Evolutionary algorithms for optimal operating parameters of vendor managed inventory systems in a two-echelon supply chain

Advances in Engineering Software 52 (2012) 47–54

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Evolutionary algorithms for optimal operating parameters of vendor managed inventory systems in a two-echelon supply chain Goh Sue-Ann a, S.G. Ponnambalam a,⇑, N. Jawahar b a b

School of Engineering, Monash University Sunway Campus, 46150 Bandar Sunway, Malaysia Department of Mechanical Engineering, Thiagarajar College of Engineering, Madurai, India

a r t i c l e

i n f o

Article history: Received 29 November 2011 Received in revised form 2 March 2012 Accepted 10 June 2012 Available online 17 July 2012 Keywords: Two-echelon Single-Vendor–MultipleBuyers Supply chain Vendor managed inventory Particle Swarm Optimization Genetic Algorithm Artificial Immune System

a b s t r a c t This paper focuses on the operational issues of a Two-echelon Single-Vendor–Multiple-Buyers Supply chain (TSVMBSC) under vendor managed inventory (VMI) mode of operation. The operational parameters for TSVMBSC model are: sales quantity and sales price that determine the channel profit of the supply chain, and contract price between the vendor and the buyer, which depends upon the understanding between the partners on their revenue sharing. The optimal sales quantity for each buyer in TSVMBC is determined using a mathematical model available in the literature. The optimal sales price, the optimal channel profit and contract price between the vendor and buyer are determined based on the optimal sales quantity determined. Particle Swarm Optimization (PSO) and a hybrid of Genetic Algorithm and Artificial Immune System (GA–AIS) are proposed to solve this TSVMBSC problem. These two algorithms are evaluated for their solution quality. The robustness of the algorithms with their parameters are also analyzed and presented. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Supply chain is defined as a network of organizations, people, activities information and resources that are involved in the movement of products from suppliers to buyers. Therefore, supply chain management is then defined as the process of integrating and utilizing suppliers, manufacturers, warehouses, and retailers to properly ensure that the products produced and delivered are right in quantities and on time, at the same time reducing costs and meeting customer requirements [1]. Supply chain modelling is normally classified as normative modelling and descriptive modelling. Normative modelling, in other words optimization models, assists the managers in making better decisions. Researchers pointed out that the thorough integration of optimization models and methods with the classical inventory theory opens up opportunities for future research. Descriptive modelling on the other hand outlines the functional relationships of a company’s supply chain. The practitioners widely address this model as it is well developed [2,3]. Economic order quantity (EOQ) formula will give an optimal solution when the vendor and buyer inventory problems are treated in isolation under the deterministic conditions. However, in certain cases, optimal EOQ for buyer is unacceptable to the vendor ⇑ Corresponding author. Tel.: +60 355146203. E-mail addresses: [email protected] (G. Sue-Ann), sgponnambalam@ monash.edu (S.G. Ponnambalam), [email protected] (N. Jawahar). 0965-9978/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.advengsoft.2012.06.003

and vice versa. Therefore, collaborative arrangement between vendor and buyer is important and could be through some contractual agreement [4]. Two-echelon Single-Vendor–Multiple-Buyers Supply chain (TSVMBSC) is an integrated inventory model of vendor managed inventory (VMI) system in a two-echelon supply chain that comprises of a single vendor and multiple buyers [5]. VMI, also known as the consignment inventory and has been widely used in various industries [6]. Stocks are kept in the buyer’s location and payments are not made to vendor until it has been sold. The typical VMI system involves the supplier monitoring the stocks in the inventory located at the buyer’s location and assumes full responsibility in replenishing it. It has been proved through a survey that VMI achieved higher penetration than just-in-time and stockless methods [6]. Evidently, in the long run, VMI benefits both side of the party, the buying company and the supplier. This is because, with this merging, there is a clear decrease in inventory related cost, and final sales volume. With such decrease, the purchase price will increase, and finally, most importantly, the channel profit, which is the measure of the supply chain success, increases. One important aspect of VMI is information sharing. It is believed that since the parties involved share sales information under VMI, less information distortion should be expected [7,8]. With this, the inventory and other production cost will likely be reduced while capacity utilization will be increased [22,23]. On the other hand, some parties involved in a VMI relationship, supplier in particular, are less certain about these potential

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W1

W2

Buyer 1 P(y1)

Buyer 2 P(y2)

W3

Buyer 3 P(y3)

Vendor

.........

