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A branch-and-price algorithm for an integrated production and inventory routing problem

A branch-and-price algorithm for an integrated production and inventory routing problem
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  A branch-and-price algorithm for an integrated production andinventory routing problem  Jonathan F. Bard a,  , Narameth Nananukul b a Graduate Program in Operations Research and Industrial Engineering, 1 University Station C2200, The University of Texas, Austin, Tx 78712-0292, USA b Optimize Science, 100 Hepburn Road, Suite 1D Clifton, NJ 070122, USA a r t i c l e i n f o Available online 25 March 2010 Keywords: Production planningLot-sizingInventory routingColumn generationBranch and price a b s t r a c t With globalization, the need to better integrate production and distribution decisions has become evermore pressing for manufacturers trying to streamline their supply chain. This paper investigates apreviously developed mixed-integer programming (MIP) model aimed at minimizing production,inventory, and delivery costs across the various stages of the system. The problem being modeledincludes a single production facility, a set of customers with time varying demand, a finite planninghorizon, and a fleet of homogeneous vehicles. Demand can be satisfied from either inventory held at acustomer site or from daily product distribution. Whether a customer is visited on a particular day isdetermined by an implicit tradeoff between inventory and distribution costs. Once the decision is made,a vehicle routing problem must be solved for those customers who are scheduled for a delivery.A hybrid methodology that combines exact and heuristic procedures within a branch-and-priceframework is developed to solve the underlying MIP. The approach takes advantage of the efficiency of heuristics and the precision of branch and price. Implementation required devising a new branchingstrategy to accommodate the unique degeneracy characteristics of the master problem, and a newprocedure for handling symmetry. A novel column generation heuristic and a rounding heuristic werealso implemented to improve algorithmic efficiency. Computational testing on standard data setsshowed that the hybrid scheme can solve instances with up to 50 customers and 8 time periods within1h. This level of performance could not be matched by either CPLEX or standard branch and price alone. &  2010 Elsevier Ltd. All rights reserved. 1. Introduction Integrating production and distribution decisions is a challen-ging problem for manufacturers trying to optimize their supplychain. At the planning level, the immediate goal is to coordinateproduction, inventory, and delivery to meet customer demand sothat the corresponding costs are minimized. Achieving this goalprovides the foundations for streamlining the logistics networkand for integrating other operational and financial components of the organization. In this paper we analyze a single productionfacility that serves a set of customers with time varying demandover a finite and discrete planning horizon. The capacity of thefacility is assumed to be limited and each day production isscheduled, a setup cost is incurred. The focus is on the case wherea routing problem must be solved daily to either restockinventory, meet that day’s demand or both. When the associateddecisions are made in a coordinated fashion, we have what isreferred to as the production, inventory, distribution, routingproblem (PIDRP).Lei et al. [1] were the first to formulate the PIDRP as a mixed-integer program (MIP) and proposed a two-phase solutionapproach that avoided the need to address lot-sizing and routingsimultaneously. Boudia et al. [2,3] developed a similar MIP andproposed both a memetic algorithm with population manage-ment (MAPA) and a reactive greedy randomized adaptive searchprocedure (GRASP) with path-relinking [4] as solution methodol-ogies. Following their lead, Bard and Nananukul [5] developed atabu search procedure that provided slightly better results. Theirmodel was based on the MIP in Boudia et al. [3] but is moreefficient in terms of variable definitions and number of con-straints. The same model is used in this paper.Manufacturers who resupply a large number of retailers on aperiodic basis continually struggle with the question of how toformulate a replenishment strategy. One popular approach is abalanced strategy in which an equal proportion of retailers isreplenished each day of the work week. This has the advantage of evening the workload at the warehouse. A second