Integrated optimization and multi-agent tehnology for ombined prodution and transportation planning Jan A. Persson 1 and Paul Davidsson 2 1 Blekinge Institute of Tehnology, Department Systems and Software Engineering, 374 24 Karlshamn, Sweden 2 Blekinge Institute of Tehnology, Department Systems and Software Engineering, 372 25 Ronneby, Sweden {Jan.Persson,Paul.Davidsson}@bth.se Abstrat In this researh projet, an integration of multi-agent tehnology and optimization tehniques is suggested for the ombined prodution and transport planning problem in a transport hain. The hain onsists of a produer, a transport operator, and a number of ustomers. Optimization deomposition tehniques use (dual) pries of resoures to oordinate the generation of different plans in the so alled s. We argue that these dual pries and generated plans an enhane the multi-agent based approah. By using the agent tehnology, we an better resemble the interations between real planners realloating resoures. A transport hain within the food industry has been seleted for validating the developed solution method. 1. Introdution The integrated prodution and transportation planning problem is a omplex problem. Partiularly, this is the ase with respet to the many resoures to be oordinated and the number of planners/deision makers involved. The planning problem onsists of a number of ombinatorial deision problems: prodution sheduling (lot-sizing), fleet management and inventory management. Large benefits an be expeted from a suessful integration of these deision problems, whih have been indiated by for example the use of VMI (Vendor Managed Inventory) approahes. For a long time mathematial optimization tehniques based on linear programming and branh and bound have been used to solve different types of resoure alloation problems, e.g., prodution and transportation planning in various industries at strategi and tatial level, [1,2,3]. Additionally one an find examples of optimization tehniques applied for short term planning (operational), e.g., ativity sheduling [4,5,6]. Solution approahes based on different optimization tehniques are promising, partiularly when used as heuristis for dynami resoure alloation problems. However, it is unlikely that these methods on their own will be the only route forward due to the high omplexity and ombinatorial aspets in this type of planning problems. Potentially onstraint programming is an alternative to (lassial) optimization tehniques. The strength of onstraint programming is the ability to handle ombinatorial variables. However, it is less suitable for handling ontinuous variables representing, for example, inventory levels, prodution levels and transportation quantities. Further, onstraint programming methods do not typially, in addition to use re-planning, enompass tehniques for handling the dynamiity (i.e. frequent hanges to the planning task). Stohasti optimization tehniques may provide means for handling the dynamiity if useful information of the unertainty is obtainable. However, sine it is generally more diffiult to use stohasti optimization problems, effiient re-planning may be a more effiient approah. Agent-based omputing has often been suggested as a promising tehnique for problem domains that are distributed, omplex and heterogeneous [7]. The investigated problem in this researh projet, possess these harateristis. A number of agent-based approahes have been proposed to solve different types of resoure alloation problems [8]. An agent-based approah supports the handling of dynamiity (by whih it is implied that the planning situation may hange rapidly and a response is required in a short amount of time). Further, the multi-agent based approah an support the situation when limited information is available for the planner to make deisions or plans. Moreover, the agent-based approah supports the designing of a solution method whih enompasses the solution approah of real planners and dispathers. The above mentioned positive features of the multiagent based approah are typially not assoiated with or are at least not aentuated in optimization tehniques. Aording to a preliminary study [9], agent-based approahes tend to be preferable when: - the problem domain is large - the probability of node or link failures is high - the time-sale of the domain is short - the domain is modular in nature 1
