Expert Systems with Applications

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1 Expert Systems with Applications 38 (2011) Contents lists available at ScienceDirect Expert Systems with Applications journal homepage: Initialization of the Benders master problem using valid inequalities applied to fixedcharge network problems Georgios K.D. Saharidis, Maria Boile, Sotiris Theofanis Center for Advanced Infrastructure and Transportation (CAIT), Department of Civil and Environmental Engineering, Rutgers The State University of New Jersey, 98 Brett Road, Piscataway, NJ , United States article info abstract Keywords: Fixedcharge network problem Benders decomposition Valid inequalities Mixed integer programming Refinery Scheduling of crude oil Network problems concern the selection of arcs in a graph in order to satisfy, at minimum cost, some flow requirements, usually expressed in the form of node node pair demands. Benders decomposition methods, based on the idea of partitioning of the initial problem to two subproblems and on the generation of cuts, have been successfully applied to many of these problems. This paper presents a novel way to reinitialize the Benders master problem for this group of problems using a series of valid inequalities. A generic presentation of the developed valid inequalities is presented as well as a case study of a refinery system is used in order to illustrate the advantage of the proposed procedure. The valid inequalities significantly restrict the solution space of the Benders master problem from the first iteration of the algorithm leading to improved convergence. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction Network design problems are central to a large number of contexts including transportation, telecommunications, power systems and chemical processing. The idea is to establish or use a network of links (roads, optical fibers, electric lines, pipelines, etc.) that enables the flow of commodities (people, data, packets, electricity, crude oil, etc.) in order to satisfy some demand characteristics. Herein, we are particularly interested in fixedcharge network problems, where, in order to use a link, one must pay a fixed cost representing, for example, the cost of using a road, or an electric line, or a pipeline, etc. A large number of practical applications may be represented by fixedcharge network models. One important area is the service network design problem which arises, for example, in airline and trucking companies, pipelines and electric lines. The idea is to maximize the profit by setting routes and schedules, given some resource constraints. For example, airline companies must determine the covered routes and the frequency of the flights considering aircraft and crew availability (Lederer & Nambimadom, 1998). Similarly, express package delivery companies must establish routes, assign aircrafts to them and decide about the flow of packages (Armacos, Barnhart, & Ware, 2002). The multicommodity, multimodel distribution planning problem is another distribution problem with fixed charge where Benders decomposition has been Corresponding author. address: (G.K.D. Saharidis). applied successfully (Cakir, 2009). Various applications can also be found in chemical processing problems. In a refinery system a schedule defines which pipeline will be used for loading and unloading of tanks in order to load the crude oil arriving to the refinery port and unload blends of crude oil toward the crude distillation units. In power systems, the fixedcharge network problem is used to plan the energy transmission from the generation plants to the consumer centers (Binato, Pereira, & Granville, 2001; Romero & Monticelli, 1994) and obtain the configuration that minimizes daily loss costs (Cavelluci & Filho, 1997) and setup cost of the system (Saharidis, Minoux, & Dallery, 2009; Saharidis & Ierapetritou, 2009). In all these cases, proper use of the system resources can result in improved operations and reduced costs. The total amount of these reductions is obviously related to each specific problem. However, the economical importance of most of the cited problems and the key role played by the use of the network in the systems operation suggest that the savings can be significant (Costa, 2005). This economical importance has prompted the development of several solution methodologies for network problems. These methodologies range from pure heuristic methods (Golias, 2007) to optimal implicit enumeration (Saharidis, Minoux, & Ierapetritou, 2010). Amongst the most successful solution approaches is the Benders decomposition method (Benders, 1962). The outline of the paper is as follows. In Section 2 we review the classical Benders algorithm and in Section 3 we give a general description of a network system as well as a case study adopting the model presented in Saharidis et al. (2009), for illustrative /$  see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi: /j.eswa
2 6628 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) Nomenclature Index i z k j t NZ NP NCDU NJ NT Data Cap E i,t ae i,j,t S k,t as k,j,t a k,j,t amin k,j,t amax k,j,t w i,j,t q k,j,t TC t L j,t first level nodes: docks intermediate level nodes: tanks final level nodes: CDU type of commodity (e.g. crude oil) period the number of tanks the number of docks the number of crude distillation units the number of various types of crude oil the number of periods the storage capacity of node z (the same for any tank) the total quantity of the crude oil available, in dock i, for period t the percentage of the crude oil (E i,t ) of type j, of the quantity available in dock i, for period t the total quantity requested from CDU k, for the period t the percentage of the crude oil (S k,t ) of type j, required by CDU k, for period t equal to 1 if the CDU k requests crude oil of type j, for period t and equal to 0 if not acceptable minimal percentage, of type j, for the blend unloading towards CDU k, for period t acceptable maximal percentage, of type j, for the blend unloading towards CDU k, for period t equal to 1 if a boat is in dock i and brings crude oil of type j, for period t; equal to zero otherwise equal to 1 if the CDU k requests crude oil of type j, for period t; equal to 0 otherwise number of different types of crude oil or blends of crude oil requested by CDUs for period t equal to 1 if the crude oil of type j is requested by CDUs for period t; equal to 0 otherwise Decision variables X i,z,j,t continuous variable which corresponds to the quantity of crude oil type j, loaded by dock i, in tank z, for period t (flow dock ) tank) Y z,k,j,t continuous variable which corresponds to the quantity of crude oil type j, unloaded by tank z, with CDU k, for period t (flow tank ) CDU) I z,j,t continuous variable which corresponds to the quantity of crude oil type j, stocked at the end of period t, in tank z. We specify that variables I z,j,0 are data corresponding to the quantities stored in the tanks at the beginning of scheduling period C i,z,t binary variable (0 1) which