Modelling and solving of a lot sizing problem for ...

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Int. J. Services and Operations Management, Vol. 20, No. 1, 2015

Modelling and solving of a lot sizing problem for multiple items and multiple periods with shortage Nasim Ganjavi* Department of Industrial Management, Najafabad Branch, Islamic Azad University, Isfahan, Iran Fax: +989112168551 Email: [email protected] *Corresponding author

Reza Tavakkoli-Moghaddam Department of Industrial and Systems Engineering, College of Engineering, University of Tehran, Iran Fax: +989121580480 Email: [email protected]

Seyyed Mojtaba Sajadi Faculty of Entrepreneurship, University of Tehran, Tehran, Iran Email: [email protected] Abstract: This paper presents a multi-period inventory lot sizing scenario, where there is multiple product and single suppliers. It constructs a goal programming model to determine the order quantities of each product in each period. In this model, three goals are considered: 1) meeting the total available periodic budget, 2) meeting the buyer’s maximum acceptable quality, 3) minimise the shortage. Constructing the approach in this way effectively reduces the risk of purchasing .The proposed approach provides flexibility to decision maker in multi-period procurement lot-sizing decisions. In other words, the purchase of any product in any period is determined as by the possibility to get closer to the goals defined in the model. It solves utilising GAMS and meta-heuristic algorithms, and analyses the efficiency of the heuristics. Efficiency of the proposed algorithm by comparing genetic algorithm in large size problems show the proposed algorithm has also better efficiency than genetic algorithm. Keywords: goal programming; lot sizing; differential evolution algorithm; shortage.

Copyright © 2015 Inderscience Enterprises Ltd.

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Reference to this paper should be made as follows: Ganjavi, N., Tavakkoli-Moghaddam, R. and Sajadi, S.M. (2015) ‘Modelling and solving of a lot sizing problem for multiple items and multiple periods with shortage’, Int. J. Services and Operations Management, Vol. 20, No. 1, pp.102–117. Biographical notes: Nasim Ganjavi graduated in Iran in 2009 with BS in Industrial Engineering. She obtained her MS in Industrial Management in 2013 from the Department of Industrial Management at Najafabad Branch, Islamic Azad University, Isfahan, Iran. She is the author of one published paper at national level in a conference. Reza Tavakkoli-Moghaddam is a Professor of Industrial Engineering at College of Engineering, University of Tehran in Iran. He obtained his PhD in Industrial Engineering from the Swinburne University of Technology in Melbourne (1998), his MSc in Industrial Engineering from the University of Melbourne in Melbourne (1994) and his BSc in Industrial Engineering from the Iran University of Science & Technology in Tehran (1989). He serves in the editorial boards of the International Journal of Engineering and Iranian Journal of Operations Research. Also, he is an executive member of Board of Iranian Operations Research Society. He is the recipient of the 2009 and 2011 Distinguished Researcher Award and the 2010 Distinguished Applied Research Award at the University of Tehran, Iran. He has been selected as National Iranian Distinguished Researcher for 2008 and 2010. He has published three books, 12 book chapters, more than 560 papers in reputable academic journals and conferences. Seyyed Mojtaba Sajadi graduated in Iran in 2003 and 2006 with BS and MS in Industrial Engineering. He obtained his PhD in Industrial Engineering in 2011 from the Amirkabir University of Technology. Tehran, Iran. Currently, he is a Faculty of the Department of Industrial Engineering at Najafabad Branch, Islamic Azad University, Isfahan, Iran. His published research articles appear in IJAMT, Energy Journal, Operation Production and others. He has presented several papers at the CIE40, ENBIS9, IJPR21 and several national conferences.

1

Introduction

Supply chain management (SCM) which is also referred to as the logistics network, consists of suppliers, manufacturing centres, warehouses, distribution centres, and retail outlets, as well as raw materials, work-in-process inventory, and finished products that flow between the facilities. SCM is turning into one of the major activities of the management. It is of great importance in competitive markets. One of the well-known subjects of SCM is supply management, which generally implies the activities regarding suppliers such as empowering, evaluation and selection, making partnerships and so on. One of the major objectives of suppliers’ evaluation and selection is to determine the optimal quota assigned to each supplier while needing to replenish an order. This problem can be modelled as a multi-objective decision making (MODM) problem (Seifbarghy and Esfandiari, 2011). Procurement of raw materials from supplier is one of the important parts of the supply chain between its activities. Purchasing processes are analysed in two stages: first stage is the selection of suppliers formally by filtering them through an evaluation process that includes both qualitative and quantitative measures.