strategyadvocated by many practitioners is synchronized replacement inwhich all retailers are replenished concurrently and goods are ARTICLE IN PRESS Contents lists available at ScienceDirectjournal homepage: Computers & Operations Research 0305-0548/$-see front matter  &  2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2010.03.010  Corresponding author. Tel.: +15124713076; fax: +15122321489. E-mail addresses: (J.F. Bard), (N. Nananukul).Computers & Operations Research 37 (2010) 2202–2217  ARTICLE IN PRESS moved into the manufacturer’s warehouse immediately prior todistribution. This creates unbalanced workloads at the warehousebut allows for cross-docking of a significant portion of the goods[6]. For just-in-time suppliers, it is common to partition thecustomers into compact groups and follow the same deliverysequence daily, skipping those locations with absent demand [7].Rather than treating the replenishment strategy in partialisolation, we take a more integrated view and develop a branch-and-price-based (B&P) scheme for solving the PIDRP as acomplement to existing metaheuristics. This is our majorcontribution. Secondary contributions include the design of methods for dealing with symmetry during branching and theuse of heuristics to achieve integrality. As part of the solutionprocess, four critical decisions have to be made: how many itemsto manufacture each day, when to visit each customer, how muchto deliver to a customer during a visit, and which delivery routesto use. Because the last decision requires the solution of a vehiclerouting problem (VRP) each day, the PIDRP is evidently NP-hard inthe strong sense, so it is not likely that large instances will betractable for more than a few time periods. With this limitation inmind, we propose several compromises to the exact algorithmthat involve the solution of a lot-sizing problem to estimatedelivery quantities in each period and the use of a VRP heuristic.In the next section, the literature related to the primarycomponents of the PIDRP is reviewed with an emphasis on recentresearch. In Section 3, a formal definition of our version of theproblem is given along with a new MIP formulation. The B&Palgorithm is described in Section 4. In Section 5, we discuss ourapproach to dealing with symmetry along with the details of anenhanced branching strategy. Heuristic ideas are outlined inSection 6, and in Section 7, computational results are presentedfor a wide range of problem instances. Section 8 offers severalideas for extending the work. 2. Literature review There is a vast quantity of literature on each component of thePIDRP so we will only highlight the most relevant work. A vendormanaged inventory replenishment (VMI) system is a goodexample of the type of integration that we wish to address (e.g.,see [8, 9]). In the VMI model, the manufacturer observes andcontrols the inventory levels of its customers, as opposed toconventional approaches where customers monitor their owninventory and decide the time and amount of product to reorder.One of the benefits of VMI is that it permits a more uniformutilization of transportation resources leading to lower distribu-tion costs. Customers benefit from higher service levels andgreater product availability due to the fact that vendors can useexisting inventory data at their customer sites to more accuratelypredict future demand [10, 11].Lei et al. [1] studied a multi-facility, heterogeneous fleetversion of the PIDRP that was motivated by a chemicalmanufacturer with international customers. In phase one of theirtwo-phase methodology, they solve a reduced model in whichtransporter routings were restricted to direct shipments betweenfacilities and customer sites. The results provided a productionschedule and the number of items to be delivered to eachcustomer in each period. In phase two, a routing heuristic wasproposed based on an extended optimal partitioning procedurethat consolidated the less-than-truckload assignments obtainedin phase one into more efficient delivery schedules. The overallapproach contained several novel features but with limitedapplicability to the general problem.As mentioned, Boudia et al. [2,3] proposed both a MAPAand a reactive GRASP with path-relinking to solve the PIDRP.Their model included a single facility and a set of customerslocated on a grid. Holding costs at the customer sites wereassumed to be negligible compared to the holding costs at thefactory and so were ignored. As in our case, the objective was tominimize the sum of production, holding and transportation costswhile ensuring that all demand was satisfied over the