Solving Scheduling Problems in Distribution Centers by Mixed Integer Linear Programming Formulations

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1 Solving Scheduling Problems in Distribution Centers by Mixed Integer Linear Programming Formulations Maria Pia Fanti Gabriella Stecco Walter Ukovich Department of Electrical and Electronic Engineering, Polytechnic of Bari, Bari, Italy, (Tel: ; Department of Industrial Engineering and Information Technology, University of Trieste, Trieste, Italy, ( Department of Industrial Engineering and Information Technology, University of Trieste, Trieste, Italy, ( Abstract: This paper considers the problem of scheduling the internal operations of a distribution center: a system to which goods arrive from several suppliers and are forwarded to several customers. Internal operations consist essentially in unloading incoming cases and loading outgoing ones. Thus, the internal operations are divided in three phases: de-consolidation, sorting and consolidation. The objective is to minimize the total operation time determining the optimal sequence of the operations in the three phases. Starting from a previous formulation, we propose two new Mixed Integer Linear Programming models assuming infinite buffer capacities, in order to deal with instances of larger size. Some preliminary results show promising prospectives for the effectiveness of the new approaches. Keywords: Scheduling; Distribution Systems; Integer programming; Optimization Problems. 1. INTRODUCTION Today companies are cutting costs by reducing inventory at every step of the operations. In order to reduce costs and improve efficiency, storage and retrieval, the two most expensive warehousing operations, can be eliminated if feasible.adistributioncenter(dc)isatypeofwarehouse where the storage of goods is limited or nonexistent [5]. In the DC under consideration, large incoming loads from different suppliers are disaggregated and combined to create consolidated outbound shipments to be sent to customers according to their requests. Inbound trucks are unloaded and the loads are unpacked. Successively, the single items are sorted according to the customer requests, packed and sent to the customers by outbound trucks. The relatedliteraturedealswith the schedulingofinbound and outbound trucks, in cross docking terminals [2] where the products do not need to be unpacked, sorted and packed but only moved from inbound to outbound docks. Nevertheless, in the literature the terms warehouse, DC and cross-docking are used as synonyms. A recent survey [1] presents the schedule of inbound and outbound trucks at the terminal. In [2] a model with a single inbound, a single outbound dock and interchangeable products is presented. The problem of minimizing the makespan is solved using different algorithms. Miao et al. [6] consider a cross-docking problem with multiple docks, a limited buffer inside and a transshipment time between each pair of docks. In this case the objective is to minimize the total cost, i.e. the operational cost and penalty cost due to the unfulfilled cargo shipments. The authors use an exact procedure and also meta-heuristic approaches. In [3] the authors study a cross-docking scheduling problem and model it as a two-machine flow shop problem with the objective of minimizing the makespan. Moreover, the authors prove that the problem is NP-hard and they solve it by Branch-and-Bound and heuristic algorithms. This paper considers the problem of scheduling the internal operations of a DC, i.e., the de-consolidation, sorting and consolidation operations. In the de-consolidation phase, incoming containers are unpacked in pallets and then pallets are unpacked in boxes. Each box contains itemsofthesametypethatareassignedtooutboundboxes according to the customers requests. In the consolidation phase, these boxes are packed in pallets and successively in containers. The objective is to determine the optimal sequence of operations in order to minimize the total operation time. In [4] the authors solved this scheduling problem by proposing an Integer Linear Programming (ILP) approach and considering limited buffer capacities. However, the proposed model turns out to be NP-hard, so only instances of very limited size could be actually solved. In order to have a more manageable problem, in this paper we relax the constraints on buffer capacities. Such an assumption is justified in the case of large capacity availability, with a limited number of jobs. For this uncapacitated problem, we present two new Mixed Integer Linear Programming (MILP) formulations: the first one is slightly different from the formulation introduced in [4], the second one is based on time indexed variables. Test Copyright by the International Federation of Automatic Control (IFAC) 8205

