Continuous Demand Distributions Research Articles

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.

Robust newsvendor problems: effect of discrete demands

Distribution-free newsvendor models often assume continuous demand distributions to facilitate analysis and computation. However, in practice, discrete demand is a natural phenomenon. So far, there exists no analytical and computational results in the literature under this setting. Thus, the goal of this paper is to investigate the newsvendor problems with partial information when the demand is discrete and solve them using the so-called discrete moment problems. Numerical results are presented to illustrate the value of discrete information.

On the multi-product newsvendor with bounded demand distributions

We consider a multi-item newsvendor problem with side constraints and common continuous demand distributions that are bounded implying that the items’ profit functions are nondifferentiable. In particular we focus on the cases of uniform and triangular distributions. These distributions naturally describe demand that is guaranteed to exceed a certain threshold – for example, consumption of basic food products or electric power consumption over any given day. Moreover, in practice, it is often difficult to estimate the demand distribution. Accordingly, the uniform and triangular distributions become default modeling choices when only information regarding the bounds and possibly the mode of the distribution is known. We generalize a previous quadratic programming model for uniformly distributed demand on [a,b] to allow a to be nonzero and to allow the order quantity to be smaller than a. We study the corrected model and propose an efficient algorithm for determining an optimal solution. The algorithm is motivated by a structural result of an upper bound on the number of guaranteed shortage products, which typically appear in multiproduct settings with a positive demand distribution lower bound. The performance of our specialized algorithm is compared to that achieved when solving our formulation with a piecewise quadratic objective using a state-of-the-art standard solver. We also extend the modeling technique to propose a nonlinear programming formulation for triangular demand distributions. A similar approach can be adopted to approximate other demand distributions with a possibly non-finite support, such as truncated normal with strictly positive lower bounds.

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.

An efficient algorithm for capacitated assortment planning with stochastic demand and substitution

The ever increasing product variety in grocery retailing is in conflict with the limited shelf space. Managing the assortment and regular order quantity for perishable and non-perishable products is therefore a core task for retailers. Retailers want to maximize profit by considering potential revenues, purchase costs, diminishing profits for products and opportunity costs for unfulfilled demand when determining assortment sizes and order volume to meet stochastic consumer demand.We considered a newsvendor-based decision model that jointly optimizes assortment size and the associated order volumes for listed items. The model considers out-of-assortment (OOA) and out-of-stock (OOS) substitution effects. The model is a structural equivalent to the model described in Kök and Fisher (2007) and solved there via a heuristic procedure. We develop an optimal solution procedure and a time-efficient heuristic that solves the model. The heuristic suggested is applicable for practically relevant, large-scale problem instances. Numerical tests reveal that the heuristic procedure produces close-to-optimal solutions and outperforms the Kök and Fisher heuristic with regard to both computational time and solution quality. A simulation study with discrete demand justifies our approach with using continuous demand distributions. The numerical analyses show that considering substitution effects in a decision-based model has a significant impact on the total profit and solution structure. Further managerial insights are presented with regard to assortment heterogeneity when shelf size, substitution and demand variability are varied.

Multi-tier binary solution method for multi-product newsvendor problem with multiple constraints

This paper considers a multi-product newsvendor problem with multiple constraints. Multiple constraints in the problem make it more challenging to solve. Previous research has attempted to solve the problem by considering two-constraint case or/and using approximation techniques or active sets methods. The solution methods in literature for solving multi-constraint problem are limited or cumbersome. In this paper, by analyzing structural properties of the multi-constraint multi-product newsvendor problem, we develop a multi-tier binary solution method for yielding the optimal solution to the problem. The proposed method is applicable to the problem with any continuous demand distribution and more than two constraints, and its computational complexity is polynomial in the number of products. Numerical results are presented for showing the effectiveness of our method.

Managing Inventories with One-Way Substitution: A Newsvendor Analysis

This paper presents a newsvendor approach to determine optimal order-up-to levels in a two-item inventory system with one-way substitution, assuming that both items are managed according to a periodic base stock order policy. The objective is to minimize the expected total cost per period, which consists of expected purchasing costs, expected inventory holding costs, expected shortage costs and expected adjustment costs. It is shown that, for any arbitrary (bivariate) continuous demand distribution, the optimal solution is unique. Moreover, the model yields useful insights on the impact of substitution on service level, the optimality of a borderline case in which the order-up-to level of the inflexible item is reduced to zero, and the pivotal role of the purchasing cost.

A binary solution method for the multi-product newsboy problem with budget constraint

Multi-product newsboy problem (MPNP) with budget constraint is a classical inventory control/management problem. However, solution methods for MPNP under general demand distributions are limited in the current literature. In this paper, by analyzing properties of the optimal solution to the MPNP with a budget constraint, we develop a solution algorithm for the constrained MPNP. The proposed algorithm is binary in nature, and is applicable to general types of demand distribution functions, discrete as well as continuous. For continuous demand distribution function, our approach can obtain the optimal or near optimal solution to the constrained MPNP with polynomial computation complexity of the o( n) order. On the other hand, for discrete demand distribution functions, it can effectively provide good approximate solution. Numerical experiments are presented to show the performance of our method.