is equal to 1 if loading pipeline is opened between the dock i and tank z, for period t and equal to 0 if not D z,k,t binary variable (0 1) which is equal to 1 if unloading pipeline is opened between the tank z and CDU k, for period t and equal to 0 if not SC i,z,t binary variable (0 1) which is equal to 1 if setup of loading pipeline is established for loading crude oil from the dock i, at the beginning of the period t and equal to zero if not SD i,z,t binary variable (0 1) which is equal to 1 if setup of unloading pipeline is established for unloading crude oil towards CDU k, at the beginning of the period t and equal to zero if not F z,j,t binary variable (0 1) which is equal to 1 if tank z contains commodity type j, for the period t and equal to zero if not c z,k,t Integer variable that takes value in the interval [0,1 0 0] and defines the percentage of the total quantity stored in the tank z unloaded towards the CDU k in period t purpose. In Section 4 we present a series of general valid inequalities which initialize the problem with significant good convergence properties as we demonstrate through numerical examples. Finally in Section 5 we present conclusions and some perspectives for further research. 2. Benders method 2.1. Benders algorithm The Benders decomposition method is based on the idea of exploiting the decomposable structure present in the formulation of an initial problem (IP) so that its solution can be converted into the solution of several smaller subproblems. Benders method is typically used when the initial model has complicating decision variables (Conejo, Castillo, Minguez, & GarciaBertrand, 2006). Complicating decision variables could be variables which make the problem nonconvex, such as integer decision variables and/ or a group of decision variables which appear in all or in most of the constraints. Decomposing the problem using these variables results in a series of subproblems which are easier to solve (e.g. continuous problems). For the fixedcharge network problem the complicating decision variables correspond to a series of binary variables which appear in many constraints. We briefly state the idea of the Benders algorithm (Benders, 1962) considering, without loss of generality, the following linear problem Initial ProblemðIPÞ : Mim c T x þ d T y st: Ax þ By 6 b x 2 R n þ ; y 2 Zq þ ; where c 2 R n ; d 2 R q ; b 2 R m, A and B are m n and m q, matrices respectively. The decision variables are partitioned into two sets x and y. For fixed y ðy ¼ yþ, IP takes the following form: Primal Slave ProblemðPSPÞ : f ðxþ ¼Mim c T x þ d T y st: Ax 6 b By x 2 R n þ : In Benders decomposition we decompose the IP into PSP, which is a restriction of IP and provides an upper bound (UB) in the case of minimization, and the following relaxation of IP, which is called the restricted master problem (RMP) and provides a lower bound (LB): Restricted Master ProblemðRMPÞ : Fðy; zþ ¼Mim z st:
3 ) v it ðb ByÞ 6 0 Benders cuts u jt ðb ByÞþd T y z 6 0 z P 0; y 2 Z q þ ; where v i is the vector that corresponds to the extreme ray i of the dual of PSP and u j is the vector that corresponds to the extreme point j. In each iteration of the Benders algorithm, the PSP is solved for a different value of ðy ¼ yþ which is updated by the optimal solution of RMP obtained in the previous iteration. Note that in the first iteration of the algorithm an arbitrary value is given to y. In practice it is not the PSP which is solved in each iteration but the dual of PSP which has the following form: Dual Slave ProblemðDSPÞ : f 0 ðuþ ¼Mim u T ðb ByÞ st: A T u P c u 2 R m : In each iteration the objective function of DSP is updated using the optimal solution of RMP obtained in the previous iteration. The solution space of DSP is always the same. In each iteration the Benders algorithm produces a cut called the Benders cut which is added to RMP. The cut is produced from the optimal extreme point (or extreme ray) of DSP solution space. Two different types of cuts can be produced in classic Benders algorithm: Case 1: if the optimal value of DSP is unbounded then the following feasibility cut is added to RMP: v it ðb ByÞ 6 0 where v i is the vector that corresponds to extreme ray i. Case 2: if the optimum of DSP is bounded and the optimality condition is not satisfied then the following optimality cut is added to RMP: u jt ðb ByÞþd T y z 6 0 where u j is the vector that corresponds to extreme point j. If we produce all possible Benders cuts using all the extreme points and extreme rays of DSP solution space then the resulting augmented RMP is an equivalent version of IP and its solution gives the same optimal solution as IP. The total number of Benders cuts, which equals the number of extreme points and extreme rays of DSP is, generally, enormous. However, it is known that at the optimum the number of RMP active constraints will never exceed the number of RMP decision variables (Minoux, 1986). The main idea of the Benders algorithm is based on the observation that the algorithm will converge, satisfying the optimality condition before the addition of all Benders cuts. The convergence criterion is satisfied when the difference between the UB obtained by the best optimal solution of PSP and the LB obtained by the solution of the last RMP is less than or equal to the parameter e (UB LB 6 e), where e is a very small number (e = 0.01 LB). Note that the finite convergence of the algorithm results from the fact that DSP has a finite number of extreme points and extreme rays. More details about the Benders decomposition algorithm are given in a number of references including the papers of Benders (1962) and Minoux (1986). For the multicommodity, multimode distribution planning problem the classical implementation of Benders algorithm gives good convergence results (Cakir, 2009). In this case, the issue to be addressed is the possibility for faster convergence of the algorithm. For the fixedcharge network problem the classical implementation of Benders does not always give good convergence results (Saharidis et al., 2009). The question in this case is whether the Benders algorithm can be made more beneficial and converge to the optimal solution faster than for the nondecomposed problem. These questions are translated to the following one: how can G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) we obtain better UP and LB in each iteration of the algorithm in order to have faster convergence of the algorithm. Beside the quality of the produced Benders cut which was the subject under study in Saharidis et al. (2010) and Magnanti and Wong (1981) the other main reason that makes Benders algorithm to converge slowly is the LB (in the case of minimization) obtain by the RMP and consecutively the UB obtained by PSP. A strategy which could initialize with a better and valid way the RMP would give the opportunity to the algorithm to start from a better LB (higher value). As a result the UB obtained from the PSP will be better and the algorithm will converge faster to the optimal solution due to the smaller difference between the UB and LB. In this paper a new approach is developed as alternative strategy which can be used separately or in combination with any other strategy already developed for the speedingup of the Benders algorithm. This new strategy initialize the RMP problem with the addition of a series of valid inequalities. Some or all of these inequalities could be used in any problem equivalent to the fixedcharge network problem. In this paper the RMP initialization procedure is applied to a fixedcharge network problem, which represents a refinery system, to illustrate its convergence advantage in Benders algorithm. In order to show the applicability of the proposed initialization procedure different variants of the same system are studied (see Section 3.4). 