A stochastic lot sizing model with partial backordering and imperfect production processes

Study’s Excerpt/Novelty This study introduces a novel economic order quantity (EOQ) model for managing non-instantaneous deteriorating items under two-phase demand rates and linear holding costs, incorporating constant partial backlogging and two-level pricing strategies within a trade credit framework. The model uniquely addresses the complexities of time-dependent quadratic demand before deterioration and constant demand afterward, providing optimal solutions for maximizing total profit by determining the best timing for positive inventory, cycle length, and order quantity. The research not only establishes the theoretical existence and uniqueness of these optimal solutions but also demonstrates practical implications through numerical examples and sensitivity analyses, highlighting the critical impact of deterioration rates on inventory management and profitability. Full Abstract This research developed an economic order quantity model for non-instantaneous deteriorating items with two-phase demand rates, linear holding cost, constant partial backlogging rate and two-level pricing strategies under trade credit policy. It is assumed that the holding cost is linear time-dependent, the unit selling price before deterioration sets in is greater than that after deterioration sets and the demand rate before deterioration sets in is considered as continuous time-dependent quadratic, after which it is considered as constant up to when the inventory is completely exhausted. Shortages are allowed and partially backlogged. The purpose of the model is to determine the optimal time with positive inventory, cycle length and order quantity such that the total profit of the inventory system has a maximum value. The necessary and sufficient conditions for the existence and uniqueness of the optimal solutions have been established. Some numerical examples have been given to illustrate the theoretical result of the model. Sensitivity analysis of some model parameters on the decision variables has been carried out and suggestions towards maximising the total profit were also given. , it is seen that the higher the rate of deterioration the lower the optimal time with positive inventory ( ), cycle length order quantity and the total profit and vice versa. This implies that the retailer needs to take all the necessary measures to avoid or reduce deterioration to maximise higher profit. Based on the results application of the model led to an increase in revenue.

An integrated production-inventory model with imperfect production processes and Weibull distribution deterioration under inflation

In this paper, we develop economic production quantity (EPQ) models to determine the optimal production lot size and backorder quantity for a manufacturer under an imperfect production process. The imperfect production process is characterised by the fraction of defective items at the time of production γ. The paper considers different cases of the EPQ model depending on (1) whether γ is known with certainty or is a random variable, and (2) whether imperfect items are drawn from inventory (a) as they are detected, (b) at the end of each production period or (c) at the end of each production cycle. Straightforward convexity results are shown and closed-form solutions are provided for the optimal order and backorder quantities for each of the cases we considered. We provide two numerical examples: one in which the defective probability follows a uniform distribution and the second which we assume follows a beta distribution, to illustrate the effects of yield variability and timing of the withdrawal of defectives on the optimal solutions. We obtain similar results for both numerical examples, which show that both the yield variability and the withdrawal timing are not critical factors.

This article discusses a vendor-buyer inventory model with permissible delay in payments and controllable lead time in which the order quantity, lead time and the number of shipments delivered from vendor to the buyer in a production cycle are decision variables. Here the production process is imperfect. The lead time is crashed and the crashing cost is an exponential function of lead time. Based on the lead time demand, two models are developed, that is, lead time demand follows a normal distribution in the first model and then distribution free approach is considered in the second model to minimize the joint total expected cost per unit time. Efficient computational algorithm is designed to find the optimal solution. Numerical examples are provided to illustrate the results obtained. Sensitivity analysis is carried out to study the changes in the effect on optimal solution and some managerial phenomena are obtained through sensitivity analysis.

In this paper, we develop and optimise the lot-sizing policy in an integrated Economic Production Quantity (EPQ) model with partial backorders and re-workable products considering linear and fixed backordering costs. Time intervals, number of lots, and lot size are the decision variables of our model. The cost function includes the set-up costs, the holding costs, the goodwill loss costs, the fixed backorder costs, the backorder costs, the costs of reworking, the production costs, the disposal costs, and the screening costs. The profit function consists of the sale profits obtained from the sold products. The goal is to minimise the cost function and maximise the profit function under stochastic constraints simultaneously. A Lexicographic method is applied for integrating the conflicting objective functions. The integrated objective function, with stochastic constraints, is a Mix Integer Nonlinear Programming (MINLP) model. Accordingly, an Outer Approximation (OA) algorithm is provided for optimal lot-sizing the integrated EPQ model. Two numerical examples and a real large-scale example demonstrate the decent and acceptable performance of the presented OA with respect to the optimality criteria such as optimum solutions, complementarity, and taken iterations.

This study establishes the competitive production-inventory models with vendor’s imperfect production processes by using game theory. The imperfect production processes can be improved by capital investment which is shared out between the vendor and the buyer jointly. In addition, the inspection process is assumed to be imperfect and Type I and Type II inspection errors occur during product quality inspection. We first develop the total cost per unit time of the buyer and vendor, respectively and then seek the optimal buyer’s order quantity, vendor’s shipping times and defective rate of the product where the system achieves a Nash equilibrium. The necessary and sufficient conditions of the existence and uniqueness of the optimal solutions for the buyer and the vendor respectively are shown. Furthermore, we develop a algorithm to find the optimal solutions. Finally, an numerical example are presented to demonstrate the solution procedure.

