Positive Lead Time Research Articles

A continuous review replenishment–disposal policy for an inventory system with autonomous supply and fixed disposal costs

In this study, we analyze an inventory system facing stochastic external demands and an autonomous supply (independent return flow) in the presence of fixed disposal costs and positive lead times under a continuous review replenishment–disposal policy. We derive the analytical expressions of the operating characteristics of the system; and, construct the objective function to minimize the total expected costs of ordering, holding, purchasing and disposal per unit time subject to a fill rate constraint. An extensive numerical analysis is conducted to study the sensitivity of the policy parameters and the benefit of employing a policy which allows for disposal of excess stock in this setting. We model the net demand process as the superposition of normally distributed external demand and inflows, which is expressed as a Brownian motion process. Our findings indicate that the disposal option results in considerable savings even (i) in the presence of non-zero fixed disposal costs, (ii) large actual demand rates with high return ratios (resulting in small net demands) and (iii) for moderate return ratios with high demand variability.

An Analysis of the Control-Algorithm Re-solving Issue in Inventory and Revenue Management

While inventory- and revenue-management problems can be represented as Markov decision process (MDP) models, in some cases the well-known dynamic-programming curse of dimensionality makes it computationally prohibitive to solve them exactly. An alternative solution, called here the control-algorithm approach, is to use a math program (MP) to approximately represent the MDP and use its optimal solution to heuristically instantiate the parameters of the decision rules of a given set of control policies. As new information is observed over time, the control algorithm can incorporate it by re-solving the MP and revising the parameters of the decision rules with the newly obtained solution. The re-solving issue arises when one reflects on the consequences of this revision: Does the performance of the control algorithm really improve by revising its decision-rule instantiation with the solution of the re-solved MP, or should an appropriate modification of the prior solution be used? This paper analyzes the control-algorithm re-solving issue for a class of finite-horizon inventory- and revenue-management problems. It establishes sufficient conditions under which re-solving does not deteriorate the performance of a control algorithm, and it applies these results to control algorithms for network revenue management and multiproduct make-to-order production with lost sales and positive lead time.

Extending capacity planning by positive lead times and optional overtime, earliness and tardiness for effective master production scheduling

In this paper we consider the problem of improving the Master Production Schedule (MPS) in make-to-order production systems when demand exceeds available resource capacity. Due to the complexity of the problem, in practice solutions are usually obtained manually. We propose an algorithm that offsets production orders guided by tardiness, earliness and overtime penalties. The intermediate tool used to determine resource utilization is Rough Cut Capacity Planning (RCCP) extended by positive lead times and options for overtime, earliness and tardiness. This approach leads to a more realistic resource loading calculation, similar to CRP but without its computational burden. The discrete optimization model is solved by a Genetic Algorithm (GA); within the GA, delays or early deliveries of each order are represented as genes of a chromosome. The method is tested against systematically developed benchmark problems and real industrial data. Improvements over the traditional RCCP procedure and an ERP's embedded routines are demonstrated by the computational results and support the applicability of the proposed approach for real-life make-to-order environments.

Analysis of the (s, S) policy for perishables with a random shelf life

A continuous review perishable inventory system operating under the (s,S) policy is considered. Assuming a random shelf life with a general distribution, renewal arrivals and a negligible replenishment lead time, exact expressions for the expected cost rate function for unit and batch demands are derived. For unit demands, it is shown that the average cost rate function is quasi-convex in (s,S). Numerical findings indicate that the loss incurred by ignoring the randomness of the shelf life can be drastic. It is observed that the shape of the shelf life distribution has a significant impact on the costs and a precise estimation of shelf life distribution may result in substantial savings. Based on the presented analytical results, a new heuristic for positive lead times is proposed. Extensive numerical studies show that the proposed heuristic performs better than an existing one suggested for fixed shelf lives in most of the cases studied.

Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale

Chiang [C. Chiang, Optimal ordering policies for periodic-review systems with replenishment cycles, European Journal of Operational Research 170 (2006) 44–56] recently proposed a dynamic programming model for periodic-review systems in which a replenishment cycle consists of a number of small periods (each of identical but arbitrary length) and holding and shortage costs are charged based on the ending inventory of small periods. The current paper presents an alternative (and concise) dynamic programming model. Moreover, we allow the possibility of a positive fixed cost of ordering. The optimal policy is of the familiar ( s, S) type because of the convexity of the one-cycle cost function. As in the periodic-review inventory literature, we extend this result to the lost-sales periodic problem with zero lead-time. Computation shows that the long-run average cost is rather insensitive to the choice of the period length. In addition, we show how the proposed model is modified to handle the backorder problem where shortage is charged on a per-unit basis irrespective of its duration. Finally, we also investigate the lost-sales problem with positive lead-time, and provide some computational results.

Comparison of Inventory Systems with Service, Positive Lead-Time, Loss, and Retrial of Customers

We analyze and compare three (s,S) inventory systems with positive service time and retrial of customers. In all of these systems, arrivals of customers form a Poisson process and service times are exponentially distributed. When the inventory level depletes to s due to services, an order of replenishment is placed. The lead-time follows an exponential distribution. In model I, an arriving customer, finding the inventory dry or server busy, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the inventory dry or server busy, returns to the orbit with probability δ and is lost forever with probability (1−δ). In addition to the description in model I, we provide a buffer of varying (finite) capacity equal to the current inventory level for model II and another having capacity equal to the maximum inventory level S for model III. In models II and III, an arriving customer, finding the buffer full, proceeds to an orbit with probability γ and is lost forever with probability (1−γ). A retrial customer in the orbit, finding the buffer full, returns to the orbit with probability δ and is lost forever with probability (1−δ). In all these models, the interretrial times are exponentially distributed with linear rate. Using matrix-analytic method, we study these inventory models. Some measures of the system performance in the steady state are derived. A suitable cost function is defined for all three cases and analyzed using graphical illustrations.

A finite horizon inventory model: An operational framework

We present a simulation based decision support system motivated by a real life situation to decide the inventory ordering policy in the context of a single commodity, multipack, and finite horizon situation. The multiple objectives include (a) minimizing the end of the season inventory, (b) maximizing the operating profit, and (c) minimizing the peak working capital requirements during the season. Stochastic demand and positive lead time add to the complexity of the problem context. In addition, multiple partners in the supply chain with distinct and conflicting set of objectives necessitate the need for a formal approach. The problem can be formulated as a multiperiod dynamic programing problem with stochastic demand with an objective to optimize the expected operating profit, subject to specific constraints on working capital requirement, service level, order fill rate, and end of the season inventory. Such a formulation is hard to solve analytically and does not lend itself to analyze several ordering policies (what-if analysis). Based on simulation experiments, we propose an ordering policy which optimizes the overall objective(s) of supply chain partners. The model is simple and easy to use and is designed with adequate flexibility to conduct what-if analysis.

Advance demand information and a restricted production capacity: on the optimality of order base-stock policies

This paper considers the optimality of order aggregation in a single-item production/inventory problem with advance demand information and a restricted production capacity. The advance demand information is modeled by introducing a positive customer order lead time. The paper proves, when customer order lead times are less than a threshold value, it is allowed to aggregate the orders over time when establishing the optimal production decision. This implies the optimality of an order base-stock policy. It shows also that in case of linear inventory cost, the positive effect of advance demand information is equal to a cost reduction that is proportional to idle time and foreknowledge horizon. The results hold for the backlogging case as well as for the lost-sales case.

Demand forecast accuracy and performance of inventory policies under multi-level rolling schedule environments

Our incentive is to assess the impact of demand forecast errors on the cost performance of several lot-sizing techniques in a multi-level context. Unpredicted changes in demand keep on plaguing consumer product companies. However, efforts to improve demand forecast accuracy may not be rewarded if lot-sizing techniques perform equally badly as soon as forecast errors affect the demand. With an extensive simulation study we show that it is always worth decreasing the error magnitude: the performance of all techniques improves when the error level is decreased. But the relationship between cost improvement and error level is not linear as bigger cost reductions are obtained when the error decrease is applied to an initial value of error that is moderate. This means that increasing forecast accuracy is more profitable for companies that already have more accurate forecasts than for those who face inadequate forecasts. Although the presence or absence of forecast errors tends to matter more than the error level itself, we show that lot-sizing rules exhibit significant differences in their behavior as the level of error is augmented. This paper also provides a clear description of the rolling procedure when applied to general product structures, demand with forecast errors within the forecast window and positive lead times.

