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     

Branch‐and‐price‐and‐cut for the manpower routing problem with synchronization constraints

In this article, we propose a branch‐and‐price‐and‐cut (BPC) algorithm to exactly solve the manpower routing problem with synchronization constraints (MRPSC). Compared with the classical vehicle routing problems (VRPs), the defining characteristic of the MRPSC is that multiple workers are required to work together and start at the same time to carry out a job, that is, the routes of the scheduling subjects are dependent. The incorporation of the synchronization constraints increases the difficulty of the MRPSC significantly and makes the existing VRP exact algorithm inapplicable. Although there are many types of valid inequalities for the VRP or its variants, so far we can only adapt the infeasible path elimination inequality and the weak clique inequality to handle the synchronization constraints in our BPC algorithm. The experimental results at the root node of the branch‐and‐bound tree show that the employed inequalities can effectively improve the lower bound of the problem. Compared with ILOG CPLEX, our BPC algorithm managed to find optimal solutions for more test instances within 1 hour. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 138–171, 2016     

Scheduling of depalletizing and truck loading operations in a food distribution system

This paper studies a scheduling problem arising in a beef distribution system where pallets of various types of beef products in the warehouse are first depalletized and then individual cases are loaded via conveyors to the trucks which deliver beef products to various customers. Given each customer's demand for each type of beef, the problem is to find a depalletizing and truck loading schedule that fills all the demands at a minimum total cost. We first show that the general problem where there are multiple trucks and each truck covers multiple customers is strongly NP‐hard. Then we propose polynomial‐time algorithms for the case where there are multiple trucks, each covering only one customer, and the case where there is only one truck covering multiple customers. We also develop an optimal dynamic programming algorithm and a heuristic for solving the general problem. By comparing to the optimal solutions generated by the dynamic programming algorithm, the heuristic is shown to be capable of generating near optimal solutions quickly. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2003     

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     

Optimal dispatching strategies for vehicles having exponentially distributed trip times

A transportation system has N vehicles with no capacity constraint which take passengers from a depot to various destinations and return to the depot. The trip times are considered to be independent and identically distributed random variables. The dispatch strategy at the depot is to dispatch immediately, or to hold any returning vehicles with the objective of minimizing the average wait per passenger at the depot, if passengers arrive at a uniform rate. Optimal control strategies and resulting waits are determined in the special case of exponentially distributed trip time for various N up to N = 15. For N ? 1, the nature of the solution is always to keep a reservoir of vehicles in the depot, and to decrease (increase) the time headway between dispatches as the size of the reservoir gets larger (smaller). For sufficiently large N, one can approximate the number of vehicles in the reservoir by a continuum and obtain analytic experession for the optimal dispatch rate as a function of the number of vehicles in the reservoir. For the optimal strategy, it is shown that the average number of vehicles in the depot is of order N^1/3. These limit properties are expected to be quite insensitive to the actual trip time distribution, but the convergence of the exact properties to the continuum approximation as N → ∞ is very slow.     

What you should know about the vehicle routing problem

In the Vehicle Routing Problem (VRP), the aim is to design a set of m minimum cost vehicle routes through n customer locations, so that each route starts and ends at a common location and some side constraints are satisfied. Common applications arise in newspaper and food delivery, and in milk collection. This article summarizes the main known results for the classical VRP in which only vehicle capacity constraints are present. The article is structured around three main headings: exact algorithms, classical heuristics, and metaheuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Vehicle routing problems with regular objective functions on a path

We investigate the problem of scheduling a fleet of vehicles to visit the customers located on a path to minimize some regular function of the visiting times of the customers. For the single‐vehicle problem, we prove that it is pseudopolynomially solvable for any minsum objective and polynomially solvable for any minmax objective. Also, we establish the NP‐hardness of minimizing the weighted number of tardy customers and the total weighted tardiness, and present polynomial algorithms for their special cases with a common due date. For the multivehicle problem involving n customers, we show that an optimal solution can be found by solving or O(n) single‐vehicle problems. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 61: 34–43, 2014     

