Discrete Demand Distribution Research Articles

The newsvendor problem with GHG emissions from waste disposal under cap-and-trade regulation

We study a newsvendor problem with greenhouse gas (GHG) emissions at the disposal stage regulated by a cap-and-trade policy. We first consider a synchronous version of the problem where the replenishment period and the emission assessment period are the same. For this problem, we propose simple algorithms to derive the optimal quantity and cost for both continuous and discrete demand distributions. We show that ignoring the carbon tax may lead to large overcost, especially when the GHG quota is small. For a normally distributed demand, we also show that the demand standard deviation has a non-linear and non-monotonic effect on the order quantity. We then investigate an asynchronous version of the problem where replenishment decisions occur at a higher frequency than the quota allocation. We formulate this lot sizing problem as a Markov decision process and solve it with a dynamic programming algorithm whose complexity is assessed. Through a numerical experimentation and theoretical results, we demonstrate how and when the flexibility of the optimal dynamic replenishment decisions allows to reduce costs. When approaching the end of the time horizon, the uncertainty reduction allows the decision maker to adopt a riskier policy and order larger quantities.

Dynamic Inventory Control with Fixed Setup Costs and Unknown Discrete Demand Distribution

When a new product has just been introduced or the economy has just entered a new phase, a firm is often at a loss as to what the underlying demand pattern has become let alone how best to respond to it. In “Dynamic Inventory Control with Fixed Setup Costs and Unknown Discrete Demand Distribution,” Davoodi, Katehakis, and Yang faced off this challenging problem by tailoring ordering decisions to empirical distributions formed out of past demand observations. In the presence of fixed setup costs, however, an (s,S) policy optimal in the conventional known-distribution setting would take many periods for its long-term benefit to be realized. Therefore, a good online policy has to balance between letting ordering decisions settle for long periods and adjusting them frequently to take advantage of newly available information. When properly balanced, such policies could indeed achieve tight bounds for the performance measure of regret.

On the estimation of the fill rate for the continuous (s, S) inventory system for the lost sales context.

In the continuous review reorder point, base-stock (s, S) policy, the replenishment order is launched when the inventory position reaches the reorder point, s. It is commonly assumed that the inventory position is exactly equal to the reorder point at the moment the order is launched, when actually it could be lower at that moment. This implies neglecting the possible undershoots at the reorder point, which has a direct impact on the calculation of the expected shortages per replenishment cycle. This article presents a method for an exact calculation of the fill rate (fraction of demand that is immediately satisfied from shelf) which takes explicit account of the existence of undershoots and is applicable to any discrete demand distribution function in a context of lost sales. This method is based on the determination of the stock probability vector at the moment the replenishment order is launched. Furthermore, neglecting the undershoots is shown to lead to an overestimation of the fill rate, particularly when we move farther away from the unitary demand assumption. From a practical point of view, this behaviour involves underestimating the base-stock level, S, when a target fill rate is set for its determination. The method proposed in this paper overcomes these shortcomings.

Optimization of regional distributed energy system under discrete demand distribution

This study break through the idealized demand-side distribution and build an optimization model of regional distributed energy systems under discrete conditions. Aiming at minimizing the system comprehensive cost, which includes the annual conversion cost of the initial investment and the annual operation and maintenance cost, the optimization model can determine the number, site, capacity of the energy station and the optimal pipe network layout path from the energy station to the load center, and make a comparative analysis of the annual costs in different situations. The main impact of the energy station site is the cost difference caused by the pipe network layout.

Flexible contracting with heterogeneous agents and stochastic demand

While the predominant pricing scheme for services is a set of long-term contracts stipulating a fixed fee for a level of usage, many service providers now offer flexible contracts. These contracts include flexibility in usage level and the option to opt out of the service in certain time periods. We study the effectiveness of flexible contracts in aligning the interests between the service provider and the customers in the presence of customers with different discrete demand distributions. This work develops a unified principal–agent framework to examine multiple versions of flexible contracts under ex post information asymmetry. We explore two types of contract variations. The first is aggregated versus differentiated contracts across an agent’s random demand values, the second is whether an agent may decline participation in a future time period after committing to a set of contract options. We find that the principal always prefers differentiation to aggregation, and under differentiation prefers to require participation at each time period. However, when demand variability is grouped together under aggregated contracts, we derive a sufficient condition for when is it more profitable to allow the agents the flexibility to opt out of participating in future time periods. Furthermore, the agents’ preference and the overall social utility depends on agents’ value functions and demand distributions. Our study provides insights that help decision makers price their services to better satisfy customers’ varying needs while improving the profitability of the service.

