Single‐warehouse multi‐retailer inventory systems with full truckload shipments

We consider a multi‐stage inventory system composed of a single warehouse that receives a single product from a single supplier and replenishes the inventory of n retailers through direct shipments. Fixed costs are incurred for each truck dispatched and all trucks have the same capacity limit. Costs are stationary, or more generally monotone as in Lippman (Management Sci 16, 1969, 118–138). Demands for the n retailers over a planning horizon of T periods are given. The objective is to find the shipment quantities over the planning horizon to satisfy all demands at minimum system‐wide inventory and transportation costs without backlogging. Using the structural properties of optimal solutions, we develop (1) an O(T²) algorithm for the single‐stage dynamic lot sizing problem; (2) an O(T³) algorithm for the case of a single‐warehouse single‐retailer system; and (3) a nested shortest‐path algorithm for the single‐warehouse multi‐retailer problem that runs in polynomial time for a given number of retailers. To overcome the computational burden when the number of retailers is large, we propose aggregated and disaggregated Lagrangian decomposition methods that make use of the structural properties and the efficient single‐stage algorithm. Computational experiments show the effectiveness of these algorithms and the gains associated with coordinated versus decentralized systems. Finally, we show that the decentralized solution is asymptotically optimal. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Selecting T for a periodic review inventory model with staggered deliveries

Consider a single‐item, periodic review, infinite‐horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. Orders can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting). Every T periods, one audits the current stock level and decides on deliveries for the next T periods, thus incurring a fixed audit cost and—when one schedules deliveries—a fixed order cost. The problem is to find a review period T and an ordering policy that satisfy the average cost criterion. The current article extends an earlier treatment of this problem, which assumed that the fixed order cost is automatically incurred once every T periods. We characterize an optimal ordering policy when T is fixed, prove that an optimal review period T** exists, and develop a global search algorithm for its computation. We also study the behavior of four approximations to T** based on the assumption that the fixed order cost is incurred during every cycle. Analytic results from a companion article (where μ/σ is large) and extensive computational experiments with normal and gamma demand test problems suggest these approximations and associated heuristic policies perform well when μ/σ ≥ 2. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 329–352, 2000     

The transportation paradox

A paradox arises when a transportation problem admits to a total cost solution which is lower than the optimum and is attainable by shipping larger quantities of goods over the same routes that were previously designated as optimal. That is, falling total costs are present in moving to the greater shipment quantities. Necessary conditions for this to occur are established and an algorithm for solving this expanded transportation problem is supplied.     

Selecting review periods for a coordinated multi‐item inventory model with staggered deliveries

Consider an N‐item, periodic review, infinite‐horizon, undiscounted, inventory model with stochastic demands, proportional holding and shortage costs, and full backlogging. For 1 ≤ j ≤ N, orders for item j can arrive in every period, and the cost of receiving them is negligible (as in a JIT setting). Every T_j periods, one reviews the current stock level of item j and decides on deliveries for each of the next T_j periods, thus incurring an item‐by‐item fixed cost k_j. There is also a joint fixed cost whenever any item is reviewed. The problem is to find review periods T₁, T₂, …, T_N and an ordering policy satisfying the average cost criterion. The current article builds on earlier results for the single‐item case. We prove an optimal policy exists, give conditions where it has a simple form, and develop a branch and bound algorithm for its computation. We also provide two heuristic policies with O(N) computational requirements. Computational experiments indicate that the branch and bound algorithm can handle normal demand problems with N ≤ 10 and that both heuristics do well for a wide variety of problems with N ranging from 2 to 200; moreover, the performance of our heuristics seems insensitive to N. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:430–449, 2001     

Algorithms for minimizing total cost,bottleneck time and bottleneck shipment in transportation problems

