An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α-cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α-cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.     

Fuzzy mixture two warehouse inventory model involving fuzzy random variable lead time demand and fuzzy total demand

This paper considers a two-warehouse fuzzy-stochastic mixture inventory model involving variable lead time with backorders fully backlogged. The model is considered for two cases—without and with budget constraint. Here, lead-time demand is considered as a fuzzy random variable and the total cost is obtained in the fuzzy sense. The total demand is again represented by a triangular fuzzy number and the fuzzy total cost is derived. By using the centroid method of defuzzification, the total cost is estimated. For the case with fuzzy-stochastic budget constraint, surprise function is used to convert the constrained problem to a corresponding unconstrained problem in pessimistic sense. The crisp optimization problem is solved using Generalized Reduced Gradient method. The optimal solutions for order quantity and lead time are found in both cases for the models with fuzzy-stochastic/stochastic lead time and the corresponding minimum value of the total cost in all cases are obtained. Numerical examples are provided to illustrate the models and results in both cases are compared.     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

An inventory model with generalized type demand,deterioration and backorder rates

This study is motivated by the paper of Skouri et al. [Skouri, Konstantaras, Papachristos, Ganas, European Journal of Operational Research 192 (1) (2009) 79–92]. We extend their inventory model from ramp type demand rate and Weibull deterioration rate to arbitrary demand rate and arbitrary deterioration rate in the consideration of partial backorder. We demonstrate that the optimal solution is actually independent of demand. That is, for a finite time horizon, any attempt at tackling targeted inventory models under ramp type or any other types of the demand becomes redundant. Our analytical approach dramatically simplifies the solution procedure.     

An inventory model for deteriorating items with stock-dependent demand rate

This paper discusses an inventory model with an inventory-level-dependent demand rate followed by a constant demand rate for items deteriorating at a constant rate θ, where the terminal condition of zero inventory at the end of the scheduling period has been relaxed. Sensitivity of the decision variables to changes in the parameter values is examined and the effects of these changes on the optimal policy are discussed in brief.     

An economic production lot-size model with shortages and time-dependent demand

In the present article, a production-inventory model is developedover an infinite plan ning horizon where the demand varies linearlywith time, unit production cost is taken as a function of theproduction rate, and shortages in inventory are permitted andare fully back-ordered. The machine production rate, which isassumed to be flexible, is treated as a decision variable. Theassociated nonlinear programming problem is modified by usingthe barrier-function method, and then a search technique isused to find the solution numerically. The analysis of the presentmodel of the production system points to optimality under conditionsthat are commonly recognized as ‘just in time’.     

Deterministic economic production quantity models with time-varying demand and cost

In today’s time-based competition, the unit cost of a high-tech product declines significantly over its short product life cycle. Consequently, in this paper, we relax the traditional economic production quantity model to allow for time-varying cost. We then prove that the optimal production schedule uniquely exists. In addition, we also show that the total cost is a convex function of the number of replenishments, which reduces the search for the optimal solution to finding a local minimum. Furthermore, we characterize the influences of both demand and cost over the length of production run time and the economic production quantity.     

An inventory model for increasing demand under two levels of trade credit linked to order quantity

During the growth stage of a product life cycle especially for high-tech products, the demand function increases with time. In this paper, we extend the constant demand to a linear non-decreasing demand function of time and incorporate a permissible delay in payment under two levels of trade credit into the model. The supplier offers a permissible delay linked to order quantity, and the retailer also provides a downstream trade credit period to its customers. The objective is to find the optimal replenishment cycle that minimizes the retailer’s annual total relevant cost per unit time. The condition for an optimal solution to the generalized model is presented and some fundamental theoretical results are established. Finally, numerical examples to illustrate the proposed model are provided. Sensitivity analysis is performed and some relevant managerial insights are obtained.     

Two storage inventory model with fuzzy deterioration over a random planning horizon

An inventory model for a deteriorating item with stock dependent demand is developed under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. For crisp deterioration rate, the expected profit is derived and maximized via genetic algorithm (GA). On the other hand, when deterioration rate is imprecise then optimistic/pessimistic equivalent of fuzzy objective function is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to maximize the optimistic/pessimistic return and finally fuzzy simulation-based GA is developed to solve the model. The models are illustrated with some numerical data. Sensitivity analyses on expected profit function with respect to distribution parameter λ and confidence levels α₁ and α₂ are also presented.     

Fuzzy profit measures for a fuzzy economic order quantity model

Changing economic conditions make the selling price and demand quantity more and more uncertain in the market. The conventional inventory models determine the selling price and order quantity for a retailer’s maximal profit with exactly known parameters. This paper develops a solution method to derive the fuzzy profit of the inventory model when the demand quantity and unit cost are fuzzy numbers. Since the parameters contained in the inventory model are fuzzy, the profit value calculated from the model should be fuzzy as well. Based on the extension principle, the fuzzy inventory problem is transformed into a pair of two-level mathematical programs to derive the upper bound and lower bound of the fuzzy profit at possibility level α. According to the duality theorem of geometric programming, the pair of two-level mathematical programs is transformed into a pair of conventional geometric programs to solve. By enumerating different α values, the upper bound and lower bound of the fuzzy profit are collected to approximate the membership function. Since the profit of the inventory problem is expressed by the membership function rather than by a crisp value, more information is provided for making decisions.     