planninghorizon. An improved version of the MAPA is presented by Boudiaand Prins [12]. Testing was done on a set of 90 benchmarkproblems.Similarly, Bard and Nananukul [5] developed a reactive tabusearch algorithm for solving the PIDRP. An essential component of their methodology was the use of an allocation model in the formof a MIP to find good feasible solutions that were used as startingpoints for the tabu search. The neighborhood consisted of swapand transfer moves. Path-relinking was also used in a post-processing phase to seek out marginal cost reductions. Testing onthe 90 benchmark instances demonstrated the effectiveness of theapproach. In all cases, improvements ranging from 10–20% wererealized in comparison of the results obtained from the above-mentioned GRASP, but often at a greater computational cost.When routing is a dominant consideration, the PIDRP is mostsimilar to the inventory routing problem, IRP [13–17] and theperiodic routing problem, PRP [18–21]. Although there has beenmuch research on these two problems, little of it carries over tothe PIDRP. The primary reason relates to the formidable complex-ity of its structure, as defined by a combination of a capacitatedlot-sizing problem (e.g., see [22]) a capacitated, multi-period VRP.The full PIDRP has so far proven to be beyond the capability of exact methods. By decoupling of the lot-sizing and routingdecisions, though, several researchers have had some success infinding good solutions with heuristics. Chandra and Fisher [23],for example, first determine a production schedule without regardto the logistics. Next, they develop a distribution schedule foreach planning period based on the results obtained from the first-stage model. This approach worked well when there was enoughinventory in the system to buffer production from the distributionoperations, but consequently led to increased holding costs.Similar to our work, Savelsbergh and Song [24] went beyondthe traditional IRP and included a production component withlimited capacity and the need to route customers over severaldays in their model. While we use out-and-back trips each day,they allowed for sleeper teams that are on the road for a week ormore to cover wide geographic areas. To find solutions, theydeveloped an integer programming-based algorithm and em-bedded it in a randomized greedy heuristic with a local searchfeature.In a prior study, Christiansen [25] presented a ship routing andscheduling problem with production and inventory considera-tions. In the model, a fleet of heterogeneous ships transport asingle product (ammonia) between harbors, some of which areproduction sites and others consumption sites. Since the quan-tities loaded and discharged are determined by the productionrates of the harbors, possible stock levels, the actual ship visitingthe harbor, and the fact that one company owns all facilities,inventory cost is not an issue. The objective is to minimizetransportation costs while meeting time window constraints andproduction targets. A multicommodity integer programmingformulation is given and solved with a B&P algorithm.The IRP and the PRP are relaxations of the PIDRP, differing inseveral ways. Neither, for example, takes the production decisionand inventory level at the factory into consideration. In addition,the PRP assumes that the delivery patterns defined by deliveryfrequencies or delivery days are given in advance and tries toselect the most suitable pattern for each customer.In their version of the PRP, Mourgaya and Vanderbeck [20]proposed a column generation-based heuristic to fix dates for  J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217   2203  ARTICLE IN PRESS customer visits and to assign customers to vehicles. The dailysequencing decisions were left to an operational model. Zhao et al.[9] studied the integration of inventory control and vehiclerouting for a distribution system in which a set of retailers withconstant rates of demand were resupplied with a single productfrom a central warehouse. The objective was to determineinventory policies and routing strategies such that the long-runaverage costs were minimized and all demand was satisfied. Intheir model, and in contrast to ours, no inventory capacityconstraints were imposed on the warehouse or on the retailers.Testing was done on problem instances with 50 and 75 retailers.By analyzing the relationship between the lower bound and theoptimal solution, they were able to establish that the proposedalgorithm in conjunction with a fixed partitioning policy waseffective and robust in almost all cases. 