Wn

Buyer n P(yn) Fig. 1. TSVMBSC model.

benefits and tend to accept VMI as a necessity due to intense global competition. There are some companies faced with the dilemma in using the VMI system. This could happen if the company intention is to have no inventory, but at the same time, constantly being replenished by the vendor [6]. Therefore, in dealing with this dilemma, companies are not convinced that VMI system could improve business process, but in turn, it would incur an increase in total administration and processing costs [9]. Fig. 1 shows a schematic representation of the Two-echelon Single-Vendor–Multiple-Buyers Supply chain (TSVMBSC) model. The TSVMBSC model belongs to the nonlinear integer programming (NIP) problem. Evolutionary algorithm like Genetic Algorithm (GA), simulated annealing algorithm (SAA), Particle Swarm Optimization (PSO) and Artificial Immune System (AIS) are evolving as promising solution methods for NIP problem [10–12]. The objective of this paper is to propose an efficient PSO algorithm and a hybrid GA–AIS algorithm for TSVMCSC model. In attempt to improve the company’s operational performance and remain competitive under the growing competition, companies takes approaches like optimization models and algorithms, decision support systems and computerized analysis tools. The scope of this work is to find the optimal sales quantity ‘yjopt ’ that maximizes channel profit. Based on the optimal sales quantity, the sales price, ‘P yjopt ’ and contract price ‘W jopt ’ are derived. 2. Literature review Many researches have been done on two-echelon supply chain integrated inventory models. Lu [4] formulated a model with the objective of minimization of the vendor’s total annual costs without considering the cost of the buyers in order to analyse its importance in the following cases such as: products critical to buyers, products with high switching costs and high supplier concentration. This model is based on the assumption that the buyers are ready to pay what ever the price fixed by the vendor. Lu [4] proposed to find the optimal solution for the one-vendor–one-buyer case before presenting a heuristic approach for the one-vendor–multi-buyer case. The relative performances of ‘Identical Delivery Quantity (IDQ)’ and ‘Deliver what is produced (DWP)’ strategies for delivering quantities using various parameters with the objective of minimizing joint average annual cost, in an integrated vendor–buyer inventory model was analyzed by Viswanathan [13]. Goyal [14] proposed a joint economic lot size (JELS) model with the objective of minimizing the total relevant cost for both vendor and buyer. An essential assumption must be made before

implementing these models, the vendor must know the buyer’s annual demand, holding cost, and ordering cost that governs the buyer’s inventory policy. However, this is often not the case, because buyers are usually unwilling to reveal the true values of their cost parameters. Goyal and Nebebe [3] proposed a methodology to determine economic production and shipment policy of a product supplied by a vendor to a single buyer with the objective of minimizing the total joint annual costs. Hoque and Goyal [15] also developed an optimal solution procedure, to minimize the total joint annual costs, for the single-vendor single-buyer production inventory system with unequal and equal sized shipments from the vendor to the buyer and under capacity constraint of the transport equipment. Cachon [16] analyzed the competitive and cooperative selection of inventory policies in a two-echelon supply chains with one supplier and N retailers facing stochastic demands. In his studies, he concluded that cooperation strategies can assist companies in improving the performance of the supply chain. Woo et al. [17] discussed and develop an analytical model to derive the optimal investment amount and replenishment decisions for both vendor and buyers. They investigated an integrated inventory system where a single vendor purchases and processes raw materials in order to deliver finished items to multiple buyers. They claimed that their model serves as a pioneering work on investigating the effects of ordering cost reduction on the integrated inventory system. Yao and Chiou [18] considered an integrated supply chain model in which one vendor supplies items for the demand of multiple buyers. The objective of this model is to minimize the vendor’s total annual cost subject to the maximum cost that the buyer may be prepared to incur. They explored the optimality structure of this integrated model and assert that the optimal cost curve is piecewise convex. Fahimnia et al. [1] presented a research on the comprehensive review and analyzed the characteristics of selected models. The complexity level of the models are outlined and considered. They also developed a mixed integer formulation for two-echelon supply network. LINGO is a NIP solution provider for smaller scale problems. Dong and Xu [6] proposed an alternative model which, can be applied when the number of buyers reduced to a single buyer, thus reducing the TSVMBSC problem to a single-vendor–single-buyer model. Nachiappan and Jawahar [5] proposed a Genetic Algorithm approach for the TSVMBSC problem. Nachiappan et al. [19] proposed a knowledge management system using GA for the TSVMBSC model. The mathematical model presented in Nachiappan and Jawahar [5] is adopted in this research. Channel profit is maximized and profit is shared among the members involved in the supply chain due to ever increasing competition. Therefore this paper deals with the price and quantity of product to increase the channel profit with relation to buyers. The objective of this paper is to propose an efficient PSO algorithm and a hybrid GA–AIS algorithm to optimize channel profit to TSVMCSC model and to evaluate these algorithms performance.