Coordination of elderly healthcare service supply chain with information asymmetry: Designs of option contracts under different demand distribution statuses

This paper studies the issue of demand information asymmetry in an elderly healthcare service (EHS) system represented by a two-echelon elderly healthcare service supply chain (EHSSC) comprising an elderly service integrator (ESI) and a service provider (ESP). The goal of the ESI is to decide on how much service capacity is required for placing orders to the ESP, who directly serves the customers. Considering discrete and continuous demand distribution statuses, a centralised model with symmetric demand information and decentralised models with asymmetric demand information are developed to analyse the optimal ordering decisions and discuss the influence of information asymmetry. Furthermore, option contracts are applied to help coordinate the supply chain under asymmetric demand information based on different demand distribution statuses. Optimal option contract menus are designed for the ESP to promote the information sharing. Results show that the option contract can coordinate the EHSSC with asymmetric demand information under both discrete and continuous demand distribution statuses. The exercise price will be higher under lower demand information than that under higher demand information and the transfer payment will be less under lower demand information than that under higher demand information. Moreover, although the ESI has demand information superiority and can make use of opportunistic behaviour to maximise its own profit, the ESP as the leader can design the option contract to incentive the ESI to achieve true information sharing, and even obtain nearly all of the channel profit.

The price-setting newsvendor with Poisson demand

The price-setting newsvendor (PSN) model has received considerable attention since it was first introduced by Whitin (1955). However, the existing publications that study this model consistently assume the existence of a continuous density function of demand. In this paper, we study the PSN model with Poisson demand — that is, a discrete demand distribution without density function. The Poisson PSN has an important property, it combines price-dependency of variance and coefficient of variation of the (standard) additive and multiplicative models: demand variance decreases and the coefficient of variation increases in the selling price. We develop an analytical solution approach that covers a broad class of demand models, including linear and logit demand, explain how to apply our approach to more general demand functions via piece-wise linear approximation, and develop analytical and numerical insights. We characterize the behavior of the optimal price and we analyze the performance gap of different price-setting heuristics. Among other insights, we observe some instances in which a significant share of profits would be lost if the discrete nature of demand were not modeled explicitly. To help companies overcome this risk, we present an easily applicable decision rule with which to determine when to use simple heuristics and when to solve the associated discrete optimization problem.

Dynamic Inventory Control with Fixed Setup Costs and Unknown Discrete Demand Distribution

We study a dynamic inventory control problem involving fixed setup costs and random demand distributions. With an infinite planning horizon, model primitives including costs and distributions are set to be stationary. Under a given demand distribution, an $(s,S)$ policy has been known to minimize the long-run per-period average cost. Out of the need to model situations involving new products or unencountered economic conditions, however, we depart from the traditional model by allowing the stationary demand distribution to be largely unknown, to the effect that it could be anywhere in a given ambiguity set. Our goal is to rein in the long-run growth of the regret resulting from applying a policy that strives to learn the underlying demand while simultaneously meting out ordering decisions based on its learning. We propose a policy that controls the pace at which a traditional $(s,S)$-computing algorithm is applied to the empirical distribution of the demand learned over time. The regret incurred from the policy has a bound of $O(T^{1/2}\cdot(\ln T)^{1/2})$.

Last Time Buy and repair decisions for fast moving parts

Spare part availability is essential for advanced capital goods with a long service period. Sourcing becomes challenging once the production of spare parts ceases, while the remaining service period is still long. In this paper, we focus on fast moving parts with repair of failed parts as an alternative supply option. We proceed from the methodology of Behfard et al. (2015) for slow movers, which assumes discrete demand distributions and therefore leads to excessive computation times for fast movers. We find that the use of continuous demand distributions requires significant modifications, both for the approximation of the performance indicators and for the optimization of the repair policy. We develop accurate heuristics to find the near-optimal Last Time Buy (LTB) quantity and the repair policy that we apply for two control policies: pull return - push repair, and push return - pull repair. We show that pull return - push repair is better to follow if return lead times are short and return costs are low. For long return lead times, we find that when the return cost exceeds 35%–40% of the part's value, push return - pull repair becomes more cost efficient. We also show that for relatively high demand of spare parts over the planning period (>300 for a 10 years planning period) the continuous model is a good approximation for the discrete model of Behfard et al. (2015). In addition, the computation time of our method is much lower then.

Local Supply Chain Control Model with Unreliable Suppliers

This paper is concerned with a model of controlling supply chain for a local warehouse with two suppliers whose trustability is different. An algorithm is obtained for design of optimal inventory control strategies. A process is described in which stationary strategies are introduced. A software package is developed in which the algorithm is numerically modeled for two varieties of Gauss-like and nearly uniform discrete demand distribution. Examples illustrate the dependence of the parameters and the cost for optimal stationary strategies on the system’s econometric characteristics, in particular, the discount factor.