3. Fixedcharge network problem 3.1. Introduction Network problems concern the selection of arcs in a graph in order to satisfy, at minimum cost, some flow requirements usually expressed in the form of nodenode pair demands. For surveys of these problems see (Costa, 2005; Magnati & Wong, 1984; McDaniel & Devine, 1977; Minoux, 1989). In the fixedcharge network problem each arc has an associated fixedcost which must be paid if the arc is part of the solution in a certain period. We consider, without loss of generality, the following fixedcharge network where in order to connect one node with another node of the following level a fixed cost should be paid (Fig. 1). Typically, a network representation is given as a graph G (N; A; J) where N is the set of nodes (i, z and k), A is the set of edges (i z and z k), and J is the set of commodities to be transferred. The commodities could be people, goods, data packets, electric power, gas, crude oil, etc. In general in this kind of systems binary decision variables are introduced and associated with the utilization of a link (i and z) or a link (z and k). Note that we call inlink of a node the link which transfers commodity to the node by another node (e.g. link i z) and outlink of a node the link which sends commodity from this node to another node (e.g. link z k). Continuous decision variables are also introduced to represent the flow of Fig. 1. General network system.
4 6630 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) commodity j going from node i to node z and from node z to node k. This general system could represent a number of systems including: (a) a transportation network which transports products from one place to another (i could be the place where the trucks load the products and z and k could represent customers, depots, or terminals); (b) a crossdocking facility or a warehouse which receives products from origin nodes and send them to the destination nodes (z corresponds to one or many crossdocking facilities and/or warehouses, i to the origin nodes and k to the destination nodes); (c) a power supply system which distributes electricity (i could be the power electricity production units and z and k two different groups of customers e.g. industry and homes); (d) a computer or a telecommunication network design system (i and k could be the customers and z the servers or telecommunication centers); (e) a refinery system to define the use of pipelines for each period of scheduling (i could be the ports, z the storage tanks and k the distillation units). The structure of all these systems which correspond to a fixedcharge network problem presents a natural decomposition scheme for the Benders approach. The variables representing the opening of the links (e.g. use of pipelines) or the connection between two nodes (eg select a route to deliver a commodity) could be considered as the RMP decision variables while the ones representing the actual flow of commodities are kept in the SP. Therefore, at each iteration the RMP solution gives a tentative network for which the SP finds the optimal flow of commodities Typical refinery system The fixedcharge network system that we use for illustration, corresponds to a classical refinery system. A classical refinery is a system composed of docks, pipelines, a series of tanks to store the crude oil (and prepare the different blends), crude distillation units (CDUs), blenders and tanks to store the raw materials and the final products. Once the quantities and the types of crude oil required are known, schedulers must decide the use of the pipelines system. There are two types of pipelines: The first one is used for the loading of crude oil transfered to the port by vessels referred to as loading pipelines and the second one for the unloading of crude oil or blends of crude oil toward the CDU referred to as unloading pipelines. The scheduling of loading and unloading pipelines (SLUP) problem is a classical fixed charged network problem where the objective is to define the opening of a link (e.g. loading or unloading pipeline) between the docks of the port and a series of tanks for the storage of the crude oil and between these tanks and the CDUs. A general refinery system is described in Fig. 2. It is very important that the crude oil is loaded and unloaded contiguously, both for security reasons and to reduce the fixed setup cost incurred when flowlink between a dock and a tank or between a tank and a CDU is reconfigured. As in any fixedcharge network problem for the use of a link, operational constraints and constraints relating to the use and the availability of resources must be taken into account. The dimensions of the system and the capacity of each resource, as well as the adopted strategy for sending the commodities from i to z (which commodity could be sent to each node: e.g. recipe preparation alternative) and from z to k (which commodity is demanded by each node: e.g. mode of blending) should also be considered. The use of a link (e.g. pipeline) for loading the quantities of crude oil at the docks and unloading the quantities of blends toward the CDUs, requires a series of operations that are expensive for the refinery. The most critical and expensive operations associated with a link setup, which corresponds to the configuration of the pipeline system, are: Before the use of loading/unloading pipeline: opening of pipelines valves (extra workload needed); configuration of the electric pumps associated with each pipeline (extra workload needed as well as limitation of the number of pumps should be considered); filling pipelines with crude oil (a dangerous and lengthy procedure); sampling of crude oil (chemical analyses needed); measuring of the crude oil stock in tank before loading/unloading (extra workload needed); starting the loading/unloading (high risk startup requiring extra control of the pipeline system). After the use of loading/unloading pipeline: closing of valves (extra workload needed); emptying of pipelines (lost of crude oil); configuration and maintenance of pumps and pipelines (high cost procedure not associated with the amount of crude oil transferred from one node to another); measuring the crude oil loaded/unloaded in the tanks. Fig. 2. System under study.