3. Assumptions and nomenclature There are many variables in the real world VMI system. The assumptions made in this research are:  There is a linear relationship between the price and the sales quantity.  Zero lead time.  No backlog and stockout allowed.

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the sales price ‘P(yj)’ corresponding to ‘Y jopt ’ and acceptable contract price ‘W jopt ’ are also found. The mathematical expressions for these parameters are discussed in this section.

Table 1 Notations used. aj bj C H bj

Intercept value for the demand pattern of the jth buyer Cost slope of the demand pattern of the jth buyer Capacity of the vendor Holding cost of the jth buyer in independent mode

Hs HjVMI j n P bj

Holding cost of the vendor in independent mode Holding cost of the vendor in VMI mode Buyer identifier (j = 1 to n) Number of buyers Profit of the jth buyer

Pc P copt Ps P sj PRj P(y) P yjopt

Channel profit Optimal channel profit Vendor profit Profit obtained by vendor when supplying products to the buyer ‘j’ Revenue share ratio between vendor and the jth buyer Sales price Optimal sales price of the jth buyer

Sbj

Setup cost of the jth buyer per order in independent mode

Ss SjVMI W Wj W jopt

Setup cost of the vendor per order in independent mode Setup cost of the vendor buyer per order in VMI mode Contract price Contract price between vendor and buyer ‘j’ Optimal contract price between vendor and buyer ‘j’

Yj Y jmin Y jmax Y jopt

Sales quantity of the jth buyer Minimum expected sales quantity of the jth buyer Maximum expected sales quantity of the jth buyer Optimal sales quantity of the jth buyer

hj Vj d

Flow cost per unit from vendor to buyer ‘j’ Transportation resource cost per unit from vendor to buyer ‘j’ Production cost per unit

4.1. Optimal sales price By substituting the optimal sales quantity ‘Y jopt y’, the optimal sales price ‘PðY jopt yÞ’, is furnished as below:

PðY jopt Þ ¼ aj  bj Y jopt

ð2Þ

4.2. Acceptable contract price By substituting the optimal sales quantity ‘yjopt ’, the acceptable contract price ‘W jopt ’, is as below: W jopt ¼

aj yjopt PRj  bj y2jopt PRj þ dyjopt þ 0:5hj y2jopt þ ½2ðHs þ Hbj ÞðSs þ Sbj Þyjopt 1=2 ð1 þ PRj Þyjopt

ð3Þ

5. Proposed heuristics The details of the PSO and GA–AIS proposed algorithms are discussed in this section. 5.1. Particle Swarm Optimization (PSO)

 Holding cost and setup cost are assumed to be the summation of both of the buyer and vendor. The development of the algorithms and the experiments are conducted using a computer with Intel Core 2, 1.83 GHz, 2 Gb RAM and Matlab 7.7.0 (R2008b). Table 1 presents the notations used in this paper. 4. The mathematical model The objective function used in this research is maximizing the channel profit of the supply chain. The mathematical expression for channel profit ‘Pc’ is as below:

Pc ¼

n n o X ; aj yj  bj y2j  dyj  0:5hj y2j  ½2ðHs þ Hbj ÞðSs þ Sbj Þyj 1=2 j¼1

ð1Þ Subjecting to: Buyer sales quantity constraints:

Y jmin 6 yj 6 Y jmax Vendor capacity constraints: n n X X Y jmin 6 C 6 Y jmax j¼1

j¼1

Nonnegative constraints and integer:

yj P 0 In this research, the aim is to determine an optimal sales quantity ‘Y jopt ’ for the maximum channel profit ‘Pc’ of the Two-echelon Single-Vendor–Multiple-Buyers Supply chain under vendor managed inventory mode of operation. After finding the channel profit, the optimal sales quantity ‘Y jopt ’ can be found. Subsequently,

