Two warehouse inventory models for single vendor multiple retailers with price and stock dependent demand

Here a single vendor multiple retailer inventory model of an item is developed where demand of the item at every retailer is linearly dependent on stock and inversely on some powers of selling price. Item is produced by the vendor and is distributed to the retailers following basic period policy. According to this policy item is replenished to the retailers at a regular time interval (T₁) called basic period (BP) and replenishment quantity is sufficient to last for the period T₁. Due to the scarcity of storage space at market places, every retailer uses a showroom at the market place and a warehouse to store the item, little away from the market place. Item is sold from the showroom and is filled up from the warehouse in a bulk release pattern. Some of the inventory parameters are considered as fuzzy in nature and model is formulated to maximize the average profit from the whole system. Imprecise objective is transformed to equivalent deterministic ones using possibility/necessity measure of fuzzy events with some degree of optimism/pessimism. A genetic algorithm (GA) is developed with roulette wheel selection, arithmetic crossover and random mutation and is used to solve the model. In some complex cases, with the help of above GA, fuzzy simulation process is used to derive the optimal decision. The model is illustrated through numerical examples and some sensitivity analyses are presented.     

Inventory model of deteriorated items with a constraint: A geometric programming approach

An inventory model for deteriorating items is built-up with limited storage space. Here, demand rate for the items is finite, items deteriorate at constant rates and are replenished instantaneously. Following EOQ model, the problem is formulated with and without truncation on the deterioration term and ultimately is converted to the minimization of a signomial expression with a posynomial constraint. It is solved by modified geometric programming (MGP) method and non-linear programming (NLP) method. The problem is supported by numerical examples. The results from two versions of the model (with and without truncation) and two methods (i.e. MGP and NLP) are compared.     

Oligopoly models for market price of electricity under demand uncertainty and unit reliability

Several oligopoly models have been proposed for representing strategic behavior in electricity markets, which include Bertrand, Cournot, and Supply Function Equilibrium (SFE). For the most part, these models are deterministic, with the exception of the SFE originally developed by Klemperer and Meyer. However, their model does not include supply side uncertainties. In this paper, we consider both load and supply side uncertainties (resulting from generator availabilities). We obtain Nash equilibrium solutions for Cournot and SFE models, in which asymmetric firms (whose generating units have different costs and capacities) submit their bids so that each firm’s expected profit is maximized.     

On the optimality of inventory models with deteriorating items for demand and on-hand inventory dependent production rate

In this paper we present two flexible production lot size inventorymodels for deteriorating items in which the production rateat any instant depends on the demand and the on-hand inventorylevel at that instant. Each of demand and deterioration rateare general continuous functions of time. For one model, shortagesare allowed but are partially backordered. Also, all cost componentsare affected by inflation and the time value of money. For eachmodel, a closed form of the per unit time total relevant costis derived and sufficient conditions that minimize this totalcost are built. Then, mathematical methods are used to showthat, under certain conditions, each of the underlying inventorysystem can attain a unique global optimal solution.     

Fuzzy pricing,marketing and service planning in a fuzzy inventory model: A geometric programming approach

In the real world markets, demand is influenced by different parameters. Recently, many researchers have been interested in integrated production and marketing planning strategies in inventory models where demand depends on different parameters such as price and/or marketing expenditure. The quality of services that are offered to customers of a product is one of the most important parameters that affects demand in the real markets and has not been considered in development of inventory models. On the other hand, the cost parameters in real inventory systems and other parameters such as price, marketing and service elasticity to demand are imprecise and uncertain in nature. So, the notion of fuzziness can be applied to cope with this uncertainty. In this paper, a new fuzzy profit maximization inventory model with shortages is proposed. The demand is considered as a power function of price, marketing expenditure and service expenditure. Furthermore, unit cost is determined as a power function of order quantity. Since the proposed model is in a fuzzy environment, a fuzzy decision should be made to meet the decision criteria, and the results should be fuzzy. Therefore, the proposed model is formulated and solved using geometric programming and fuzzy optimization techniques to derive an approximation of the results’ membership functions. The model is illustrated with a numerical example and finally a case study is provided for evaluation and validation of the results of model.     

A multi-product continuous review inventory system with stochastic demand,backorders, and a budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary or storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when orders arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. This paper models a multi-item stochastic inventory system with backordered shortages when estimation of marginal backorder cost is available, and payment is due upon order arrival. The budget constraint can easily be converted into a storage constraint.     