An interactive method for inventory control with fuzzy lead-time and dynamic demand

Normally, the real-world inventory control problems are imprecisely defined and human interventions are often required to solve these decision-making problems. In this paper, a realistic inventory model with imprecise demand, lead-time and inventory costs have been formulated and an inventory policy is proposed to minimize the cost using man–machine interaction. Here, demand increases with time at a decreasing rate. The imprecise parameters of lead-time, inventory costs and demand are expressed through linear/non-linear membership functions. These are represented by different types of membership functions, linear or quadratic, depending upon the prevailing supply condition and marketing environment. The imprecise parameters are first transformed into corresponding interval numbers and then following the interval mathematics, the objective function for average cost is changed into respective multi-objective functions. These functions are minimized and solved for a Pareto-optimum solution by interactive fuzzy decision-making procedure. This process leads to man–machine interaction for optimum and appropriate decision acceptable to the decision maker’s firm. The model is illustrated numerically and the results are presented in tabular forms.     

An integrated production and inventory model to dampen upstream demand variability in the supply chain

We consider a two-echelon supply chain: a single retailer holds a finished goods inventory to meet an i.i.d. customer demand, and a single manufacturer produces the retailer’s replenishment orders on a make-to-order basis. In this setting the retailer’s order decision has a direct impact on the manufacturer’s production. It is a well known phenomenon that inventory control policies at the retailer level often propagate customer demand variability towards the manufacturer, sometimes even in an amplified form (known as the bullwhip effect). The manufacturer, however, prefers to smooth production, and thus he prefers a smooth order pattern from the retailer. At first sight a decrease in order variability comes at the cost of an increased variance of the retailer’s inventory levels, inflating the retailer’s safety stock requirements. However, integrating the impact of the retailer’s order decision on the manufacturer’s production leads to new insights. A smooth order pattern generates shorter and less variable (production/replenishment) lead times, introducing a compensating effect on the retailer’s safety stock. We show that by including the impact of the order decision on lead times, the order pattern can be smoothed to a considerable extent without increasing stock levels. This leads to a situation where both parties are better off.     

An inventory model for stock with advertising sensitive demand

K. S. Chaudhuri Department of Mathematics, Jadavpur University, Kolkata-700 032, West Bengal, India Corresponding author. Email: shib_sankar{at}yahoo.com Email: k_s_chaudhuri{at}yahoo.com The effect of advertising media and sales effort on future demandof merchandize is considered. In an oligopolistic marketingsystem, it is quite natural to boost sales by using advertisingand sales effort to earn more money from a business sector.Determination of demand and costs due to advertising and saleseffort is quite difficult. The approach in this paper is toconcentrate on investment for the purpose of advertising andincreasing sales effort in order to maximize profit. The paperextends an inventory model over finite and infinite time horizonswhere demand is influenced by the level of stock, advertisingmedia and sales effort. The associated profit maximization problemis solved by the ‘Euler–Lagrange method’.The model is illustrated with a numerical example.     

Optimal inventory policies for an economic order quantity model with decreasing cost functions

In this paper, three total cost minimization EOQ based inventory problems are modeled and analyzed using geometric programming (GP) techniques. Through GP, optimal solutions for these models are found and sensitivity analysis is performed to investigate the effects of percentage changes in the primal objective function coefficients. The effects on the changes in the optimal order quantity and total cost when different parameters of the problems are changed is also investigated. In addition, a comparative analysis between the total cost minimization models and the basic EOQ model is conducted. By investigating the error in the optimal order quantity and total cost of these models, several interesting economic implications and managerial insights can be observed.     

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.     

An EOQ model with two-parameter Weibull distribution deterioration and price-dependent demand

An Inventory replenishment policy is developed for a deteriorating item and price-dependent demand. The rate of deterioration is taken to be time-proportional and the time to deterioration is assumed to follow a two-parameter Weibull distribution. A power law form of the price dependence of demand is considered. The model is solved analytically and is illustrated with a numerical example.     

Deterministic lot-size inventory models with shortages and deterioration for fluctuating demand

We establish various inventory replenishment policies. We then analytically identify the best alternative among them based on the minimum total relevant costs. Finally, we prove that the relevant cost is convex with the number of replenishments. Consequently, the search for the optimal replenishment number is reduced to finding a local minimum.     

An inventory model for non-instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy

noindent In this paper, we propose an appropriate inventory model for non-instantaneous deteriorating items over quadratic demand rate with permissible delay in payments and time dependent deterioration rate. In this model, the completely backlogged shortages are allowed. In several existing results, the authors discussed that the deterioration rate is constant in each cycle. However, the deterioration rate of items are not constant in real world applications. Motivated by this fact, we consider that the items are deteriorated with respect to time. To minimize the total relevant inventory cost, we prove some useful theorems to illustrate the optimal solutions by finding an optimal cycle time with the necessary and enough conditions for the existence and uniqueness of the optimal solutions. Finally, we discuss the numerical instance and sensitivity of the proposed model.