Particle Swarm Optimization (PSO) is a population based algorithm where each particle represents a potential solution. This algorithm is first invented by Kennedy and Eberhart [24]. According to Haupt and Haupt [20], the thought process behind the algorithm was inspired by the social behavior of animals such as bird flocking or fish schooling. PSO is very similar to Genetic Algorithm (GA), this is due to the fact they both have fitness values to evaluate the population, searches the solution space for feasible and optimal solutions and updates the populations. A swarm in PSO is very much similar to the population in the analogy of evolutionary computation systems such as the GA. The particle on the other hand is the individual. They both do not guarantee success. However, PSO has no evolution operations such as crossover and mutation. In each of the PSO particles, it contains important features in obtaining the optimal results. They are:  Velocity (flying direction).  Position (sequence and fitness value).  Knows it’s current position and objective function value for this position.  Remembers previous best position.  Know its neighbors, previous best position and objective function value. Particles are fired into a multidimensional solution search space. This is where the position of each particle is adjusted based on their own experiences and the neighbors. Each of these particles represents a possible solution in the search space. These particles move around to search for the optimum solution. They cooperate and exchange information about positions they have visited. The particles move to a new position by adjusting the velocity. Then, they update their respective velocities and positions based on the local and global best solution. The two main components of PSO algorithm are the position update and velocity update.

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Fig. 2. PSO particle representation.

5.1.1. Velocity update

Start

Vðt þ 1Þ ¼ VðtÞ þ C 1 ðParticle best  PðtÞÞ þ C 2 ðGlobal best  PðtÞÞ:

ð4Þ

where C1 and C2 are the local and global fitness (C1 + C2 = 1).

Initialize PSO parameters. (swarm size, max iterations, c1, c2)

5.1.2. Position update

Sðt þ 1Þ ¼

1 þ1 ev ðtÞ

ð5Þ

where

Generate swarm of binary particles randomly. Initialize velocity vectors for each particle

if S more than 0, set p(t) = 1. if S less than 0, set p(t) = 0.

For each Particle: Evaluate particle’s fitness. Update Particle best.

5.2. PSO implementation Fig. 3 shows the program flowchart of the proposed PSO. The implementation of the model is explained in this section. Swarm size of the problem depends on the feasible solution space of the problem. It is normally based on the number of decision variables [5]. In this model, it is set to be double the number of buyers, that is

Popsize ¼ 2n

ð6Þ

The particles in the swarm population are randomly generated and these particles represents the sales quantity for all buyers. The jth particle in the notation ‘yj’ indicates the sales quantity for buyer ‘j’. Each particle is represented by a set of 10 bits binary number presented in Fig. 2. Each particles in the swarm are to be converted to provide a feasible solution for the system. These particles are interpolated to fit the range of ymin < yj < ymax using the following equation:

Set best of Particle best as global best

Update velocity and position for all partcles

Is maximum iteration reached?

yj ¼ yjmin

! 169ðcorresponding decimal value of binary stringÞ ðyjmax þ yjmax

Return the Global Best Solution

 yjmin Þ ð7Þ The objective function considered in this research is maximizing the channel profit, ‘Pc’. The particles are evaluated with this objective function. The PSO and GA–AIS proposed evolve an optimal or nearer-to-optimal sales quantity ‘Y jopt ’ to maximize the channel profit. Eq. (1) is used to calculate the channel profit. The particle best and global best values are updated in each generation. After particle best and global best solutions are updated in every generation the velocity and position updates of each particles are to be done using Eqs. (4) and (5). The global best solution is obtained when the algorithm is terminated. The global best solution obtained provided the best generated channel profit. The corresponding optimal sales quantity, ‘Y jopt ’ is noted down. Finally, using the optimal sales quantity, ‘Y jopt ’, the optimal sales price, ’Pyjopt ’, and acceptable contract price,’W jopt ’ are calculated for each buyer using Eqs. (2) and (3) respectively. The PSO parameters used in this paper are given in Table 2.

End Fig. 3. Flowchart of PSO.