3. Model formulation We are given a set of   n  customers geographically dispersed ona grid and a single facility for producing a unique item. Distancesare calculated with the Euclidean metric and accordingly satisfythe triangle inequality. Over a planning horizon of   t  periods, eachcustomer  i  has a fixed nonnegative integral demand  d it   in period  t  that must be fulfilled; i.e., shortages are not permitted (for adiscussion of backordering, see, e.g. [26]). If production takesplace at the facility in period  t  , then a setup cost  f  t   is incurred, t  ¼ 0,1, y , t  1. A limited number of items (up to  C  ) can beproduced in each time period and a limited number  I  P  max  can bestored at a unit cost of   h P  . The definition of a period depends onthe context of the problem, and might be a shift, a day, or a week.We use a day in the description of the model.In constructing delivery schedules, each customer can bevisited at most once per day and each of   y  homogeneous vehiclescan make at most one trip per day. A limited amount of inventory, I  C  max , i , can be stored at customer  i ’s site at a unit cost of   h C i  , buttransshipments between customers are not permitted (cf. [27]).An additional assumption is that all deliveries take place at thebeginning of the day and arrive in time to satisfy demand for atleast that day if necessary. All production on day  t   is available fordelivery only on the following morning (this is common in foodproduction and distribution; e.g., see [28]) and all inventory ismeasured at the end of the day. Demand on day  t   can be met outof deliveries from the factory on day  t   and ending inventory onday  t   1 at the customer site. Initial customer inventory on day 0simply reduces demand on subsequent days, while initialinventory at the factory must be eventually routed. At the endof each day it is assumed that all vehicles return to the factoryempty, and at the end of the planning horizon all inventory isrequired to be zero. While the latter requirement was imposed toreduce the number of parameters in the experimental design,practical considerations often argue for positive, final inventorylevels, especially for short planning horizons. Nevertheless,restricting the final level to zero has no effect on algorithmiccomplexity or performance with respect to runtimes.Our goal is to construct a production plan and deliveryschedule that minimize the sum of all costs while ensuring thateach customer’s demand is met over the planning horizon.Implicit in the solution will be a daily distribution of surplusitems to be placed in inventory. Under the common assumptionthat unit production costs are constant, the decision to over-produce and hold items in inventory will be a function of thesetup cost at the factory, system holding costs, production andstorage capacities, vehicle routes, and daily demand (foran analysis of synchronized ordering under fixed order intervals,see [6]).In the development of the model, we make use of the followingnotation. Indices and setsi ,  j  indices for customers, where 0 corresponds to thefactory t   index for periods or days N   set of customers;  N  0 ¼ N  [ {0}; | N  | ¼ nT   set of days in the planning horizon;  T  0 ¼ T  [ {0} and | T  | ¼ t Parametersd it   (integral) demand of customer  i  on day  t D max it   upper bound on the maximum amount to be delivered tocustomer  i  on day  t D max t   upper bound on the maximum amount that can beloaded on a vehicle on day  t  y  number of available vehicles Q   (integral) capacity of each vehicle I  P  max  maximum inventory that can be held at the factory I  C  max , i  maximum inventory that can be held by customer  iC   (integral) daily production capacity of the factory c  ij  cost of going from customer  i  to customer  j f  t   setup cost at factory on day  t h P  unit holding cost at the factory h C i  unit holding cost at customer  i  site Decision variables x ijt   1 if customer  i  immediately precedes customer  j  on adelivery route on day  t  ; 0 otherwise  y it   load on a vehicle immediately before making a deliveryto customer  i  on day  t  p t   production quantity on day  t  z  t   1 if there is production on day  t  ; 0 otherwise I  P t   inventory at the production facility at the end of day  t I  C  it   inventory at customer  i  at the end of day  t w it   amount delivered to customer i on day  t Model f IP  ¼ Minimize X t  A T  X i A N  0 X  j A N  0 c  ij  x ijt   þ X t  A T   f  t   z  t  þ X t  A T  0 \ f t g h P  I  P t   þ X t  A T  \ f t g X i A N  h C i  I  C  it  ð 1a Þ subject to  I  P t   ¼ I  P t   1 þ  p t   X i A N  