- the struture of the domain hanges frequently - there are sensitive information that should be kept loally and mathematial optimization tehniques when: - the ost of ommuniation is high - the domain is monolithi in nature - it is important that a system optimal (or near optimal) solution is found - it is important that the quality of the solution an be guaranteed We propose a solution method apitalizing on some of the relative merits of optimization tehniques and some of the relative merits of agent-based approahes. In partiular, we explore for the optimization tehniques, the ability to ahieve system optimality or near optimality with a quality assurane; and for the agent-based approah, we explore, the ability to handle a large problem domain and a short time-sale of the domain. The proposed solution method utilizes the solutions and dual pries obtained from a deomposition optimization tehniques in order to strengthening an agent-based approah. The solutions onstitute potential good plans for the agents to implement. The dual pries onstitute values of resoures to be used in the ooperation between the agents for finding good solutions from a system perspetive. In this paper, we first desribe the problem domain with some speifis of a studied transport hain in Setion 2. In Setion 3, we introdue the roles of the planners involved and map this to a rather generi agent-based desription of the planning in a transport hain. The problem is formalized into an optimization formulation suitable for deomposition in Setion 4. A deomposition based approah is outlined in Setion 5. The suggested integration of the deomposition approah and the agentbased approah is outlined in Setion 6 and onlusions are presented in setion 7. 2. Problem Desription The type of transport hain onsidered onsists of one produer, a fleet of truks, and a number of ustomers, se Figure 1. Figure 1 A transport hain onsisting of a produer, a fleet of truks, and ustomers. The produer has limited prodution apaity and limited storage apaity of finished produts. The produer an produe a number of different produts. The transport operator uses a fleet of truks for transporting the produts to the ustomers. The truks may also be used for other transports outside the studied transport hain when opportunities arise. However, the spot market is onsidered to be rather limited (due to the speial requirements of food transports in the studied ase). The transport operator may send its truk to one or more ustomers with respet to an outbound transport from the produer. At the ustomer, whih may be a produer, there is often a limited inventory apaity available. These harateristis of a transport hain an be found in a hain onsisting of the produer Karlshamns AB, the transport operator FoodTankers, and a typial ustomer to Karlshamns AB, Proordia Foods. An important ategory of produts produed at the Karlshamns AB is Oils & Fats, whih is mainly sold on the Sandinavia market. We fous on bulk produts whih typially are transported by FoodTankers. FoodTankers uses about forty-five truks for transporting oils and fat produts. The transport quantities to a partiular ustomer are rarely less than 5 tons and rarely more than two ustomers are visited on an outbound trip from the produer. In the studied transport hain, it is ustomer orders that drive the transportation and prodution planning. The ustomer orders in turn are based on the ustomer s prodution planning typially omputed using MRP-logi, i.e. foreasted demand, available produts and safety stok levels, lead time et. At many ustomers of Karlshamns AB, produts may be stored before they are proessed further. Many of these ustomer orders to the produer are based on long-term (yearly) ontrats of how muh to order and to what pries. Further, the ontrats inlude speifiations of transport osts depending on the shipping quantities. Initements for ordering full truk loads are typially inluded in the ontrats. The ontrats also speify a minimum lead time between order and expeted delivery. The prodution and transport planning has a rather short-term planning horizon (up to two weeks) and the part of the planning horizon normally onsidered for replanning is 2-4 days. The (short-term) planning and re-planning is initiated by a ustomer orders. Currently the first planning step is to deide whether a new order an be aepted or not. If it is within the boundaries of ontrats between produer and ustomer it is almost always aepted (else a penalty is inurred). The subsequent prodution (re-)planning is about deiding when and in whih order the produts should be produed; and about planning the inventory storage. Lotsizing may apply by integrating different ustomer orders in the prodution. In this planning step, a bakward 2
