 2 3955584 46981 3 h 08 min 49 se Data 3 237120 614 6 min 27 se Data 4 197370 124 4 min 17 se Data 5 16319572 37682 6 h 13 min 53 se Data 6 698365 1499 15 min 18 se Data 7 154632 410 4 min 41 se Data 8 240990 886 7 min 49 se Data 9 70741707 124958 27 h 14 min 59 se Data 10 1528372 4536 35 min 35 se Table 1: Numerial results obtained by using the branh and bound algorithm 4.2 Heuristi approah The developed heuristi approah is a hybrid iterative method whih breaks up the strategi network design problem into several subproblems and solves them using various tehniques. This method is based on a oupling between a random loal searh tehnique and an exat method. Figure 3 desribes the logi of our algorithm. The heuristi approah fixes, for eah iteration, the state of the prodution and the distribution failities. It onsiders, among the opened sites, several alloations of the neessary apaity and determines, for eah of them, the resulting material flows in the supply network. The determination of the state of the sites and their apaity onstitutes the master problem. It is solved by using an iterative improvement tehnique (Hoos and Stützle, 2005). In order to settle on the material flows assoiated with the onsidered apaity alloation and to evaluate the after-tax profit assoiated with this andidate solution, a linear program is solved using the simplex method. In our sheme the determination of the flows is the slave problem. The resolution of slave problem initiates an iterative improvement method that searhes for new solutions with improving values for the obetive funtion. The proposed approah has the advantage of providing, at eah iteration, a feasible network design. Let us note that in our algorithm, we broke up the problem so as to keep a global vision of the supply hain from the suppliers to the ustomers. In the following setion we briefly desribe the prinipal steps of the heuristi approah.
Figure 3: Desription of the funtioning of the heuristi approah program is solved using the simplex method. In our sheme the determination of the flows is the slave problem. The resolution of slave problem initiates an iterative improvement method that searhes for new solutions with improving values for the obetive funtion. The proposed approah has the advantage of providing, at eah iteration, a feasible network design. Let us note that in our algorithm, we broke up the problem so as to keep a global vision of the supply hain from the suppliers to the ustomers. In the following setion we briefly desribe the prinipal steps of the heuristi approah. 4.2.1 Definition of the deision variables for the master problem In the desription of the funtioning of the algorithm, we will regularly refer to two matrixes: the matrix State = (State t ) and the matrix Capaity = (Capaity t ). The former matrix ontains the state of eah site for eah period of time. The element State t indiates the state of the site during the period t (open = 1 and losed = 0). The latter matrix ontains the available apaity in eah site of the enterprise for eah period of time. The element Capaity t indiates the available apaity in site during the period t. This value limits the quantities of produts flowing out of site during the period t. 4.2.2 Determination of the failities states This step onsists of solving the relaxed linear program assoiated with the mixed integer program and of fixing the values of the matrix State aording to the obtained solution. If we let site_relax t denote the state of site during period t obtained by solving the relaxed program, then the value of the elements of the matrix State are as follows:
State t 1 = 0 indeterminate if site _ relax if site _ relax Otherwise where δ is a parameter fixed by the user (0 δ < 0, 5). t t > 0,5 + δ < 0,5 δ One the relaxed linear program is solved, we hek that eah line of the matrix State ontaining no indeterminate value respets the onstraints that make it forbidden for a faility to be opened or losed several times during the planning horizon. If these onstraints are not satisfied or if the onsidered line ontains at least one indeterminate value, the site state is fixed to indeterminate for all periods of the planning horizon. Note that the lines having all their elements different from indeterminate and respeting the above onstraint are definitively fixed. 4.2.3 Fixation of the failities states This step onsists of assigning a value to the lines of the matrix State ontaining at least one indeterminate value. In order to take into aount the onstraints forbidding that a faility opens or loses several times during the planning horizon, we established a list of the possible ombinations of state. The fixing of the state is done by randomly seleting a ombination in the list of the possible ombinations of state. 