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Second stage is the order allocation where the order amounts for each supplier are determined (Sencer Erdem and Gocen, 2012). Multi-period procurement lot-sizing decision seeks best trade-offs among multiple cost objectives to determine appropriate lot size and its timing to minimise total cost over the decision horizon (Choudhary and Shankar, 2011). Large companies that have successfully implemented SCM, like Dell, Cisco Systems and HP, are sometimes faced with inventory shortage (Walsh, 2010; Gollner, 2008). There are several causes for inventory shortage, including part variations, miss-operation, and inventory reduction (Jiang et al., 2010), small number of suppliers, conservative production plan of suppliers (Xu, 2010) and supplier’s service level. This paper applied an integer linear programming (LP) approach to solve multiperiod procurement lot-sizing problem for multi-product and single supplier considering shortage. The purpose of this paper is to: •

Develop a mathematical model to establish tradeoffs among cost objective, shortage, defective items and determine appropriate lot-size to procure and its period to minimise total cost over the decision horizon.

•

Present solution approaches and investigate efficiency of the algorithm by run in different size of problems.

The paper is further organised as follows. Section 2 presents a brief literature review of the existing quantitative approaches related to procurement lot-sizing problem. In Section 3, an integer LP formulation is developed for multi period procurement lot-sizing problem. Section 4 and 5 explain parameters setting, performance, strategies and quality of differential evolutionary algorithm. Section 6 presents numerical example and investigate the effectiveness of the proposed approach. Conclusions are provided in Section 7. Finally, Section8 presents recommendations.

2

Literature review

To procure an optimal lot-sizing, various approaches have been utilised, including goal programming (GP), LP, non-linear programming, mixed-integer programming and multiobjective optimisation models. By its nature, order allocation problem includes several targets to be reached and thus, GP is widely used in order allocation for the selected suppliers. Buffa and Jackson (1983) presented a schedule purchase for a single product over a defined planning horizon via a GP model considering price, quality and delivery criteria. Sharma et al. (1989) proposed a non-linear, mixed integer, GP model. They considered price, quality, delivery and service as goals with demand and budget constraints. Weber et al. (2000) implemented the data envelopment analysis (DEA) for selecting suppliers and determining their quota allocations. Two multi-objective mixed integer non-linear models are developed for multi-period lot- sizing problems involving multiple products and multiple suppliers. Karpak et al. (2001) used GP to identify the best suppliers and how to allocate orders among them, thereby analysing trade-offs among multiple goals such as cost, quality and delivery simultaneously. Gao and Tang (2003) proposed a multi-objective LP model for decisions related to purchasing of raw materials in a largescale steel plant in China. Kumar et al. (2006) presented a rational approach to decision-