Table 2 Operating parameters for PSO. Popsize Bits C1 C2

10 30 0.09 1  C1

5.3. Hybrid Genetic Algorithm and Artificial Immune System (GA–AIS) Many constraints handling techniques has been introduced throughout the years. Genetic Algorithm is an optimization technique itself; however, it is an unconstrained search technique. In

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G. Sue-Ann et al. / Advances in Engineering Software 52 (2012) 47–54

feasible solutions and ‘antibody’ for infeasible solutions. The matching magnitude is calculated, representing the fitness of the antibodies pool. This fitness is the similarity of antibodies with a set of antigens. A sample of antibodies are selected at random. Next, an antigen is selected randomly from the pool of antigens. Each antibody in the selected sample is compared with the antigen. The fitness is calculated by the comparison of the antibody and antigen where each matching ‘1’s or ‘0’s are taken as a single match. The fitness value, Z is calculated by using Hamming distance, where it is the sum of total matches of the ‘1’s and ‘0’s. A higher Z value denotes that it is a high fitness value. Based on the calculated fitness, the antibodies population is reproduced using the traditional GA evolution method of crossover and mutation. The process is repeated until convergence, or until the maximum number of iteration has been reached. Convergence is when the mean and the maximum fitness in the population are practically the same. When convergence or maximum iteration is reached, the process returned to outer GA loop. Binary tournament selection is used in the algorithm. The objective function for this problem is maximizing channel profit, ‘Pc’. The pool of solution returned from the inner loop still contains feasible and infeasible solution. Therefore, in binary tournament selection, the special rules as proposed by Coello and Cortés [21] are used.

this section, a hybrid Genetic Algorithm and Artificial Immune System (GA–AIS) is presented. This approach incorporates an emulation of the immune system. The proposed GA–AIS has two GA loops. The inner loop is inspired by the nature of the Artificial Immune System. The internal GA loop acts as a local search mechanism where it guides the GA algorithm to reach the feasible solution space in a more efficient manner. Antibodies are introduced to attain the antigen specificity in the recognition process. The GA–AIS structure proposed by Coello and Cortés [21] is adopted in this paper. Fig. 4 shows the program flowchart of the GA–AIS algorithm. The implementation of the model is explained in this section. Both the vendor’s and buyers’ data are fed as input into the system. All variables are declared globally. Population size of this problem in GA–AIS is set to be 12. The population size of 12 is chosen to ensure that the range of the buyers’ quantity are covered. This is done by checking that 212 = 4096, which in this context is the maximum number of buyers for 3 buyers case. The initial population is randomly generated. Similar to in PSO, each buyer is also represented with a string of 10 binary bits called the chromosome. After the generation of the initial population, the algorithm enters the inner GA loop. The initial population will contain both feasible and infeasible solutions. The population is checked for its solution’s feasibility, and segregation of them into two groups called ‘antigen’ for

Outer GA

Generate random initial population

Check for feasibility

Inner GA No

Yes is string feasible?

Antibody (infeasible)

Calculate matching fitness (Z)

No

Antigen (feasible)

Convergence ?

Yes Binary tournament selection

Evolution operations (crossover & mutation)

Yes

No Maximum generations?

Fig. 4. GA–AIS flowchart.

end

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G. Sue-Ann et al. / Advances in Engineering Software 52 (2012) 47–54 Table 3 Operating parameters for GA–AIS. Popsize Bits Probability crossover Probability mutation Crossover operation Mutation operation

Table 5 Vendor related data. 12 30 0.6 0.03 Single point crossover Uniform mutation

Hs

Ss

C

D

9

150

6150

40

Table 6 Performance of PSO and GA–AIS compared with LINGO, DX, GA and KMS.

Each and every solution are compared with one another using special rules as follows:  One feasible and one infeasible chosen, feasible wins.  Both feasible chosen, highest fitness value between the both wins.  Both infeasible chosen, the one with the least constraint violation wins. After the selection, the individuals are evaluated. Then the top 3% of the chromosomes in the population are identified as antigens, and the remaining chromosomes are identified as antibodies. The matching process is repeated again and the newly calculated matching score is used as the fitness in the traditional GA evolution process. Crossover and mutation are applied to the antibodies in a conventional way. The chromosome are selected based on the fitness calculated, where higher fitness chromosomes are more likely to be selected. The probability of cross over used in this paper is 0.6 Single point crossover is used in this paper. The probability of mutation used is 0.03. A uniform mutation operation is used in this paper. The evolved offsprings are then placed into a new population of antibodies. The entire process is repeated until the maximum number of iterations is reached. Upon reaching the termination criterion the optimal or nearer-to-optimal sales quantity, ‘Y jopt yjopt’ and the corresponding channel profit  value  are determined. Finally,  using  the optimal sales quantity Y jopt , the  optimal sales price Pjopt , and acceptable contract price W jopt are calculated for each buyer using Eqs. (2) and (3) respectively. The parameters used in GA– AIS are given in Table 3. 6. Performance evaluation of the heuristics The proposed algorithms are analyzed based on the case study presented by Nachiappan and Jawahar [5]. LINGO Optimization software can be used to obtain optimal parameters for a small sized TSVMBSC problem. The performance of PSO and GA–AIS are compared with existing algorithms reported in the literature. 6.1. Performance on 1-vendor 1-buyer case When the model is reduced to a single-vendor single-buyer problem, the iterative heuristic DX generates the best solution [6]. The data for 1-vendor 1-buyer is presented in Tables 4 and 5. The performance of PSO and GA–AIS with the results reported in the literature on 1-vendor 1-buyer case model is presented in Table 6. It is found that the proposed PSO generates a solution comparable to the solutions generated by DX, GA and optimal solution of