w it  , 8 t  A T   ð 1b Þ I  C  it   ¼ I  C i , t   1 þ w it   d it  ,  8 i A N  , t  A T   ð 1c Þ X i A N  w it  r I  P t   1 ,  8 t  A T   ð 1d Þ  p t  r Cz  t  ,  8 t  A T  0 \ f t g ð 1e Þ  p 0 Z X i A N  ð d i 1  I  C i 0 Þ I   p 0  ð 1f  Þ X  j A N  0  j a i  x ijt  r 1 ,  8 i A N  ,  t  A T   ð 1g Þ X i A N  0 i a  j  x ijt   ¼ X i A N  0 i a  j  x  jit  ,  8  j A N  ,  t  A T   ð 1h Þ X  j A N   x 0  jt  r y ,  8 t  A T   ð 1i Þ  y  jt  r  y it   w it  þ D max t   ð 1   x ijt  Þ ,  8 i A N  ,  j A N  0 ,  t  A T   ð 1j Þ  J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217  2204  ARTICLE IN PRESS w it  r D max it  X  j A N  0  x ijt  ,  8 i A N  ,  t  A T   ð 1k Þ 0 r I  P t  r I  P  max ,  0 r I  C  it  r I  C  max , i ,  8 i A N  ,  t  A T  \ f t g ;  I  P  t  ¼ I  C i t  ¼ 0 ,  8 i A N  ð 1l Þ  x ijt  A f 0 , 1 g ,  0 r  y it  r Q  ,  w it  A Z n  t þ  ,  8 i a  j A N  0 ,  t  A T  ;  z  t  A f 0 , 1 g ,  p t  A Z t þ  ,  t  A T  0 \ f t g ,  p t  ¼ 0  ð 1m Þ where  D max it   ¼ min  Q  , X t l  ¼  t  d il ( )  and  D max t   ¼ min  Q  , X i A N  X t l  ¼  t  d il ( ) The objective function in (1a) minimizes the sum of transporta-tion costs, production setup costs, holding costs at the factory, andholding costs at the customer sites. Because all demand must bemet, the variable component of the production costs, which areassumed to be linear and constant, can be omitted, as can anyinitial inventory costs. Constraints (1b) and (1c) are inventoryflow balance equations in which it is assumed that the initialinventories  I  P  0  and  I  C i 0  are given for all customers  i A N  . Constraints(1d) limit the total amount available for delivery on day  t   to theamount in inventory at the factory on day  t   1. The specificamount delivered to customer  i  is limited by the parameter  D max it  in (1k), which is the smaller of the vehicle capacity  Q   or thecumulative demand from day  t   to the end of planning horizon  t .When customer  i  is not visited on day  t  , the right-hand side of (1k)will be zero so no delivery is possible.Constraint (1e) bounds production on day  t   to the capacity of the factory. A simple way to tighten this constraint is to replace  C  with  C  t   ¼ min f C  , I  P  max , D max t  þ 1 g , where the third term is the demand of all customers for the remainder of the planning horizon. Theassumption that items produced on day  t   are only available fordelivery on day  t  +1 implies that  p t  r I  P  max  and  p t ¼ 0. It is possibleto strengthen the latter inequality by subtracting from  I  P  max  thereduction in inventory due to deliveries on day  t   to get  p t  r I  P  max ð I  P t   1  P i A N  w it  Þ , but this constraint is dominated by(1b). To ensure that demand on day 1 can be met, it is necessary toinclude (1f) which allows production on day 0. If   I  P  0  ¼ I  C i 0  ¼ 0, then  p 0 Z P i A N  d i 1  or the problem is infeasible.Constraints (1g)–(1j) represent the routing aspect of theproblem. Constraints (1g) and (1h) ensure that if customer  i  isserviced on day  t  , then it must have a successor on its route,which may be the factory. Route continuity is enforced by (1h);i.e., if a vehicle arrives at customer  j  on day  t  , then a vehicle mustdepart customer  j  on day  t  . Logic, and the fact that the fleet ishomogeneous, requires that it be the same vehicle.The number of vehicles that leave the factory on any day  t   islimited to the number available,  y , as indicated by (1i). Finally, (1j)keeps track of the load on the vehicles and guarantees that on day t  , if customer  i  is the immediate predecessor of customer  j  on aroute, then the load on the vehicle before visiting customer  j  mustbe less than or equal to the load just before visiting customer  i minus the amount delivered, which is represented by the variable w it  . The value of the parameter  D max t   is specified to be as small aspossible while ensuring that (1j) is always feasible. Because theload on each vehicle is monotonically decreasing as customers arevisited, (1j) provides the added benefit of eliminating subtoursthat do not include the factory. If a vehicle returned to apreviously visited customer  j  on day  t  , the result would be that  y  jt  o  y  jt  , a logical inconsistency. After all deliveries are made, thefleet returns to the factory empty so  y 0 t   can be set to 0 for all  t  A T  .To conclude the formulation, variables are defined in (1l) and(1m). It is straightforward to generalize model (1), to includemultiple facilities and multiple products.The size of model (1) is determined largely