sheduling is used respeting expeted time required for delivery to the ustomers. Inventory storage apaities of finished produts are rather limited and the produts are often proessed in the last prodution step at the same day as of delivery. Another motive for proessing and deliver the produts on the same day is requirements on the freshness of the produt. However, no hard onstraints for proessing and deliver in the same day exist. The transport planning onerns meeting the transport requirements speified by ustomer orders obtained via the produer. Currently not muh flexibility exists with respet to altering the delivery times and quantities of transport orders. However, this may hange partly as a result of this researh projet, whih may show on the potential benefits from a system perspetive of inreasing this flexibility. A hallenge in the transport planning is to meet the requirements with existing transport apaity, without using overtime or hiring extra apaity. Further, the hallenge is to explore additional transport options enabling a higher utilization of the truks by inreasing the loads when moving from ustomers bak to the produer or loations lose to the produer. When transport orders appear whih require high ost solutions, the transport planner may suggest hanges to the produer whih in turn hek for suh hanges with the ustomer. Suh ontats may also be initiated when relatively small hanges to the time and quantities of transport orders an deliver signifiant ost savings. However, urrently it is rather hard to identify suh potential ost savings. The transport planning (fleet management) is arried out by a number of different planners eah responsible of a partiularly ustomer area. Eah truk is normally assigned to suh a ustomer area, but frequent temporary hanges are done to this assignment. The transport planners are ooperating losely and are sitting in the same room with frequent ontats, whih is helpful when reassigning truk to different ustomer areas. Prodution planner Transport hain oordinator Transport planner Customer Customer Figure 2 Deision making system of agents handling the transport hain. This is in large a simplifiation of the general framework suggested in [10]. Here we onsider a less general transport hain. Storage is only allowed at the produer or at the ustomer and only a single produer is onsidered. However, the transport planning of a single transport operator s fleet may here be handled by a number of agents and not just by a single agent. We have inluded a number of agents for representing the transport planners eah involved in the transport planning for a partiular ustomer area. This may not be the ase in a general setting of the problem. The roles of the agents are further desribed in Table 1. 3. Roles for Planning in an Agent Perspetive In an agent perspetive, we have identified a number of important roles involved in the proess of planning the transports and prodution. These are: - Customer agent - Transport hain oordinator agent - Prodution planner agent - Transport planner agent These roles are onneted to the transport hain as illustrated in Figure 2. 3
Table 1 The roles of the agents in the deision system. Deision Deisions and Based on Goal maker ations Customer agent Transport hain oordinator Prodution planner Transport planner Makes requests of produts with respet to quantities, time of delivery (or time window). Deides how muh should be shipped and when. Makes requests to prodution and transport planners. Provides the best offer to the prodution request. Gives prodution orders to the produer. Provides the best offer to the transport request. Assigns transport tasks to transport vehiles Antiipated ustomer demand and inventory levels at the ustomer. Requests from the ustomer agents, and on the answers to the request of transportation and prodution. Prodution apaity, storage levels at the prodution site. Transport apaity and status (availability, position, et) of the transport vehiles ontrolled by the operator. Mediate ustomer requirements inluding osts for different delivery options. Satisfy the ustomer requirements at the lowest possible system ost. Minimize prodution osts. Minimize transport osts. The prodution planner agent and transport planner agents should provide answers to the requests of the transport hain oordinator agent without implementing them. The transport hain oordinator agent will inform whih requests and answers that should be implemented. The suggested hierarhial design of the deision making system of agents allows for the study of different levels of ooperation. In an extreme, but rather ommon, ase, the agents have pure loal objetives (loal ost minimizer) with virtually no sharing of information. In another extreme ase, they are fully ooperative with the objetive of minimizing total ost of the system. Sine we want to improve the performane of the system by reduing total osts, we strive for the ase when the agents are fully ooperative. It is assumed that eah agent has a plan of ations to take (urrently and in the future). Then, inline with Table 1, agents are ooperative and will aept a loal inrease of osts if it is assured that ost dereases as muh or more in other parts of the system. Sine we strive for system optimality it is important that the deision agents an handle rather omplex offers implying hanges to urrent plan. For example, the prodution planner may be given