Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA, and WCA

Highlights

• A multi-product EPQ model with several technical and physical constraints is developed.

• Shortages are allowed and partially backordered with fixed and linear costs.

• SQP, SFS, SA, and WCA are utilized for solution.

• Ninety numerical examples in small, medium, and large sizes are solved to evaluate the performances.

• The results show the better performance of WCA.

Abstract

A multi-product economic production quantity model with several real-world technical and physical constraints is developed in this paper. The cost function includes ordering, holding, backordering, lost sale, and the cost caused by unused space in the warehouse. The goal is to minimize the total inventory cost, where shortages are allowed and partially backordered with fixed and linear costs. The aim is to determine the length of the inventory cycle, the length of positive inventory period, and the backordering rates of the products during the shortage period in order to minimize the total inventory costs while satisfying all constraints. Due to complexity and non-linearity of the proposed model, sequential quadratic programming (SQP), stochastic fractal search (SFS), simulated annealing (SA), and water cycle algorithm (WCA) are utilized for solution. Ninety numerical examples in small, medium, and large sizes are solved to evaluate the efficiency of the solution methods. The performances of the solution methods are compared statistically. Besides, sensitivity analysis is performed to determine the effect of change in the main parameters of the problem on the objective function value and decision variables.

Introduction

The economic order quantity (EOQ) and the economic production quantity (EPQ) are the two classical inventory control models that were first introduced by Harris [10] and Taft [33], respectively. Although both are the most applicable inventory models, some of their inherent assumptions make them unrealistic. That is why many researchers relaxed the limitation involved to develop more realistic and applicable EOQ and EPQ models in the last decades. For instance, they assumed the shortages can appear in the forms of backorders or lost sales. This assumption forces the EOQ or EPQ model to fully backorder the shortage or to consider it as lost sales. Besides, many researchers paid attention to the partial backordering in recent years, where most of them showed that shortages in the form of partial backorders results in the total inventory cost to decrease.

Hadley and Whitin [9] and Montgomery et al. [15] were the pioneers to study EOQ and EPQ models with partial backordering. Their works were extended by Rosenberg [23], Park [19], Abad [1], Pentico et al. [22], and Yang et al. [45], who studied EOQ and EPQ models with partial backordering as well. Wee [42] proposed an inventory model with constrained backordering and lost sales fractions to ensure the convexity of the total cost function, in order to use a Hessian matrix to propose a simple and stable solution for the problem. Later, Wee [43] developed an EPQ model for deteriorating items with partial backordering. Pentico and Drake [21] employed a different approach to model a deterministic EOQ model with partial backordering. They proposed formulae which were comparable to the ones in the classical EOQ model. Zeng [47] considered a Poisson demand and an exponential production time in a queuing model to determine the cost-effectiveness of the partial backordering. Abad [2] used a new approach to model backordered demands. He considered pricing and lot-sizing problems for perishable goods under finite production horizon and partial backordering. Abad [3] modeled pricing and lot-sizing problems for a single item with general deterioration rate and partial backordering. San José et al. [27] studied a continuous review inventory model with known and constant demand where shortages were allowed and partially backordered.

In addition to the works cited above, many researchers assumed exponential or time-proportional backordering rate to develop more applicable EOQ and EPQ models. San José et al. [28] studied an inventory system in which the partial backordering followed an exponential function. They proposed a solution procedure to find the optimal solution. Dye and Ouyang [4] extended the model proposed by Padmanabhan and Vrat [18] by considering a time-proportional backordering rate for perishable items under stock-dependent selling rates to achieve more applicable results in the EOQ model. Sarkar and Sarkar [30] proposed an inventory model with deteriorating items with stock-dependent demand. Their model included time varying backordering and time varying deterioration rate. They aimed to determine the optimal cycle length of each product to minimize total inventory costs. Zhang et al. [48] developed an applicable EOQ model with partial backordering considering the cross-selling effect.

In some works in EOQ and EPQ environments, partial backordering is considered for perishable items [11]. For instance, Sana [29] studied an EOQ model with infinite planning horizon for perishable items under price-dependent demand, partial backordering, and time-dependent deterioration rate. Ghosh et al. [7] proposed an EOQ model for perishable items with price dependent demand, lost sales, and partial backordering depended on the waiting-times. In some other cases, researchers considered two costs associated with the partial backordering. For example, Sphicas [32] investigated EOQ and EPQ models with fixed and linear backordering costs. Wee et al. [44] studied EPQ models with fixed and linear backordering costs associated with partial backordering. Omar et al. [17] proposed an approach to transform the EOQ and EPQ models into a better form to obtain the optimal solution.

Recent works on partial backordering involve allowable delay in payments. For instance, Taleizadeh et al. [37] developed an EOQ model with a certain sale price and partial backordering. In addition, Taleizadeh et al. [38] proposed an EOQ model with partial payments in three situations of (a) partial backordering, (b) not allowing shortages, and (c) full backordering of the shortages. They proposed a new solution method that minimizes the convex cost function of the problem. Taleizadeh et al. [39] developed an EOQ model with partial backordering and payments where a fraction of purchasing cost could be paid at the beginning of the cycle. Taleizadeh and Pentico [35] proposed EOQ models with price increase and partial backordering under two different assumptions on the occurrence time of the price increase. In addition, Taleizadeh [34] developed an EOQ model for an evaporating item allowing prepayments for partial backordering costs. Taleizadeh and Pentico [36] proposed an EOQ model and a solution method under all-unit discounts and partial backordering assumptions. They used multi-factor experiment to improve the performance of their proposed method.

Although the importance of the partial backordering has been highlighted in many inventory models including extended versions of EOQ and EPQ models, the vacuity in the literature is that almost all of them were developed for only one item under specific assumptions without considering any technical and physical constraints. Therefore, developing a constrained multi-product stochastic EPQ model with partial backordering would be beneficial. In addition, previous works considered backordering rate of the demand during shortage period as an input parameter, while, considering backordering rates as decision variables can lead to a significant decrease in total inventory costs by finding their optimal values. In other words, the contributions of this paper are as follows:

• A multi-product EPQ model with partial backordering is developed.

• Some decision variables are added to the total cost function to determine the optimal backordering rate during the shortage period.

• Various constraints are considered to propose a closer to reality and a more applicable EPQ model.

• Due to uncertainty involved in real-world environments, the budget constraint is considered stochastic.

• The costs of unused free space in the warehouse are added to the EPQ model.

• An exact method and three meta-heuristic algorithms are developed to solve the complex constrained non-linear programming (NLP) problem.

• The performance of the solution methods are evaluated using different measures such as percentage relative Error, relative percentage deviations, standard deviation of the obtained solutions and average computation time.

• All solution methods are evaluated statistically to determine the best solution algorithm in small, medium and large size test problems, separately.

The remainder of this paper is organized as follows. In Section 2, the definition of the problem and the assumptions are presented. Section 3 contains the parameters, variables, and the mathematical model of the problem. In Section 4, the solution methods are explained. Ninety test problems in different sizes of small, medium, and large are solved in Section 5 using the utilized solution methods, based on which the performances of the solution methods are statistically compared in terms of three performance measures. In Section 6, sensitivity analyses are performed to determine the effects of changes in the values of the critical parameters of the model on the objective function value. Finally, Section 7 concludes the paper.