Publication in the conference proceedings of EUSIPCO, Lisbon, Portugal, 2014

This paper considers the number of defective items in a lot to be a random variable and derives a modified (s, Q) model with stochastic demand and constant lead time. The paper develops explicit results for the case of exponential and uniform demand during lead time when the number of defective items in a lot follows a binomial process. It also investigates the effect of lower setup costs on the operating characteristics of the model. A numerical example is given and sensitivity analysis is performed.

In today's competitive business situation, the supplier frequently offers his or her retailers a permissible delay period to stimulate sales. In addition, the capacity of any warehouse is limited in practice; thus, the retailer needs an additional rented warehouse to store the excess units when the order quantity exceeds the capacity of the own warehouse. Furthermore, with the globalization of the marketing policy, the supplier may provide the retailer a discounted price if the quantity of purchase is large enough. Green inventory management reduces the environmental impacts of a business without lacking its profit. Considering all of the factors mentioned above, in this paper, we study a green economic order quantity model with capacity constraint under order-size-dependent trade credit and all-units discount along with minimizing carbon for a cleaner environment. The paper discusses all the potential cases, which may occur in green inventory models with carbon emission cost under different allowable delay in payments. Shortages are allowed and partially backordered. The main objective is to determine the sustainable optimal ordering strategies for retailers and decide whether a rented warehouse (RW) is to obtain ordering and replenishment policies for a retailer such that the retailer’s annual profit is maximized. First, we prove that the conditions of the objective functions have interior minimizer value and the closed-form optimal solution is found. Next, an algorithm is developed to determining the global optimal solution of the problem. Finally, some management insights are drawn by observing the applicability of the developed algorithm and also by performing sensitivity analyses on different parameters.

This paper deals with a lot-sizing problem in the economic production quantity model with imperfect production processes and failure in repair where a Brownian motion process governs the inventory level variation. We assume that the unsatisfied demand is fully backlogged. In addition, we assume that all imperfect items produced are stocked separately and then reworked. Reworking and shortage costs are included in the proposed mathematical modelling and analysis. The aim of this research is to determine the optimal production lot size and backorder level such that the long-run total expected cost of the system is minimised. For the proposed model, we prove closed-form expressions for the inventory process and for the backlog costs. Moreover, a computer code using software package for the numerical work is furnished to derive the optimal solution. Some numerical examples and sensitivity analysis are provided to illustrate the validity of the proposed model.

This study develops an improved inventory model to help the enterprises to advance their profit increasing and cost reduction in a single vendor single‐buyer environment with general demand curve, adjustable production rate, and imperfect reworking process under permissible delay in payments. For advancing practical use in a real world, we are concerned with the following strategy determining, which includes the buyer_s optimal selling price, order quantity, and the number of shipments per production run from the vendor to the buyer. An algorithm and numerical analysis are used to illustrate the solution procedure.

Economic production lot size model for deteriorating items with partial back-ordering

EOQ models with stockouts for disassembly systems

A production inventory model with Just-In-Time (JIT) set-up cost has been developed in which inflation and time value of money are considered under an imperfect production process. The demand rate is considered to be a function of advertisement cost and selling price. Unit production cost is considered incorporating several features like energy and labour cost, raw material cost and development cost of the manufacturing system. Development cost is assumed to be a function of reliability parameter. Considering these phenomena, an analytic expression is obtained for the total profit of the model. The model provides an analytical solution to maximize the total profit function.A numerical example is presented to illustrate the model along with graphical analysis. Sensitivity analysis has been carried out to identify the most sensitive parameters of the model.

Several algorithms were developed in the past for blind equalization. The two most common ones are the constant modulus algorithm (CMA) and the reduced constellation algorithm (RCA). RCA is simple to implement but is not very reliable. CMA is very reliable but has a high complexity. In this thesis, an improved blind algorithm, called multimodulus algorithm (MMA), is proposed. This new algorithm is more reliable than RCA and less complex than CMA. MMA makes better usage of the knowledge of the statistics of the data signal and employs multiple moduli to achieve initial convergence. The use of multiple moduli makes MMA very flexible and easily adaptable to applications using two-dimensional transmission schemes with nonsquare or very dense signal constellations. Most blind equalization algorithms minimize a cost function. Our investigation has shown that the minimum (residual) value of this cost function has a great influence on the reliability and speed of convergence of the corresponding blind algorithm. For example, for CMA, RCA and MMA, this residual value increases when the bandwidth efficiency of the transmission system increases, and the blind algorithms suffer a corresponding decrease in performance. Two generalized versions of MMA are proposed to circumvent the above problem. One is called generalized MMA (GMMA). The basic version of MMA uses multiple moduli primarily for non-square constellations when nonuniform symbol distributions are utilized. Multiple moduli are also used with GMMA but the additional purpose there is to decrease the residual value of the cost function. A similar effect is achieved by using another blind algorithm, called windowed MMA (WMMA). With WMMA, only one modulus is used, but the tap coefficients of the filter are only updated with some of the data, which leads to a reduction of the residual value of the cost function. The new MMA algorithm has been extensively tested with computer simulations. Most simulation results presented here were obtained with the 155 Mb/s 64-CAP transceiver specified in the ATM LAN standard for category 3 unshielded-twisted-pair office wiring. However, our results are equally applicable to other applications, such as the 52 Mb/s 16-CAP transceiver defined in DAVIC's specification for fiber-to-the-curb networks. Some preliminary experimental results obtained in the laboratory with such a 16-CAP transceiver are presented in this thesis.