Efficiency in shortage reduction when using a more expensive common component

The paper considers the contribution of a common component in minimizing the expected units shortage subject to a budget constraint. A two-level product structure in an assemble-to-order system, where demands for two end products follow independent Erlang distributions, is assumed. We present closed form expressions for the objective function under various scenarios and provide efficient algorithms to compute the corresponding optimal inventory stock levels. It is found that, for all demand patterns considered, the relative reduction in the expected units shortage can be substantial when the inventory budget is large unless the cost of the common component is very high. When the budget is small, any potential benefit from employing commonality can be insignificant even if the premium for the common component is tiny. Scope and purpose Using common components in different products is an appealing idea. However, if a common component is more expensive than the unique components that it replaces, then it will require careful analysis to decide whether it is beneficial to employ the common component. Here we consider the contribution of a more expensive common component in minimizing the expected units shortage for meeting demands of two end products, subject to a budget constraint. Under a general cost structure, for models with or without commonality, we show that the objective function is convex for any unbounded continuous demand distributions. Applying this result and assuming Erlang demand distributions, we show that a more expensive common component can be beneficial when the inventory budget is large but commonality will not be helpful if the budget is small.

Supplier diversification: effect of discrete demand

Diversification under supply uncertainty has been adopted by manufacturers in order to improve performance. In the presence of multiple suppliers, it is essential to develop an operational policy in order to utilize the services of different suppliers effectively. Anupindi and Akella (Manage. Sci. 39 (8) (1993) 944–963) consider two suppliers differing in cost and reliability and develop the optimal inventory policy for the manufacturer under continuous demand distribution. The policy obtained by them indicates that the manufacturer should never order products from the more expensive supplier alone. In this paper, we consider the case where demand is discrete and provide examples to show that ordering products from the more expensive (and more reliable) supplier alone is optimal. We also provide sufficient conditions under which it is optimal to order a larger share from the more expensive supplier when demand is discrete.

Common component inventory problems with a budget constraint: Heuristics and upper bounds

This paper addresses a so-called assemble-to-order environment. There are a number of end items each of which requires an assembly of several components. Some of the components are unique to specific end items while others are common to two or more final products. Components must, due to the leadtimes, be ordered prior to knowing the end item demand levels (assumed poisson distributed), but final assemblies can be completed after knowing the demands. The objective is to maximize the expected profit contribution subject to a budget constraint on the component inventory investment. The resulting two-stage stochastic programming problems obtained are very difficult, if not impossible, to solve. Heuristics and bounding procedures that give good results have therefore been suggested. The proposed heuristics can easily be extended to problems with continuous demand distributions.

An Inventory Model with Limited Production Capacity and Uncertain Demands II. The Discounted-Cost Criterion

This paper considers a single-item, periodic-review inventory model with uncertain demands. We assume a finite production capacity in each period. With stationary data, a convex one-period cost function and a continuous demand distribution, we show (under a few additional unrestrictive assumptions) that a modified basic-stock policy is optimal under the discounted cost criterion, both for finite and infinite planning horizons. In addition we characterize the optimal base-stock levels in several ways.

An Application of the Theory of Weak Convergence to the Dynamic Perishable Inventory Problem with Discrete Demand

This paper develops a technique for analyzing dynamic inventory problems with discrete demand using the theory of weak convergence. We treat the dynamic fixed life perishable inventory problem with discrete demand. The optimal structure has been established assuming continuous demand. We construct an infinite sequence of continuous demand distributions that converge weakly to a discrete distribution, and show that the key results carry over in the limit.

Computing optimal (s, S) policies in inventory models with continuous demands

Special algorithms have been developed to compute an optimal (s, S) policy for an inventory model with discrete demand and under standard assumptions (stationary data, a well-behaved one-period cost function, full backlogging and the average cost criterion). We present here an iterative algorithm for continuous demand distributions which avoids any form of prior discretization. The method can be viewed as a modified form of policy iteration applied to a Markov decision process with continuous state space. For phase-type distributions, the calculations can be done in closed form.

Computing optimal (s, S) policies in inventory models with continuous demands

Special algorithms have been developed to compute an optimal (s, S) policy for an inventory model with discrete demand and under standard assumptions (stationary data, a well-behaved one-period cost function, full backlogging and the average cost criterion). We present here an iterative algorithm for continuous demand distributions which avoids any form of prior discretization. The method can be viewed as a modified form of policy iteration applied to a Markov decision process with continuous state space. For phase-type distributions, the calculations can be done in closed form.