2 results highlight the difference of the two formulations in terms of computation time and number of variables and constraints. As a result, the practical applicability of the proposed approach is sensibly improved. The paper is organized as follows. Section 2 describes the problem and Section 3 presents two MILP formulations of the problem with some properties. Hence, in Section 4 some preliminary results are presented. A conclusion section with future research directions closes the paper. 2. PROBLEM DESCRIPTION Goods arrive to the DC from suppliers within containers. Each inbound container collects pallets and each pallet contains boxes. Containers and pallets are multi-product, on the contrary inbound boxes contain items of the same type. In the DC inbound boxes must be unpacked to sort items in order to prepare new boxes to be shipped to the customers.subsequently, thesenew boxesareconsolidated with other ones in pallets and are sent to customers within containers. Schematically, the sequence is the following: de-consolidation: container pallet box sorting: box item box consolidation: box pallet container. In this paper the word case is used to generically denote a container, a pallet or a box. Moreover, we distinguish between inbound and outbound cases: inbound cases are containers, pallets and boxes that arrive to the DC; outbound cases are boxes, pallets and containers that are prepared in the DC for shipment. The objective of this scheduling problem is determining the optimal sequenceofoperationsinthe de-consolidation, sorting and consolidation phases in order to minimize the totaloperationtime(makespan)i.e.,thetimeelapsedfrom the start of the first operation in the DC till the end of the last operation. In the de-consolidation phase, containers coming from suppliers areunpacked and the unloaded pallets are stored in a buffer area with unlimited capacity. We assume that all the inbound containers are available at time zero and that their content is known. We assume that only one container at a time can be unloaded. Successively, all the unloaded pallets of a container are moved in the pallet buffer area and the corresponding operation time is equal to a number of t.u. equal to the number of pallets in the container. For instance, if a container is unpacked at time 1 and it contains 2 pallets, the pallets are both available to be unpacked at time 3. We do not define an unpacking sequence of a container but we suppose that all the pallets of a container are moved in the pallet buffer area at the same moment. In this way, the two pallets are available to be unpacked at time 3. In the pallet buffer area each pallet is subsequently unpacked and only one pallet can be unpacked at a time. The obtained boxes move in the box buffer area. Moreover, the operation time associated to the storage operation (in t.u.) is equal to the number of boxes of the pallet. The boxes reach the sorting machine. In the sorting phase the items of the inbound boxes are assigned to new boxes that are loaded up with the right quantity of items. The sorting phase is performed by a sorting machine that includes a conveyor with unlimited capacity and chutes where the outbound boxes wait for the items. We assume that the box loading time (in t.u.) is equal to the number of items of the box. Moreover, only one item can be sorted at a time. When an outbound box is ready, it is moved in the outbound box buffer area. During the consolidation phase, only one pallet can be consolidated at a time and the operation time associated to a pallet consolidation is equal (in t.u.) to the number of boxes on the pallet. Furthermore, the pallets are stored in the buffer until they are loaded in a container. The time to load all the pallets of a container is equal (in t.u.) to the number of pallets necessary to fill up the container. Only one containerat a time can be loadedup. The processgoes to an end when the last container is ready. 3. MATHEMATICAL FORMULATIONS OF THE PROBLEM This Section presents an ILP formulation (Formulation 1) and a MILP formulation (Formulation 2) of the problem presented in Section 2. The first one is similar to the one introduced in [4] and the other one is a timed two-indexed formulation. We want to compare the two formulations to highlight what is the most promising in term of computing time, and number of variables and constraints. 