#### Abstract

We address the most general case of multiperiod, multiproduct network planning problems, where we allow spoilage on arcs and storage at nodes. In our models, all network parameters change over time and products. The minimum-cost flow problem in the discrete-time model with varying network parameters is investigated when we allow storage and/or spoilage, and some reformulation techniques employing polyhedrals are developed to obtain optimal solutions for a predefined horizon. Our methods rely on appropriate definitions of polyhedrals and matrices that lead to LP problems comprising a set of sparse subproblems with special structures. Knowing that computational expenses of solving such a large-scale planning problem can be decreased by using decomposition techniques, the special structure of polyhedrals is utilized to develop algorithmic approaches based on decomposition techniques to handle the global problem aiming to save computational resources.

#### 1. Introduction

Static network flows are very common in the literature [1]. However, they fail to capture the time-varying property of many practical applications, such as production-distribution systems, traffic planning, communication networks, and evacuation planning [2–10]. A static flow cannot properly consider the evolution of such systems over time. The need for more realistic models led to the development of multiperiod and dynamic network flow formulations which have been applied to a variety of applications. In such applications, flow values on arcs are not constant but may change over time and not only the amount of flow to be transmitted but also the time needed for the transmission plays an essential role. Capacitated dynamic networks arise in a variety of relevant decision making problems (Aronson [2], Cai et al. [3], Hoppe [4], Wayne [5], Lozovanu [6], Hosseini [7], Newman et al. [8], Pantelides [9], Groß and Skutella [10, 11], and Stefansson et al. [12]). In some practical applications the network structure and problem parameters may be time-varying (time-dependent network flow) [11, 13, 14], and there is no guarantee for the flow to be conserved [5, 15–17].

Motivated by time-dependent multi-item distribution planning problems, we study an extension of this class of problems as network problems which generalize the problems in Hosseini and Saridarq [14, 15] by including* horizon capacities*,* time-commodity varying capacities, time-commodity varying costs*, and* time-commodity varying loss*/*gain factors* over a finite horizon. This study focuses on the* minimum cost dynamic flows* (MCDF) on* multiperiod multiproduct networks* (MMN) where spoilage over arcs and storage on nodes are allowed. In such networks, an arc is assigned a nonnegative time-commodity varying* gain*/*loss factor*, two nonnegative time and time-commodity varying capacity functions, a nonnegative horizon capacity, and a nonnegative time-commodity varying cost function. In our setting, a positive time-commodity varying factor represents the fraction of flow that remains when it is sent at a specific time period. MMNs have many interesting applications [18–23].

There are some approaches to address multiperiod network problems such as state-task network [23] and resource-task network [9], with important differences with respect to the assumption of continuity or discreteness of the time horizon. Jackson et al. [18, 19] use temporal and spatial Lagrangean decompositions to solve the multisite multiperiod planning problems. Chen and Pinto [24] use Lagrangean-based decomposition techniques for solving the temporal decomposition of a continuous flexible process network; they use subgradient methods to solve the decomposed problem. Neiro and Pinto [25] use temporal Lagrangean decomposition to solve a multiperiod mixed-integer nonlinear programming planning problem under uncertainty concerning a petroleum refinery.

We consider nonsimultaneous shipment of commodities from production sites (sources) to markets (sinks) in a distribution network with known production capacities where demand should be satisfied from available supply with time-commodity dependent shipments during a planning horizon represented in discrete time periods. Hence, the problem is a decision problem that finds an optimal dynamic flow minimizing a predefined nonnegative distribution cost function. We formulate a MCDF problem on a MMN with special structures that permit efficient computation of its solution and help save storage requirements. Our approach relies on appropriately defining polyhedral sets. We develop some decomposition-based approaches to solve the minimum-cost flow problem employing polyhedral sets embedded within the underlying network.

The organization of this paper is as follows. We discuss the necessary notations and time representations in Section 2; we introduce and analyze the minimum cost flow problem on multiperiod multiproduct distribution planning networks including spoilage in Section 3. Section 4 discusses the same problem while spoilage and/or storage become the key elements of the problem formulation. In Section 5, we discuss some special cases of the problem and propose some alternative approaches. Section 6 reports a real-life application and presents some computational experiments.

