Application of the NSGA-II algorithm to a multi-period inventory-redundancy allocation problem in a series-parallel system

Highlights

• An inventory control system employing an all-unit discount policy is considered in the proposed model.

• The proposed model considers limited total budget, storage space, transportation capacity, and total weight. Moreover, a penalty function is used to penalize infeasible solutions.

• The overall goal is to find the optimal number components purchased for each subsystem so that the total costs including ordering cost, holding cost and purchasing cost are minimized and the system reliability are maximized, simultaneously.

• A NSGA-II algorithm is derived where a multi-objective particle swarm optimization and a multi-objective harmony search algorithm are used to evaluate the NSGA-II results.

Abstract

In this paper, we formulate a mixed-integer binary non-linear programming model to study a series-parallel multi-component multi-periodic inventory-redundancy allocation problem (IRAP). This IRAP is a novel redundancy allocation problem (RAP) because components (products) are purchased under an all unit discount (AUD) policy and then installed on a series-parallel system. The total budget available for purchasing the components, the storage space, the vehicle capacities, and the total weight of the system are limited. Moreover, a penalty function is used to penalize infeasible solutions, generated randomly. The overall goal is to find the optimal number of the components purchased for each subsystem so that the total costs including ordering cost, holding costs, and purchasing cost are minimized while the system reliability is maximized, simultaneously. A non-dominated sorting genetic algorithm-II (NSGA-II), a multi-objective particle swarm optimization (MOPSO), and a multi-objective harmony search (MOHS) algorithm are applied to obtain the optimal Pareto solutions. While no benchmark is available in the literature, some numerical examples are generated randomly to evaluate the results of NSGA-II on the proposed IRAP. The results are in favor of NSGA-II.

Introduction

Redundancy allocation problems (RAPs) are commonly observed in complex telecommunication, safety, transportation, satellite, and electrical power systems where strict system reliability requirements are needed [22]. Several RAP structures and configurations are considered in the literature for series [19], parallel [4], series-parallel [11], [12], and k-out-of-n [7] systems among others. Series-parallel RAP systems are generally more common than other types [11], [12], [21], [24], [9].

Traditionally, the RAP literature has focused heavily on system reliability with little attention to the preparation and transformation of the system components. However, there are some studies in the literature which have considered the production process of the components prior to system installation. Sadeghi et al. [19] considered a two-objective vendor managed inventory and redundancy allocation problem in a vendor-retailer supply chain problem. The goal of the problem is to optimize the number of machines working in series to manufacture a single product. Xie et al. [23] optimized the system reliability of a repairable k-out-of-n RAP where the spare parts inventory in addition to the redundancy allocation of components were taken into account.

In this study, a multi-objective series-parallel RAP is modeled in a multi-component multi-period inventory control problem. The multi-product multi-period inventory problem is a common problem often studied in the literature. Mousavi et al. [14] formulated a multi-item multi-period inventory control problem in which both all unit and incremental quantity discounts, in addition to interest and inflation factors were considered. In addition, Mousavi et al. [15] modeled a fuzzy multi-product multi-period inventory control problem with backorder and lost sales, where the budget was restricted and the items were delivered in pre-specified boxes. Gholamian et al. [5] considered a multi-objective multi-period multi-site multi-product inventory problem in a fuzzy multiple-echelon supply chain problem where two different linearization methods were applied to solve the proposed problem. Ghoniem and Maddah [6] studied a multi-product multi-period selling horizon pricing-inventory control problem which was modeled as a mixed-integer nonlinear mathematical programming. Mousavi et al. [13] formulated a multi-periodic multi-item inventory control problem modeled as a mixed integer problem. Shortages were allowed and in case of shortage, a fraction of demand was considered as backorders and a fraction as lost sales. Pasandideh et al. [17] proposed a mixed-integer nonlinear mathematical programming model for a seasonal multi-product multi-period inventory control problem in which the total budget and the total storage space were restricted. They used the genetic algorithm and memetic algorithm to optimize their proposed problem. In this study, a non-dominated sorting genetic algorithm-II (NSGA-II) along with MOPSO and MOHS are utilized to solve the proposed IRAP.