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making process for vendor selection problem. They used the multi-objective model contained three fuzzy goals and some crisp constraints. They also applied the GP approach for solving the problem. Wang et al. (2004) considered a second goal that minimises the total cost of purchase. The resulted pre-emptive GP determines the optimal order quantity from the chosen suppliers considering as constraints vendor capacities and demand requirements. Narasimhan et al. (2006) proposed a mathematical programming model which contained five different goals and allocated the optimum order quantities of multi-products to selected suppliers. Bilsel and Ravindran (2011) presented studies a stochastic multi-objective sequential supplier allocation problem and presented several techniques like GP to model and solve the problem. The first objective is to minimise total cost, second objective is to maximise quality items and third objective is to minimise delivery times. Rezaei and Davoodi (2011) in their paper two multi-objective mixed integer non-linear models developed for multi-period lot-sizing problems involving multiple products and multiple suppliers model is represented. This problem in situations where shortage is not allowed while in the second model, all the demand during the stock-out period is backordered. Considering the complexity of these models on the one hand, and the ability of genetic algorithms (GAs) to obtain a set of Pareto-optimal solutions. Raut et al. (2011) proposed a combined MCDM methodology. Fuzzy analytic hierarchy process (AHP) and LP. Wu and Hwang (2011) investigated the supplier inventory policies for a supplier facing multiple retailers in a two-echelon supply chain given the retailers implement EOQ policy. Choudhary and Shankar (2011) used integer LP approach to solve a multi-period procurement lot-sizing problem for a single product that is procured from a single supplier considering rejections and late deliveries under all-unit quantity discount environment. Sencer Erdem and Göçen (2012) developed an AHP and a GP model for order allocation among suppliers. Kumar and Babu (2012) used a DEA-based procedure and AHP-PGP formulation to identify the best source for the given targets of the outsourcer. Senyigit and Soylemez (2012) formed The MMDSCN system with the mixed integer LP by LINDO. The heuristics and MMDSCN system were modelled by ARENA 4.0. Choudhary and Shankar (2012) proposed an integer LP model to simultaneously determine the timings of procurement, lot-sizes, suppliers and carriers to be chosen so as to incur the least total cost over the planning horizon. Khalili-Damghani et al. (2013) proposed a GP approach for project portfolio selection that embraces conflicting fuzzy goals with imprecise priorities. TOPSIS is used to reduce the multi-objective problem into a bi-objective problem. The resulting bi-objective problem is solved with fuzzy GP (FGP). Nazari-Shirkouhi et al. (2013) proposed model to solve a supplier selection problem under multi-price level and multi-product using interactive two-phase fuzzy multi-objective LP to order allocation. Kannan et al. (2013) in their paper presented an integrated approach, of fuzzy multi attribute utility theory and multi-objective programming, for rating and selecting the best green suppliers according to economic and environmental criteria and then allocating the optimum order quantities among them. Lee et al. (2013) in their paper a mixed integer programming (MIP) model constructed .first to solve the lot-sizing problem with multiple suppliers, multiple periods and quantity discounts. No inventory shortage is allowed in the system, GA is proposed to solve the problem. In the light of the literature review conducted, and GP model is formulated in the next section.

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Model formulation

Purchasing department gives periodical orders to the suppliers and allocating this order from the suppliers is one of the main Concerns of this department. In this section a GP model is developed to determine the annual order quotas for the successful supplier selected. However, there are concerns about procuring lot-sizing that these concerns are in the form of either goals or restrictive constraints. In this model, three goals are identified among which the company should make a selection and identify target levels. The details of the order allocation model are given below.

3.1 Model notations Notations used in the formulation of the model are as follows:

Indices i = 1, 2 . . . n index of products t = 1, 2 . . . T index of time periods j = 1, 2 . . . K index of goals.

Parameters dit

Demand of product i in period t

Dti

Total demand

hit

Holding cost of product i in period t

oit

Ordering cost for product i in period t

qit

Expected defect rate for product I in period t

Qit

Buyer’s maximum tolerable defect rate of product i in period t

Bt

Buyer’s total budget in period t

Iit

Inventory of the product i, carried over from period t to period t + 1

Ci

Purchasing price of the product i in period t

Decision variables xit

Number of product i ordered in period t

Yit

1 if an order is placed on product i in time period t, 0 otherwise

d +jt

Positive deviation from the target value of jth goal in period t

d −jt

Negative deviation from the target value of jth goal in period t

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3.2 Model assumptions a

the model is developed for multi item and multi period

b

time horizon is finite

c

shortages are allowed and lost sales

d

there is no quantity discount

e

the costumers’ demands for all products are deterministic

f

lead time is negligible.

3.3 Mathematical formulation With above parameters and decision variables, a mathematical formulation may be stated as follows: n

Min z =

∑w d j

ji

+

; t = 1, 2,… , T

(1)

j =1

∑ n

∑

n

cx i =1 i it

+

∑

n

qit xit + d 2−t − d 2+t =

i =1

+

oY i =1 it it

∑

n

h I i =1 it it

+ d 1−t − d 1+t = Bt ; t = 1, 2,… , T

(2)

n

∑Q D ; it

it

t = 1, 2,… , T  

(3)

i =1

n

∑b

it

+ d3−t − d3+t = 0 t = 1, 2,… , T  

(4)

i =1

xit + I it −1 − (dit − bit ) = I it ; i = 1, 2,… n, xit ≥ 0 Yit ∈ {0,1} d ji ≥ 0

i = 1, 2,… n, i = 1, 2,… n, j = 1,… 3,

t = 1, 2,… , T t = 1, 2,… , T   i = 1, 2,… n

t = 1, 2,… , T

(5) (6) (7) (8)