an offer to realloate the prodution of a number of prodution orders to new dates for potentially ahieving prodution plans whih better math the transport requirements. There are many possible mappings between organizations and the different deision making agents, see [10] for some examples. In the extreme ase, all deision makers belong to the same organization for a transport hain, e.g., petroleum ompanies. Another extreme, is where all deision makers belongs to different organizations. Also, intermediate arrangements exist suh as in the studied transport hain. In the studied transport hain, the transport hain oordinator agent and prodution planner agent are assoiated with the produing ompany. The ustomer agent and the transport planner agents are assoiated with the ustomer and the transport operating ompany, respetively. In the studied ase, the funtion of the transport hain oordinator is mainly arried out by a ustomer order reeiver, whih ross-hek for problems with prodution and transportation. In ase of problems, e.g. hard to meet the ustomer request, the prodution planner and transport planner may ooperate diretly for finding solutions. Although not ommon pratie, the negotiations of hanges to the ustomer orders (quantity and time of delivery) with the ustomer may also involve the prodution planner and transport planner diretly. 4. Optimization Model Next we formulate the integrated prodution and transportation planning problem. We hoose the objetive of minimize total ost inluding penalties for not meeting ustomer requests. Sine the aim of this paper is not to explore solution methods for the details of the problems, suh as lot-sizing and fleet management, the formulation is somewhat simplified. Thus, we will present the problem and a solution method at general level for illustrating the potential of integrating agent-based and deomposition approahes. In order to formulate the problem we use the notation p for a produt in the set P, t for a time period in the ordered set T, and j for a ustomer in the set J. Further, we use the index for denoting suh ativities assoiated with a partiular ustomer luster C, where C is the set of ustomer lusters. The ustomers in a luster are represented by the set and a ustomer j belongs to a single suh luster. The problem an be formulated using the following variables: - x pt, prodution of produt p in time period t, - z, quantity of produt p dispathed from the produer in time period t to ustomer j. - y, quantity delivered to ustomer j of produt p in time period t, - o pt, quantity in storage at the produer of produt p in time period t, - u, quantity in storage at ustomer j of produt p in time period t, and the parameters: 4
- d, quantity demanded by ustomer j of produt p in time period t. The reason for introduing both the variables z, and y, is that the delivery to ustomer may our in a time period later than the time period when the produt was loaded into a truk at the produer. For onveniene of notation we refer to the equivalent vetor variables by dropping the indies. We introdue a general vetor variable v, whih represents the usage of truks. We introdue this variable sine the transport hain oordinator may onsider a number of transport options. We avoid introduing an exat definition of the variable v sine this allows for different deomposition approahes introdued later. We assume we have expressions of the ost funtions aording to: - f 1 (x), prodution ost of produing the produts at the produer, - f 2 (v,y,z), transportation osts, - f 3 (o), inventory osts at the produer, - f 4 (u), inventory osts at the ustomers inluding penalty osts for not meeting the required demand. Note that in order to ompute these osts, a orresponding optimization problem may need to be solved, inluding additional variables and onstraints, whih we leave out for reasons of onveniene. We denote, for short hand notation, variables onneted to a ustomer luster C, with the index aording to u, v, z, and y. Here we assume the transportation osts represented by funtion f 2 (v,y,z), an be separated for eah ustomer luster, suh that, C f ( v, y, z ) 2 = f 2 ( v, y, z). The same assumption applies for the separation of the ost funtion for inventories at the ustomers (f 4 (u)). Additionally we use the notation, X, V, O, and U to denote the feasible set of the variables aording to: x X, (v,y,z) V, o O, u U. The integrated transport and prodution planning problem an be formulated aording to: min {f 1 (x) + f 2 (v,y,z) + f 3 (o) + f 4 (u)} s.t. o p,t-1 + x pt z = o pt j J p P,, (1) u jp,t-1 + y d = u j J, p P,, (2) x X, (v,y,z) V, o O, u U. The onstraints (1) model the inventory balane between time periods at the produer of the different produts. In onstraints (2), the equivalent inventory balanes at the ustomers are modeled. The sets, X, V, O, and U, may ontain rather