We consider the transportation problem of determining nonnegative shipments from a set of m warehouses with given availabilities to a set of n markets with given requirements. Three objectives are defined for each solution: (i) total cost, TC, (ii) bottleneck time, BT (i.e., maximum transportation time for a positive shipment), and (iii) bottleneck shipment, SB (i.e., total shipment over routes with bottleneck time). An algorithm is given for determining all efficient (pareto-optimal or nondominated) (TC, BT) solution pairs. The special case of this algorithm when all the unit cost coefficients are zero is shown to be the same as the algorithms for minimizing BT. provided by Szwarc and Hammer. This algorithm for minimizing BT is shown to be computationally superior. Transportation or assignment problems with m=n=100 average about a second on the UNIVAC 1108 computer (FORTRAN V)) to the threshold algorithm for minimizing BT. The algorithm is then extended to provide not only all the efficient (TC, BT) solution pairs but also, for each such BT, all the efficient (TC, SB) solution pairs. The algorithms are based on the cost operator theory of parametric programming for the transportation problem developed by the authors.     

Economic lot sizing with constant capacities and concave inventory costs

This article studies the classical single‐item economic lot‐sizing problem with constant capacities, fixed‐plus‐linear order costs, and concave inventory costs, where backlogging is allowed. We propose an O(T³) optimal algorithm for the problem, which improves upon the O(T⁴) running time of the famous algorithm developed by Florian and Klein (Manage Sci18 (1971) 12–20). Instead of using the standard dynamic programming approach by predetermining the minimal cost for every possible subplan, we develop a backward dynamic programming algorithm to obtain a more efficient implementation. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

An extension of the (szwarc) truck assignment problem

In this journal in 1967. Szware presented an algorithm for the optimal routing of a common vehicle fleet between m sources and n sinks with p different types of commodities. The main premise of the formulation is that a truck may carry only one commodity at a time and must deliver the entire load to one demand area. This eliminates the problem of routing vehicles between sources or between sinks and limits the problem to the routing of loaded trucks between sources and sinks and empty trucks making the return trip. Szwarc considered only the transportation aspect of the problem (i. e., no intermediate points) and presented a very efficient algorithm for solution of the case he described. If the total supply is greater than the total demand, Szwarc shows that the problem is equivalent to a (mp + n) by (np + m) Hitchcock transportation problem. Digital computer codes for this algorithm require rapid access storage for a matrix of size (mp + n) by (np + m); therefore, computer storage required grows proportionally to p². This paper offers an extension of his work to a more general form: a transshipment network with capacity constraints on all arcs and facilities. The problem is shown to be solvable directly by Fulkerson's out-of-kilter algorithm. Digital computer codes for this formulation require rapid access storage proportional to p instead of p². Computational results indicate that, in addition to handling the extensions, the out-of-kilter algorithm is more efficient in the solution of the original problem when there is a mad, rate number of commodities and a computer of limited storage capacity.     

A general treatment of upper unbounded and bounded hitchcock problems

This paper is designed to treat (a) the problem of the determination of the absolute minimum cost, with the associated assignments, when there is no limit, N, on the number of parcels available for shipment in a modified Hitchcock problem. This is accomplished with the use of a transformed cost matrix. C*, to which the so-called transportation paradox does not apply. The general Hitchcock solution using C* gives the cost T*, which is the absolute minimum cost of the original problem, as well as sets of assignments which are readily transformed to give the general assignments of the original problem. The sum of these latter assignments gives the value of N_u, the unbounded N for minimum cost. In addition, this paper is designed to show (b) how the method of reduced matrices may be used, (c) how a particular Hitchcock solution can be used to determine a general solution so that one solution using C* can provide the general answer, (d) how the results may be modified to apply to problems with fixed N, and hence (e) to determine the function of the decreasing T as N approaches N_u, and finally (f) to provide a treatment when the supplies at origin i and/or the demands at destination j, are bounded.     