The objective function of the GP model (1) is developed to minimise the total deviation cost from the selected target levels where the parameters wj (j = 1, 2, 3) are the weights reflecting preferential objective functions. The first goal, budget goal, is minimising the total purchasing cost of the order allocation as in equation (2). Consists of three parts: the purchasing cost, ordering cost, and the inventory holding cost for the remaining inventory in each period the total cost should meet the periodic budgets. Shortage cost considered infinity. The second goal in equation (3), quality goal, is related with the quality performance of the suppliers. Buyer sets an acceptable rate of quality for each product in per period. The expected number of defective items of product i in period t should be less than the buyer’s maximum acceptable defective items of product i in period t. Third goal in equation (4), shortage goal is minimising the shortage of product i in period t. As u knows, the shortage causes a lot of problems in the system. So, the ideal of this goal is

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defined zero. The system constraint (5) provides balance for inventory flow from the previous period (t_1) into the current period t. Finally, constrains (6)–(8) are used to force non-negative values and binary restrictions in the model.

4

Differential evolutionary algorithm quality

Differential evolutionary algorithm (DE) was introduced for the first time by storm and price in 1995 (Bergey and Ragsdale, 2005). Its usage in solving different problems, made it more accepted by research societies in which are the witness of ability and simplicity (Bergey and Ragsdale, 2005). It has so many application in solving science problems and research grounds like curve fitting, for example, curve fitting of non-linear function to the photoemission data’s and etc. Different researchers mix it to other optimisation solution approaches in order to increase algorithm’s applicability. DE algorithm is very capable of solving limited optimisation problems and problems by non-linear objective function and non-derivative. On the other hand, algorithm’s variables are real number. Comparing the result to other evolutionary algorithm and even other heuristic algorithms shows DE is comparable to other algorithms in the case of speed to achieve good result (Storn and price, 1996). Today, new areas of applications of DE are being developed, some of which include: •

curve fitting

•

photoemission

•

heat transfer

•

information fusion

•

gear design

•

constraint satisfaction problems.

In 2000, utility of this algorithm in the different fields such as electrical power distribution, pharmacology, aviation science and so was the growing. In 2002, entered the field of medical sciences. In 2003, attention was drawn to its use in solving a multi-criteria. Also, DE is used in solving problems related to plant layout, scheduling, reverse logistics and theory of games that acceptable results for this new evolutionary algorithm have brought.

5

Parameters setting, performance and strategies in differential evolutionary algorithm

DE is also particularly easy to work with; having only a few control variables which remain fixed throughout the entire optimisation procedure. DE is a parallel direct search method which utilises NP D-dimensional parameter vectors. As a population for each generation G, i.e., for each iteration of the optimisation. NP does not change during the minimisation process. The initial population is chosen randomly and should try to cover the entire parameter space uniformly. As a rule, we will assume a uniform probability distribution for all random decisions unless otherwise

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stated. The crucial idea behind DE is a scheme for generating trial parameter vectors. Basically, DE generates new parameter vectors by adding the weighted difference between two population vectors to a third vector. If the resulting vector yields a lower objective function value than a predetermined population member, the newly generated vector will replace the vector with which it was compared in the following generation; otherwise, the old vector is retained. This basic principle, however, is extended when it comes to the practical variants of DE. For example an existing vector can be perturbed by adding more than one weighted difference vector to it. In most cases, it is also worthwhile to mix the parameters of the old vector with those of the perturbed one. The performance of the resulting vector is then compared to that of the old vector. We will describe two variants of DE which have proven to be useful. The first step in DE is to coding problem variables in vector form or chromosome carrying answer and in better words are ‘the method of displaying answer’. Based on this definition, we could define fitting function according to problem’s goal. Components and the stage of DE algorithm include: 1

creating population initialisation

2

mutation

3

cross over

4

parent vector selection

5

survivors selection for the next generation

6

term of algorithm stop.

As it is clear, the efficiency of an algorithm is dependent to its parameters. Whereas different parameter would produce different answer by different qualities. Thus, if parameters would not be adjusted correctly, we could not reach to the best answer. For setting proposed algorithm parameters in this paper, we divide them in large and small size and then for each one, estimate the best parameters. In this paper, we have used of response surface methodology. This method has been used for estimating the best parameters influential in a procedure. In this method, we have used of regression equation method for evaluating different level of parameters. The method is to evaluate a series of different level of influential parameters of algorithm based on input indicators (usually we use of objective function). Then we suggest the best amount for setting parameters by fitting the best regression equation on different surfaces.