4.2.4 Initialization of the apaity alloation to the opened failities This step aims at alloating the neessary apaity among the opened failities. It starts evaluating the apaity needed at eah ehelon of the supply network. This evaluation is done during the first iteration based on the ustomers demands and respeting the bills of material. We assume that the available apaity is a multiple of ap whih refers to a number of units of omponents or finished produts. When the neessary apaity in a given ehelon of the supply hain is evaluated, we alloate this apaity among the open failities at this ehelon. Doing that requires us to determine a apaitated minimum ost flow on a apaitated network with nodes representing the ehelons of the supply hain and the links representing the failities. Eah link value refers to the minimum apaity (lower bound), the maximum apaity (upper bound), the initial available apaity, the ost of owning the faility, and the investment and disinvestment osts. We solve the problem using a linear program. 4.2.5 Determination of the material flows When we ome to this step, all the binary as well as the integer variables of the initial mixed-integer program are fixed. The determination of the material flows requires solving the resulting linear program where the matries State and Capaity, previously fixed, are input data. 4.2.6 Evaluation of the supply hain design When the elements of the matries State and Capaity are determined, it is easy to evaluate for eah period of time the openings and losings of the failities, the apaity investments and disinvestments as well as the osts assoiated with these hanges. The solution of the linear program assoiated with the determination of the material flows stage provides a first estimation of the profit of the multi-site enterprise. By subtrating the evaluated ownership and apaity osts from this profit, and by multiplying the result by a fator relating to taxation, we obtain the after-tax profit of the multi-site enterprise. If this final value is higher than the value of the best
stored solution, the urrent solution beomes the best one and the iterative proess goes on onsidering new alloations of apaities. The best solution is the one that the algorithm returns when it stops. 4.2.7 Modifiation of the apaity alloation To modify the alloation of the apaity among the open failities, we selet randomly a supply hain ehelon, two open failities of this ehelon and a time period. From this period of time and for the ommon opening period, we add ap units to the available apaity of one of the seleted failities and we subtrat the same ap units from the available apaity of the other faility. We hek that the new apaity alloation of the seleted failities respets the permitted interval variation defined by upper and lower bounds for the onsidered sites. Note that the modifiation of the alloation of the apaity always takes plae from the last matrix whih improved the value of the obetive funtion. 4.2.8 Stopping rule of the heuristi proess The heuristi approah inludes two nested loops: the first fixes the state of the failities (for those whose state is not fixed in a definitive way at the first stage), and the seond looks for several alloations of the neessary apaity. It is the user who fixes the number of iterations to do in eah one of them. The heuristi approah stops when it arried out the speified number of iterations. 4.2.9 Numerial results The heuristi approah is oded in Delphi and alls LINGO solver to solve the various linear programs. In order to investigate its effetiveness, this approah was applied to the same test problem we used to evaluate the branh and bound algorithm. More preisely, we seleted, among the data sets employed to evaluate this proedure, 8 sets for whih the branh and bound algorithm found the optimal solution in a omputational time higher than 13 hours. We arbitrarily fixed the number of iterations whih roughly orrespond to a omputational time of 3 hours. As some steps of the heuristi approah imply random seletions, we made 20 repliations for eah data set. Table 2 ontains the numerial results obtained. We indiate the omputational time of the branh and bound algorithm, the mean and the standard deviation of the gap and the mean of the omputational time for the heuristi approah. The gap is evaluated like this: gap = (Profit(Optimal solution) Profit(Best solution)) / Profit(Optimal solution). We an observe, for eah data set, that the heuristi approah finds solutions lose to the optimal ones. The results show that the heuristi approah an be a good alternative to the branh and bound algorithm. 5 Conlusions In this paper, we presented a multi-ommodity, multi-ehelon and multi-period mixedinteger program for the strategi network design problem in the ontext of multi-site enterprises. This model is generi. It an be used to solve the problem for networks with various strutures. This model enables deision-makers to simultaneously determine the opening, the losing and the state of eah faility, the material flows in the network and the apaity investments and disinvestments. The main obetive is to maximize the after-tax profit of the multi-site enterprise. To solve it, we first used the branh and bound algorithm. As the solution proess requires a onsiderable amount of omputational time for different instanes of the supply hain design