3.1 Notation The notation used in the model formulations is introduced in the following. De-consolidation phase Inthisphasethreetypesofcases are considered: containers, pallets and boxes. Indexing Sets S 1 = {c c = 1,...,n 1 } Set of inbound containers; S 2 = {p p = 1,...,n 2 } Set of inbound pallets; S 3 = {b b = 1,...,n 3 } Set of inbound boxes. Parameters Integer parameters describe the composition of each case: F 1c Number of inbound pallets in container c; F 2p Number of inbound boxes in pallet p. { 1 if inbound pallet p is in inbound container c, A pc := { 1 if inbound box b is in inbound pallet p, A bp := Sorting phase In this phase we describe the assignment of items from inbound boxes to outbound boxes. Indexing Sets S 7 = {i i = 1,...,n 7 } Set of items; G = {g g = 1,...,n g } Set of types of items; S 4 = {b b = 1,...,n 4 } Set of outbound boxes. Parameters E gb Number of g type items contained in box b; E gb Number of g type items required by box b. 8206

3 { 1 if item i is in inbound box b, A ib := { 1 if item i is of type g, Q ig := { 1 if item i is of type g and is in box b, D igb := Consolidation phase Intheconsolidationphasethereare three types of cases: boxes, pallets and containers. Indexing Sets A set for each type of case is defined: S 5 = {p p = 1,...,n 5 } Set of outbound pallets; S 6 = {c c = 1,...,n 6 } Set of outbound containers. Parameters F 4p Number of outbound boxes in pallet p ; F 5c Number of outbound pallets in container c. A b p := { 1 if outbound box b is in pallet p, A p c := { 1 if outbound pallet p is in container c, 3.2 A heuristic algorithm for the upper bound computation InthisSectionwepresentaheuristicalgorithmtocompute an upper bound T for the total operation time of the DC that is used in the two formulations. More precisely, in Formulation 1 T determines the number of variables and in Formulation 2 T is used to set an upper bound of each time indexed variable. The computation of the upper bound for the total operation time can be performed on the basis of the following assumptions: since there are not buffer constraints, we can assume that first all the containers are unpacked and successivelypalletsandboxesaremanaged.toensurethat all the cases are available, we suppose that the unpack operation for a type of case starts when all the cases of the previous type have been unpacked. Moreover, when all the items of an outgoing box are assigned, such box can be packed. The closing time is equal to the time the last item is sorted in such a box. When all outgoing boxes are ready, we can pack them in pallets and so on. The succession of the following steps allows us to evaluate the upper bound of the total operation time and an upper bound U k for each type of case. The unpacking time of all inbound containers is U 1 = n 1. Let suppose that the last unpacked container is the one with maximum number of pallets, then all the pallets are available to be unpacked at time U 1 + max c S1 F 1c 1. All the inbound pallets are unpacked, at the latest, at time U 2 = U 1 +max c S1 F 1c 1+n 2. All the inbound boxes are unpacked, at the latest, at time U 3 = U 2 +max p S2 F 2p 1+n 3. All the items are sorted, at the latest, at time U 7 = U 3 +max b S3 g G E gb 1+n 7. All the outgoing boxes are ready to be packed at time U 4 = U 7. The time to pack all the pallets is equal to U 5 = U 4 + max p S 5 F 4p 1. The time to pack all the containers is T = U 6 = U 5 + max c S 6 F 5c Formulation 1 In this Section we present a slightly different formulation from the one introduced in [4]. Since we consider the uncapacitated problem, the buffer constraints can be removed.also the variablesand the connected constraints, that determine the opening time of an outgoing box, are removed since the number of chutes is unlimited. The variables that determine the packing time of an outgoing boxarealsoremoved.however,it is necessaryto addother constraints to link the items with the outgoing pallets. As in [4], t is used to index the t.u., so that t = 1,2,...,T. Variables For each type of case, except for the outgoing boxes, a two indexes binary variable is defined: { 1 if case j is unpacked/packed at time t, x jt := A triple index binary variable for the items is defined: { 1 if item i is assigned to box b at time t, r ib t := Let f R express the packing time of the last outbound container. The problem can be formulated as follows: s.t. f minf (1) t x jt j S 6 (2) x jt = 1 j S k,k = 1,2,3,5,6 (3) j S k x jt 1 t = 1,...,T,k = 1,2,3,5,6 (4) r ib t = 1 i S 7 (5) b S 4 r ib t 1 t = 1,...,T (6) b S 4 i S 7 i S 7 b S 4 i S 7 b S 3 t x jt D igb r ib t = E gb b S 3,g G (7) D igb r ib t = E gb b S 4,g G (8) t x lt +F kl j S k+1,l S k,a jl = 1, k = 1,2 (9) 8207