5 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) Costs associated with these operations require that the use of the pipelines for loading and unloading should be completed with the minimum number of link setups. As we stated before, the minimization of the number of setups is required also for security reasons, as the above operations are complicated and with each start of a new loading and unloading, the system is strained. In summary, the problem under consideration is to schedule the transfer of crude oil from the docks to the tanks and from the tanks to the CDUs minimizing the setup cost of the existing pipeline network General model for scheduling of loading and unloading pipelines In a typical refinery there are 1 10 tanks for the storage of the crude oil. The storage capacity of each tank can range from 80,000 m 3 to 150,000 m 3. Crude oil flows from the docks to the tanks, and its flow depends on the unloading capacity of the vessels, ranging from 1000 m 3 /h to 5000 m 3 /h. Duration of a vessel s unloading is typically defined a priori by contract. After the loading of tanks from the vessels, crude oil flows from the tanks towards the CDU. This flow corresponds to the distillation capacity of each unit. Each CDU has a distillation capacity that can range from 200 m 3 /h to 2000 m 3 /h. A typical refinery is composed of 1 4 CDUs and 1 or 2 docks, which can accommodate 4 5 vessels per month. The data and the parameters of the system are: dimensions of the refinery, arrival dates of vessels based on mediumterm planning, demand of crude oil by the CDU, initial conditions (quantity and composition in each tank) and the system s capabilities (operational time, tank capacity and flow limit). Moreover, the number of tanks available for storage and their storage capacity are known. The production rates are determined prior to the selection of the recipe preparation alternative and the blending mode, which are known as well. The modeling of this problem involves both continuous and binary decision variables. The continuous variables correspond to flows from docks to tanks, flows from tanks to CDUs, and quantities stored in the tanks. Binary variables are used to specify connections between docks and tanks, connections between tanks and CDUs, and also the availability and setup for the use of loading and unloading pipelines. The constraints necessary to describe the system include: (a) satisfying operations rules; (b) satisfying material balances; (c) meeting storage capacity constraints; (d) satisfying blend properties; (e) establishing a connection between docks and tanks; (f) establishing connections between tanks and CDUs; and (g) setup of tanks for loading or unloading. In this section, we describe our model for the scheduling of loading and unloading pipelines for all modes of blending. In the following nomenclature, we present the parameters, the data and the decision variables which are used in each type of blending mode. After the presentation of the general features of the model we provide the specific additional data, decision variables and constraints associated with its main variants, depending on the specific context of application in each system s setup. The objective in all cases is the minimization of the fixedcharge cost associated with the use of a link, which corresponds to the use of loading/ unloading pipelines. In general, we have four groups of constraints which guarantee certain operational conditions. We have constraints which guarantee that the quantities loaded in tanks are equal to the quantities available to docks in each period and constraints guaranteeing that the quantities unloaded towards CDUs are equal to the quantities required by them. We also have constraints expressing the material balance, and others which guarantee that quantity stored in a tank is not greater than its storage capacity. X NZ X NZ X i;z;j;t ¼ E i;t ae i;j;t 8i; j; t ð1þ X NJ Y z;k;j;t ¼ S k;t 8k; t ð2þ amin k;j;t a k;j;t S k;t 6 XNZ Y z;k;j;t 6 amax k;j;t a k;j;t S k;t 8k; j; t ð3þ I z;j;t ¼ I z;j;t 1 þ XNP X i;z;j;t XNCDU Y z;k;j;t 8z; j; t ð4þ 0 6 XNJ C i;z;t 6 XNJ i¼1 I z;j;t 6 Cap 8z; t ð5þ D z;k;t 6 XNJ X i;z;j;t 6 MC i;z;t ; 8i; z; t ð6þ Y z;k;j;t 6 MD z;k;t ; 8z; k; t ð7þ C i;z;t þ D z;k;t 6 1 8i; z; k; t ð8þ C i;z;t 1 þ SC i;z;t P C i;z;t 8i; z; t ð9þ D z;k;t 1 þ SD z;k;t P D z;j;t 8z; k; t ð10þ Constraint (1) is a loading constraint and guarantees that the sum of the quantities loaded through loading pipelines (inlinks) in tanks, from dock i, for period t, is equal to the quantity available at the dock. Constraint (2) is an unloading constraint and guarantees that the sum of the quantities unloaded through unloading pipelines (outlinks), towards CDU k, for period t, is equal to the quantity required by the CDU k.constraint (3) guarantees that the sum of quantities type j unloaded by all tanks, for period t, towards CDU k satisfies the upper and lower percentage of the quantity required by this CDU k. Constraint (4) is the material balance constraint expressing that the quantity of crude oil, type j, which is stored in tank z, at period t, is equal to the quantity which was stored at period t 1, plus the sum of quantities of the same type that are transferred by loading pipelines for this period by all docks i, less the sum of the quantities unloaded by the outlinks of this tank towards all CDUs. The capacity constraint is defined by (5) where the sum of different types of crude oil should be less than or equal to the total storage capacity of the tanks (node capacity constraint). Constraints (6), (7) are associated with operation rules and express the opening of a link if a flow between a tank z and dock i or CDU k is established using loading or unloading pipelines. (Note that constant M takes a value equal to the storage capacity of the tanks). In addition to constraints (6), (7), constraint (8) guarantees that loading and unloading does not take place at the same time. Constraints (9), (10) are the setup constraints. These constraints guarantee that when using a loading or unloading pipeline, a setup cost is charged at the beginning of the loading or unloading period. Finally the objective function for the developed model is to minimize the total number of loading and unloading pipeline openings necessary for the loading and unloading of the crude oil. This objective is expressed by the following function: MinZ ¼ XNP i¼1 X NZ X NT t¼1 SC i;z;t þ XNZ XNCDU X NT t¼1 SD z;k;t : ð11þ