Many researchers have studied series-parallel RAPs. Mousavi et al. [12] studied a fuzzy multi-state RAP in a fuzzy environment where the homogenous components were installed in each subsystem. Yeh [24] used an orthogonal simplified swarm optimization to solve a series-parallel RAP with a mix of components in which the aim was to maximize the system reliability. Soltani et al. [21] solved a series-parallel redundancy allocation problem with uncertainty in the reliability of the components. Hsieh and Yeh [9] employed an artificial bee colony algorithm to optimize a series-parallel RAP where a penalty strategy was used to remove the equalities in the constraints and also to find the infeasible solutions, generated randomly. Mousavi et al. [11] considered an improved fruit fly optimization algorithm to solve a series-parallel RAP in a fuzzy environment in which identical components were allowed to be installed within each subsystem. The aim was to obtain the optimal number of identical components in each subsystem so that the system reliability was maximized. Cao et al. [1] provided a multi-objective series-parallel RAP where an exact method i.e. a decomposition-based approach was applied to solve the problem. Khalili-Damghani and Amiri [10] used both the epsilon-constraint approach and data envelopment analysis to solve a binary-state multi-objective series-parallel RAP in which an epsilon-constraint was employed to generate Pareto fronts.

Multi-objective RAP has been of interest to many researchers in the recent literature. Zhang and Chen [25] provided a multi-objective RAP with imprecise components in an interval environment where the total cost and the system reliability were optimized simultaneously. They solved their problem using an improved multi-objective particle swarm optimization (MOPSO) algorithm. Mousavi et al. [12] solved a multi-objective multi-state series-parallel RAP in a fuzzy environment where the components in each subsystem were selected from the identical type. They applied two controlled elitism non-dominated ranked genetic algorithms and a non-dominated sorting genetic algorithm (NSGA-II) to optimize their problem. Ghorabaee et al. [7] utilized NSGA-II to solve a bi-objective k-out-of-n RAP in which both objectives of the total cost and the system reliability were optimized simultaneously. Dolatshahi-Zand and Khalili-Damghani [2] used a multi-objective RAP to design a SCADA water source management control center where both MOPSO and TOPSIS algorithms were applied. Garg et al. [3] modeled a fuzzy multi-objective series-parallel RAP where a MOPSO and a bi-objective particle swarm optimization were employed to solve the proposed RAP. Cao et al. [1] used NSGA-II to solve a decomposition-based approach of multi-objective RAP for series-parallel systems. Ghorabaee et al. [8] solved a k-out-of-n multi-objective series-parallel RAP using NSGA-II where the objectives were to optimize both the total cost and the system reliability simultaneously. Rahmati et al. [18] optimized a multi-objective location model within a multi-server queuing framework in which facilities were based on M/M/m queues where both NSGA-II and NRGA algorithms were applied to solve the problem.

There is almost no research in the literature which considers both the inventory and transportation components of a series-parallel RAP. The main contribution of this work is to fill this gap. A bi-objective binary-state RAP is formulated in a multi-component (product) multi-period inventory control problem with restrictions on the total storage space, transportation vehicle capacity and the total budget available to purchase the components. To solve the proposed IRAP, a NSGA-II algorithm is derived where both MOPSO and MOHS are also used to validate the results obtained by the NSGA-II.

A real world example of the proposed problem would be an airplane where several engines are connected in series, each containing parts which are installed in parallel (see Feller 1950). The components of each engine work in parallel, meaning that the engine subsystem operates if at least one of the components operates. The components are manufactured by a wide range of companies storing their products in a special warehouse and are sold to satisfy the required demand. The manufacturer determines the price and reliability for each component. It is assumed that the airplane manufacturing companies order the identical type of the components for each engine in order to profit from the discount provided by the manufacturer.

Another real world example is to consider the case of the engines installed on each wing of an airplane as a subsystem. The components (engines) of each wing work in parallel, meaning that the engine subsystem operates if at least one of the engines of a wing operates. In order to make sense of the contribution of the current study, a comparison of the current work with the recent literature is shown in Table 1.

The rest of the paper is organized as the follows. The next section describes the notations, parameters and the decision variables applied in this article to formulate the IRAP under investigation. In Section 3, the proposed IRAP is explained and modeled. The explanation of the NSGA-II, MOPSO and MOHS algorithms comes in Section 4. In Section 5, some numerical illustrations are generated to evaluate the performance of the provided algorithms on the IRAP. The conclusions and recommendations for future research are given in Section 6.