4 t x jt t x lt t x lt +F 5j j S 6,l S 5,A lj = 1 (10) t r ijt +F 4l l S 5,i S 7,j S 4 b S 4 t r ib t (11) A jl = 1 t x bt + g GE gb b S 3,i S 7, A ib = 1 (12) The objective function minimizes the variable f (1). Assuming that the first operation starts at time t = 1 t.u., the total operation time is equal to the time the last operation goes to an end. Constraints (2) set the value of f greater than or equal to the packing time of the last outbound container. Constraints (3) ensure that each case (except for the outbound box) is unpacked/packed and constraints (4) ensure that only one case for each type (except for the outbound box) can be unpacked/packed at a time. Analogously, constraints (5) and (6) impose that the sorting machine sorts all the items and only one at a time. Constraints (7) ensure that the total number of items of product type g that transfer from inbound box b to all the outbound boxes is exactly the same as the number of items of product type g that is initially loaded in inbound box b. Analogously, constraints (8) ensure that all the items of product type g that are in all the inbound boxes are moved in the right quantity to each outbound box b. Constraints (9) ensure that the unpacking time of incoming pallets/boxes must be greater than or equal to the unpacking time of incoming containers/pallets plus the time to unpack. Analogously, constraints (10), (11) and (12) impose the unpacking time of outgoing containers, of outgoing pallets and of items, respectively. This formulation has 1+T (n 1 +n 2 +n 3 +n 7 n 4 +n 5 +n 6 ) binary variables and n 1 +2n 2 +2n 3 +2n 5 +2n 6 +2n 7 + n 3 n g +n 4 n g +n 4 n 7 +6T constraints. 3.4 Some tricks for Formulation 1 InordertostrengthenFormulation1somevariablescanbe fixed.usingtheupperboundsu k calculatedinsection3.2, some variables can be set equal to zero. For instance, since the last container is unpacked, at the latest, at time U 1, we can set x jt = 0 for t = U 1 +1,...,T. The same setting is imposed to the others cases using the appropriate upper bound. 3.5 A timed two-indexed formulation (Formulation 2) In this Section we introduce a new formulation with differentvariables.adrawbackofformulation1isthatifa large instance with a high upper bound is considered, then T binary variables for each case have to be introduced. Hence, in Formulation 2 we use binary variables that do not depend on the upper bound T.Moreprecisely, foreach pair of cases of the same type and items we use a binary variablethat defines the sequenceofthe operations.in this waythe cardinalityof the binary variable set depends only on the cardinality of the case set. Moreover, as explained in Formulation 1, it is not necessary to consider the variables associated with the outgoing boxes since the packingtime of an outgoingpallet is grater than or equal to the sorting time of the last item assigned toaboxofthepalletplusthenumberofboxesofthepallet. In this way, we compute directly the packing time of the pallets. Since the new variables define the sequence of the operations, we have to introduce two dummy cases/items for each type: 0 and n k + 1 for k = 1,2,3,5,6,7. More precisely, for each type of case/item the dummy case 0 is the first case/item that is unpacked/packed/sorted and n k + 1 is the last case for that group that is unpacked/packed/sorted. Variables For each pair of cases of the same type (except for the outgoingboxes)anditems(includedthetwodummycases) a binary variable x jl and a continuous variable t jl R + are defined (j S k {0},l S k {n k + 1} for k = 1,2,3,5,6,7): 1 if case/item l is unpacked/packed/sorted x jl := immediately after case/item j, t jl { > 0 if xjl = 1, := 0 if x jl = 0. where, if l is unpacked/packed/sorted immediately after case/item j, then t jl > 0 denotes the unpacking/packing/sorting time of case/item l else t jl = 0. We remark that x 0l = 1 if case/item l is the first case/item to be unpacked/packed/sorted. Analogously x l,nk +1 = 1 if l is the last case/item to be unpacked/packed/sorted. For each pair of items and outbound pallets, we define the following binary variable: { 1 if item j is assigned to outbound pallet l, y jl := The problem can be formulated as follows: min t j,n6+1 (13) j S 6 s.t. n k +1 x lj = 1 l S k {0},k = 1,2,3,5,6,7 (14) j=1 n k x jl = 1 l S k {n k +1},k = 1,2,3,5,6,7 (15) j=0 n k +1 j=1 n k+1 i=0 t lj t il n k n k t ml +1 l S k,k = 1,2,3,5,6,7 (16) t mj +F kj l S k+1, j S k,a lj = 1, k = 1,2 (17) 8208