The dynamic transportation problem: A survey

The dynamic transportation problem is a transportation problem over time. That is, a problem of selecting at each instant of time t, the optimal flow of commodities from various sources to various sinks in a given network so as to minimize the total cost of transportation subject to some supply and demand constraints. While the earliest formulation of the problem dates back to 1958 as a problem of finding the maximal flow through a dynamic network in a given time, the problem has received wider attention only in the last ten years. During these years, the problem has been tackled by network techniques, linear programming, dynamic programming, combinational methods, nonlinear programming and finally, the optimal control theory. This paper is an up-to-date survey of the various analyses of the problem along with a critical discussion, comparison, and extensions of various formulations and techniques used. The survey concludes with a number of important suggestions for future work.     

New heuristic methods for the capacitated multi‐facility Weber problem

In this paper we consider the capacitated multi‐facility Weber problem with the Euclidean, squared Euclidean, and ?_p‐distances. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the distance between them. We first present a mixed integer linear programming approximation of the problem. We then propose new heuristic solution methods based on this approximation. Computational results on benchmark instances indicate that the new methods are both accurate and efficient. © 2006 Wiley Periodicals, Inc. Naval Research Logistics 2006     

A single‐period inventory placement problem for a serial supply chain

This article addresses the inventory placement problem in a serial supply chain facing a stochastic demand for a single planning period. All customer demand is served from stage 1, where the product is stored in its final form. If the demand exceeds the supply at stage 1, then stage 1 is resupplied from stocks held at the upstream stages 2 through N, where the product may be stored in finished form or as raw materials or subassemblies. All stocking decisions are made before the demand occurs. The demand is nonnegative and continuous with a known probability distribution, and the purchasing, holding, shipping, processing, and shortage costs are proportional. There are no fixed costs. All unsatisfied demand is lost. The objective is to select the stock quantities that should be placed different stages so as to maximize the expected profit. Under reasonable cost assumptions, this leads to a convex constrained optimization problem. We characterize the properties of the optimal solution and propose an effective algorithm for its computation. For the case of normal demands, the calculations can be done on a spreadsheet. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48:506–517, 2001     

Spatial pricing and product allocation in online retailing

Spatial pricing means a retailer price discriminates its customers based on their geographic locations. In this article, we study how an online retailer should jointly allocate multiple products and facilitate spatial price discrimination to maximize profits. When deciding between a centralized product allocation ((i.e., different products are allocated to the same fulfillment center) and decentralized product allocation (ie, different products are allocated to different fulfillment centers), the retailer faces the tradeoff between shipment pooling (ie, shipping multiple products in one package), and demand localization (ie, stocking products to satisfy local demand) based on its understanding of customers' product valuations. In our basic model, we consider two widely used spatial pricing policies: free on board (FOB) pricing that charges each customer the exact amount of shipping cost, and uniform delivered (UD) pricing that provides free shipping. We propose a stylized model and find that centralized product allocation is preferred when demand localization effect is relatively low or shipment pooling benefit is relatively high under both spatial pricing policies. Moreover, centralized product allocation is more preferred under the FOB pricing which encourages the purchase of virtual bundles of multiple products. Furthermore, we respectively extend the UD and FOB pricing policies to flat rate shipping (ie, the firm charges a constant shipping fee for each purchase), and linear rate shipping (ie, the firm sets the shipping fee as a fixed proportion of firm's actual fulfillment costs). While similar observations from the basic model still hold, we find the firm can improve its profit by sharing the fulfillment cost with its customers via the flat rate or linear rate shipping fee structure.     

A two-echelon inventory model with purchases,dispositions, shipments,returns and transshipments

This paper presents a one-period two-echelon inventory model with one warehouse in the first echelon and n warehouses in the second echelon. At the beginning of the period the stock levels at all facilities are adjusted by purchasing or disposing of items at the first echelon, returning or shipping items between the echelons and transshipping items within the second echelon. During the period, demands (which may be negative) are placed on all warehouses in the second echelon and an attempt is made to satisfy shortages either by an expedited shipment from the first echelon to the second echelon or an expedited transshipment within the second echelon. The decision problem is to choose an initial stock level at the first echelon (by a purchase or a disposition) and an initial allocation so as to minimize the initial stock movement costs during the period plus inventory carrying costs and system shortage costs at the end of the period. It is shown that the objective function takes on one of four forms, depending on the relative magnitudes of the various shipping costs. All four forms of the objective function are derived and proven to be convex. Several applications of this general model are considered. We also consider multi-period extensions of the general model and an important special case is solved explicitly.     