Multi-item inventory control with full truckloads: A comparison of aggregate and individual order triggering

In this paper, we consider the stochastic joint replenishment problem in an environment where transportation costs are dominant and full truckloads or full container loads are required. One replenishment policy, taking into account capacity restrictions of the total order volume, is the so-called QS policy, where replenishment orders are placed to raise the individual inventory positions of all items to their order-up-to levels, whenever the aggregate inventory position drops below the reorder level. We first provide a method to compute the policy parameters of a QS policy such that item target service levels can be met, under the assumption that demand can be modeled as a compound renewal process. The approximation formulas are based on renewal theory and are tested in a simulation study which reveals good performance. Second, we compare the QS policy with a simple allocation policy where replenishment orders are triggered by the individual inventory positions of the items. At the moment when an individual inventory position drops below its item reorder level, a replenishment order is triggered and the total vehicle capacity is allocated to all items such that the expected elapsed time before the next replenishment order is maximized. In an extensive simulation study it is illustrated that the QS policy outperforms this allocation policy since it results in lower inventory levels for the same service level. Although both policies lead to similar performance if items are identical, it can differ substantially if the item characteristics vary.     

A multi-item mixture inventory model involving random lead time and demand with budget constraint and surprise function

This study deals with a multi-item mixture inventory model in which both demand and lead time are random. A budget constraint is also added to this model. The optimization problem with budget constraint is then transformed into a multi-objective optimization problem with the help of fuzzy chance-constrained programming technique and surprise function. In our studies, we relax the assumption about the demand, lead time and demand during lead time that follows a known distribution and then apply the minimax distribution free procedure to solve the problem. We develop an algorithm procedure to find the optimal order quantity and optimal value of the safety factor. Finally, the model is illustrated by a numerical example.     

Coordinating a three-echelon supply chain under price and quality dependent demand with sub-supply chain and RFM strategies

The paper considers a three-echelon supply chain which consists of one supplier, one manufacturer and one retailer for trading a single product. The market demand at the retailer is influenced by the retail price and the quality of the product. The quality of the finished product at the manufacturer depends on the supplier’s raw material quality. We analyze the model for both deterministic and stochastic demand patterns. We first study the centralized and decentralized systems, and then the decentralized system with a sub-supply chain coordination strategy (where the manufacturer chooses to merge with either the supplier or the retailer and then acts as a single entity) and the two-level retail fixed mark-up (RFM) strategy. In the case of the two-level RFM strategy, the manufacturer and the retailer use fixed mark ups over the supplier’s wholesale price. The proposed models are demonstrated through numerical examples. It is observed from the numerical study that the two-level RFM strategy is superior to the sub-supply chain coordination strategy. Further, the two-level RFM strategy in the stochastic demand scenario is not as effective as in the deterministic demand scenario.     

Synchronization of inventory and transportation under flexible vehicle constraint: A heuristics approach using sliding windows and hierarchical tree structure

This paper investigates the integrated inventory and transportation planning under flexible vehicle constraint. To offer better services at lower prices, more and more companies turn to outsource transportation functions to other professional service providers, namely 3rd party logistics companies. Under these vehicle rental arrangements, the number of vehicles is a decision variable instead of a fixed number, and the transportation cost includes not only the delivery cost but also the cost of vehicle rental that is proportional to the number of vehicles rented in a given planning horizon. In this paper, the problem is formulated as a mixed integer programming problem. A heuristic algorithm is developed, in which sliding windows are applied to approximate the problem by repeatedly solving a series of overlapping short-term subproblems, and a hierarchical tree structure is used to evaluate the closeness of different groups of retailers. Numerical experiments show that a better tradeoff between the inventory cost and transportation cost can be achieved through the proposed heuristic algorithm.     

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, the usual rank, and the 2-number (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Revised version: 6 August 2001 / Published online: 4 April 2002     

Inventory model for deteriorating items with freshness and price dependent demand: Optimal discounting and ordering policies

A major part of retail industry deals with items whose freshness declines with time, resulting in lower demand at the same price. The item may later begin to deteriorate, when it is customary to offer discount in order to boost sales. A discounting policy may bring many benefits for the retailer, if correctly chosen. Motivated by this we have developed and analyzed an inventory model when demand for a deteriorating item depends initially only upon its selling price and later also on the freshness condition. We consider general demand function and general deterioration distribution for an inventory model with lost sales shortage. It is shown that net profit is a concave function of the period with positive inventory and conditionally concave function of discount. Important managerial insights obtained from sensitivity analysis suggest some policies counter to those commonly practiced by the retailers while others are in concurrence with the strategies in vogue.     