Optimal pricing and scheduling control of product shipping

In this article, we seek to understand how a capacity‐constrained seller optimally prices and schedules product shipping to customers who are heterogeneous on willingness to pay (WTP) and willingness to wait (WTW). The capacity‐constrained seller does not observe each customer's WTP and WTW and knows only the aggregate distributions of WTP and WTW. The seller's problem is modeled as an M/M/N_s queueing model with multiclass customers and multidimensional information screening. We contribute to the literature by providing an optimal and efficient algorithm. Furthermore, we numerically find that customers with a larger waiting cost enjoys a higher scheduling priority, but customers with higher valuation do not necessarily get a higher scheduling priority. © 2015 Wiley Periodicals, Inc. Naval Research Logistics 62: 215–227, 2015     

Computing demand properties at the wholesale warehouse level

Computational formulas are given for the mean, variance, and autocorrelation function of the demand process at an upper-echelon facility (warehouse). The demand process at the warehouse is induced by the aggregated inventory replenishment processes of N independently operated lower-echelon facilities (stores) in parallel. Each store, we assume, employs an (s,S) inventory replenishment policy with complete backlogging to satisfy its own random, independently and identically distributed demand. The formulas result from an analysis of the stochastic replenishment process at a single store. Examples of the properties of the demand process at the upper-echelon facility are presented for several lower-echelon environments.     

Distance-directed augmenting path algorithms for maximum flow and parametric maximum flow problems

Until recently, fast algorithms for the maximum flow problem have typically proceeded by constructing layered networks and establishing blocking flows in these networks. However, in recent years, new distance-directed algorithms have been suggested that do not construct layered networks but instead maintain a distance label with each node. The distance label of a node is a lower bound on the length of the shortest augmenting path from the node to the sink. In this article we develop two distance-directed augmenting path algorithms for the maximum flow problem. Both the algorithms run in O(n²m) time on networks with n nodes and m arcs. We also point out the relationship between the distance labels and layered networks. Using a scaling technique, we improve the complexity of our distance-directed algorithms to O(nm log U), where U denotes the largest arc capacity. We also consider applications of these algorithms to unit capacity maximum flow problems and a class of parametric maximum flow problems.     

Approximate evaluation of batch-ordering policies for a one-warehouse,N non-identical retailer system under compound poisson demand

Considered is a two-level inventory system with one central warehouse and N retailers facing different independent compound Poisson demand processes. The retailers replenish from the warehouse and the warehouse from an outside supplier. All facilities apply continuous review installation stock (R, Q) policies with different reorder points and batch quantities. Presented is a new approximate method for evaluation of holding and shortage costs, which can be used to select optimal policies. The accuracy of the approximation is evaluated by comparison with exact and simulated results. © 1995 John Wiley & Sons, Inc.     

Split delivery routing

This article examines a relaxed version of the generic vehicle routing problem. In this version, a delivery to a demand point can be split between any number of vehicles. In spite of this relaxation the problem remains computationally hard. Since only small instances of the vehicle routing problem are known to be solved using exact methods, the vehicle route construction for this problem version is approached using heuristic rules. The main contribution of this article to the existing body of literature on vehicle routing issues in (a) is presenting a new vehicle routing problem amenable to practical applications, and (b) demonstrating the potential for cost savings over similar “traditional” vehicle routing when implementing the model and solutions presented here. The solution scheme allowing for split deliveries is compared with a solution in which no split deliveries are allowed. The comparison is conducted on six sets of 30 problems each for problems of size 75, 115, and 150 demand points (all together 540 problems). For very small demands (up to 10% of vehicle's capacity) no significant difference in solutions is evident for both solution schemes. For the other five problem sets for which point demand exceeds 10% of vehicle's capacity, very significant cost savings are realized when allowing split deliveries. The savings are significant both in the total distance and the number of vehicles required. The vehicles' routes constructed by our procedure tend to cover cohesive geographical zones and retain some properties of optimal solutions.     