by constraint (1j)and the number of binary variables,  x ijt  , both of which grow at arate proportional to  O ( n 2 t ). The majority of the other constraintsonly grow at a rate proportional to  O ( n t ). A small problem with 3vehicles, 30 customers, and a 5-day planning horizon containsroughly 5200 constraints, 4600 binary variables, and 500continuous variables. Initial attempts to solve instances of thissize with CPLEX 8.1 were not encouraging. When a time limit of 90min was imposed, optimality gaps between 7% and 10% werethe norm. 4. Solution methodology  This experience led first to development of a tabu searchalgorithm [5] and then to an exact method based on branch andprice (B&P), which we discuss in this section. In simple terms, B&Pcombines (i) Dantzig–Wolfe (D–W) decomposition extended toaccommodate integer variables, and (ii) standard branch andbound (B&B) [29,30]. Below we outline how initial feasiblesolutions are obtained and then describe the principal compo-nents of our B&P algorithm. 4.1. Initial solutions All exact optimization methods depend on good bounds tospeed convergence. As a forerunner to this work, we developed athree-phase methodology centered on tabu search to solve thePIDRP. In phase 1, an initial solution is found by solving arelaxation of (1a)–(1m) called the  allocation model . In thisformulation the routing variables (  x ijt  ) and constraints (1g)–(1j)are removed and an aggregated vehicle capacity constraint isadded for each time period. Because the actual cost of making adelivery to customer  i  on day  t   cannot be determined withoutincluding the routing constraints, the cost term P ijt  c  ij  x ijt   in (1a) isreplaced with a lower bounding approximation.The solution to the allocation model provides customerdelivery quantities  w it   for all  i ¼ 1, y , n  and  t  ¼ 1, y , t . In phase 2,these values become the demand for  t  independent routingproblems. An efficient CVRP subroutine also based on tabu search[31] is called to find solutions. In phase 3, a full tabu searchalgorithm is used to improve the allocations and routings found inphase 2. 4.2. Column generation D–W decomposition like Lagrangian relaxation is mosteffective when the feasible region can be partitioned into a setof complicating or aggregated constraints and a set of easy ordisaggregated constraints. For the current problem, it is natural toseparate the production and inventory constraints (1b)–(1f),which cut across two time periods each, from the routing anddelivery constraints, which can be written separately for eachtime period. We use the former along with the objective functionin (1a) to create the D–W master problem  ð MP  Þ .After removing the production and inventory constraints, weare left with (1g)–(1k) and the variable definitions, whichconveniently decompose by time period giving  t  subproblems.Points in the feasible region of subproblem  t   ð SP  t  Þ  correspond toall feasible schedules on day  t  , where a schedule is a set of   y  orfewer tours specifying the delivery sequence and the amount tobe delivered if a customer is serviced on that day. In thereformulated model, each column in  MP   corresponds to afeasible schedule for the  n  customers. In a solution, at most onecolumn can be selected for each day.  J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217   2205  ARTICLE IN PRESS Informally speaking, the idea behind B&P is to have thesubproblems act as schedule generators guided by the values of the dual variables associated with the LP solution to the restricted MP  ; i.e., one that contains only a subset of feasible schedules. Ateach major iteration, an optimality check is made by implicitlypricing out  MP  . This is done by solving the subproblems to seewhether one or more schedules can be identified that hasnegative reduced cost. When added to the restricted  MP  , suchschedules in the form of columns will improve the current LPsolution, which serves as a lower bound on the solution to thesrcinal MIP; i.e., model (1). If all reduced costs are nonnegativebut some variable values are fractional in the MP   solution, B&B isperformed. 