An application of real-coded genetic algorithm (RCGA) for mixed integer non-linear programming in two-storage multi-item inventory model with discount policy

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Abstract

The purpose of this research work is to solve mixed-integer non-linear programming problem with constraints by a real-coded genetic algorithm (RCGA). This GA is based on Roulette wheel selection, whole arithmetic crossover and non-uniform mutation. Here, mutation is carried out for the fine-tuning capabilities of the system by non-uniform operator whose action depends on the age of the population. This methodology has been applied in solving multiple price break structure and implemented for multi-item deterministic inventory control system having two separate storage facilities (owned and rented warehouse) due to limited capacity of the existing storage (owned warehouse). Also, demand rate is a linear function of selling price, time and non-linearly on the frequency of advertisement. The model is formulated with infinite replenishment and shortages are not allowed. The stocks of rented warehouse (RW) are transported to the owned warehouse (OW) in bulk-release rule. So, the mathematical model becomes a constrained non-linear mixed-integer problem. Our aim is to determine the optimal shipments, lot size of the two warehouses (OW and RW), shipment size and maximum profit by maximizing the profit function. The model is illustrated with numerical example and sensitivity analyses are performed with respect to different parameters.

Introduction

Generally, the basic assumption of classical inventory model is that the management purchases or produces a single item. However, in many real-life situations, this assumption is not correct. Instead of a single item, many companies or enterprises or retailers are motivated to store several items in their show-room for more profitable business affair. Another cause of their motivation is to attract the customers to purchase several items in one show room/shop. Multi-item classical inventory models under different resource constraints such as available floor space/shelf space, capital investment and average number of inventory, etc. are presented in the well-known books by Churchman and Ackoff [1], Silver and Peterson [2], etc., of this subject. Padmanabhan and Vrat [3] developed a multi-item multi-objective inventory model of deteriorating items with stock dependent demand by a non-linear goal programming method. Considering two constraints on available space and budget, Ben-Daya and Raouf [4] discussed a multi-item inventory model with stochastic demand. Abou-et-ata and Kotb [5] formulated and solved a multi-item inventory model with varying holding cost under two restrictions with the help of geometric programming. Recently, Guria et al. [6] studied multi-item EOQ model with storage facilities for uniform demand.

Now-a-days, the inventory systems with quantity discount are of growing interest due its practical importance in purchasing and material control. In the third world countries, with the introduction of open market system and advent of multi-nationals, there is a stiff competition amongst the companies to capture the maximum possible market. It is a common practice on the procurement in inventory systems that the suppliers (whole sellers) offers price discount to the retailers for purchase orders of large sizes. In general, there are two types of discount-All Unit Discount (AUD) and Incremental Quantity Discount (IQD). In AUD, the discount is available for every unit purchased where as in the incremental quantity discount system, the discount applies only to the additional units beyond the quantity over which the discount is given. Among these two types of discount, AUD is more popular and is usually utilized by the retailers.

The basic technique to solve the quantity discount models dates back to the early days of operational research. The basic model of EOQ under price breaks has been extensively analyzed in Hadley and Whitin [7] and reported in other books. Later, several authors have made extensions of the above model. Benton [8] considered quantity discount for MRP lot sizing, Majewicz and Swanson [9] for dynamic lot sizing, Goyal [10], Monahan [11], Kim and Hwang [12] for integrated decision making by supplier and buyer, Pirkul and Arkas [13] for multi-item inventory, Das [14] for generalized discount structure unifying the IQD and AUD policies. Also, Rubin et al. [15] proposed some computational simplifications and Das [16] presented a complete graphical solution to the discount problems. The purpose of the quantity discount is to offer a lower price which motivates retailers to increase order quantities and thereby reduce the total purchase cost. Therefore, quantity discount models always demand to buy a large number of items for which existing warehouse may not be sufficient to store these items. In the existing literature, it is found that the classical inventory models generally deal with a single storage facility. The basic assumption in these models is that the management has a storage with unlimited capacity. However, it is not true (e.g., in an important supermarket, the storage space of showroom is very limited) in the field of inventory management. Due to attractive price discount for bulk purchase or some problems in frequent procurement or very high demand of items, management decides to purchase a huge quantity of items at a time. These items cannot be stored in the existing storage (owned warehouse, OW) with limited capacities. So, for storing the excess items, one (some time more than one) warehouse is hired on rental basis. The rented warehouse RW is located near the OW or little away from it. Usually, the holding cost in RW is greater than the same in OW. Further, the items of RW are transported to OW in bulk fashion to meet the customer’s demand until the stock level of RW is emptied.