Transportation mode selection with positive manufacturing lead time

In this paper we present a dual supply model taking into account that in many industries the replenishment cycle does not only involve the physical distribution of goods, but also the manufacturing of products. Further, we suggest postponing the transportation mode decision until the products are available for shipping. We investigate a class of order-up-to policies and show how to compute the optimal policy parameters. Our model enables us to quantify the value of the postponement of the transportation decision and the value of using an additional slow mode instead of only using the existing fast mode. In a numerical study the influence of various model parameters is investigated and it is shown that the use of a slow mode can be economically beneficial, especially for low value items.

An alternative to safety stock policies for multi-level rolling schedule MRP problems

We study the impact of positive lead times on the multi-level lot-sizing problem in a rolling schedule environment. We show how stockout situations may arise even in a context of deterministic demand. We therefore develop a procedure to avoid such stockouts and we compare its performance through a simulation study to a safety stock strategy. Simulation results show the superiority of the proposed procedure.

Optimal machine capacity expansions with nested limitations under stochastic demand

AbstractThis paper studies capacity expansions for a production facility that faces uncertain customer demand for a single product family. The capacity of the facility is modeled in three tiers, as follows. The first tier consists of a set of upper bounds on production that correspond to different resource types (e.g., machine types, categories of manpower, etc.). These upper bounds are augmented in increments of fixed size (e.g., by purchasing machines of standard types). There is a second‐tier resource that constrains the first‐tier bounds (e.g., clean room floor space). The third‐tier resource bounds the availability of the second‐tier resource (e.g., the total floor space enclosed by the building, land, etc.). The second and third‐tier resources are expanded at various times in various amounts. The cost of capacity expansion at each tier has both fixed and proportional elements. The lost sales cost is used as a measure for the level of customer service. The paper presents a polynomial time algorithm (FIFEX) to minimize the total cost by computing optimal expansion times and amounts for all three types of capacity jointly. It accommodates positive lead times for each type. Demand is assumed to be nondecreasing in a “weak” sense. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.

Replenishment Strategies for Distribution Systems Under Advance Demand Information

Customers with positive demand lead times place orders in advance of their needs. A portfolio of customers with different demand lead times gives rise to what we call advance demand information. We develop effective inventory policies for a distribution system to account for this information. In particular, we study a centralized system with one warehouse serving multiple retailers under advance demand information. The inventory manager replenishes the warehouse from an outside supplier. Units arriving to the warehouse are allocated to the retailers. To control this system, we develop a lower bound and proposed a close-to-optimal heuristic for which the optimality gap is on average 1.92%. We also provide a closed-form solution to approximate the system-wide inventory level. Using this explicit solution, the model and the heuristic, we investigate (1) the benefit of advance demand information, and its impact on allocation decisions, (2) the joint role of risk pooling and advance demand information, and (3) the system performance with respect to supplier and retailer lead times. We illustrate how advance demand information can be a substitute for lead times and inventory, and how it enhances the outcome of delayed differentiation.

Optimal control of a one product recovery system with leadtimes

In this paper a recovery system for a single product is investigated. Besides a remanufacturing and a manufacturing facility the system consists of one inventory for returned and recoverable items and one for serviceable items. Demands are satisfied from serviceable inventory, which can be replenished by remanufactured returned items, which are as good as new, or by new produced items. Additionally, there is the possibility of disposing of returned items. We determine the cost optimal manufacturing, remanufacturing and disposal rates for the system under the assumptions of a linear cost structure, a finite planning horizon and deterministic and dynamic demand and return rates. Thereby we study two classes of policies, one where backorders are forbidden and another one where they are allowed. In contrast to the existing literature positive leadtimes are considered.