5 n 6 t il i=0 n 7 i=0 t il n 5 n 3 t mj +F 5l l S 6, j S 5,A jl = 1 (18) t mj + g G E gj l S 7, j S 3,A lj = 1 (19) t lj T x lj l S k {0},j S k {n k +1}, k = 1,2,3,5,6,7 n 5 l=0 n 7 i=1 n 5 Q ig y ij = j=1 n 4 m=1 (20) A mj E gm g G,j S 5 (21) y ij = 1 i S 7 (22) t lj T y ij +T n 7 t mi +F 4j i S 7,j S 5 (23) The objective function (13) minimizes the total operation time that is equal to the packing time of the dummy outbound container n Constraints (14) and (15) ensure that each case/item is unpacked/packed/sorted and that each case/item is unpacked/packed/sorted before another case/item (14) and after another case/item (15) of the same type. Constraints (16) set the time and the sequence in each type of case/item. For each case/item the unpacking/packing/sortingtime is greaterthan orequal to the time the previous case/item has been unpacked/packed/sorted plus one. Constraints (17) set the time and the sequence between incoming containers and pallets and between palletsandboxes.moreover,constraints(18)and(19) set the time between outgoing pallets and containers and between incoming boxes and items, respectively. Constraints(20) ensurethat if a caseofacertain type is not unpacked/packed/sortedimmediately after another case of the same type the related time is equal to zero. Constraints (21) ensure that the right quantity of a type of item is assigned to the right outbound pallet. Constraints (22) ensure that each item is assigned to a pallet. Constraints (23) set the time and the sequence between items and outbound pallets. The packing time of an outbound pallet must be greater than or equal to the sorting time of an item that is assigned to that pallet plus the number of outgoing boxes in that pallet. This formulation has n 2 1 +n 1+n 2 2 +n 2+n 2 3 +n 3+n 2 7 +n 7+ n 2 5 +n 5 +n 2 6 +n 6 +n 7 n 5 binary variables and n 2 1 +n 1 + n 2 2+n 2 +n 2 3+n 3 +n 2 7+n 7 +n 2 5+n 5 +n 2 6+n 6 continuous variables. The constraints are n n 1 +n n 2 +n n 3 +n n 5+n n 6+n n 7+n g n 5 +n 7 n Some tricks for Formulation 2 Since there are multiple solutions with the same objective value we use in Formulation 2 some tricks to differentiate the solutions. We order the items in each incoming box. The first item of each box is the first one to be sorted. In this way,onlythe firstitem ofeachboxcanbe sortedafter the dummy item 0 and only the last item of each box can be sorted before the dummy item n Moreover, we add some constraints on the sorting time of the items. For instance, the sorting time of the second item of a box must be grater than or equal to the sorting time of the first item of the same box plus one. If two items of the same incoming box are assigned to the same outgoing pallet, then the two items are consecutive according to the sequence of items in the box. We add to the objective function a quantity in order tochooseasolutionwithsomecharacteristics.among all the optimal solutions we want to choose those in which the items of an incoming box are sorted sequentially. To do this, we add to the objective function the variables that denote the sequence of items in an incoming box. For instance, considering a box with 3 items (1, 2 and 3), we add to the objective function x 12 and x 23. In this way a solution with these two variables equal to one is preferred among all the optimal solutions. Since we add a term to the objective function we have to multiply the other variables of the objective function ( j S 6 t j,n6+1) by a term that ensure that we choose a solution that minimize the old objective function. In particular, we multiple for one plus the maximum number of items in an incoming box times the number of incoming boxes. We add extra constraints to set lower bounds to the continuous variables. For instance for incoming pallets we write: t ij x ij (1+ min c S 1 F 1c ) 0 i,j S 2 In this way the unpacking time of a pallet must be greater than or equal to the minimum time needed to unpack a container. 4. PRELIMINARY RESULTS The formulations areimplemented in C++ by using ILOG Concert 2.9 and CPLEX 12.1 and they are run on a 2.7 GHz Intel Xeon E5520 with 16 GB of memory. Branching priorities are used in order to give higher priority to the variables x 0i and x i,n7+1 for all i S 7. In this way we give higher priority to the sorting of the first and last items. The MILP formulations are tested on four sets of randomly generated instances. As preliminary test, we generate three instances for each set. Table 1 shows the four data sets. In particular, the first column denotes the name of the instances, the second till the seventh columns report the number of inbound containers, pallets and boxes, outbound containers, pallets and boxes, respectively. Moreover, the last column shows the maximum number of items in an incoming box. In addition, the content (number of items and type) of incoming and outgoing boxes is randomly generated. 8209