omplex restrition on the variables. Further, they may impliitly inlude additional variables representing issues assoiated with prodution, storage, and transportation. However, these issues an be seen as hidden in separate parts representing prodution, transportation, and onsumption with the feasible sets X, V, O, and U. The set X may ontain restrition and variables handling the sequening of prodution of different produts. An important issue is how the transportation (v) is modeled and onneted to dispathing (z) and delivery (y). For instane, the set V onnets the quantities delivered from the produer to the ustomer whih may ontain a model of the possible routes for the truks. In the feasible set U (inventory levels at the ustomer) and in the assoiated ost f 4 (u), penalties for not meeting inventory requirements and osts assoiated with deliveries at non-preferred dates and times may be inluded. 5. Deomposition Deomposition of optimization problems is generally motivated by the algorithmi performane of the approah. Speifi motives an, for example, be onneted to strengthening the linear relaxation, to handle a large problem in terms of variables and onstraints, or to generate good solutions heuristially. For a given optimization problem there are typially a number of ways to design the deomposition. It is ruial to hoose a suitable deomposition approah for ahieving algorithmi effiieny. In the present work, the main motive is to outline how the elements of the deomposition an resemble the roles and agents identified above and to get valuable input to how the different agents may ooperate in an effiient way. For further reading on deomposition we suggest [1,11]. A possible deomposition approah is in effet to Lagrange relax onstraints (1), and (2), using Lagrange multipliers µ, and λ, respetively. In the following we refer to the multipliers as dual pries sine we view them in a Dantzig-Wolfe deomposition setting. By relaxing these two groups of onstrains, a for the prodution, a for transportation to eah ustomer luster C, and ustomer inventory, an be obtained. The following problem represents the produers problem: min {f 1 (x ) + f 3 (o ) + ( o + x )) s.t. x X, o O. ( pt p, t 1 pt-o pt p P µ } For eah ustomer luster C, we have a transport optimization : 5
min{f 2 (v,y,z )- y λ } s.t. (v,z,y ) V. µ + pt z For eah ustomer luster C, we have a ustomer inventory : min { f 4 (u)+ λ ( u jp, t 1 u ) } s.t. u U An important and entral part in the problem above, is the part modeling the transports inluding dispathing (z) and delivery (y), that is the set V. Here, we suggest a rather rudimentary, but onrete, modeling of the transportation problem. For this purpose we introdue the variable v t, whih is the number of truks alloated to ustomer luster in time period t and parameter e t, whih represents the number of truks available in time period t. In addition, we use g to denote the truk apaity (here assumed to be equal for all truks). Further, we introdue an indiator, ξ j, representing the number of time periods required for transportation between dispathing at the produer and delivery to a ustomer j. Then we an onsider onstraint: C v <=e t, (3), t and the optimization problem min {f 1 (x) + f 2 (v,y,z) + f 3 (o) + f 4 (u)} s.t (1 ), (2) and (3), whih onstitutes the base for the oordinating problem (or master problem), where the Lagrange multiplier τ is used for onstraint (3). For eah ustomer luster C, we now have the transport optimization : min{f 2 (v,y,z )- λ y + t T µ + pt z τ tv t } s.t. z = y jp,t+ξj j J, p P,, j J p P z gv t (v,z,y ) V., In the above s the onstant ontributions to the objetive funtions have been ignored (e t τ ), sine these ontributions do not affet the plans generated, represented by variables: x, o, v, z, y, u. However, in a Dantzig-Wolfe deomposition, these ontributions must be aounted for when deiding on whether to terminate the solution proedure or not. In this rudimentary ase, the set V may simply inlude non-negativity onstraints on the variables and integer onstraints on v t. In a more detailed representation of the transportations, a set of truk shedules an be pregenerated and from whih eah truk has to selet suh a shedule in eah time period. Additionally one may use deomposition approah where shedules are generated for the truks in the s and the seletion of whih shedule to use is made in the oordination problem (see [12] for a similar approah). For illustration purpose, we assume that the transportation problem is modeled rudimentary as suggested above in the following. Figure 3 outlines the different optimization problems and the information sent between the s and a oordination problem. The oordination problem and the s are solved interhangeably (all the subproblems in parallel). The role of the oordination problem (the restrited master problem) is to