In the last two decades, a good number of two warehouses inventory models have discussed by several researchers. This type of problem was first developed by Hartely [17] with the assumption of uniform demand of items. After Hartely [17], one may refer to the works of Sarma [18], [19], Dave [20], Goswami and Choudhuri [21], Bhunia and Maiti [22], [23], Pakkala and Achary [24], Benkherouf [25], Lee and Ma [26], Kar et al. [27] and others.

As an inventory problem is a decision-making problem which can be formulated as constrained/unconstrained non-linear optimization problem, there is a question: How it can be solved? Generally, most of the optimization problems of different inventory system are non-convex or non-concave optimization problems. In these problems, both local and global optimal solutions may exist. Then, special methods for global optimization are needed in order to solve these problems. Global optimization methods can be divided into deterministic and stochastic ones. Deterministic methods are usually based on some special assumptions on the problem to be solved, whereas stochastic methods utilize randomness. Because of their general nature, stochastic methods work even with discontinuous functions. Genetic algorithm (GA) represents this type of method. It is a robust technique, based on the natural selection and genetic production mechanism. It processes a group or population of possible solutions within a search space. This search is probability guided and stochastic, rather than deterministic or random searching which distinguish it from traditional methods. The basic idea behind the genetic algorithms is to artificially imitate the evaluation process of nature. The algorithms are based on the evaluation of a set of solutions, called a population. The population is upgraded by genetic operators in each iteration (generation). At each iteration (generation), the population consists of a number of individuals, i.e., possible solution of the problem. Typically, the population initialized by randomly generated individuals.

When individuals are encoded using real numbers the corresponding methods are called real-coded genetic algorithm. Each individual is a vector of variables where each variable is a real number. The suitability of an individual is determined by the value of a so-called fitness function based on the objective function. The population of next generation is created by these genetic operators: selection, crossover and mutation.

The selection operation chooses some offspring for survival according to their genetic diversity and fitness. The crossover operation generates offspring from two or more chosen individuals in the population by exchanging their genetic materials. The offspring thus inherit some characteristics from each parent. The mutation operation generates offspring by randomly changing one or several genes in an individual. Offspring may thus possess different characteristics from their parents. Mutation prevents local searches of the search space and increases the probability of finding global optima. Recently, GA has been successfully applied to a wide variety of problems such as Travelling salesman problems [28], Scheduling problems [29], Numerical Optimization [30], etc. Till now, only a very few researchers have applied it to solve the problem in the field of inventory control system. Among them, one may refer to the work of Khouja et al. [31], Sarkar and Charles [32], Mandal and Maiti [33], Pal et al. [34] among others.

In this research paper, we develop a multi-item two storage (OW and RW) multiple price breaks deterministic inventory model with a discount policy. The model is formulated as constrained non-linear mixed integer model and is solved by real-coded genetic algorithm with advanced GA operators. Also, demand is a function of selling price, time and frequency of advertisement. Shortages are not allowed, the stocks of RW are transported to OW in bulk-release fashion. Our objective is to determine the optimal shipments, lot-size of the OW and RW, shipment size and maximum profit by maximizing the profit function. Numerical examples illustrate that the above approaches are feasible and efficient.