Table 4 Buyer related data. Hb1

Sb1

A1

B1

Y1min

Y1max

H1

PR1

9

300

80

0.01

1000

2000

0.005

1

Heuristics

Y jopt

P yopt

P copt

LINGO Dong and Xu (DX) GA KMS PSO GA–AIS

1535 1535 1532 1537 1536 1029

64.65 64.65 64.68 64.62 64.64 69.71

26960.49 26960.49 26960.42 26960.42 26960.5 23841.62

Table 7 Buyer related data. Yj

Hbj

Sbj

aj

bj

yjmin

yjmax

hj

PRj

1 2 3

7 8 9

10 20 30

20 19 18

0.003 0.005 0.008

2000 500 500

4000 3000 1500

0.004 0.006 0.008

1 1.2 1.2

Table 8 Vendor related data. Hs

Ss

C

d

9

15

5750

7

LINGO. GA–AIS on the other hand is not generating a comparable solution. 6.2. Performance on 1-vendor 3-buyers case The case study done by Nachiappan and Jawahar [5] are used to evaluate the performance of PSO and GA–AIS. The case study data (1-vendor 3-buyer case) are presented in Tables 7 and 8. The performance of PSO and GA–AIS is presented in Table 9, it is observed that the proposed PSO generates solution closer to the optimal solution of LINGO. In the case of GA–AIS, even with minor variation on the sales quantity obtained for the 3-buyers, a significant deviation of channel profit is observed. 7. 1-Vendor n-buyers model sensitivity with PSO This section discusses the sensitiveness of various model parameters such as sales quantity, demand slope, holding cost and set up cost. This will help the decision makers to understand the implications of them on channel profit. 7.1. Effect of change in limits of sales quantity The limits of sales of commodities are subjected to governmental regulations, seasonal effects, competition and price war between competitors are liable for revision. The effects of the sales limit play a vital role in testing the sensitivity of the model. Slight changes this sales limit will reflect greatly on sales profit. In this section, we analyze the effect of the changes in the sales limits to better understand the model. Tables 10 and 11 show the effects of the limits on the sales quantity on channel profit. Based on the information in these tables, we can deduce that by decreasing

53

G. Sue-Ann et al. / Advances in Engineering Software 52 (2012) 47–54 Table 9 Performance of PSO and GA–AIS compared with LINGO and GA. Heuristics

LINGO

GA

PSO

J

1

2

3

1

2

3

1

2

3

1

2

3

Y jopt

2000

657

500

2002

673

500

2000

729

500

2072

736

513

W jopt

14.28

14.98

15.59

13.3

13.96

13.36

13.82

13.08

5

12.739

13.07

12.85

P yopt

14

15.7

14

14

15.6

14

13

15.3514

14

11.71

15.32

12.232

P copt

5982.1

5978.2

5979.8

Table 10 Effect of change in lower limit (ymin). J

yjopt Level 1

Level 2

Level 3

3

1 2 3

1755 727 400

2000 724 502

2105 685 942

1 2 3 4

1900 730 537 1364

2000 520 500 1952

2425 628 602 2041

1 2 3 4 5

1943 401 422 1360 539

2000 656 1004 1700 512

2450 609 628 2086 601

5

P copt Level 1 8471.2

Level 2 5980.1

N

J

Level 1

Level 2

Level 3

3

1 2 3

2000 737 500

2000 724 502

2000 724 501

1 2 3 4

2112 501 501 1700

2000 520 500 1952

2175 501 675 1705

1 2 3 4 5

2183 529 505 1722 535

2000 656 1004 1700 512

2350 597 825 1700 813

5

yjopt

N

J

Level 1

Level 2

Level 3

3

1 2 3

2000 627 750

2000 724 502

2016 813 501

1 2 3 4

2250 998 501 1782

2000 520 500 1952

2000 524 633 1704

1 2 3 4 5

2363 812 502 1700 567

2000 656 1004 1700 512

2000 501 578 1847 563

Level 3 1698.9 4

14680

10872

3982.2 5

15911.2

10923.1

5255.9

Table 11 Effect of change in upper limit (ymax).