6 c p b b p c g max item 01-1/ / / / Table 1. Characteristics of the four data sets Formulation 1 Formulation 2 cpu cons. vars. Nodes cpu cons. vars. Nodes * * * ,3 607,8 2875, ,9 265,9 911,6 536, ,8 Table 2. Preliminary results The preliminary results of the two MILP problem solutions, obtained in a maximum of 60 min of CPU time, are reported in Table 2: for each instance and for each formulation the columns report the cpu time spent to obtainthesolution(inseconds),thenumberofconstraints, the number ofvariables,and the number ofnodes explored by CPLEX. An asterisk preceding the CPU time indicates that CPLEX has reached the time limit before proving optimality. The last row of the table reports the average for all instances that are solved to optimality by both formulations (then but for 02-3, 03-2 and 04-2). As an example, a representation of an optimal solution for the instance 01-1 is reported in Fig. 1. It shows the schematic process performed in the DC: the inbound container contains 2 pallets that contain two boxes each. During the de-consolidation phase the container, the pallets and the boxes are unpacked obtaining 3 items from two boxes and one item from the other two boxes. Hence, in the sorting phase the outbound boxes are prepared with thesuitablemixofproducts(herethetypesofproductsare denoted by symbols and ). Moreover, we assume that it is necessary to obtain four multi-product boxes, that are grouped in two pallets and are sent in a container. The cases in Fig. 1 are labeled by the time instant in which the operation is scheduled: the inbound container is unpacked at time 1, the pallets are unpacked at time 3 and 4 respectively, and so on. Furthermore, the label on the arrows in the sorting phase denote the sorting time of an item: at time 8 the items start to be sorted. Finally, the outbound container is ready at time 18. The results (Tab. 2) show that the new formulations allow dealing with instances of larger size with respect to the formulation proposed in [4]. Moreover, the second formulations has less variables and more constraints than the first formulation. However, the analysis of the cpu times points outthat it is not possibleto establishwhether oneofthetwoformulationshastobepreferred.indeed,the results suggest to run the two MILP problems in parallel for a same instance in order to get advantage of the fastest one to reach the optimal solution Fig. 1. An optimal solution for instance CONCLUSIONS This paper presents two Mixed Integer Linear Programming (MILP) models to address the scheduling problem involving the internal operations in Distribution Centers. Moreover, for each formulation some tricks to strengthen the defined MILP problems are presented. Some preliminary results show promising prospective for the effectiveness of the two approaches and suggest to run the two MILPmodelsinparallelforeachinstanceofthescheduling problem. Future research will be directed on more computational experiments and on other tools to strengthen the formulations. In addition, a study of other properties of the problem and of heuristic algorithms will be carried out. REFERENCES [1] N. Boysen and M. Fliedner, Cross dock scheduling: Classification, literature review and research agenda, Omega, vol. 38, 2010, pp [2] N. Boysen, M. Fliedner and A. Scholl, Scheduling inbound and outbound trucks at cross docking terminals, OR Spectrum, vol. 32, 2010, pp [3] F. Chen and C.-Y. Lee, Minimizing the makespan in a two-machine cross-docking flow shop problem, European Journal of Operational Research, vol. 193, 2009, pp [4] M.P. Fanti, G. Stecco and W. Ukovich, Scheduling the Internal Operations in Distribution Centers with Buffer Constraints, in 6th annual IEEE Conference on Automation Science and Engineering, Toronto, CA, [5] J.K. Higginson and J.H. Bookbinder, Distribution centres in supply chain operations, in A. Langevin and D. Riopel Editors, Logistics Systems: Design and optimization, Kluwer, Norwell, M.A., [6] Z. Miao, A. Lim and H. Ma, Truck dock assignment with operational time constraint within crossdocks, European Journal of Operational Research, vol. 192, 2009, pp