Section snippets

Assumptions and notations

The following notations are used for the proposed model:

    n

    number of items

    W

    storage area or volume in RW

    j

    any cycle of proposed inventory system (j = 1, 2, …)

Parameters are used for the ith (i = 1, 2,  ,n) item in the jth cycle (j = 1, 2, …)

    Qj,i

    initial inventory units (decision variable)

    Wj,i

    storage capacity of OW

    mj.i

    mark-up rate

    wi

    storage area or volume required for each item (m2)

    Tj,i

    total time period

    Pj,i

    unit selling price

    pj,i(=pj.i1, pj,i2,  , pj,in)

    purchase cost per unit

    L1-system

    single storage/warehouse system

    L2

Model description and analysis

Initially, a company purchases Qj,i units of ith item of which Wj,i units are kept in OW and (Qj,i  Wj,i) units are kept in RW. The stocks of OW are used to meet the customer’s demand until the stock level of OW drops to (Wj,i  ki) units at the time of t1,ji. At this stage, ki(ki  Wj,i) units are transported from RW to OW to restore the inventory into original level and to meet the further customer’s demand. This process is continued until the stock of RW is fully exhausted. After the last

The inventory cost function

The total cost in the jth cycle consists of the following components:

(a) Transportation cost (Ctj,i), (b) Advertisement cost (CAdj,i), (c) Holding cost (Chj,i), (d) Purchase cost (pj,i), (e) Set-up cost (C3j,i).

  • (a)

    Transportation cost: The transportation cost for transferred the items/goods from RW to OW in ni shipments is given byCtj,i=ni[ai+bi(ki-Si)]forki>Si,=niai,otherwise.

  • (b)

    Advertisement cost: The total advertisement cost per replenishment isCAdj,i=μj,iQj,iPj,iNj,i.

  • (c)

    Holding cost: The inventory

Genetic algorithm

Genetic algorithms are heuristic search process for optimization that resembles natural selection. In most cases, they can find the global optimum solution with a high probability. They mimic the process of natural selection and is based on Darwin’s survival of the fittest principles. In this algorithm, a population of individuals (potential solutions) undergo a sequence of unary (mutation type) and higher order (crossover type) transformations. These individuals select the next generation.

The algorithm of AUD for multi-price break

The quantity Qj,i which maximizes the average total profit for the above model can be determined by the following procedure which is similar to that proposed by Wee [35]:

  • Step 1.

    Starting with the lowest unit cost (in our case pj,in),

    • Step 1.1.

      calculate the optimal ni and Qj,i values using the proposed procedure;

    • Step 1.2.

      calculate the respective ki from Eq. (9) and check if Qj,i is valid quantity;

    • Step 1.3.

      continue until the first valid ordered quantity, Qj,i is obtained and calculate the optimum Z(Qj,i, ni).

  • Step 2.

    Calculate the optimal

Numerical examples

To illustrate the developed model, an example has been considered. Though the values of the model parameters have not been selected from any case study, the values considered here are feasible. Here, we have considered two items and four-price break. The input values are given in Table 1 and optimum results are displayed in Table 2.For the1stitemFor the2nd itempj,1=$15.00,0<Qj,1<50,$14.00,50Qj,1<75,$13.25,75Qj,1<150,.$12.00,Qj,1150.pj,2=$12.00,0<Qj,2<25,$11.25,25Qj,2<70,$10.5070Qj,2<100,$

Sensitivity analysis

The earlier numerical example is used to study the effect of under or overestimation of various parameters on optimal cycle length, maximum net profit, optimal number of shipments from RW to OW and optimal units transported in each shipment of the inventory system. Here, we employ ΔTj,i=(Tj,i-Tj,i)/Tj,i×100%, ΔZ = (Z  Z)/Z × 100%, Δni=(ni-ni)/ni×100% and Δki=(ki-ki)/ki×100% as a measure of sensitivity, where ni, ki, Z and Tj,i are the true values and ni, ki, Z′ and Tj,i′ the estimated values.

Conclusion

Here, for the first time, a multi-item two-storage inventory model with AUD has been formulated with a resource constraint and successfully solved by real-coded GA with crossover and mutation for integer and non-integer variables, developed for this purpose. Due to the complexities, till now, none has attempted this type of multi-item discounted problem with space constraint by conventional price-break methodology. Though the model has been illustrated with only four price breaks, the GA

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