A two-echelon inventory system with transportation capacity constraint

Consider a one-warehouse multi-retailer system under constant and deterministic demand, which is subjected to transportation capacity for every delivery period. To search for the best stationary zero inventory ordering (ZIO) policy, or the best power-of-two policy, or the best nested policy, the problem is formulated as a 0–1 integer linear program in which the objective function comprises of a fixed transportation cost whenever a delivery is made and the inventory costs for both the warehouse and retailers. To overcome the transportation capacity limitation, we extend the policies to allow for staggering deliveries. It is shown that with transportation capacity constraint the non-staggering policy can have its effectiveness close to 0% from the best staggering policy and the power-of-two policy with staggering allowed can have its effectiveness close to 0% from the optimal policy. Nevertheless in general, the power-of-two policy fairs well on a number of randomly generated problems. To solve the large distribution network problem, an efficient heuristic based on the power-of-two policy with staggering of deliveries is suggested.     

Constrained continuous-time Markov decision processes with average criteria

In this paper, we study constrained continuous-time Markov decision processes with a denumerable state space and unbounded reward/cost and transition rates. The criterion to be maximized is the expected average reward, and a constraint is imposed on an expected average cost. We give suitable conditions that ensure the existence of a constrained-optimal policy. Moreover, we show that the constrained-optimal policy randomizes between two stationary policies differing in at most one state. Finally, we use a controlled queueing system to illustrate our conditions. Supported by NSFC, NCET and RFDP.     

Minimizing the Total Cost in an Integrated Vendor—Managed Inventory System

In this paper we consider a complex production-distribution system, where a facility produces (or orders from an external supplier) several items which are distributed to a set of retailers by a fleet of vehicles. We consider Vendor-Managed Inventory (VMI) policies, in which the facility knows the inventory levels of the retailers and takes care of their replenishment policies. The production (or ordering) policy, the retailers replenishment policies and the transportation policy have to be determined so as to minimize the total system cost. The cost includes the fixed and variable production costs at the facility, the inventory costs at the facility and at the retailers and the transportation costs, that is the fixed costs of the vehicles and the traveling costs. We study two different types of VMI policies: The order-up-to level policy, in which the order-up-to level quantity is shipped to each retailer whenever served (i.e. the quantity delivered to each retailer is such that the maximum level of the inventory at the retailer is reached) and the fill-fill-dump policy, in which the order-up-to level quantity is shipped to all but the last retailer on each delivery route, while the quantity delivered to the last retailer is the minimum between the order-up-to level quantity and the residual transportation capacity of the vehicle. We propose two different decompositions of the problem and optimal or heuristic procedures for the solution of the subproblems. We show that, for reasonable initial values of the variables, the order in which the subproblems are solved does not influence the final solution. We will first solve the distribution subproblem and then the production subproblem. The computational results show that the fill-fill-dump policy reduces the average cost with respect to the order-up-to level policy and that one of the decompositions is more effective. Moreover, we compare the VMI policies with the more traditional Retailer-Managed Inventory (RMI) policy and show that the VMI policies significantly reduce the average cost with respect to the RMI policy.     

Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length

In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale

In this paper, we formulate an analytical model for the joint determination of an optimal age-dependent buffer inventory and preventive maintenance policy in a production environment that is subject to random machine breakdowns. Traditional preventive maintenance policies, such as age and periodic replacements, are usually studied based on simplified and non-realistic assumptions, as well as on the expected costs criterion. Finished goods inventories and the age-dependent likelihood of machine breakdowns are usually not considered. As a result, these policies could significantly extend beyond the anticipated financial incomes of the system, and lead to crises. In order to solve this problem, a more realistic analysis model is proposed in this paper to consider the effects of both preventive maintenance policies and machine age on optimal safety stock levels. Hence, a unified framework is developed, allowing production and preventive maintenance to be jointly considered. We use an age-dependent optimization model based on the minimization of an overall cost function, including inventory holdings, lost sales, preventive and corrective maintenance costs. We provide optimality conditions for the manufacturing systems considered, and use numerical methods to obtain an optimal preventive maintenance policy and the relevant age-dependent threshold level production policy. In this work, this policy is called the multiple threshold levels hedging point policy. We include numerical examples and sensitivity analyses to illustrate the importance and the effectiveness of the proposed methodology. Compared with other available optimal production and maintenance policies, the numerical solution obtained shows that the proposed age-dependent optimal production and maintenance policies significantly reduce the overall cost incurred.     