Branh and bound algorithm Computational time Mean (gap) (%) Heuristi approah Standard deviation (gap) (%) Mean (Computational time) DH.1 13 h 57 min 33 se 0,04 0,02 3 h 02 min 30 se DH.2 162 h 32 min 51 se 0,19 0,03 2 h 59 min 01 se DH.3 243 h 18 min 47 se 0,16 0,02 2 h 56 min 36 se DH.4 29 h 55 min 46 se 0,17 0,02 2 h 57 min 58 se DH.5 50 h 34 min 10 se 0,15 0,03 2 h 55 min 54 se DH.6 27 h 14 min 59 se 0,17 0,03 2 h 58 min 26 se DH.7 41 h 21 min 27 se 0,16 0,03 2 h 58 min 31 se DH.8 12 h 58 min 26 se 0,16 0,03 2 h 56 min 05 se Table 2: Numerial results obtained by using the heuristi approah (n=10 and m=50) problem, we developed a heuristi approah and onduted a series of experimental tests. The numerial results show that the heuristi approah an be a good alternative when the branh and bound algorithm fails to find the optimal solution in a reasonable omputational time. In the future, we would like to validate our mathematial model with other network design problems or with real industrial ases. We also would like to improve our heuristis approah so that it finds a solution lose to the optimal ones more rapidly. We envisage a hybridization between a metaheuristi and linear programming. The mathematial model presented in this paper is used as pivotal element within the framework of an integrated approah (Pirard, 2005). In this approah, the mathematial model is assoiated, in a omplementary way, with a simulation model that onsiders additional aspets of the supply hain like random demand distributions, transportation onstrains, lead time requirements, tatial elements and management poliies (e.g., inventory poliy, prodution poliy, prodution order assignment). The simulation model reprodues the dynamis of the entire supply hain and enables the deision maker to evaluate in-depth the effetiveness of the network design provided by the mathematial model. 6 Aknowledgements This study is partly supported by Interuniversity Attration Pole proet of the Walloon Region, Belgium 7 Referenes Amiri A. (2006). Designing distribution network in a supply hain system: Formulation and effiient solution proedure. European Journal of Operational researh, 171, 567-576. Arntzen B. C., G. G. Brown, T. P. Harrison, and L. L. Trafton (1995). Global supply hain management at Digital Equipment Corporation. Interfaes, 25(1), 69-93. Beamon B. M. (1998). Supply hain design and analysis: Models and methods. International Journal of Prodution Eonomis, 55, 281-294. Eskigun E., R. Uzsoy, P. V. Prekel, G. Beauon, S. Krishnan and J. D. Tew (2005). Outbound supply hain network design with mode seletion, lead times and apaitated vehile distribution enters. European Journal of Operational researh, 165, 182-206.
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Appendix A: Mathematial model formulation Indexes We use the following sets of indexes to define our model: Supplier: \ = {1 I} Finished produt: = {1 P} Site: ] = {1 J} Component: V = {1 C} Customer: ^ = {1 K} Time period: g = {1 T} From these sets, we define the following subsets: \ : The set of suppliers delivering the omponent. Vin : The set of omponents used in site. Vout : The set of omponents manufatured in site. V : The set of omponents used or manufatured in site (V = Vin Vout ). : The set of finished produts manufatured in site. ]in : The set of sites using the omponent. ]out : The set of sites manufaturing the omponent. ] : The set of sites using or manufaturing the omponent (] = ]in ]out ). ] p : The set of sites manufaturing the finished produt p. Data To formulate the model, the following data are required: d pkt : Number of units of finished produt p required by ustomer k in period t. bom : Number of units of omponent ' required to make one unit of omponent. bomfp p : Number of units of omponent required to make one unit of finished produt p. qbs it : Maximum number of units of omponent that the supplier i an deliver during the period t. biap : Lower bound on the available apaity available in site. bsap : Upper bound on the available apaity available in site. ap: Number of units of omponents or finished produts that an be manufatured or stored in addition (respetively, in less) when one unit of apaity is invested (respetively, disinvested). The following parameters desribe the initial state of the sites of the multi-site enterprise: initsite : Initial state of site (open = 1, losed = 0). initap : Available apaity in site at the beginning of the first period. To define the profit of the enterprise, the following osts, reeipts and tax rate are required: opsite t : Fixed ost assoiated to the opening of site in period t. site t : Fixed ost assoiated to the ownership of site in period t. lsite t : Fixed ost assoiated to the losing of site in period t. inap t : Fixed ost assoiated to the investment of one unit of apaity in site in period t. ap t : Fixed ost assoiated to the ownership of one unit of apaity in site in period t. deap t : Fixed ost assoiated to the disinvestment of one unit of apaity in site in period t. dfs it : Unit distribution ost of omponent from supplier i to site in period t. dfpss p t : Unit distribution ost of finished produt p from site to site ' in period t. dss t : Unit distribution ost of omponent from site to site ' in period t. dfps pkt : Unit distribution ost of finished produt p from site to ustomer k in period t. prodfp pt : Unit prodution ost of finished produt p in site in period t. prod t : Unit prodution ost of omponent in site in period t.