Minimizing logistic costs in multistage supply chains

We consider the problem in which a set of products has to be shipped from a common origin to a common destination through one or several intermediate nodes with the objective of minimizing the sum of inventory and transportation costs when a set of possible shipping frequencies is given on each link. From the theoretical point of view, the main issue is the computation of the inventory cost in the intermediate nodes. From the computational point of view, given that the simpler single link problem is known to be NP-hard, we present four classes of heuristic algorithms. The first two classes are based on the decomposition of the sequence in links, the third class on the adaptation of the EOQ-type solution known for the continuous case, and the fourth on the optimal solution of a simpler problem through dynamic programming techniques. Finally, we compare them on a set of randomly generated problem instances. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 399–417, 1999     

Capacitated warehouse location model with risk pooling

In this article, we introduce the capacitated warehouse location model with risk pooling (CLMRP), which captures the interdependence between capacity issues and the inventory management at the warehouses. The CLMRP models a logistics system in which a single plant ships one type of product to a set of retailers, each with an uncertain demand. Warehouses serve as the direct intermediary between the plant and the retailers for the shipment of the product and also retain safety stock to provide appropriate service levels to the retailers. The CLMRP minimizes the sum of the fixed facility location, transportation, and inventory carrying costs. The model simultaneously determines warehouse locations, shipment sizes from the plant to the warehouses, the working inventory, and safety stock levels at the warehouses and the assignment of retailers to the warehouses. The costs at each warehouse exhibit initially economies of scale and then an exponential increase due to the capacity limitations. We show that this problem can be formulated as a nonlinear integer program in which the objective function is neither concave nor convex. A Lagrangian relaxation solution algorithm is proposed. The Lagrangian subproblem is also a nonlinear integer program. An efficient algorithm is developed for the linear relaxation of this subproblem. The Lagrangian relaxation algorithm provides near‐optimal solutions with reasonable computational requirements for large problem instances. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Optimal outbound dispatch policies: Modeling inventory and cargo capacity

In this paper, we present an optimization model for coordinating inventory and transportation decisions at an outbound distribution warehouse that serves a group of customers located in a given market area. For the practical problems which motivated this paper, the warehouse is operated by a third party logistics provider. However, the models developed here may be applicable in a more general context where outbound distribution is managed by another supply chain member, e.g., a manufacturer. We consider the case where the aggregate demand of the market area is constant and known per period (e.g., per day). Under an immediate delivery policy, an outbound shipment is released each time a demand is realized (e.g., on a daily basis). On the other hand, if these shipments are consolidated over time, then larger (hence more economical) outbound freight quantities can be dispatched. In this case, the physical inventory requirements at the third party warehouse (TPW) are determined by the consolidated freight quantities. Thus, stock replenishment and outbound shipment release policies should be coordinated. By optimizing inventory and freight consolidation decisions simultaneously, we compute the parameters of an integrated inventory/outbound transportation policy. These parameters determine: (i) how often to dispatch a truck so that transportation scale economies are realized and timely delivery requirements are met, and (ii) how often, and in what quantities, the stock should be replenished at the TPW. We prove that the optimal shipment release timing policy is nonstationary, and we present algorithms for computing the policy parameters for both the uncapacitated and finite cargo capacity problems. The model presented in this study is considerably different from the existing inventory/transportation models in the literature. The classical inventory literature assumes that demands should be satisfied as they arrive so that outbound shipment costs are sunk costs, or else these costs are covered by the customer. Hence, the classical literature does not model outbound transportation costs. However, if a freight consolidation policy is in place then the outbound transportation costs can no longer be ignored in optimization. Relying on this observation, this paper models outbound transportation costs, freight consolidation decisions, and cargo capacity constraints explicitly. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 531–556, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10030     

