Advances in Operations Research

Volume 2017 (2017), Article ID 3601217, 18 pages

https://doi.org/10.1155/2017/3601217

## Population Based Metaheuristic Algorithm Approach for Analysis of Multi-Item Multi-Period Procurement Lot Sizing Problem

Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, Mangalore, Karnataka, India

Correspondence should be addressed to Prasanna Kumar

Received 20 March 2017; Revised 6 August 2017; Accepted 7 November 2017; Published 20 December 2017

Academic Editor: Paolo Gastaldo

Copyright © 2017 Prasanna Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This research study focuses on the optimization of multi-item multi-period procurement lot sizing problem for inventory management. Mathematical model is developed which considers different practical constraints like storage space and budget. The aim is to find optimum order quantities of the product so that total cost of inventory is minimized. The NP-hard mathematical model is solved by adopting a novel ant colony optimization approach. Due to lack of benchmark method specified in the literature to assess the performance of the above approach, another metaheuristic based program of genetic algorithm is also employed to solve the problem. The parameters of genetic algorithm model are calibrated using Taguchi method of experiments. The performance of both algorithms is compared using ANOVA analysis with the real time data collected from a valve manufacturing company. It is verified that two methods have not shown any significant difference as far as objective function value is considered. But genetic algorithm is far better than the ACO method when compared on the basis of CPU execution time.

#### 1. Introduction

In the present day competitive markets, it is imperative for the organizations to manage their supply chain as efficiently as possible to sustain their market share and improve profitability. Optimized inventory control is the integral part of supply chain management. Harris [1] had suggested a classical inventory model which was later on extended by several research works to incorporate the different practical constraints under different business scenario. Main extensions which had attracted the attention of many researchers were multi-item and multi-period planning inventory model. The multi-period inventory lot sizing scenario with a single product was introduced by Wagner and Whitin in [2], where a dynamic programming solution algorithm was proposed to obtain feasible solutions to the problem.

Another extension of EOQ model was worked out by Das et al. [3] by considering the multi-item inventory model with constant demand under the restrictions on storage area, total average shortage cost, and total average inventory investment cost. In another research work, Nenes et al. [4] proposed a model to manage inventory of a product with irregular demand in multiple periods. A nonlinear goal programming model was developed by Panda et al. [5] for lot sizing for a multi-item inventory problem using penalty functions in a decision-making environment. A combination of inventory control and distribution planning was suggested by D. Kim and C. Kim [6] for a multi-period environment. Another additional practical consideration which was suggested into the classical inventory model was the simulation of all unit discount (AUD) and incremental quantity discount (IQD) policy.

On this research front, A. K. Maiti and M. Maiti [7] modeled a multi-item inventory control system of items with AUD and IQD and a combination of these discounts. Sana and Chaudhuri [8] extended this work by considering the delayed payments and their effect on the lot sizing.

In this research, the multi-item multi-period inventory model is extended to incorporate varying but deterministic periodic demand rate for optimizing the procurement lot size so that total inventory cost is minimized. The model approaches more realistic business situation by considering the all unit discount, overall storage capacity limit, and budget constraint. The aim of the study is to arrive at the optimal order quantities of all items in different periods such that the total inventory cost including the ordering, holding, and the purchasing costs is minimized. The mixed binary integer model that is developed is overly NP-hard. Therefore, two metaheuristic algorithms of GA and ACO have been proposed for its solution. The two algorithms are used to validate and compare the quality of near-optimum solution and ease of their application.

Many of recent research works have seen increased use of Taguchi approach and its hybrid methods with other statistical and metaheuristic techniques for optimization due to their versatility and adaptiveness. Hybrid Taguchi-cuckoo search (HTCS) algorithm has been used extensively in the different fields of study. In this research study, Taguchi optimization technique has been employed to enhance the performance of GA by calibrating the parameters [9, 10].

The application of AI technique to multi-item multi-period procurement lot sizing problem with novel GA and ACO is main innovation deployed in this research work. The model usability is enhanced by approaching closer to more practical situations considering the storage area and budget constraints and all unit discount scenario. Also very few applications have been sighted in the research literature where Taguchi experimental design has been deployed for the GA parameters optimization.