Optimization of mid-block pedestrian crossing network with discrete demands

• The quantity, location and signalization of pedestrian crosswalk network are optimized simultaneously. • Pedestrian and vehicle cost are integrated in a multi-objective optimization framework. • More crosswalks may not contribute much additional improvement in pedestrian crossing cost under some conditions. • Two-stage crosswalks are more favorable than one-stage ones for both pedestrian and vehicles. In many cases, pedestrian crossing demands are distributed discretely along an arterial segment. Demand origins, destinations and crosswalks comprise a pedestrian crossing network. An integrated model for optimizing the quantity, locations and signal settings of mid-block crosswalks simultaneously is proposed to best trade-off the operational performances between pedestrians and vehicles. Pedestrian behavior of choosing crosswalks is captured under a discrete demand distribution. Detour distance and delay at signalized crosswalks are formulated as a measure of pedestrian crossing cost. Maximum bandwidths are modeled in analytical expressions as a measure of vehicular cost. To solve the proposed model, the Non-dominated Sorting Genetic Algorithm II (NSGA II) based algorithm is designed and employed to obtain the Pareto frontier efficiently. From the numerical study, it is found that there exists an optimal number of mid-block crosswalks. Excess available crosswalks may make no contributions to improvement in pedestrian cost when the constraint of the minimum interval between crosswalks and vehicular cost are taken into account. Two-stage crosswalks are more favorable than one-stage ones for the benefits of both pedestrian and vehicles. The study results show promising properties of the proposed method to assist transportation engineers in properly designing mid-block crosswalks along a road segment.

Sequential Resource Allocation for Nonprofit Operations

This paper studies a sequential resource allocation problem motivated by distribution operations of a nonprofit organization. The alternate objectives that arise in nonprofit (as opposed to commercial) operations lead to new variations on traditional problems in operations research and inventory management. Specifically, we consider the problem of distributing a scarce resource to meet customers' demands that are observed sequentially. An allocation policy that seeks to maximize profit may lead to inequitable distributions among customers. Our work in a nonprofit setting solves the sequential resource allocation problem with an objective function aimed at equitable and effective service. We define service in terms of fill rate (the ratio of the allocated amount to observed demand) and develop an objective function to maximize the expected minimum fill rate among customers, which balances equity in fill rates with effectiveness in the use of resources (low waste). Through a dynamic programming framework, we characterize the structure of the optimal allocation policy for a given sequence of customers when demand follows continuous probability distributions. We use that optimal structure to develop a heuristic allocation policy for instances with discrete demand distribution. In addition, we identify customer demand properties to consider when sequencing customer visits to optimize the fill rate objective. For both inventory allocation and customer sequencing decisions, the proposed heuristic methods yield near-optimal solutions.

Sequential resource allocation with constraints: Two-customer case

We study a two-customer sequential resource allocation problem with equity constraint, which is reflected by a max–min objective. For finite discrete demand distribution, we give a sufficient and necessary condition under which the optimal solution has monotonicity property. However, this property never holds with unbounded discrete distribution.

When is it feasible to model low discrete demand by a normal distribution?

Inventory control systems used in practice are quite often modeling the lead-time demand by a normal distribution. This may result in considerable errors when the real demand is low and discrete. For such demand, it is usually better to use a discrete demand distribution. However, this will increase the computational effort. A natural question is under what circumstances a normal approximation is feasible. This paper analyzes this question in a numerical study. Our study indicates that a normal approximation works reasonably well when the average lead-time demand is something like 10 or higher and the coefficient of variation is bounded by something like 2. The normal approximation works better for a high backorder cost or, equivalently, a high service level.

On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns

The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.

Adaptive Data-Driven Inventory Control with Censored Demand Based on Kaplan-Meier Estimator

Using the well-known product-limit form of the Kaplan-Meier estimator from statistics, we propose a new class of nonparametric adaptive data-driven policies for stochastic inventory control problems. We focus on the distribution-free newsvendor model with censored demands. The assumption is that the demand distribution is not known and there are only sales data available. We study the theoretical performance of the new policies and show that for discrete demand distributions they converge almost surely to the set of optimal solutions. Computational experiments suggest that the new policies converge for general demand distributions, not necessarily discrete, and demonstrate that they are significantly more robust than previously known policies. As a by-product of the theoretical analysis, we obtain new results on the asymptotic consistency of the Kaplan-Meier estimator for discrete random variables that extend existing work in statistics. To the best of our knowledge, this is the first application of the Kaplan-Meier estimator within an adaptive optimization algorithm, in particular, the first application to stochastic inventory control models. We believe that this work will lead to additional applications in other domains.