Scheduling a maintenance activity on parallel identical machines

We study a problem of scheduling a maintenance activity on parallel identical machines, under the assumption that all the machines must be maintained simultaneously. One example for this setting is a situation where the entire system must be stopped for maintenance because of a required electricity shut‐down. The objective is minimum flow‐time. The problem is shown to be NP‐hard, and moreover impossible to approximate unless P = NP. We introduce a pseudo‐polynomial dynamic programming algorithm, and show how to convert it into a bicriteria FPTAS for this problem. We also present an efficient heuristic and a lower bound. Our numerical tests indicate that the heuristic provides in most cases very close‐to‐optimal schedules. © 2008 Wiley Periodicals, Inc. Naval Research Logistics 2009     

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.     

Service levels in distribution systems with random customer order size

An approximate method for measuring the service levels of the warehouse-retailer system operating under (s, S) policy is presented. All the retailers are identical and the demand process at each retailer follows a stationary stuttering Poisson process. This type of demand process allows customer orders to be for a random number of units, which gives rise to the undershoot quantity at both the warehouse and retailer levels. Exact analyses of the distribution of the undershoot quantity and the number of orders place by a retailer during the warehouse reordering lead time are derived. By using this distribution together with probability approximation and other heuristic approaches, we model the behavior of the warehouse level. Based on the results of the warehouse level and on an existing framework from previous work, the service level at the retailer level is estimated. Results of the approximate method are then compared with those of simulation. © 1995 John Wiley & Sons, Inc.     

Discrete equal‐capacity p‐median problem

This paper examines the discrete equal‐capacity p‐median problem that seeks to locate p new facilities (medians) on a network, each having a given uniform capacity, in order to minimize the sum of distribution costs while satisfying the demand on the network. Such problems arise, for example, in local access and transport area telecommunication network design problems where any number of a set of p facility units can be constructed at the specified candidate sites (hence, the net capacity is an integer multiple of a given unit capacity). We develop various valid inequalities, a separation routine for generating cutting planes that are specific members of such inequalities, as well as an enhanced reformulation that constructs a partial convex hull representation that subsumes an entire class of valid inequalities via its linear programming relaxation. We also propose suitable heuristic schemes for this problem, based on sequentially rounding the continuous relaxation solutions obtained for the various equivalent formulations of the problem. Extensive computational results are provided to demonstrate the effectiveness of the proposed valid inequalities, enhanced formulations, and heuristic schemes. The results indicate that the proposed schemes for tightening the underlying relaxations play a significant role in enhancing the performance of both exact and heuristic solution methods for this class of problems. © 2000 John & Sons, Inc. Naval Research Logistics 47: 166–183, 2000.     

Optimization of order-up-to-s policies in two-echelon inventory systems with periodic review

We consider an inventory system with one warehouse and N retailers. Transportation times are constant and the retailers face independent Poisson demand. Each facility applies a periodic review order-up-to-S policy. In case of shortages at the warehouse, orders for individual units are filled in the same order as the original demand at the retailers, i.e., according to a so-called virtual allocation scheme. Using that the considered policy is very similar to a continuous review one-for-one ordering policy, we are able to provide simple recursive procedures for exact evaluation of holding and shortage costs. © 1993 John Wiley & Sons, Inc.     

On the uncapacitated K‐commodity network design problem with zero flow‐costs

Extending Sastry's result on the uncapacitated two‐commodity network design problem, we completely characterize the optimal solution of the uncapacitated K‐commodity network design problem with zero flow costs for the case when K = 3. By solving a set of shortest‐path problems on related graphs, we show that the optimal solutions can be found in O(n³) time when K = 3, where n is the number of nodes in the network. The algorithm depends on identifying a list of “basic patterns”; the number of basic patterns grows exponentially with K. We also show that the uncapacitated K‐commodity network design problem can be solved in O(n³) time for general K if K is fixed; otherwise, the time for solving the problem is exponential. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004