Performance bounds on optimal fixed prices

We consider the problem of selling a fixed stock of items over a finite horizon when the buyers arrive following a Poisson process. We obtain a general lower bound on the performance of using a fixed price rather than dynamically adjusting the price. The bound is 63.21% for one unit of inventory, and it improves as the inventory increases. For the one-unit case, we also obtain tight bounds: 89.85% for the constant-elasticity and 96.93% for the linear price-response functions.     

Integrated production and allocation policies with one direct shipping option

We consider a single period inventory problem in which a supplier faces stochastic demands and customer specific waiting costs from multiple customers. The objective is to develop integrated production, allocation, and distribution policies so that the total production and customer waiting costs are minimized. We present an optimal policy for the two customer problem and derive a heuristic for a general problem based on the structural results of the two customer case. We show, numerically, that the heuristic performs very well with error bounds of less than 2% on average, while typical approximations may lead to significant sub-optimality.     

A non-stationary periodic review production–inventory model with uncertain production capacity and uncertain demand

In this paper we consider a nonstationary periodic review dynamic production–inventory model with uncertain production capacity and uncertain demand. The maximum production capacity varies stochastically. It is known that order up-to (or base-stock, critical number) policies are optimal for both finite horizon problems and infinite horizon problems. We obtain upper and lower bounds of the optimal order up-to levels, and show that for an infinite horizon problem the upper and the lower bounds of the optimal order up-to levels for the finite horizon counterparts converge as the planning horizons considered get longer. Furthermore, under mild conditions the differences between the upper and the lower bounds converge exponentially to zero.     

Polyhedral analysis for the two-item uncapacitated lot-sizing problem with one-way substitution

We consider a production planning problem for two items where the high quality item can substitute the demand for the low quality item. Given the number of periods, the demands, the production, inventory holding, setup and substitution costs, the problem is to find a minimum cost production and substitution plan. This problem generalizes the well-known uncapacitated lot-sizing problem. We study the projection of the feasible set onto the space of production and setup variables and derive a family of facet defining inequalities for the associated convex hull. We prove that these inequalities together with the trivial facet defining inequalities describe the convex hull of the projection if the number of periods is two. We present the results of a computational study and discuss the quality of the bounds given by the linear programming relaxation of the model strengthened with these facet defining inequalities for larger number of periods.     

Multi-item single source ordering problem with transportation cost: A Lagrangian decomposition approach

As a part of supply chain management literature and practice, it has been recognized that there can be significant gains in integrating inventory and transportation decisions. The problem we tackle here is a common one both in retail and production sectors where several items have to be ordered from a single supplier. We assume that there is a finite planning horizon to make the ordering decisions for the items, and in this finite horizon the retailer or the producer knows the demand of each item in each period. In addition to the inventory holding cost, an item-base fixed cost associated with each item included in the order, and a piecewise linear transportation cost are incurred. We suggest a Lagrangean decomposition based solution procedure for the problem and carry out numerical experiments to analyze the value of integrating inventory and transportation decisions under different scenarios.     

Selective and periodic inventory routing problem for waste vegetable oil collection

We consider a biodiesel production company that collects waste vegetable oil from source points that generate waste in large amounts. The company uses the collected waste as raw material for biodiesel production. The manager of this company needs to decide which of the present source points to include in the collection program, which of them to visit on each day, which periodic routing schedule to repeat over an infinite horizon and how many vehicles to operate such that the total collection, inventory and purchasing costs are minimized while the production requirements and operational constraints are met. For this selective and periodic inventory routing problem, we propose two different formulations, compare them and apply the better performing one on a real-world problem with 36 scenarios. We generate lower bounds using a partial linear relaxation model, and observe that the solutions obtained through our model are within 3.28% of optimality on the average. Several insights regarding the customer selection, routing and purchasing decisions are acquired with sensitivity analysis.     