4

5015.42

Table 12 Effect of change in demand slope.

N

4

GA–AIS

yjopt

P copt Level 1 9199.3

Level 2 5980.1

Level 3 1429.83

15915.02

10872

5974.1

17572.02

10923.1

7008.4

Table 13 Effect of change in holding cost. P copt Level 1 5979.6

Level 2 5980.1

N

J

Level 1

Level 2

Level 3

3

1 2 3

2014 615 522

2000 724 502

2063 812 500

1 2 3 4

2016 501 751 1700

2000 520 500 1952

2158 578 538 1712

1 2 3 4 5

2016 850 512 1925 703

2000 656 1004 1700 512

2250 812 502 1702 501

Level 3 5978.2 4

10876.5

10872

9348.2 5

12219.2

10923.1

7520.52

the lower limit of sales quantity (level 1), the sales optimal quantity decreases. Channel profit increases with the decrease, because of the increase in prices in all location. Price is quantity dependent. Therefore, in any cases where there is more demand on a particular product and quantity is low, prices will increase. When the lower limit is increase (level 3), there is a significant increase in sales quantity, therefore causing the product price to decrease and subsequently decreasing the channel profit. This happens when vendor tries to distribute the stocks to location with minimum requirement of the product. 7.2. Effect of change in demand slope Table 12 shows the effect of change in demand slope on channel profit. Increasing the demand slope (level 3) will cause the cost slope to be inelastic. However, due to the random nature of PSO, the decrease in sales quantity decreases the channel profit. When the demand slope decreases, there is less variation. The range would be smaller, thus leading to a leaner demand. In any case, managers of supply chain could use this analysis to determine

yjopt

P copt Level 1 6714.45

Level 2 5980.1

11312.12

10872

13478.34

10923.1

Level 3 4885.49

9936.94

11094.21

the price changes and decisions can be made based on this analysis. 7.3. Effect of change in holding cost One of the major costs incurred on companies at any time is the holding cost. Table 13 shows the effect of change in holding cost on channel profit. Based on the study of Table 13, it is clear that by reducing the holding cost (level 1), it will increase channel profit. This is due to the fact that holding cost is inversely proportional to channel profit. In other words, the higher the holding cost, the lower the profit. This occurs because due to the increase in holding cost, the sales profit margin becomes relatively low and the turnover is slower. Therefore, any changes in holding cost will reflect clearly on the channel profit. 7.4. Effect of change in setup cost Setup costs include the administrative cost done by the company, purchase order, invoice and receipt processing cost, any

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Table 14 Effect of change in setup cost. N

J

Level 1

Level 2

Level 3

3

1 2 3

2000 519 500

2000 724 502

2027 511 505

1 2 3 4

2125 656 500 1703

2000 520 500 1952

2063 656 500 1981

1 2 3 4 5

2156 656 500 1700 500

2000 656 1004 1700 512

2252 617 626 1700 563

4

5

yjopt

P copt Level 1 5975.99

Level 2 5980.1

11507.13

10872

13485.5

10923.1

Level 3 5336.03

9556.73

10632.29

Fig. 5. Effect of% change in parameter on P copt .

inspection costs, and in some cases, setup cost includes inbound freights. These costs are calculated based on frequency, not quantities of the order. Similar to holding cost, any variations in setup cost will show in the channel profit. The more setup costs are incurred on the company; the profit decreases. This is solely based on the understanding of break-even point in any business, where if there is more initial costs incurred, the longer it takes for the business to break even. Table 14 provides the effect of set up cost on channel profit. Based on 14, the decrease in setup cost increases the channel profit significantly. Fig. 5 consolidated report of the effects of percentage change in parameters on channel profit. 8. Conclusions Particle Swarm Optimization (PSO) and hybrid Genetic Algorithm and Artificial Immune System (GA–AIS) are proposed to obtain optimal parameters for VMI system in a two echelon supply chain. The LINGO, DX and GA models reported in the literature are used as comparison. It is found that the proposed PSO algorithm performs better than GA and DX, providing values closer to the optimal solution provider, LINGO. GA–AIS on the other hand, is not performing better.