6

Numerical example and measuring the importance of proposed algorithm

In this section, we have designed a few numerical problems to test the model and evaluate the proposed heuristics. Initially, we have designed a basic problem; other problems are generated by changing the values of some parameters of the basic problem. Suppose an hospital is going to buy some amount of blood in three different groups of B+, O–, AB+, for the four next week. It is possible to confront some limitation in supplying hospital’s requirements. As a result, hospital is confronting limitation and its

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cost is infinite, because it deals with patients’ living and reducing this limitation is the best ideal for hospital. Minimising the shortage is the most important goal for hospital, so idea weights that hospital determined for each goal will be as follow: (Wi = 0.35, W2 = 0.25, W3 = 0.4)

There are three products that in four periods should be procured. Assume that the demand matrix is as follows:

⎛ 10000 14000 9000 10500 ⎞ ⎜ ⎟ Dit = ⎜ 4000 4400 5800 5000 ⎟ ⎜ 23100 22500 23000 20600 ⎟ ⎝ ⎠ The inventory holding costs for each product in each period is as follows: ⎛ 2 2 2.2 2.2 ⎞ ⎜ ⎟ 1 1 ⎟ H it = ⎜ 1 1 ⎜ 5 5.2 5.4 5.6 ⎟ ⎝ ⎠ The ordering cost from each supplier in all four periods is assumed to be constant, and is presented in the following matrix: ⎛ 200 200 200 200 ⎞ ⎜ ⎟ Qit = ⎜ 150 150 150 150 ⎟ ⎜ 210 210 210 210 ⎟ ⎝ ⎠ The order cost is incurred whenever the buyer places at least one product order on a supplier, i.e., ordering one product or three products from a supplier in a specific period incurs the same ordering cost. The expected defect rate of production process of product i in period t, is presented in following matrix: ⎛ 0.003 0.004 0.003 0.002 ⎞ ⎜ ⎟ qit = ⎜ 0.0045 0.0055 0.006 0.0065 ⎟ ⎜ 0.005 0.0055 0.0055 0.005 ⎟ ⎝ ⎠ And the buyer’s maximum acceptable defect rate of each product in each period is as follows: ⎛ 0.002 0.004 0.003 0.002 ⎞ ⎜ ⎟ Qit = ⎜ 0.002 0.004 0.005 0.003 ⎟ ⎜ 0.003 0.004 0.005 0.005 ⎟ ⎝ ⎠ Assume that purchasing price of product I is the same in all four periods as follows: Ci = (20, 18, 17)

The buyer’s total available budget in each period for this purchasing decisions i Bt = ( 950000, 990000, 1020000, 1060000 )

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The mathematical model in this study is solved by using software GAMS 23.6 that the result is showed in Table 1. Solutions obtained by solving the model are reasonable and acceptable, but there is some limitation for the size of the model, number of variables, integer variables and constraints in software GAMS 23.6. Table 1

Decision-making variables values of numerical example utilising GAMS solver Period t

Product i Xi,t

Ii,t

bi,t

1

2

3

4

1

14885.456

9114.544

9000

10500

2

13358.946

616.695

2916.667

2307.792

3

23100

22500

23000

20600

1

4885.45

0

0

0

2

9358.946

0

0

0

3

3

0

0

1

0

4885.456

2

0

3783.305

2883.333

2692.308

3

0

0

0

0

0 0

For showing the efficiency of solution approach, problems by different size have been suggested for measuring the importance of proposed algorithm. Here, the GAMS software could not reach to determined answer in rational time for the problems by the size of more than intermediate. But according to the result in small size, DE algorithm produce good answer in rational computation time. In this section, at first show a problem by its amount of feasible constraint and also the basic and new parameter influence on model. Efficiency and performance of algorithm have been achieved based on average criterion of objective function, the average time of algorithm computation, the average of repetition in any problem and split measure. The result of comparing proposed algorithm to the exact answer in different sizes is according to Table 2. Whereas, it is clear GAMS software is not capable of solving problem in more than intermediate sizes, in small problems proposed algorithm have has accepted errors. Therefore, in large size problems, we could use proposed algorithm certainly. Table 2

Compare proposed algorithm with GAMS software

Problem no.