A transportation type aggregate production model with bounds on inventory and backordering

We consider a certain T period aggregate production planning model, where the two sources of production are regular and overtime. The model allows for time varying production, holding and backordering costs and includes bounds on inventory and backorders. We show that the problem has a rather interesting network structure and exploit this structure to develop a greedy algorithm to solve the problem. The procedure is easy to implement and has a computational complexity of O(T²). We report computational experience with the greedy procedure and demonstrate its superiority to a well known network simplex code, Gnet, implemented on the classical network formulation of the problem.     

Minimizing maximum lateness of jobs in inventory constrained scheduling

This paper focuses on a machine scheduling problem having applications in truck scheduling at transshipment terminals. Jobs increase and decrease, respectively, the level of a central inventory. Naturally, jobs decreasing the inventory level can be processed only if the level of the inventory is high enough not to drop below zero. We consider the problem to find a schedule for jobs such that the maximum lateness among all jobs is minimized. We develop properties of optimal solutions, lower bounds, and heuristic methods in order to find upper bounds. These are incorporated in four branch and bound algorithms that are based on fixing sequences of jobs in forward or backward direction in two different types of representations. By means of a computational study, we compare these approaches with each other in order to show their efficiency.     

An algorithm for single-item economic lot-sizing problem with general inventory cost, non-decreasing capacity, and non-increasing setup and production cost

This paper considers an economic lot-sizing model with non-decreasing capacity constraint, non-increasing setup cost and production cost, and a general inventory cost. We prove that when periodic starting inventory is not less than a certain critical value, it is optimal to produce nothing; this critical value can be computed easily which results in a new effective algorithm.     

Production planning with time-dependent capacity bounds

In this paper, we examine the problem of finding minimum-cost production schedules that satisfy known demands over a finite planning horizon. A dynamic programming algorithm is developed to find these schedules for cases in which production in each period is constrained by a time-dependent capacity bound. The costs considered are production and inventory holding costs, and all cost functions are assumed to be nondecreasing and concave. The algorithm is an extension of Florian and Klein's method developed for problems in which capacity bounds are the same in all periods. Although the problem with time-dependent bounds is NP-complete, the algorithm is shown to be efficient when the capacity bounds are integer multiples of a common divisor and the largest multiplier is small. Hence, it is useful in applications in which production capacity is periodically increased by adding facilities of the same size.     

Economic lot sizing problem with inventory bounds

This paper studies a economic lot sizing (ELS) problem with both upper and lower inventory bounds. Bounded ELS models address inventory control problems with time-varying inventory capacity and safety stock constraints. An O(n²) algorithm is found by using net cumulative demand (NCD) to measure the amount of replenishment requested to fulfill the cumulative demand till the end of the planning horizon. An O(n) algorithm is found for the special case, the bounded ELS problem with non-increasing marginal production cost.     

Optimal production inventory policy for defective items with fuzzy time period

In this paper, an optimal production inventory model with fuzzy time period and fuzzy inventory costs for defective items is formulated and solved under fuzzy space constraint. Here, the rate of production is assumed to be a function of time and considered as a control variable. Also the demand is linearly stock dependent. The defective rate is taken as random, the inventory holding cost and production cost are imprecise. The fuzzy parameters are converted to crisp ones using credibility measure theory. The different items have the different imprecise time periods and the minimization of cost for each item leads to a multi-objective optimization problem. The model is under the single management house and desired inventory level and product cost for each item are prescribed. The multi-objective problem is reduced to a single objective problem using Global Criteria Method (GCM) and solved with the help of Fuzzy Riemann Integral (FRI) method, Kuhn–Tucker condition and Generalised Reduced Gradient (GRG) technique. In optimum results including production functions and corresponding optimum costs for the different models are obtained and then are presented in tabular forms.     