In this paper, the sensitiveness of the model under various parameters is studied using the PSO algorithm. Based on the studies, it is found that the VMI operation mode increases the channel profit as compared to the current independent operation mode. Through the sensitivity test, it is clear that the lower limit on sales quantity should be decreased to obtain maximum channel profit. Due to the randomness of the PSO algorithm, small changes in the demand slope affect the channel profit. In further studies, it is found that the holding and setup costs are inversely proportional to channel profit. Any changes in these costs will affect the channel profit significantly. It is advisable for supply chain managers to lower these costs to be able to maximized profit. References [1] Fahimnia B, Luong L, Marian R. An integrated model for the optimisation of a two-echelon supply network. J Achievements Mater Manuf Eng 2008;31(2):477–84. [2] Sharprio JF. Modelling the supply chain. Thompson Duxbury Publishers. [3] Goyal SK, Nebebe F. Determination of economic production shipment policy for a single vendor buyer system. Eur J Oper Res 2000;121:38–42. [4] Lu L. A one-vendor multi-buyer integrated inventory model. Eur J Oper Res 1995;81:312–23. [5] Nachiappan SP, Jawahar N. A genetic algorithm for optimal operating parameters of VMI system in a two-echelon supply chain. Eur J Oper Res 2007;182:1433–52. [6] Dong Y, Xu K. A supply chain model of vendor managed inventory. Transp Res Part E 2002;38:75–92. [7] Lee H, Padmanabhan P, Whang S. Information distortion in a supply chain: the bullwhip effect. Manage Sci 1997;43(4):546–58. [8] Chen F, Drezner Z, Ryan JK, Simchi-Levi D. Quantifying the bullwhip effect in a simple supply chain: the impact of forecasting, lead times and information. Manage Sci 2000;46(3):436–43. [9] Gamble RH. Curse of the consignment sales: carrying customers’ inventory weighs down cash flow. Corporate Cash Flow 1996;15(10):30–2. [10] Yokota T, Gen M, Li YX. Genetic algorithm for nonlinear mixed integer programming problems and its applications. Comput Ind Eng 1996;30(4):905–17. [11] Costa L, Oliveira P. Evolutionary algorithms approach to the solution of mixed integer nonlinear programming problems. Comput Chem Eng 2001;25:257–66. [12] Wu DJ. Software agents for knowledge management: coordination in multiagent supply chains and auctions. Expert Syst Appl 2001;20:51–64. [13] Viswanathan S. Optimal strategy for integrated vendor–buyer inventory model. Eur J Oper Res 1998;105(1):38–42. [14] Goyal SK. An integrate inventory model for a single supplier–single customer problem. Int J Prod Res 1977;15:107–11. [15] Hoque MA, Goyal SK. An optimal policy for single vendor single buyer integrated production inventory system with capacity constraint of transport equipment. Int J Prod Econ 2000;65(3):305–15. [16] Cachon GP. Stock wars: inventory competition in a two-echelon supply chain with multiple retailers. Oper Res 2001;49(5):658–74. [17] Woo YY, Hsu SL, Wu S. An integrated inventory model for a single vendor and multiple buyers with ordering cost reduction. Int J Prod Econ 2001;73:203–15. [18] Yao MJ, Chiou CC. On a replenishment coordination model in an integrated supply chain with one vendor and multiple buyers. Eur J Oper Res 2004;159:406–19. [19] Nachiappan SP, Gunasekaran A, Jawahar N. Knowledge management system for operating parameters in two-echelon VMI supply chain. Int J Prod Res 2007;45(11):2479–505. [20] Haupt RL, Haupt SE. Practical genetic algorithm. Hoboken, New Jersey: John Wiley & Sons Inc.; 2004. [21] Coello CA, Cortés NC. Hybridizing a genetic algorithm with an artificial immune system for global optimization. Taylor Francis Group 2004;36(5):607–34. [22] Xu K, Dong Y, Evers PT. Towards better coordination of the supply chain. Transport Res Part E Logist Transport Rev 1995;37(1):35–54. [23] Waller M, Johnson ME, Davis T. Vendor-managed inventory in the retail supply chain. J Business Logist 1999;20(1):183–203. [24] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of IEEE international conference on neural networks, vol. IV; 1995. p. 1942–48.