Sourcing decision under disruption risk with supply and demand uncertainty: A newsvendor approach

Multi-sourcing is considered as a common practice to hedge against supply disruption risk. In this context, this paper proposes two models for optimal order allocation in newsvendor setting, where both supply and demand are uncertain. The first model considers a risk neutral decision maker who maximizes the total expected profit under disruption risk. The second one is for a risk averse decision maker who does so under service level constraints. Analytical closed form solutions for both the models are derived. To overcome the computational complexity of the exact optimal solution, two algorithms are developed to generate optimal order quantity and the corresponding set of suppliers. The solutions with exact optimization algorithms and the proposed ones are illustrated and compared with numerical examples. The results show that the proposed algorithms give the exact optimal solution while being tractable. Finally, a case study is used to illustrate the applicability of the proposed model.     

Optimal exit and valuation under demand uncertainty: A real options approach

This paper considers both the optimal exit strategy and the valuation of stochastic cash flows of a firm facing demand uncertainty and potential excess supply. By relying on the standard theory of linear diffusions and ordinary nonlinear programming, we derive the value of the rationally managed firm, and state the necessary condition for optimal exit. In contrast to the standard approaches in the real options literature, our analysis is completely independent of both dynamic programming and the smooth-fit principle. I demonstrate that irreversible exit is optimal only when the value of the future productive opportunities becomes smaller than the value of irreversibly exercising the option to exit and in this way avoid further cumulative losses. I also present the comparative static properties of the optimal exit threshold and demonstrate that increased uncertainty may increase or decrease the optimal exit threshold depending on the sign of the net convenience yield.     

Effect of declining selling price: profit analysis for a single period inventory model with stochastic demand and lead time

In this paper, considering the empirical trend for sales and price of fashion apparels as prototype, optimal ordering policy for a single period stochastic inventory model is investigated. The impact of the presence of random lead time and declining selling price on the profitability of the retailer is explored. Existence of unique optimal solutions for net profit functions is proved. Numerical examples are presented to illustrate the method of identifying profitable levels of inventory holding and penalty costs. Percentage profit per unit investment in inventory is obtained in order to assist managers in taking business decisions, specifically to the extent of whether or not to take up a particular business under known constraints. It is demonstrated that the optimal inventory policy in the absence of price decline and lead time differs considerably from that when lead time and price decline are simultaneously considered.     

An inventory model for a deteriorating item with displayed stock dependent demand under fuzzy inflation and time discounting over a random planning horizon

An inventory model for a deteriorating item (seasonal product) with linearly displayed stock dependent demand is developed in imprecise environment (involving both fuzzy and random parameters) under inflation and time value of money. It is assumed that time horizon, i.e., period of business is random and follows exponential distribution with a known mean. The resultant effect of inflation and time value of money is assumed as fuzzy in nature. The particular case, when resultant effect of inflation and time value is crisp in nature, is also analyzed. A genetic algorithm (GA) is developed with roulette wheel selection, arithmetic crossover, random mutation. For crisp inflation effect, the total expected profit for the planning horizon is maximized using the above GA to derive optimal inventory decision. On the other hand when inflationary effect is fuzzy then the above expected profit is fuzzy in nature too. Since optimization of fuzzy objective is not well defined, the optimistic/pessimistic return of the expected profit is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to determine this optimistic/pessimistic return. Finally a fuzzy simulation based GA is developed and is used to maximize the above optimistic/pessimistic return to get optimal decision. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.     

Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory

Marketing researchers and practitioners have long recognized the demand of many retail items is proportional to the amount of inventory displayed. Recently, two distinct types of inventory control models reflecting this relationship have appeared in the literature, models in which the demand rate of an item is a function of the initial inventory level and those in which it is dependent on the instantaneous inventory level. We present a comprehensive overview of this literature and demonstrate the equivalence of the two types of models through the use of a simple, periodic-review model. An alternative approach to sensitivity analysis for inventory models with inventory-level-dependent demand is also presented.     

A class of one-dimensional geometric variational problems with energy involving the curvature and its derivatives: a geometric measure-theoretic approach

The notions of Legendrian and Gaussian towers are defined and investigated. Then applications in the context of one-dimensional geometric variational problems with the energy involving the curvature and its derivatives are provided. Particular attention is paid to the case when the functional is defined on smooth boundaries of plane sets.      

A comprehensive note on: An inventory model under two levels of trade credit and limited storage space derived without derivatives

In 2006, Huang proposed an inventory model with two warehouses when the supplier offers the retailer a permissible delay of M periods, and the retailer also provides its customers a permissible delay of N periods. He then solved it without derivatives. In this note, we extend his model to complement the shortcomings of his model. In contrast to the complicated and tedious quadratic–algebraic method suggested by Huang, we propose a simple arithmetic–geometric method to solve the inventory problem. Finally, we run computer programs for several numerical examples to illustrate the proposed model and obtain some managerial implications.