Serum tumor marker CA 125 for monitoring ovarian cancer during follow-up

CA 125 is currently widely applied in the management of patients with ovarian cancer. However, a change in results of CA 125, which should be considered significant, has not been defined. The aim of this study was to investigate the ability of CA 125 to signal progressive ovarian cancer during follow-up after firstline chemotherapy. The study patients were selected retrospectively among 255 patients with stage IC-IV ovarian cancer. The evaluation of the CA 125 information was based on the analytical imprecision, the normal intra-individual biological variation, the sampling interval, and the cut-off value. Additionally, the utility of a new assessment criterion based upon an increment of 2.5 times the baseline CA 125 concentration confirmed by a third measurement was investigated. The efficiency of CA 125 to identify progression and non-progression during follow-up varied between 76.5 and 79.9%, depending on the applied time limit for an acceptable positive lead time. The median lead time for true positive results was 95-99.5 days. Using the new elaborated criterion, the efficiency of CA 125 for identifying progression and non-progression varied between 75.7 and 78.5%, depending on the applied time limit for an acceptable positive lead time. The median lead time for true positive results was 91-95.5 days. CA 125 provided early and reliable information about progressive disease during follow-up. The applied criteria can therefore be recommended in further studies assessing the clinical utility of serological tumor markers in patients with ovarian cancer.

Integrating Replenishment Decisions with Advance Demand Information

There is a growing consensus that a portfolio of customers with different demand lead times can lead to higher, more regular revenues and better capacity utilization. Customers with positive demand lead times place orders in advance of their needs, resulting in advance demand information. This gives rise to the problem of finding effective inventory control policies under advance demand information. We show that state-dependent (s, S) and base-stock policies are optimal for stochastic inventory systems with and without fixed costs. The state of the system reflects our knowledge of advance demand information. We also determine conditions under which advance demand information has no operational value. A numerical study allows us to obtain additional insights and to evaluate strategies to induce advance demand information.

Continuous review perishable inventory systems: models and heuristics

We consider continuous review perishable inventory models with renewal batch demands. Assuming zero lead time, we construct an embedded Markov chain. With a probabilistic approach, we derive a closed-form long run average cost function under the (s, S) replenishment policy. Numerical analysis is then used to identify the properties of the cost function and to demonstrate the impacts of changing batch sizes and other system parameters. Based on these results, a heuristic approach to handle problems with positive lead times is proposed. The performance of the heuristics against simulation results demonstrates its effectiveness and accuracy

Continuous Review Perishable Inventory Systems: Models and Heuristics

We consider continuous review perishable inventory models with renewal batch demands. Assuming zero lead time, we construct an embedded Markov chain. With a probabilistic approach, we derive a closed-form long run average cost function under the (s,S) replenishment policy. Numerical analysis is then used to identify the properties of the cost function and to demonstrate the impacts of changing batch sizes and other system parameters. Based on these results, a heuristic approach to handle problems with positive lead times is proposed. The performance of the heuristics against simulation results demonstrates its effectiveness and accuracy.

(T, S) policy for coordinated inventory replenishment systems under compound Poisson demands

This paper considers the coordinated 'periodic-review, order-up-to-level' policy, called (T, S) policy, for multi-item inventory systems. The mathematical model of (T, S) policy for single-item inventory systems is extended to the more realistic multi-item inventory systems with compound Poisson demands, positive replenishment lead times, as well as service level constraints. Numerical comparisons with 'can-order' policy (S, c, s) with different treatments show that the (T, S) policy outperforms the extensively studied (S, c, s) policy. Especially when the lead time is not too short, considerable cost reductions can be achieved through an inventory system employing the (T, S) policy.

Coordination of Production/Distribution Networks with Unbalanced Leadtimes

We investigate multiproduct, multilocation production/distribution networks with deterministic, stationary demands. Most research on such systems assumes zero leadtimes. Methods using fixed reorder intervals that are power-of-two multiples of a base planning period have proved to be very successful for such systems. The same methods apply to networks with positive leadtimes, provided the leadtimes are balanced in a specific sense. We explore networks with general, unbalanced leadtimes. A key result is a lower bound on the cost of any feasible policy. Other results concern policy heuristics and their performance. For a large class of networks, we construct a policy whose cost is within 45% of the lower bound. For general networks, the performance guarantee is 1:02√(1+η), where η is a number dependent on the network topology only. In general, the best performance bound is obtained by systematically reducing the order intervals derived from the corresponding zero-leadtime system.