find the best solution given the solutions generated so far. While doing so, it will ahieve its most important role: to generate new dual pries to be ommuniated to the subproblems. µ Prodution Coordination problem µ, λ,τ v, z, y, µ λ x, τo, Transport Figure 3 Outline of the information sent in a deomposition approah of the onsidered problem. In our appliation, we have assumed that the ustomer s prodution plan has been fixed, and we are onsidering the ustomer demand d as a parameter. This implies that there is little flexibility in the ustomer inventory ; it is only the inventory levels that an be planned. Then it might be useful to onsider the ustomer inventory levels together with transportation planning. In suh ase, for eah ustomer luster C, we have a transport and ustomer inventory optimization subproblem: λ u Cust. Inventory 6
τ,o, µ λ, Proeedings of the 38th Hawaii International Conferene on System Sienes - 2005 min {f 2 (v,y,z ) + f 4 (u) - λ ( jp, t 1 u + y u ) + s.t. z = y jp,t+ξj j J, p P,, j J p P z gv t (v,z,y ) V, u U., µ + pt z τ } tv t In Figure 4, the ase of ombined transport and ustomer inventory s are outlined. µ Prodution x Coordination problem µ, λ,τ 6. Enhaned Agent-based Deisions by Deomposition The basis for the planning of the studied problem is the agent-based deision making system, presented in Figure 12. For this system to strive towards system optimality, suitable requests and answers need to be sent between the agents. Information provided by the deomposition approah an assist in this proess. Either suh information an be obtained after termination of the deomposition but more effiiently during the progress of the proess. In partiular, the transport hain oordinator agent an from the oordinator problem obtain information about urrent best offers of prodution and transportation. Also, the agents an reeive, by the dual pries, approximate information about prodution ost of the different produts ( µ ), ost of meeting ustomer demand at the different ustomers ( λ ), and ost of transport resoures (τ ), for the different time periods. These pries are obtained from the deomposition approah (e.g. the Dantzig-, µ λ v, z τ, y, u, Transp.& Inv Figure 4 Outline of the information sent in a deomposition approah in ase of ombined transport and ustomer inventory sup-problems. Figures 3 and 4 represent the solution proedure needed for solving a type of linear relaxation to the problem. The oordination problem (e.g. a Dantzig-Wolfe restrited master problem formulation) and the subproblems will iteratively be solved until the proess onverges, i.e. until no improving solutions an be found. The oordination problem provides new updated dual pries µ, λ, and τ, based on previous values of variables for prodution, transportation, and inventory levels (i.e. based on solutions generated). In the prodution, a prodution plan is developed given the values of inventories of produts in different time periods, i.e. given by µ. Hene the prodution planning is simplified ompared to the real prodution planning problem sine in the the exat quantities of transports an be ignored. In the real prodution planning problem, the transport quantities are given by bakward sheduling of ustomer orders. Issues of sequening and start-ups of prodution may be aounted for in the by introduing them in the feasible set X. In the transport, transports are onsidered and expliit routing of the truks an be onsidered. It is simplified ompared to real-life transport planning and fleet management, sine the quantities to transport are not ditated by the prodution and ustomer requirements expliit, but by urrent values/osts of the produts in different time periods, i.e. the values µ and λ. The values of τ, represent ost for using truks in different time periods. In this a number of rules onerning the truks and drivers, suh as, rules for drivers driving hours an be inluded in the feasible set V. The ustomer inventory is a rather trivial problem, when the demand is fixed. It is about deiding optimal inventory levels with respet to values of inventory levels at different time periods and inventory onstraints. Alternatively, this an be aounted for in an integrated transport and ustomer inventory, as suggested above. In order to solve this problem using the above sheme for Dantzig-Wolfe deomposition, a strategy for handling the ombinatorial aspets is typially required. A single prodution and a single transport plan for eah ustomer must be hosen among those generated in the subproblems. However, the output from the Dantzig-Wolfe deomposition approah onsists of an optimal onvex ombination of the different plans. For ahieving optimality, a branh and bound approah or at least a limited tree searh must be applied on some suitable harateristi of the problem. Suh a harateristi an onern the exat time period a ustomer should be visited. Conerning a limited tree searh, one may implement a approah of fixing more and more of the prodution and transportation plans until all parts of the plans are deided, see [12] for an example. 7