6 6632 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) Due to the objective function (11) and constraints (9) and (10), the binary decision variables SC i,z,t and SD z,k,t are relaxed to continuous [0,1] decision variables Preparation of commodity In general, the commodity of a fixedcharge network could be goods, data, packets, gas, petrol, etc. For the general refinery system presented herein, the commodities represent the crude oil and/or the mixtures of different types of crude oil. The preparation of the final commodity requested by the final node (e.g. CDUs) is done based on the blending mode adopted by the managers of the refinery. Typically, there are three different blending modes: (a) blending made just before the CDU in a place called manifold (b) blending made in the tanks and (c) the combination of the previous two blending modes. For the first mode, the blend is made just before the distillation units through the use of unloading pipelines. In this case, only one type of crude oil can be stored in each tank at a time. The second mode is to prepare the blends required by the CDUs in the tanks themselves using the loading pipelines. In this case, a quantity of a given type of crude oil is already loaded in a tank then stored and kept on standby until a quantity of another type of crude oil is unloaded into the same tank, in order to produce the required blend. The third distillation mode allows both options of blending preparation Varian 1: commodity preparation at node k (e.g CDUs) When the blend is prepared just before the CDU in the manifold using unloading pipelines, the following decision variables and constraints should be added. The additional decision variable (F z,j,t ) is a binary variable (0 1) which is equal to 1 if tank z contains type j, for the period t and equal to 0 if not. The additional constraints, guaranteeing that no blend is allowed in the tanks, are: X NJ F z;j;t 6 1 8z; t ð12:1þ F z;j;t 6 I z;j;t 6 MF z;j;t 8z; j; t: ð13:1þ Constraint (12.1) guarantees that only one type j of the crude oil can be stored in a tank z, in the period t and constraint (13.1) ensures the coherence between decision variables I z,j,t and F z,j,t Varian 2: commodity preparation at node z (e.g. tanks) When the blends are prepared in the tanks, we have to introduce a new term: the acceptable blend. This term is introduced to linearize constraints that are referred to as nonlinear in the literature (Saharidis et al., 2009). A blend is referred to as acceptable if and only if it satisfies the constraints determined by the recipe preparation alternative for the percentages of the components for each required blend. For a tank to unload its contents towards a CDU, the blend inside must be acceptable for the CDU. This situation is described by the constraints presented below, which are added to the main problem. For a tank z to be unloaded towards CDU k, at period t, the quantity of the j type must satisfy the upper and lower bounds of the component j required by the CDU k. That means that the component j must satisfy the lower bound amin k,j,t as k,j,t and the upper bound amax k,j,t as k,j,t, which implies the following two constraints: 2 3 4I z;j;t XNJ I z;j 0 ;tamax k;j 0 ;tas k;j 5 0 ;t 6 M½1 D z;k;t Š 8k; z; j; t ð12:2þ 2 j 0 ¼1 4I z;j;t XNJ j 0¼1 I z;j 3 0 ;tamin k;j 0 ;tas k;j 5 0 ;t P M½1 D z;k;t Š 8k; z; j; t ð13:2þ Table 1 Difference between nondecomposed and Benders implementation. Examples CPU time non decomposed implementation Varian 3: commodity preparation at nodes z and/or k (e.g. tanks and CDUs) When the blends could be prepared using either of the above cases, the following nonlinear constraints have to be introduced. These constraints consider the simultaneous unloading of all types of crude oil (because we cannot separate a blend) stored in a tank. When a percentage of the crude oil is unloaded from the tank z, to the CDU k, in period t then all the types of crude oil are unloaded simultaneously in a quantity which satisfies the proportion of the mixture. Y z;k;l;t ¼ I z;j;t c z:k:t =100 8z; k; j; t: ð12:3þ We have to notice that for the exact linearization of constraint (12.3) the technique presented in Petersen (1971), Glover (1975), and Floudas (1995) is used First numerical examples CPU time classical Benders implementation Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Relative difference (%) The scheduling of loading and unloading pipeline (SLUP) problem presented in the previous section is a classical fixedcharge network problem and the applicability of Benders algorithm is appropriate because the developed mixedinteger problem is a largescale problem with complicating decision variables. The structure of the model presents a natural decomposition scheme for the Benders approach. The decision variables of the initial model are split in two subgroups: (a) the variables representing the use of the loading and unloading pipelines are considered as the RMP decision variables, and (b) the ones representing the actual flow of commodities between nodes are kept in the SP. Therefore, at each iteration the RMP solution gives a tentative opened pipeline network for which the SP finds the optimal flow of crude oil and/or blends of crude oil. The constraints (7) and (12.1) are added to the RMP problem and the rest are kept to SP problem where the binary decision variables (C i,z,t D z,k,t F z,j,t ) have fixed values. The objective function of the initial problem is the objective function of the SP. As we mention in the end of Section 3.3 the SC i,z,t SD z,k,t decision variables are relaxed to continuous [0,1] resulting to a continuous linear SP. The SLUP problem was solved with the classical way (nondecomposed approach) and also using Benders method (decomposed approach) for 15 examples presented in Saharidis et al., (2009). In Table 1 we display the CPU solution time needed until exact optimality is reached for nondecomposed problem and for the classical Benders method. The relative difference between the two approaches is given in column 4. The gain obtained by applying the classical Benders algorithm is poor (maximum 11%) and in