The rest of paper is organized as follows. In the next section, problem under study is explained in detail. Mathematical formulation of considered multi-item multi-period procurement model is presented. Two metaheuristic algorithms GA and ACO and their novelty application for solving our NP-hard problem are discussed in Section 3. This section also contains the discussion on Taguchi method application for GA parameter. Computational results are drawn out in Section 4 and two algorithms are compared on the basis of their performance parameters. Section 5 presents the conclusions and directions for future work.

#### 2. Problem Definition

The multi-item multi-period procurement lot sizing problem as applied to scenario under discussion is defined here. It is assumed that a company maintains the inventory of many items for satisfying its customers’ requirement under varying but known demand rates. Normal assumption of constant demand rate is foregone to make the model more practical. The known demand rates may change in different periods within a finite planning horizon having periods. The initial inventory of all items is the reserve stock. It is assumed that only one order is placed for a particular item in a given period. Lower limit and upper limit of order quantity are also specified as boundary condition. The ordered quantities of items are received in batch sizes where batch is not permitted to be split. All quantity discount regime is considered. Price discount breakpoints are defined so that if an item is ordered in price break quantity level 1, then no discount will be offered. But if an item is ordered in price break 2 then 5% discount will be offered and if an item is ordered in price break 3 then additional 5% discount will be offered and so on. Moreover, the storage space available to hold the total inventory for each period is constrained. There is a restriction on the maximum budget available for the procurement for each period. The aim is to find out the optimum values of order quantities of all the items for each period such that the total cost of the inventory is minimized and the constraints are satisfied. The above defined problem models many real world inventory control systems. The mathematical formulation of the problem is presented in the next section. However, before doing the mathematical formulation, the necessary notations which are to be used are listed here. : number of components or items ordered : number of periods or time horizons over which procurement is made : number of price breaks for consideration of price discount : requirement or demand rate of the item at period : ordering cost of the th component at the beginning of an interval : ordering quantity of th component in interval : number of price discount breakpoints : th discount breakpoint of th component : total storage space available : warehouse space occupied by one unit of the th component : purchasing cost of the th component at the breakpoint “” **H**_{x}: per unit holding cost of the component **P**_{u}: purchasing cost of th component paid at the start of the interval OC: total ordering cost PC: total purchasing cost HC: total holding cost IC: total inventory cost : total budget available for the procurement : upper band for : initial inventory of the component “” in interval “” : reserve stock for the item : a binary decision variable, set equal to one if component is purchased at price breakpoint in period , and zero otherwise : a binary decision variable, set equal to one if a purchase of a component is made in period , and zero otherwise : total warehouse space : warehouse space for th component

##### 2.1. Mathematical Formulation

The total cost of inventory is sum of the ordering cost, holding cost, and purchase cost. This is represented by the following equation:where TC is the total inventory cost, OC is the ordering cost, HC is holding cost, and PC is the purchase cost.

The individual components of the total cost are analyzed as follows.

###### 2.1.1. Ordering Cost

Consider that components are ordered over time horizons, and total ordering cost OC is obtained as in the following equation:

###### 2.1.2. Holding Cost

Holding cost can be calculated by multiplying the per unit holding cost with the average inventory during that period as shown in the following equation: The inventory of the component at the beginning of time period can be related to ordered quantity and final inventory during the period . The beginning inventory of component in period is equal to its initial inventory plus the purchased quantity minus its demand, all in interval , as related in the following equation:Using (3) and (4),

###### 2.1.3. Purchase Cost

is a binary decision variable, set equal to one if component is purchased at price breakpoint in period , and zero otherwise. As a result, the total purchasing cost will be as specified in the following equation:

###### 2.1.4. Total Cost

As a result, the complete mathematical model of the inventory control system is as follows. Minimize subject to the constraints: for , , and , Constraint (9) ensures that the quantity should be bought at only one price break. Constraint (10) is the mathematical representation of following equation: Inventory brought forward to next period = inventory brought forward to this period + quantity ordered in this period − demand (or consumption) in this period. Constraint (11) ensures that the inventory brought forward should be greater than or equal to the reserve stock. Equation (12) ensures that budget constraint should be satisfied for each period. Equation (13) ensures that warehouse area constraint should be satisfied for each period.