Wolfe). Note that these dual pries represent values/osts from a system perspetive of produts or resoures in different time periods. These dual pries reates redibility with respet the ost onsequenes to the whole system of the solution sine alternative solution an approximately be evaluated by all agents whih have aess to these dual pries. Knowing the potential effet of hanges to the plan at other part of the system, at least by approximate osts, an failitate a benefiial utilization of flexibility (e.g. prodution an be set to our earlier or later for mathing transport requirements). If the planning situation suddenly hanges (e.g. new ustomer orders appear), the previous dual pries might be aurate enough for assisting the agent to find new good solutions. Hene, we propose to run the agent-based deision making system in parallel with the Dantzig-Wolfe deomposition approah as outlined in Figure 5. Agent-based deision system Partial fixation of the plans µ,λ,τ x, o, v, z, y, u Dantzig-Wolfe deomposition Figure 5 Interation between the Agent-based deision system and the Dantzig-Wolfe deomposition approah. In a basi setting, the agent-based deision system sends information on whih fixations that are made in the plan. These fixations represent parts of the plans whih not easily an be re-planned from a real planning perspetive. In a more advaned version, it ould send some potential but not neessary fixations and with different penalties for undoing the fixations, in order more thoroughly explore the potential improvements of urrents solution/plan. In the proposed basi setting of the Dantzig-Wolfe deomposition approah, it obeys the urrent fixations but ignore the ombinatorial requirements. It generates new olumns until new improving olumns annot be found. Whenever requested by the agent-based deision system, it delivers the urrently best dual pries and plans. It may also suggest a few of the different extreme plans making up the best solution found so far. In a more advaned version, it may be useful within the Dantzig-Wolfe deomposition approah, to handle some or all of the ombinatorial issues. Then some additional fixing of the plans ould our in the master and thereby potentially provide the agent-based deision system with solutions of higher quality. The solution methods required for solving the subproblems an also be used in the agents of the deision making system. Then the agents an at, at least temporally, more independently and is not dependent on other more advaned approahes. 7. Conlusion and Future Work A deomposition approah has been suggested whih should support the agent-based deision system. The deomposition approah should provide the deision system with useful partial plans (e.g. prodution or transportations). Further it should provide the system with pries or resoures assisting the negotiations and inrease the aeptane of loally deteriorating solution. Based on experienes from [10] and [12] experiments an ommene of the studied ase. Then an interesting issue is to study different strategies for taking advantage of the information from the deomposition approah more losely. One may also test approahes for handling the planning within the deision system by mainly relying on loal searh approahes ompared to the in this paper suggested approah based on deomposition. Aknowledgements This work is primarily arried out within the projet Integrated Prodution and Transportation Planning within Food Industry (see www.ipd.bth.se/fatplan) with founding from the Swedish Knowledge Foundation (www.kks.se). Further VINNOVA, the Swedish Ageny for Innovation Systems, is in part finanially supporting this researh. Referenes [1] L.S. Lasdon. Optimization Theory for Large Systems, Mamillan. (1970) [2] C. Barnhart, E. L. Johnson, G. L. Nemhauser, M. W. P. Savelsbergh and P. H. Vane, Branh-and-Prie: Column Generation for Solving Huge Integer Programs, Operations Researh, Vol. 46. (1998) [3] L.A. Wolsey, Integer Programming, Wiley. (1998) [4] U.S. Karmarkar and L. Shrage. The deterministi dynami produt yling problem, Operations Researh, Vol. 33, 326-345. (1985) [5] P. Brandimarte and A. Villa, Advaned Models for Manufaturing Systems Management, CRC Press, In. (1995) [6] L.A. Wolsey, MIP modelling of hangeovers in prodution planning and sheduling problems, European Journal of Operational Researh, Vol. 99. (1997) [7] M. Wooldridge: An Introdution to MultiAgent Systems, Wiley, ISBN: 0-471-49691-x (2002) [8] H.V.D. Parunak. Industrial and Pratial Appliations of DAI, Multiagent Systems, (editor G. Weiss). MIT Press (1999) 8
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