#### 2. Minimum-Cost Flow Problem on Multiperiod Multiproduct Networks

Many planning problems arising in large-scale systems can be formulated as a minimum-cost multiperiod (dynamic) multiproduct network flow problem [2–6]. Towards this goal, the single-commodity dynamic flow problem formulations can be extended to consider a multiproduct network. In the MCDF problem on MMN, there exists a set of products that are manufactured in several multiproduct production sites (sources) and shipped to a set of markets (sinks) where they are sold. The aim is to find a time-dependent distribution plan to nonsimultaneously ship the products from source nodes to sink nodes through a network honoring the arc capacities (time-varying, time-commodity varying, and horizon capacities) at a minimal cost during a finite-length planning horizon. denotes a distribution (directed) network where is the set of production and demand sites (nodes), is the set of all possible connections between sites (arcs), is the set of products, and represents the length of the planning horizon. With dynamic flow decision variable as the vector of flow rates of commodity entering arc at time period , the formulation for the MCDF on MMN becomes
where is the nonnegative cost function with respect to product and denotes the predefined supply/demand capacities at node over the entire planning horizon. Constraint (2) involves the flow conservation constraints for each commodity. We refer to (3) as* horizon capacity* constraints; horizon capacity of an arc limits the amount of total flow (of all commodities) on the arc throughout the entire planning horizon. Constraint (4) represents the maximum possible amount of total flow that can enter at time : it is referred to as the* moment/period capacity* constraint. Constraint (5) is the time-commodity varying capacity constraint for each commodity at each moment.

The problem formulation in (1)–(5) represents MCDF in a* continuous-time setting*. However, as an approximation to this setting, time may be represented in discrete increments [25]. The continuous-time problem seeks for the multiproduct flow distributed continuously over time within period while the discrete case determines the flow rates over discrete time periods. For a finite-length planning horizon , we denote the time horizon in the discrete-time model. There is a natural transformation of continuous-time dynamic (multiproduct) flow to a discrete flow of the same horizon and vice versa [3, 4]. Let represent the total amount of flow sent into arc during time interval ; then
where , , and .

The last equality follows as and are nonnegative continuous functions (see [26–28]). For any discrete time period , any commodity , and time horizon , it is easy to verify that flow conservation constraints hold and satisfies all such constraints and also the flow cost is preserved as

#### 3. Multiperiod Multiproduct Network Flows with Spoilage (SMMN)

Any of the models discussed up to now has a fundamental assumption: the flow has to be conserved on any arc with respect to any commodity. However, some practical applications do not satisfy such a conservation assumption [1, 17]. In the transmission of a volatile gas, for example, we may lose some portion of the flow due to evaporation; or, in the transmission of liquids such as raw petroleum crude, some flow may be lost due to leakage [5]. The SMMN problem is a natural generalization of the problems stated in [14, 15]. When spoilage on arcs is also considered, each arc has a time-commodity varying nonnegative* gain/loss factor * with respect to each time period time and commodity . When units of flow of commodity are sent from node via arc at time , units of flow arrive at node at the same time. If , the arc is* lossy*; if the arc is* gainy* on that time with respect to that commodity*.* Therefore, if there is spoilage (loss) of flow on an arc during all periods with respect to all commodities, the mathematical model may be modified only by assigning a* loss factor* to the related arc. Then,

Such a production-distribution planning problem in discrete-time setting has an underlying graph consisting of three capacity functions: , , and , a predefined nonnegative cost function , and a predefined nonnegative time-commodity varying gain/loss function . Therefore, a* discrete feasible dynamic flow* is a nonnegative function satisfying (9)–(13), and the discrete-time minimum-cost dynamic flow problem becomes
where we suppose that the arc capacity is an upper bound on the -flow (flow of commodity ) sent from node at time period , not on the flow that becomes available at node . Similarly, should be interpreted as the cost for each unit of flow which is sent from node . In order to benefit from an efficient decomposition-based solution method by transforming the formulation structure, we introduce an unrestricted variable , and the formulation becomes
where denotes the difference between the outflow and the inflow of commodity at node at period . It is easy to check that conditions (15) and (16) together are equivalent to condition (10). To develop our polyhedral-based approach we need to define the node-arc incidence matrix including the time component of the problem. Given an underlying SMM network and predefined loss/gain factors, we introduce an auxiliary matrix for a time period and for a product as
where is the node-arc incidence matrix of the underlying distribution network (which remains unchanged during the planning horizon) and is a -diagonal matrix whose elements are the predefined arc factors (at time with respect to ) in the same order that arcs appear in -vector ( is the number of arcs of the network). For a time period and a commodity , we construct as
where and represent the th elements of the matrices and , respectively. We refer to matrix as the* t-q-node-arc incidence matrix* of the SMM network, and it represents the node-arc incidence matrix of SMMN at time for commodity . Due to the changes in arc factors over time and commodity, incident matrices are not necessarily the same during the complete planning horizon with respect to each product. However, since the time-commodity varying gain/loss functions are predefined, all matrices can be computed off-line.