Asymptotically optimal production policies in dynamic stochastic jobshops with limited buffers

We consider a production planning problem for a jobshop with unreliable machines producing a number of products. There are upper and lower bounds on intermediate parts and an upper bound on finished parts. The machine capacities are modelled as finite state Markov chains. The objective is to choose the rate of production so as to minimize the total discounted cost of inventory and production. Finding an optimal control policy for this problem is difficult. Instead, we derive an asymptotic approximation by letting the rates of change of the machine states approach infinity. The asymptotic analysis leads to a limiting problem in which the stochastic machine capacities are replaced by their equilibrium mean capacities. The value function for the original problem is shown to converge to the value function of the limiting problem. The convergence rate of the value function together with the error estimate for the constructed asymptotic optimal production policies are established.     

Bounds on stochastic convex allocation problems

We consider a stochastic convex program arising in a certain resource allocation problem. The uncertainty is in the demand for a resource which is to be allocated among several competing activities under convex inventory holding and shortage costs. The problem is cast as a two–period stochastic convex program and we derive tight upper and lower bounds to the problem using marginal distributions of the demands, which may be stochastically dependent. It turns out that these bounds are tighter than the usual bounds in the literature which are based on limited moment information of the underlying random variables. Numerical examples illustrate the bounds.     

A dynamic rationing policy for continuous-review inventory systems

Stock rationing is an inventory policy that allows differential treatment of customer classes without using separate inventories. In this paper, we propose a dynamic rationing policy for continuous-review inventory systems, which utilizes the information on the status of the outstanding replenishment orders. For both backordering and lost sales environments, we conduct simulation studies to compare the performance of the dynamic policy with the static critical level and the common stock policies and quantify the gain obtained. We propose two new bounds on the optimum dynamic rationing policy that enables us to tell how much of the potential gain the proposed dynamic policy realizes. We discuss the conditions under which stock rationing – both dynamic and static – is beneficial and assess the value of the dynamic policy.     

Network revenue management with inventory-sensitive bid prices and customer choice

We develop an approximate dynamic programming approach to network revenue management models with customer choice that approximates the value function of the Markov decision process with a non-linear function which is separable across resource inventory levels. This approximation can exhibit significantly improved accuracy compared to currently available methods. It further allows for arbitrary aggregation of inventory units and thereby reduction of computational workload, yields upper bounds on the optimal expected revenue that are provably at least as tight as those obtained from previous approaches. Computational experiments for the multinomial logit choice model with distinct consideration sets show that policies derived from our approach can outperform some recently proposed alternatives, and we demonstrate how aggregation can be used to balance solution quality and runtime.     

Four equivalent lot-sizing models

We study the following lot-sizing models that recently appeared in the literature: a lot-sizing model with a remanufacturing option, a lot-sizing model with production time windows, and a lot-sizing model with cumulative capacities. We show the equivalence of these models with a classical model: the lot-sizing model with inventory bounds.     

基于网络控制的供应链库存系统的稳定性分析

针对一类供应链库存系统,研究了系统的闭环稳定性问题.创新点在于通过引入网络控制的思想,把供应链库存系统建模成网络时滞控制系统.在网络诱导时滞的上下界和丢包率已知的前提下,通过网络控制器的设计,分析了供应链库存系统的稳定性,并利用自由权矩阵和Lyapunov稳定性理论,给出了供应链库存系统渐近稳定的充分条件,并通过数值仿真验证了所提控制方案的可行性和有效性.     

Stochastic inventory problem with piecewise quadratic holding cost function containing a cost-free interval

In this paper, we consider a periodic-review stochastic inventory model with an asymmetric or piecewise-quadratic holding cost function and nonnegative production levels. It is assumed that the cost of deviating from an ideal production level or existing capacity is symmetric quadratic. It is shown that the optimal order policy is similar to the (s, S) policies found in the literature, except that the order-up-to quantity is a nonlinear function of the entering inventory level. Dynamic programming is used to derive the optimal policy. We provide numerical examples and a sensitivity analysis on the problem parameters.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A5872. The authors wish to thank an anonymous referee for very helpful comments on an earlier version of this paper.