Optimal policies for M/M/m queue with two different kinds of (N,T)‐policies

In this paper, two different kinds of (N, T)‐policies for an M/M/m queueing system are studied. The system operates only intermittently and is shut down when no customers are present any more. A fixed setup cost of K > 0 is incurred each time the system is reopened. Also, a holding cost of h > 0 per unit time is incurred for each customer present. The two (N, T)‐policies studied for this queueing system with cost structures are as follows: (1) The system is reactivated as soon as N customers are present or the waiting time of the leading customer reaches a predefined time T, and (2) the system is reactivated as soon as N customers are present or the time units after the end of the last busy period reaches a predefined time T. The equations satisfied by the optimal policy (N*, T*) for minimizing the long‐run average cost per unit time in both cases are obtained. Particularly, we obtain the explicit optimal joint policy (N*, T*) and optimal objective value for the case of a single server, the explicit optimal policy N* and optimal objective value for the case of multiple servers when only predefined customers number N is measured, and the explicit optimal policy T* and optimal objective value for the case of multiple servers when only predefined time units T is measured, respectively. These results partly extend (1) the classic N or T policy to a more practical (N, T)‐policy and (2) the conclusions obtained for single server system to a system consisting of m (m ≥ 1) servers. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 240–258, 2000     

A note on a stochastic production-maximizing transportation problem

A stochastic production-maximizing problem with transportation constraints is considered where the production rates, R_ij, of man i — job j combinations are random variables rather than constants. It is shown that for the family of Weibull distributions (of which the Exponential is a special case) with scale parameters λ_ij and shape parameter β, the plan that maximizes the expected rate of the entire line is obtained by solving a deterministic fixed charge transportation problem with no linear costs and with “set-up” cost matrix ‖λ_ij‖.     

Multiperiod capacity expansion and shipment planning with linear costs

In this paper we consider a multiperiod deterministic capacity expansion and shipment planning problem for a single product. The product can be manufactured in several producing regions and is required in a number of markets. The demands for each of the markets are non-decreasing over time and must be met exactly during each time period (i.e., no backlogging or inventorying for future periods is permitted). Each region is assumed to have an initial production capacity, which may be increased at a given cost in any period. The demand in a market can be satisfied by production and shipment from any of the regions. The problem is to find a schedule of capacity expansions for the regions and a schedule of shipments from the regions to the markets so as to minimize the discounted capacity expansion and shipment costs. The problem is formulated as a linear programming model, and solved by an efficient algorithm using the operator theory of parametric programming for the transporation problem. Extensions to the infinite horizon case are also provided.     

Algorithms for the minimax transportation problem

In this paper, we consider a variant of the classical transportation problem as well as of the bottleneck transportation problem, which we call the minimax transportation problem. The problem considered is to determine a feasible flow x_ij from a set of origins I to a set of destinations J for which max_(i,j)εIxJ{c_ijx_ij} is minimum. In this paper, we develop a parametric algorithm and a primal-dual algorithm to solve this problem. The parametric algorithm solves a transportation problem with parametric upper bounds and the primal-dual algorithm solves a sequence of related maximum flow problems. The primal-dual algorithm is shown to be polynomially bounded. Numerical investigations with both the algorithms are described in detail. The primal-dual algorithm is found to be computationally superior to the parametric algorithm and it can solve problems up to 1000 origins, 1000 destinations and 10,000 arcs in less than 1 minute on a DEC 10 computer system. The optimum solution of the minimax transportation problem may be noninteger. We also suggest a polynomial algorithm to convert this solution into an integer optimum solution.