Substituting for (), considering relations (20)-(21), with appropriately vectors and matrices, the problem formulation, indeed, yields the following: where and are defined as the vector of free variables at time with respect to commodity and the vector of supply/demand numbers with respect to commodity . and are the -vectors of flow and capacities in for commodity . is the -vector of horizon capacities and is the -vector of period capacities with respect to period . Let represent the vector of arc costs at for commodity . Please see Appendix A for further details.

With our modeling approach, we can formulate any MCDF problem on a SMMN as a problem which possesses the* block angular* structure. In order to demonstrate this, we defineAccording to the newly introduced notation, the LP formulation of the problem becomes

Block diagonal structure is desired to speed up the solution of a sparse linear programming problem [29]. We may also conveniently exploit decomposition procedures or generalized upper bounding (GUB) techniques to solve such problems efficiently [30–34]. Interestingly enough, in our approach, the master constraint has the same matrix for any set of variables * *. In order to use the Dantzig-Wolfe decomposition in our problem, the constraint matrix should be exploited by splitting the original problem into smaller* subproblems* and a connecting constraint,* master constraint* (the one containing the master matrix )*.* The structure of the time-varying block-angular system admits a natural decomposition into a set of independent well-structured smaller parts instead of solving the original problem whose size and complexity are beyond what can be solved within a reasonable amount of time and then adjust the solution to take into account the interconnections. In general, it is not necessary for either set of constraints to have a special structure, but when available, it helps speed up the solution method. Thus, we may apply the block diagonal decomposition techniques to solve the foregoing problem to achieve the desired effect. Let us consider an application of a decomposition algorithm to the problem: define polyhedral sets for each as

Considering Minkowski’s representation theorem, any can be expressed as a convex combination of a finite number of extreme points of as
where and are extreme points and extreme directions (if any) of polyhedral . The original problem can be reformulated as the* master problem* under Minkowski’s mapping as follows:

Due to having a huge number of extreme points for each polyhedron, enumerating all the extreme points and solving this problem directly seem to be impossible. Rather, we should find a reasonable approach without enumerating all the extreme points. This is where we suggest the use of decomposition techniques, especially due to the special structure of our problem that greatly intensifies the efficiency of the decomposition methods. Next, we develop the most general form of the minimum-cost flow problem formulation for a SMMN as

The formulation of the SMMN problem shows a much simpler constraint structure than the usual matrix form. It possesses only constraints rather than in the earlier formulation. Problems of this type are well amenable by many decomposition algorithms and column generation methods [35–37]. As a result, the computational advantage of the algorithm depends on the efficiency of the decomposition methods. We propose Dantzig-Wolf decomposition (or Benders algorithm for the dual) and so, the same analysis is applied for this case. For more information, one may refer to [38]. When the problem has many thousands of rows and is unsolvable in a reasonable amount of time, however, our approach suggests a method to convert the large-scale problem (high dimensional problem) into one or more appropriately coordinated smaller sparse problems of manageable sizes.

Considering Minkowski’s mapping and feasibility of the problem, it follows that any obtained optimal basis will detect one arc set for every time step and for each commodity that transports a positive amount of flow. Moreover, the values of for each , , and will be determined at any basis. It is immediately understood that the optimal arc sets for every time step and for any commodity are not necessarily the same. In other words, the optimal solution (corresponding to the optimal basis) determines a set of original variables of form for each time step with respect to each commodity conveying a positive flow [35, 38, 39].