Optimal control of a dual service rate M/M/1 production-inventory model

We analyse a dual-source, production-inventory model in which the processing times at a primary manufacturing resource and a second, contingent resource are exponentially distributed. We interpret the contingent source to be a subcontractor, although it could also be overtime production. We treat the inventory and contingent sourcing policies as decision variables in an analytical study and, additionally, allow the primary manufacturing capacity to be a decision variable in a subsequent numerical study. Our goal is to gain insight into the use of subcontracting as a contingent source of goods and whether it can fulfill real-world managers' expectations for improved performance. We prove that a stationary, non-randomised inventory and subcontracting policy is optimal for our M/M/1 dual-source model and, moreover, that a dual base-stock policy is optimal. We then derive an exact closed-form expression for one of the optimal base stocks, which to our knowledge is the first closed-form solution for a dual-source model. We use that closed-form result to advantage in a numerical study from which we gain insight into how optimal capacity, subcontracting, and inventory policies are set, and how effectively a contingent source can reduce total cost, capacity cost, and inventory cost. We find that (i) the contingent source can reduce total cost effectively even when contingent sourcing is expensive and (ii) contingent sourcing reduces capacity cost more effectively than it does inventory cost.     

Costing minimal-repair replacement policies over finite time horizons

An integral-equation technique is used to evaluate the expectedcost of maintaining a system functioning over the period (O,t] using two minimal-repair replacement policies. These costfunctions provide appropriate criteria to determine T^*, theoptimal scheduled replacement period over this finite time horizon.For both policies, it is shown that significant cost savingscan be achieved by using the T^* values predicted by the newmodels with a finite time horizon rather than those obtainedfrom the established asymptotic formulations. An adaptive finiteminimal-repair replacement policy is also formulated using dynamicprogramming, and the expected cost of this policy is shown tobe only slightly less than that of the best stationary policy.     

Can-order policy for the periodic-review joint replenishment problem

In this paper we study the stochastic joint replenishment problem. We compare the class of periodic replenishment policies and the class of can-order policies for this problem. We present a method, based on Markov decision theory, to calculate near-optimal can-order policies for a periodic-review inventory system. Our numerical study shows that the can-order policy behaves as well as, if not better than, the periodic replenishment policies. In particular, for examples where the demand is irregular, we find cost differences up to 15% in favour of the can-order policy.     

Supply chain models for an assembly system with preprocessing of raw materials

In this paper, we investigate the material procurement and delivery policy in a production system where raw materials enter into the assembly line from two different flow channels. The system encompasses batch production process in which the finished product demand is approximately constant for an infinite planning horizon. Two distinct types of raw materials are passed through the assembly line before to convert them into the finished product. Of the two types of raw materials, one type requires preprocessing inside the facility before the assembly operation and other group is fed straightway in the assembly line. The conversion factors are assigned to raw materials to quantify the raw material batch size required. To analyze such a system, we formulate a nonlinear cost function to aggregate all the costs of the inventories, ordering, shipping and deliveries. An algorithm using the branch and bound concept is provided to find the best integer values of the optimal solutions. The result shows that the optimal procurement and delivery policy minimizes the expected total cost of the model. Using a test problem, the inventory requirements at each stage of production and their corresponding costs are calculated. From the analysis, it is shown that the rate and direction change of total cost is turned to positive when delivery rates per batch reaches close to the optimal value and the minimum cost is achieved at the optimal delivery rate. Also, it is shown that total incremental cost is monotonically increasing, if the finished product batch size is increased, and if, inventory cost rates are increased. We examine a set of numerical examples that reveal the insights into the procurement-delivery policy and the performance of such an assembly type inventory model.     

Allocating flexible servers in serial systems with switching costs

Consider a firm that operates a make-to-order serial production system and employs a cross-trained workforce. We model such a firm as a tandem queuing system in which flexible servers can be allocated across stations, and assume that a switching cost is charged when servers move between stations. We show that even in the two-station two-server case the optimal policy follows a complex state-dependent structure that may be difficult to implement in practice. We propose three alternate heuristic policies and assess their performance. We show that a simpler policy which only moves one server can achieve close to optimal results.     

Optimal maintenance of a production-inventory system with idle periods

In this paper we consider a production-inventory system in which an input generating installation supplies a buffer with a raw material and a production unit pulls the raw material from the buffer with constant rate. The installation deteriorates in time and the problem of its optimal preventive maintenance is considered. It is assumed that the installation after the completion of its maintenance remains idle until the buffer is evacuated. Under a suitable cost structure it is shown that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it prescribes a preventive maintenance of the installation if and only if its degree of deterioration is greater than or equal to a critical level. Using the usual regenerative argument, the average cost of a control-limit policy is computed exactly and then, the optimal control-limit policy is determined. Furthermore, the stationary probabilities of the system under the optimal policy are computed.     

Heuristics with guaranteed performance bounds for a manufacturing system with product recovery

We consider a manufacturing system with product recovery. The system manufactures a new product as well as remanufactures the product from old, returned items. The items remanufactured with the returned products are as good as new and satisfy the same demand as the new item. The demand rate for the new item and the return rate for the old item are deterministic and constant. The relevant costs are the holding costs for the new item and the returned item, and the fixed setup costs for both manufacturing and remanufacturing. The objective is to determine the lot sizes and production schedule for manufacturing and remanufacturing so as to minimize the long-run average cost per unit time. We first develop a lower bound among all classes of policies for the problem. We then show that the optimal integer ratio policy for the problem obtains a solution whose cost is at most 1.5% more than the lower bound.     