#### 3. Solution Methodology

Since the mathematical model proposed is NP-hard, two metaheuristic approaches ant colony optimization (ACO) and genetic algorithm (GA) have been proposed to solve the problem, with the objective of verifying the solutions obtained and comparing the performances.

##### 3.1. Ant Colony Optimization

The phenomenon of pheromone communication has been the drive behind ant colony optimization (ACO) metaheuristics algorithms. Artificial ants simulate the real life ants which communicate their experience while optimizing their search for food in nature through trails of pheromone.

Candidate solutions to the optimization problem are constructed by individual ants by interactively adding solution components to initialized empty solution. A complete solution is generated by the ants using the two components: pheromone information which is the accumulated experience and heuristic information which is problem specific data.

Which ants are allowed to modify the pheromone information and how they modify is governed by the update strategy. Usually better solution components will receive higher amount of pheromone and will have higher probability of being used by other ants in the subsequent iterations of algorithm.

###### 3.1.1. Ant Cycle Model

Ant cycle model is adopted in this research work. In this model, the trails are globally updated during each cycle by all ants. The amount of pheromone deposited by each ant is a function of the solution quality. As per the flowchart of proposed ant colony optimization algorithm, first the ant based initial solution construction method is executed. Next values of pheromone are set based on ant based initial solution construction method. Thus the values of pheromone are set accurately after the execution of ant based initial solution construction method. Followed by this, ant based solution construction method will be executed in each cycle which will utilize the values of pheromone in order to construct better solutions [11].

This research work adopts ant cycle model where the trails are uniformly updated during each cycle by all ants. The solution quality is determined by the amount of pheromone deposited by each ant. The initial solution construction method is executed. Then the ant based solution construction method is executed which will employ the values of pheromone to construct better solutions. Each ant represents a solution to the optimization problem. In our case of application to inventory management, it represents the order quantity of different items in different periods so that total inventory cost is reduced.

The novel concept is implemented for ant based solution construction in each cycle as follows. First each ant will construct the solution according to the probabilistic rule. After constructing the solution according to the probabilistic rule, ant 1 will discard its order quantity values of all periods for item 1 and item 3 and the values of order quantity of all periods for item 1 and item 3 of ant 2 will be copied into the solution generated by ant 1. Similarly, ant 2 will discard its order quantity values of all periods for item 1 and item 3 and the values of order quantity of all periods for item 1 and item 3 of ant 3 will be copied into the solution generated by ant 2. This discard-copy step is repeated for all ants. Last ant will discard its order quantity values of all periods for item 1 and item 3 and the values of order quantity of all periods for item 1 and item 3 of ant 1 will be copied into the solution generated by last ant. This novel concept represents the direct communication between ants in each cycle in order to further improve the solution constructed by individual ant separately.

In this research work, ant based solution is constructed by using problem specific knowledge so that it does not violate constraint (3) and constraint (4), specified under mathematical modeling under Section 2. Once the value of order quantity is generated for all components for all periods, it will be verified that whether the generated values of order quantity will meet the budget constraint and warehouse area constraint or not. If either of the constraints is not met, then the generated values of order quantity (solution) will be discarded. If both the budget constraint and the warehouse area constraint are met, then the solution is valid and it is stored for further processing.

The ACO algorithm has been implemented on JAVA platform and the program can be run with Net Beans IDE.

###### 3.1.2. Determination of ACO Parameter

To determine the optimum value for number of cycles, which is an important ACO algorithm parameter, ACO is run several times for one of the data sets, changing the number of cycles each time and recording the objective value function and CPU processing time. ACO run results are tabulated in Table 1. From Figure 1 which shows the variation of objective function value with number of cycles in the ACO model, it is clear that objective function reaches optimum value at 600 cycles. After that increasing the number of cycles will only increase the CPU time without improving the objective function value.