asf it : Unit purhasing ost of omponent by site to supplier i in period t. afpss p t : Unit purhasing ost of finished produt p by site to site ' in period t. ass t : Unit purhasing ost of omponent by site to site ' in period t. vl pkt : Unit sales prie of finished produt p from site to ustomer k in period t. vsfp p t : Unit sales prie of finished produt p from site to site ' in period t. vs t : Unit sales prie of omponent from site to site ' in period t. taxe : Tax rate in the nation where site is loated. Deision variables The following deision variables are used in the model: site t : State of site in period t (open = 1, losed = 0). siteop t : Binary variable indiating the opening of site in period t. sitel t : Binary variable indiating the losing of site in period t. in t : Integer variable indiating the number of units of apaity invested in site in period t. de t : Integer variable indiating the number of units of apaity disinvested in site in period t. dis t : Integer variable indiating the available apaity in site in period t. qfs it : Number of units of omponent shipped from supplier i to site in period t. qfpss p t : Number of units of finished produt p shipped from site to site ' in period t. qss t : Number of units of omponent shipped from site to site ' in period t. qfps pkt : Number of units of finished produt p shipped from site to ustomer k in period t. qfp pt : Number of units of finished produt p manufatured in site in period t. q t : Number of units of omponent manufatured in site in period t. Model The mathematial model is as follows: with Maximize f ( dfpss p' t + afpss p' t ). qfpss p' t ( dss' t + ass' t ). qss' t t p P ' J p t ' J f = [(1 taxe ).( ( vl pkt dfps pkt ). qfps pkt + vsfp p' t. qfpss p' t + vs' t. qss' t J t p P k K t p P ' Jp t ' J ( opsite t. siteop t + site t. site t + lsite t. sitel t ) ( inap t. in t + ap t. dis t + deap t. de t ) t t prodfp pt. qfppt prodt. qt ( dfsit + asfit ). qfsit )] t p P t t i I The mathematial model aims to maximize the after-tax profit of the multi-site enterprise. This profit is the differene between the reeipts oming from the sales of the omponents and the finished produts to the ustomers and the sites, and the osts related to the prourement, the manufaturing, the distribution, the ownership, the opening and the losing of the failities and the ownership, the investment and the disinvestment of the apaities. subet to Satisfy the ustomer's demand: qfps pkt = d pkt J p p P k K Respet the limits on the quantities of the omponents oming from the suppliers: (1)
qfsit qbsit J i I (2) Respet the available apaities in the failities: dis1 = initap + ap. in 1 ap. de 1 J (3) dis t = dis t 1 + ap. in t ap. de t 2 (4) biap.site t dis t bsap.site t (5) qt + qfppt dis t out p P J The onstraints (3) and (4) evaluate the available apaity in eah prodution or distribution faility. The onstraint (5) imposes that the available apaity in a faility is between the boundaries ditated for this site. This equation also onstraints to disinvest in apaity when the site loses. The onstraint (6) limits the quantities of the finished produts and the omponents that an be manufatured in a faility. This limit depends on the available apaity. Respet the onservation of flows: qfsit + qss + + ' t qss' t bom'. q' t bomfpp. qfppt Jin (7) i I ' J ' Jin ' out p P qt + qfsit + qss + + ' t qss' t bom'. q' t bomfpp. qfppt Jout (8) i I ' Jout ' J ' out p P qfp pt + qfpss p + ' t qfpss p' t qfps pkt p P J p (9) ' J ' J k K p p The onstraint (7) onerns the onservation of flows for the failities using the omponent. The onstraint (8) is related to the onservation of flows for the sites manufaturing the omponent. Finally, the onstraint (9) guarantees the onservation of flows for the plants manufaturing the finished produt p. Respet the openings and the losings of the failities: Cst.site t ( qt + qfppt ) 0 out p P J Cst.site t qfsit 0 i I (11) Cst.site t ( qss' t + qfpss p' t ) 0 J ' (12) p P Cst.site t qfps pkt 0 J k K (13) p P The onstraints (10), (11), (12) and (13) guarantee that the quantities of the manufatured omponents and finished produts and the inoming and outoming material flows are zero when a faility is losed. Cst is a large onstant. Detet the openings or the losings of the failities: initsite + ( 1 site 1) + siteop 1 1 J (14) site t 1 + (1 site t ) + siteop t 1 2 (15) (1- initsite ) + site 1 + sitel 1 1 J (16) (1- sitet-1) + site t + sitel t 1 2 (17) The onstraints (14) and (15) onern the openings and the onstraints (16) and (17) are related to the losings of the failities. (6) (10)
Respet the onstraints of the dependene between the variables: site t + sitel t 1 (18) site siteop 0 (19) t t The onstraint (18) forbids that the state of a faility is open when the losing of this site has been deteted. The onstraint (19) exludes that the state of a faility is losed when the opening of this plant has been deteted. Respet a set of additional onstraints: siteop t q J (20) t sitel t r J (21) t The onstraint (20) forbids more than q openings of the same site during the onsidered time horizon. The onstraint (21) forbids more than r losings of the same site during the onsidered time horizon. These two onstraints were added to the model in order to redue the size of the solution searh spae. q and r are two onstants. In our appliation, we fixed q and r at 1. Respet the onstraints of integrality and non negativity: { 0,1} site t, siteop t and sitel t (22) in t, de t and dis t are integer (24) q fs it 0 i I (24) q fpssp' t 0 p P J ' (25) q ss' t 0 J ' (26) q fps pkt 0 p P J k K (27) q fp pt 0 p P (28) q t 0 (29)