On the exact calculation of the mean stock level in the base stock periodic review policy

Purpose: One of the most usual indicators to measure the performance of any inventory policy is the mean stock level. In the generalized base stock, periodic review policy, the expected mean stock during the replenishment cycle is usually estimated by practitioners and researchers with the traditional Hadley-Whitin approximation. However it is not accurate enough and exact methods suggested on the related literature focus on specific demand distributions. This paper proposes a generalized method to compute the exact value of the expected mean stock to be used when demand is modelled by any uncorrelated, discrete and stationary demand pattern. Design/methodology/approach: The suggested method is based on computing the probability of every stock level at every point of the replenishment cycle for which it is required to know the probability of any stock level at the beginning of the cycle and the probability transition matrix between two consecutive periods of time. Furthermore, the traditional Hadley-Whitin approximation is compared with the proposed exact method over different discrete demand distributions Findings: This paper points out the lack of accuracy that the Hadley-Whitin approximation shows over a wide range of service levels and discrete demand distributions. Research limitations/implications: The suggested method requires the availability of appropriate tools as well as a sound mathematical background. For this reason, approximations to it are the logical further research of this work. Practical implications: The use of the Hadley-Whitin approximation instead of an exact method can lead to underestimate systematically the expected mean stock level. This fact may increase total costs of the inventory system. Originality/value: The original derivation of an exact method to compute the expected mean stock level for the base stock, periodic review policy when demand is modelled by any discrete function and backlog is not allowed.

On the exact calculation of the mean stock level in the base stock periodic review policy

Purpose: One of the most usual indicators to measure the performance of any inventory policy is the mean stock level. In the generalized base stock, periodic review policy, the expected mean stock during the replenishment cycle is usually estimated by practitioners and researchers with the traditional Hadley-Whitin approximation. However it is not accurate enough and exact methods suggested on the related literature focus on specific demand distributions. This paper proposes a generalized method to compute the exact value of the expected mean stock to be used when demand is modelled by any uncorrelated, discrete and stationary demand pattern. Design/methodology/approach: The suggested method is based on computing the probability of every stock level at every point of the replenishment cycle for which it is required to know the probability of any stock level at the beginning of the cycle and the probability transition matrix between two consecutive periods of time. Furthermore, the traditional Hadley-Whitin approximation is compared with the proposed exact method over different discrete demand distributions Findings: This paper points out the lack of accuracy that the Hadley-Whitin approximation shows over a wide range of service levels and discrete demand distributions. Research limitations/implications: The suggested method requires the availability of appropriate tools as well as a sound mathematical background. For this reason, approximations to it are the logical further research of this work. Practical implications: The use of the Hadley-Whitin approximation instead of an exact method can lead to underestimate systematically the expected mean stock level. This fact may increase total costs of the inventory system. Originality/value: The original derivation of an exact method to compute the expected mean stock level for the base stock, periodic review policy when demand is modelled by any discrete function and backlog is not allowed.

An optimal constant level rationing policy under service level constraints

We analyze a periodic review inventory system where inventory of a single item is used to serve high and low priority customers. High priority customers demand a higher service level than low priority customers and both types of customers can have quite general discrete demand distributions. The inventory is controlled by a (S, CL)-policy, where S denotes the base stock level and CL specifies the critical level, i.e., the amount of inventory reserved for high priority demand. We consider a policy, where the critical level is constant over time, which is not optimal, but makes the model analytically tractable. Unlike previous research, we clear backorders optimally and find the optimal solution. We also derive structural results.

A stochastic approximation method for the single-leg revenue management problem with discrete demand distributions

We consider the problem of optimally allocating the seats on a single flight leg to the demands from multiple fare classes that arrive sequentially. It is well-known that the optimal policy for this problem is characterized by a set of protection levels. In this paper, we develop a new stochastic approximation method to compute the optimal protection levels under the assumption that the demand distributions are not known and we only have access to the samples from the demand distributions. The novel aspect of our method is that it works with the nonsmooth version of the problem where the capacity can only be allocated in integer quantities. We show that the sequence of protection levels generated by our method converges to a set of optimal protection levels with probability one. We discuss applications to the case where the demand information is censored by the seat availability. Computational experiments indicate that our method is especially advantageous when the total expected demand exceeds the capacity by a significant margin and we do not have good a priori estimates of the optimal protection levels.