7 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) some cases negative (ex. 7) making the application of Benders algorithm no always beneficial. We observed that the main reason that the classical implementation of Benders method did not give the best convergence results was the initial large gap between the lower bound obtained from the RMP and the upper bound obtained from the SP. For almost all numerical examples the difference between the LB and UB was more than 90% during the first iterations of the algorithm. For the first example the RMP problem started with a lower bound equal to 1 while the best solution obtained by SP was 21 (a gap equal to 95%) with optimal solution equal to 14 setups of the pipeline system. Based on this observation, a strategy which initializes the RMP more efficiently, restricting as much as possible its initial solution space can solve this problem by significantly decreasing the gap between the lower and the upper bound. For the initialization of the RMP, a series of valid inequalities are developed and presented in the following section with comparative numerical results. The numerical results show that the better initialization of RMP gives rise to a better lower bound (with higher values) resulting better upper bounds and better convergence properties. 4. Initialization of the Benders master problem 4.1. Introduction In this section we present a series of valid inequalities which could be used in many equivalent fixedcharge network problems for the initialization of the RMP problem in order to improve the convergence of the algorithm. The best way to initialize RMP is to develop a series of valid inequalities which use the decision variables representing the links. These decision variables are binary variables and belong to the RMP problem. Adding these valid inequalities, an overall positive effect is obtained due to the restriction of RMP solution space giving rise to a decrease of the total number of algorithm s iterations. In the following section we present a number of general valid inequalities followed by specific explanations for their application on the SLUP problem. In general, we developed two groups of valid inequalities: the first one contains 8 general valid inequalities applicable in any fixedcharge network problem and 7 valid inequalities which are valid if some rules are applicable. In the following, the inlinks of intermediate nodes are the outlinks of the previous level nodes (e.g. origin node) and the outlinks of the intermediate nodes are the inlinks of the next level nodes (e.g. destination nodes) First group of valid inequalities The first set of valid inequalities restricts the decision variables corresponding to the establishment of an inlink or outlink when there is no flow on this link. In general, these inequalities are satisfied by the objective function of the problem, where we minimize the fixedcharge for the establishment of a connection between nodes, but a restriction of this type eliminates a series of infeasible solutions. For the SLUP problem the following two valid inequalities are developed: SC i;z;t 6 C i;z;t 8i; z; t ðvi1þ SD z;k;t 6 D z;k;t 8z; k; t: ðvi2þ These valid inequalities restrict the decision variables corresponding to the establishment of the inlink between dock i and tank z, during t for loading of crude oil and the establishment of the outlink for the unloading of crude oil to the CDU k, from a tank z, during t. The second set of valid inequalities is based on the constitution of a node. A node can be empty or have a single type of commodity or multitype of commodities during a period. If a node does not have any commodity during a period then this node cannot send any type of commodity to any connected node. This condition requires that there are no available outlinks for the empty node during the period under study. For the SLUP problem this condition is described by the following two valid inequalities: NCDU XNJ X NZ F z;j;t P XNCUD D z;k;t 8z; t ðvi3þ C i;z;t 6 ME i;t 8i; t: ðvi4þ If tank z is empty for period t, no unloading can take place by this tank (VI 3). If there is no vessel at dock i, at a period t, no inlink is opened (VI 4). Another valid inequality of this set expresses the sum of all intermediate nodes having a certain type of commodity during the period under study. The nodes which have a certain type of commodity are greater than or equal to 1 if this type of commodity is demanded from a destination node and less than or equal to the total number of outlinks divided by the number of destination nodes minus the different types of commodity demanded form the destination nodes plus, one if this type of commodity is requested. For the SLUP problem this valid inequality is expressed by the following doubleside inequality: L j;t 6 XNZ F z;j;t 6 ðnz TC t þ 1Þ 8j; t ðvi5þ where L j,t = 1 when commodity j is requested by at least one destination node (e.g. CDUs). (VI 5) restricts the sum over all tanks of decision variable F z,j,t to be less than or equal to the number of tanks minus the maximum number of different types of crude oil required by all CDUs in period t, plus 1, because among all the types required by CDUs, there is likely one which is stored in tank z. The lower bound of this inequality is greater than or equal to 1 if type j is required by CDUs, in period t. We have to notice that in the case of SLUP problem the total number of outlinks divided by the number of destination nodes is equal to the number of tanks (NZ). The third set of valid inequalities expresses the condition where a certain type of commodity appears in a node during a certain period and there is no node with demand for commodity during the period under study. For the SLUP problem this condition is expressed by two valid inequalities: F z;j;t 6 capð1 D z;k;t Þþ2a k;j;t cap 2 q k;j;t þ 4 XNJ j 0 ¼1 3 ðvi6þ F z;j 0 ;t F z;j;t 5 P D z;k;t 8z; k; j; t ðvi7þ where q k,j,t = 1 when as k,j,t 0 and otherwise q k,j,t =0.(VI 6) and (VI 7) indicate that if crude oil type j is stored in tank z (F z,j,t 0) during the period t and CDU k does not request crude oil of type j or request a mixture which is not composed of crude oil type j, then tank z could not be unloaded towards CDU k (D z,k,t = 0). Finally, the fourth set of valid inequalities connects a certain period and its following period based on the constitution of a node. In general, it describes the constitution of a node at the period t+1 knowing its constitution in the previous period t. With this valid inequality we guarantee that if nothing occurs in the period t+1 in a node, the node preserves the constitution that it had in the period t. For the SLUP problem this condition is expressed by the following inequality:
8 6634 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) X NCDU ð D z;k;tq k;j;t ÞþF z;j;t 6 F z;j;tþ1 6 F z;j;t þ XNP ðc i;z;tþ1 w i;j;tþ1 Þ 8z; j; tðt NTÞ ðvi8þ (VI 8) forces the variable F z,j,t+1 to take a value equal to zero when tank z does not have stored crude oil of type j, in period t (F z,j,t =0) and there is no vessel transferring crude oil of type j in period t+1 (w i,j,t+1 = 0). We should notice that the term C i,z,t+1 is added so that we can take into account the case where commodity is available at origin node (a vessel arrives with crude oil of type j) but it is not unloaded into tank z. The F z,j,t term guarantees that if nothing happens P NP i¼1 C i;z;t ¼ P NCDU D z;k;t ¼ 0, the tank keeps the same constitution for period t+1. Finally, in order to give the freedom to the decision variable F z,j,t+1 to take any value, we add the term P NCDU D z;k;tq k;j;t for the case where a node z (where commodity type j is stored) has an opened outlink in period t Second group of valid inequalities i¼1 The second group of valid inequalities is developed based on some specific rules that may are applicable in the fixedcharge network under study. The first rule referred to as Rule 1 describes the possible composition of commodities in a node. Two general cases are applicable: (a) only one type of commodity at a time could appear in a node, (b) multitype of commodities could appear in node during a period. The restriction to only one type of commodity during a certain period is applied in a lot of network problems where the node represents for example a depot or a tank which has special structure or special use and it cannot receive more than one type of commodity at the time. In some cases (e.g. refinery system) the links of the system (e.g. pipelines) or another node (e.g. CDU) could be used in order to mix the commodities. In some other cases (e.g. crossdocking facilities) the links of the system (e.g. the route of a truck delivering a commodity) could not be used in the same time period for transferring more than one type of commodity because the connected node cannot receive more than one type of commodity (e.g. only food). The second rule, referred to as Rule 2, which is applicable in many network problems guarantees that a commodity stays at least for one period in a node to get some treatment and leaves the node during one of the next periods of the time horizon. Finally the third rule referred to as Rule 3 restricts the simultaneous opening of an inlink and an outlink of a node even if the commodity received by the inlink is different than the commodity sent by the outlink. We have to notice that this rule is applicable in many network problems due to security or strategy reasons and operational limitations. For example, in a refinery, the simultaneous loading and unloading of a tank is prohibited because of a series of reasons, presented in Section 3.2. The first set of valid inequalities of this group is based on the data of the system. In general, the data of the network system are the dimension of the system (e.g. the number of links and the number of nodes), the input data (e.g. the availability of commodity) and output data (e.g. the demand) of each node for each time period. The initial condition of the system is known, as well as the system capabilities (e.g. transferring capacity of links and/ or receiving capacity of a node). Based on these data some minimum and maximum bounds can be defined restricting the solution space of the RMP problem. This set of valid inequalities restricts the minimum and maximum number of available inlinks and outlinks during a period. The bounds of inlinks are defined based on the minimum number of outlinks needed and based on the available amount of commodity of previous level nodes. The complimentary bounds to inlinks define the bounds of the outlinks. More specifically, the amount of the commodity available at an origin node defines the minimum number of inlinks needed for transferring this commodity to the intermediate nodes. Each link has a transferring capacity, based on its own capacity or on the receiving capacity of the nodes it connects. If a is the available amount of commodity in an origin node i and b is the capacity of any link connecting i and all intermediate nodes of the following level of the system, then the minimum number of inlinks needed for the transfer of this commodity is defined by the following equality: 8 a < if a b b c integer ¼ no : integer quotient of a þ 1 if a no integer b b Based on this function c gives the minimum number of inlinks needed during a certain period. For SLUP problem the minimum number of loading pipelines is calculated as c i,t where a = E i,j and b = cap. The maximum number of inlinks is equal to the total number of inlinks divided by the total number of destination nodes. We can restrict more this upper bound using Rule 3. The upper bound is restricted more by the minimum number of outlinks needed in the period under study. For SLUP problem the maximum number of inlinks is equal to the number of tanks minus the number of different types of crude oil requested by the CDUs or the different types of blends requested by the CDUs (TC t ). This implies the following doubleside valid inequality: c i;t 6 XNZ C i;z;t 6 NZ TC t 8i; t: ðvi9þ A complimentary valid inequality to (VI 9) defines the minimum and maximum number of outlinks. The minimum number of outlinks is equal to the different types or sets of commodities asked from the destination nodes. The maximum number is equal to the total number of outlinks divided by the total number of destination nodes. In an equivalent way as in the case of the upper bound of the total number of inlinks, we can further restrict the maximum number of outlinks using Rule 3. The upper bound is restricted more by the minimum number of inlinks needed during the period under study. For the SLUP problem these conditions result in the following valid inequality: TC t 6 XNZ D z;k;t 6 NZ c i;t 8k; t ðvi10þ The second set of valid inequalities is developed based on Rule 1 and contains three valid inequalities. The first valid inequality of this set implies that for the use of an inlink for the transfer of a commodity of a certain type into a certain node, the node should be empty or should contain the same type of commodity. This set of valid inequalities for the SLUP problem is expressed with the following inequality: ð1 w i;j;t XNJ F z;j 0 ;t F z;j;t AA P C i;z;t 8i; j; t; z ðvi11þ j 0 ¼1 Where w i,j,t = 1when ae i,j,t 0 otherwisew i,j,t =0.(VI 11) defines the candidate tanks for loading. In order to load crude oil of type j, in period t, in tank z (C i,z,t 0), this tank z should be empty P NJ j F z;j;t ¼ 0 or contains the same type of the crude oil. Moreover, one tank which contains crude oil of type j 0 in period t ðf z;j 0 ;t 0Þ, cannot store any other type of crude oil in this period. The second valid inequality of this set results from the total number of empty nodes and expresses the availability of nodes. For the SLUP problem the following doubleside inequality is implied:
9 G.K.D. Saharidis et al. / Expert Systems with Applications 38 (2011) TC t 6 XNZ X NJ F z;j;t 6 NZ XNP c i;t 8t ðvi12þ i¼1 If in period t, the total available quantity of a commodity at an origin node is equal to P NP i¼1 E i;t then the sum across all tanks and over all types of crude oil of decision variable F z,j,t should be greater than or equal to TC t and less than or equal to the total number of inlinks divided by the number of docks (which corresponds to the total number of tanks) minus the minimum number of inlinks necessary for the storage of P NP i¼1 E i;t. Finally, the last valid inequality of this set connects period t and t + 1 with a different way than the (VI 8) does. If the commodity available in an origin node and the commodity stored in an intermediate node are not of the same type, the opening of an inlink is denied. For SLUP problem this condition is expressed by the following valid inequality: 1 XNJ j 0 ¼1 F z;j 0 ;t þ F z;j;t þ XNCDU D z;k;t P XNP ðc i;z;tþ1 w i;j;tþ1 Þ 8z; j; t ðvi13þ If in period t crude oil of type j is stored intank z and in period t+1, P a vessel arrives with crude oil of type j 0 NP i¼1 w i;j 0 ;tþ1 0 the decision variable C i,j,t+1 takes a value equal to 0. The same valid inequality also guarantees that if in period t, tank z is empty P NJ F z;j;t ¼ 0, and in period t+1 a vessel arrives to the dock i transferring crude oil of any type, then the tank z can be loaded with this type of crude oil. The third set of valid inequalities is generated based on Rule 2. This set of inequalities contains an extension of (VI 3) presented in the previous section. The (VI 3) is extended to the following period because based on Rule 2 we need one period to load some commodity to the node in order to be ready after that period to use an outlink. For the SLUP problem this condition implies the following inequality: 2 NCDU XNJ i¼1 F z;j;t P XNCUD D z;k;t þ XNCUD D z;k;tþ1 8z; t ðvi14þ Finally the fourth set of valid inequalities is introduced based on Rule 3 where the simultaneous opening of an inlink and outlink of a node is prohibited. This set of inequalities guarantees that if an intermediate node is empty and a destination node requests certain type of commodity, there is not outlink of the intermediate node which could be opened during the period under study. For the SLUP problem this condition is expressed by the following inequality: D z;k;tþ1 6 1 TC k;tþ1 as k;j;tþ1 þ XNJ j 0 ¼1 Table 2 Initial solution of RMP effect of valid inequalities. Examples Initial RMP solution F z;j 0 ;t þ F z;j;t 8z; k; j; t ðvi15þ Initial RMP solution with VI Optimal Solution Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Improvement of LB (%) Table 3 Impact of valid inequalities on CPU solution time. Examples Where if in period t, tank z is empty P NJ j F z;j;t ¼ 0 and in period t+1 a CDU k requests crude oil of type j 0 (q z,j,t 0) then D z,k,t+1 = Numerical examples For the same 15 examples presented in the previous section, we applied the 15 groups of valid inequalities. Column 2 and 3 of Table 2 present the initial solution of RMP problem obtained by the classical Benders algorithm and by the addition of the valid inequalities respectively. Column 4 gives the optimal solution of each example and column 5 the improvement of the lower bound initializing the RMP. The numerical examples show that the initial lower bound of RMP obtained after the initialization of RMP increased significantly from 36% up to 83% compared to the lower bound obtained by the classical implementation of Benders algorithm. The significant reduction of the solution space of the initial RMP in turn led to a significant reduction in the CPU solution time as shows Table 3. Comparative computational results between the two approaches are presented in the last two columns of Table 3 where the advantage of applying this strategy of initialization of RMP is clear. The number of iterations of Benders algorithm decreased significantly (48 65%) resulting in an important reduction of CPU solution time, which varies between 26% and 76%. 5. Conclusions Benders without valid inequalities CPU time Number of iterations Benders with valid inequalities CPU time Number of iterations Relative difference CPU time (%) Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Number of iterations (%) A new strategy for Benders method has been discussed and applied in a classical fixedcharge network problem corresponding to the problem of scheduling of loading and unloading pipeline for the transfer of crude oil. The presented examples illustrate the applicability and efficiency of the new strategy which initialize the master problem of Benders algorithm using a series of valid inequalities which are generally applicable to many fixedcharge network systems. All the presented examples illustrate that better initialization of the RMP problem results in a significant increase of the lower bound of Benders algorithm (in the case of minimization) and in a significant decrease of the number of iterations as well as the CPU solution time. Future research should investigate the further generalization of the groups of valid inequalities to other problems where Benders decomposition is an appropriate solution approach. Finally, we note that the development of a strategy where some of the valid inequalities which are used for the initialization of the RMP are deleted in a certain iteration of the
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