Base stock policies for production/inventory problems with uncertain capacity levels

In this paper we consider a single item, stochastic demand production/inventory problem where the maximum amount that can be produced (or ordered) in any given period is assumed to be uncertain. Inventory levels are reviewed periodically. The system operates under a stationary modified base stock policy. The intent of our paper is to present a procedure for computing the optimal base stocl level of this policy under expected average cost per period criterion. This procedure would provide guidance as to the appropriate amount of capacity to store in the form of inventory in the face of stochastic demand and uncertain capacity. In achieving this goal, our main contribution is to establish the analogy between the class of base stock production/inventory policies that operate under demand/capacity uncertainty, and the G/G/1 queues and their associated random walks. We also present example derivations for some important capacity distributions.     

Joint Economic Ordering Policies of Multiple Wholesalers and a Single Manufacturer with Price-dependent Demand Functions

The economic ordering policies for multiple regional wholesalers and the production lot-sizing policy for a single manufacturer have been studied in a joint analysis under the assumption that the yearly demands of each region are functions of their respective retail proces. We obtained optimum EOQs for both linear and constant price elasticity demand functions. Although normally the wholesalers would order in quantities equal to their EOQs, they are encouraged to purchase in different quantities by the producer providing compensation to offset the wholesalers' increased costs. The production lot-size is determined to minimize the overall production cost.     

Characterizing optimal empty container reposition policy in periodic-review shuttle service systems

This paper considers a periodic-review shuttle service system with random customer demands and finite reposition capacity. The objective is to find the optimal stationary policy of empty container reposition by minimizing the sum of container leasing cost, inventory cost and reposition cost. Using Markov decision process approach, the structures of the optimal stationary policies for both expected discounted cost and long-run average cost are completely characterized. Monotonic and asymptotic behaviours of the optimal policy are established. By taking advantage of special structure of the optimal policy, the stationary distribution of the system states is obtained, which is then used to compute interesting steady-state performance measures and implement the optimal policy. Numerical examples are given to demonstrate the results.     

Asymptotic properties of constrained Markov Decision Processes

We present in this paper several asymptotic properties of constrained Markov Decision Processes (MDPs) with a countable state space. We treat both the discounted and the expected average cost, with unbounded cost. We are interested in (1) the convergence of finite horizon MDPs to the infinite horizon MDP, (2) convergence of MDPs with a truncated state space to the problem with infinite state space, (3) convergence of MDPs as the discount factor goes to a limit. In all these cases we establish the convergence of optimal values and policies. Moreover, based on the optimal policy for the limiting problem, we construct policies which are almost optimal for the other (approximating) problems. Based on the convergence of MDPs with a truncated state space to the problem with infinite state space, we show that an optimal stationary policy exists such that the number of randomisations it uses is less or equal to the number of constraints plus one. We finally apply the results to a dynamic scheduling problem.This work was partially supported by the Chateaubriand fellowship from the French embassy in Israel and by the European Grant BRA-QMIPS of CEC DG XIII     

Control Policy for a Manufacturing System with Random Yield and Rework

We develop a production policy that controls work-in-process (WIP) levels and satisfies demand in a multistage manufacturing system with significant uncertainty in yield, rework, and demand. The problem addressed in this paper is more general than those in the literature in three aspects: (i) multiple products are processed at multiple workstations, and the capacity of each workstation is limited and shared by multiple operations; (ii) the behavior of a production policy is investigated over an infinite-time horizon, and thus the system stability can be evaluated; (iii) the representation of yield and rework uncertainty is generalized. Generalizing both the system structure and the nature of uncertainty requires a new mathematical development in the theory of infinite-horizon stochastic dynamic programming. The theoretical contributions of this paper are the existence proofs of the optimal stationary control for a stochastic dynamic programming problem and the finite covariances of WIP and production levels under the general expression of uncertainty. We develop a simple and explicit sufficient condition that guarantees the existence of both the optimal stationary control and the system stability. We describe how a production policy can be constructed for the manufacturing system based on the propositions derived.     

Production/Clearing Models Under Continuous and Sporadic Reviews

We consider production/clearing models where random demand for a product is generated by customers (e.g., retailers) who arrive according to a compound Poisson process. The product is produced uniformly and continuously and added to the buffer to meet future demands. Allowing to operate the system without a clearing policy may result in high inventory holding costs. Thus, in order to minimize the average cost for the system we introduce two different clearing policies (continuous and sporadic review) and consider two different issuing policies (“all-or-some” and “all-or-none”) giving rise to four distinct production/clearing models. We use tools from level crossing theory and establish integral equations representing the stationary distribution of the buffer’s content level. We solve the integral equations to obtain the stationary distributions and develop the average cost objective functions involving holding, shortage and clearing costs for each model. We then compute the optimal value of the decision variables that minimize the objective functions. We present numerical examples for each of the four models and compare the behaviour of different solutions.AMS 2000 Subject Classification: 90B05 Inventory, storage, reservoirs; 90B22 Queues and service; 90B30 Production models