#### 4. A Solution Approach for MM Networks with Storage

In certain practical problems, the intermediate storage policy is another important issue that has to be considered in the models. It may be necessary that the -flow (flow of commodity ) is delayed at some nodes in various applications such as batch process scheduling, traffic routing, evacuation planning, energy transmission, inventory, and telecommunications [2]. This affects the complexity of the problem. When there are no storage equipments, nonintermediate storage policy is assumed. It is also common in the industry for a process to have different storage policies for different intermediates that is called mixed intermediate storage policy. In our model, we let the -flow be stored at some (or all) intermediate nodes (or demand nodes) for only one time period with finite (or infinite) storage capacity depending on a predefined capacity function for each , , and . Time and commodity dependent capacity functions allow different storage policies in different time periods with respect to different products, keeping the time lag as one period. However, the staircase structure, which will be discussed later, may have longer time lags. We represent the storage of flow by introducing loops as shown in Figure 1.

Flow storage leads to a slightly different notion of flow conservation. If we decompose the set of vertices into three subsets as , , and , respectively, comprising source nodes, intermediate nodes, and sink nodes with respect to each , is a* dynamic feasible flow* if and only if it satisfies constraints (2)–(5) together with
We may allow limited (or unlimited) flow storage at nodes but prohibit any deficit by constraint (29). As earlier, all -demands must be met, and flow must not remain on the network after time while each source/ must not exceed its forecasted supply/demand. A* discrete-time dynamic flow* on denoted by is said to be* feasible* if it satisfies the following constraints:
If is the cost of storage at node at period with respect to , the total cost of a discrete dynamic flow is defined by
With denoting the difference between -outflow and -inflow at node at time period , we may reformulate the problem as
where and denote the amount and cost of stored flow at node in period of product . By setting , our problem formulation is reduced to a standard LP formulation in* matrix form* as shown in Appendix B, whose special structure enables efficient solution of the problem. The resulting formulation is as follows:
where and are the vectors of flow and storage at time period and and are the vectors of flow capacities and storage at , respectively. , , , and are defined as earlier. is the vector of predefined storage costs and is the node-arc incidence matrix of the underlying network. A standard LP problem formulation is obtained by decomposing into nonnegative vectors and . is the vector of predefined supply/demand of nodes with respect to commodity . A further refinement of the matrix-form formulation shows that we can formulate a minimum-cost problem on a multiperiod dynamic network (with storage) in a* staircase structured matrix form*. For this purpose, we defineAccordingly, we obtain the formulation as

We may now generalize the problem to consider MMN with storage at nodes and spoilage in arcs (SSMM), by allowing the -flow to leak and be stored at the same time. The continuous time setting is then formulated as The discrete time setting formulation is obtained by replacing (14) and (15), respectively, by

To develop the matrix form of the formulation, we again use the --node-arc incidence matrix introduced in (20)-(21). As earlier, we let denote the --node-arc incidence matrix at step with respect to . With similar transformations, the minimum-cost SSMMN flow problem can be formulated by replacing each with the corresponding .

#### 5. An Alternative Approach for MMN without Period Capacities

To formulate MMN flow problem without period capacities, we slightly change the definition of the master matrix and decision variable vectors. Our approach brings more flexibility to modify the problem setting in various ways, but at the same time, it increases the number of subproblems (to be obtained by a decomposition) from to .

*Case 1 (MMN with spoilage). *We define decision variable vectors be time-commodity dependent; each will consist of one set of nonnegative variables in the th position for each and and in the th and th position, respectively, as
Analogously, we redefine the node-arc incidence matrices with respect to each and as
We also need to redefine our master matrix as
We can extract a block-angular structure with each block representing the constraint matrix for each time and commodity. In this respect, the flow conservation conditions turn out to have the following from:
We define the following two matrices:
Then, the formulation for the minimum-cost MMN flow problem with spoilage is
We now define time-commodity dependent polyhedrals and use Minkowski’s mapping as
where and are extreme points and extreme directions (if any). Then, the same analysis as that in Section 3 can be applied.

*Case 2 (MMN with storage). *To formulate the flow problem with storage, we redefine the time-commodity varying decision variable vectors asSimilarly, we change our node-arc incidence matrices with respect to each and as
We should redefine our master matrix as follows:
Then, the flow conservation constraints are written as
To obtain the staircase structure form, we again define
Then the formulation for MMN flow problem with storage becomes
Note that we assume for every and