4.2.1. Master problem We first derive  MP   and then show how its solution isused to construct the objective function of the pricing subpro-blems. Let  k  be the column index and let all previous notation bethe same. The following new symbols will be used in theformulation of   MP  . Parameters and setsc  kt   cost of the schedule for day  t   associated with column  kW  k it   amount delivered to customer  i  on day t in the scheduleassociated with column  k X  k ijt   mapping parameter; 1 if customer  i  is the immediatepredecessor of customer  j  on day  t   in the scheduleassociated with column  k , 0 otherwise K  ( t  ) set of columns for day  t Decision variables l kt  (binary) 1 if the schedule associated with column  k  on day t   is selected, 0 otherwise Master problem  ð MP  Þ f IP  ¼ Minimize X t  A T  X k A K  ð t  Þ c  kt  l kt   þ X t  A T  0 \ f t g  f  t   z  t   þ X t  A T  0 \ f t g h P  I  P t   þ X t  A T  \ f t g X i A N  h C i  I  C  it  ð 2a Þ subject to X k A K  ð t  Þ X i A N  W  k it  l kt   þ I  P t    I  P t   1   p t   ¼ 0 ,  8 t  A T   ð 2b Þ X k A K  ð t  Þ W  k it  l kt   I  C  it   þ I  C i , t   1  ¼ d it  ,  8 i A N  ,  t  A T   ð 2c Þ X k A K  ð t  Þ X i A N  W  k it  l kt   I  P t   1 r 0 ,  8 t  A T   ð 2d Þ X k A K  ð t  Þ l kt  r 1 ,  8 t  A T   ð 2e Þ  p t   Cz  t  r 0 ,  t  A T  0 \ f t g ð 2f  Þ  p 0 Z X i A N  ð d i 1  I  C i 0 Þ ,  I   p 0  ð 2g Þ l kt  A f 0 , 1 g ,  8 t  A T  ,  k A K  ð t  Þ ;  z  t  A f 0 , 1 g ,  p t  Z 0 ,  8 t  A T  0 \ f t g ð 2h Þ 0 r I  P t  r I  P  max ,  0 r I  C  it  r I  C  max , i ,  8 i A N  ,  t  A T  \ f t g ;  I  P  t  ¼ I  C i t  ¼ 0 ,  8 i A N  ð 2i Þ The objective function in (2a) minimizes the total production,inventory, and distribution cost over the planning horizon. Theonly change from (1a) is that the routing cost has been replacedwith the cost of a schedule but these are really the same.Constraints (2b)–(2d), (2f)–(2g) are equivalent of (1b)–(1f),respectively, but written in terms of the schedule variables  l kt  instead of the srcinal variables ( w it  ,  x ijt  ). The parameters  ð W  k it  ,  X  k ijt  Þ are derived from the solution of the pricing subproblems, whichare presented next. The convexity constraint (2e) ensures that atmost one delivery schedule is selected for each day.When the integrality constraint on the  l kt   variables in (2h) isrelaxed, the solution to (2) provides a lower bound on the optimalsolution to (1). With the D–W procedure, this bound is usuallytighter than the bound obtained by solving the LP relaxation of (1)directly. 4.2.2. Pricing subproblems To formulate the objective function for  SP  t  , let  a 1 t   be the dualvariable associated with the factory inventory constraints (2b),  a 2 it  the dual variable associated with customer  i ’s inventory con-straints (2c),  a 3 t   the dual variable associated with the deliveryconstraints (2d), and  a 4 t   the dual variable associated with theconvexity constraints (2e). Only  a 3 t   and  a 4 t   are required to be non-negative. For each day  t  , the reduced cost of column  k  in  MP   is c  kt   ¼ c  kt    a 1 t  X i A N  W  k it   X i A N  a 2 it  W  k it   þ a 3 t  X i A N  W  k it   þ a 4 t   ð 3 Þ where c  kt   ¼ X i A N  0 X  j A N  0 c  ij  X  k ijt  To put (3) into a form that is usable in  SP  t  , the parameters ð W  k it  ,  X  k ijt  Þ  must be expressed in terms of the original problemvariables. Making the appropriate substitutions, collecting terms,and removing the column index  k , gives c  t   ¼ X i A N  0 X  j A N  0 c  ij  x ijt   X i A N  ð a 1 t   þ a 2 it   a 3 t   Þ w it   þ a 4 t   ð 4 Þ The first term on the right-hand side of (4) represents therouting cost on day  t   while the remaining terms are theadjustments imposed by the  MP   dual values. In formulatingthe subproblems, we retain all the routing and delivery con-straints (1g)–(1k) of the srcinal model.Subproblem t  ð SP  t  Þ f SP t   ¼ Minimize X i A N  0 X  j A N  0 c  ij  x ijt   X i A N  ð a 1 t   þ a 2 it   a 3 t   Þ w it   þ a 4 t   ð 5a Þ subject to X  j A N  0  j a i  x ijt  r 1 ,  8 i A N   ð 5b Þ X i A N  0 i a  j  x ijt   ¼ X i A N  0 i a  j  x  jit  ,  8  j A N   ð 5c Þ X  j A N   x 0  jt  r y  ð 5d Þ  y  jt  r  y it   w it  þ D max t   ð 1   x ijt  Þ ,  8 i A N  ,  j A N  0  ð 5e Þ w it  r D max it  X  j A N  0  x ijt  ,  8 i A N   ð 5f  Þ  x ijt  A f 0 , 1 g , 0 r  y it  r Q  , w it  Z 0 and integer ,  8 i a  j A N  0  ð 5g Þ Although it is common to try to solve (5) to optimality, thismay be too time consuming for large instances. All that is reallyneeded, though, is a feasible solution whose objective functionvalue in (5a) is negative. If one is found, then the correspondingcolumn is added to the set  K  ( t  ) in  MP  . Alternatively, if thecomputations are terminated before feasibility is achieved or if the smallest reduced cost of a feasible solution is nonnegative,then no column is generated by the subproblem.  J.F. Bard, N. Nananukul / Computers & Operations Research 37 (2010) 2202–2217  2206
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