n

T

1

5

2

GAMS

DE

Obj

Time

Obj

Time

Std

Gap

6

15.02

27

15.02

32

1.52

0.00

5

8

14.98

27

14.98

31

1.47

0.00

3

5

10

15.62

27

15.62

33

1.50

0.00

4

6

6

14.20

34

14.20

36

1.48

0.00

5

6

8

13.97

34

13.97

35

2.21

0.00

6

6

10

15.23

34

15.23

38

2.11

0.00

112 Table 2

N. Ganjavi et al. Compare proposed algorithm with GAMS software (continued)

Problem no.

n

T

7

8

8 9

GAMS

DE

Obj

Time

Obj

Time

Std

Gap

6

35.98

34

35.98

37

1.42

0.00

8

8

36.92

982

37.10

96

1.436

0.5

8

10

38.90

982

39.13

102

1.4.48

0.6

10

10

8

45.41

982

45.41

1

1.4.83

0.00

11

10

10

46.70

982

47.08

100

1.4.09

1.05

12

10

12

45.83

1247

46.16

109

1.5.81

0.93

13

15

8

76.89

1247

77.04

111

1.5.89

0.42

14

15

10

72.47

1247

72.97

110

1.4.98

1.54

15

15

12

74.29

1247

74.70

115

1.4.98

1.22

16

20

10

-

-

105.40

198

6.67

-

17

20

15

-

-

127.82

191

6.34

-

18

20

20

-

-

97.40

201

6.97

-

19

30

10

-

-

107.60

193

4.22

-

20

30

15

-

-

97.24

212

7.17

-

21

30

20

-

-

114.39

220

6.88

-

22

40

20

-

-

97.80

209

8.35

-

23

40

30

-

-

103.27

210

6.02

-

24

40

40

-

-

877.29

352

12.12

-

25

50

20

-

-

929.89

347

18.23

-

26

50

30

-

-

821.35

350

22.49

-

27

50

40

-

-

859.80

349

26.96

-

28

60

40

-

-

793.60

371

21.89

-

29

60

50

-

-

846.80

369

20.45

-

30

60

60

-

-

787.17

366

23.01

-

31

70

40

-

-

804.91

375

22.44

-

32

70

50

-

-

234.65

589

34.14

-

33

70

60

-

-

253.87

597

34.02

-

On comparing efficiency of proposed algorithm in problems by large size, we use of GA and based on Table 3. It is clear proposed algorithm have has more efficiency than GA. It is said to in Table 3 the error of genetic has been determined related to proposed algorithm. Finally, behaviour of proposed algorithm in different repetition and for different problems has been provided in Figure 1, 2 and 3.

Modelling and solving of a lot sizing problem for multiple items Table 3

113

Compare proposed algorithm with GA

Problem no.

n

T

16

20

17

20

18

20

19

30

20

30

21

30

22

40

23

40

24 25

DE

GA

Obj

Time

Std

Obj

Time

Std

Gap

10

105.40

198

6.67

108.47

218

7.60

2.92

15

127.82

191

6.34

131.37

213

6.10

2.78

20

97.40

201

6.97

100.43

213

6.20

3.12

10

107.60

193

4.22

111.39

216

6.35

3.53

15

95.24

212

7.17

99.98

232

5.19

4.98

20

114.39

220

6.88

119.45

233

7.11

4.43

20

97.80

209

8.35

103.18

237

9.54

5.51

30

103.27

210

6.02

107.25

232

8.66

3.86

40

40

877.29

352

12.12

941.50

391

19.80

7.32

50

20

929.89

347

18.23

1000.46

389

31.73

7.59

26

50

30

821.35

350

22.49

878.68

390

28.57

6.98

27

50

40

859.80

349

26.96

928.75

389

27.80

8.02

28

60

40

793.60

371

21.89

851.77

410

26.36

7.33

29

60

50

846.80

369

20.45

902.01

412

27.58

6.52

30

60

60

835.17

366

23.01

848.41

419

31.26

7.78

31

70

40

804.91

375

22.44

870.99

426

28.01

8.21

32

70

50

234.65

589

34.14

256.92

630

41.42

9.49

33

70

60

253.87

597

34.02

279.20

634

39.19

9.98

Figure 1

Convergence of evolutionary algorithms for the problem (6) (see online version for colours) 3

10

2

10

1

10

0

200

400

600

800

1000

1200

1400

1600

1800

2000

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Figure 2

Convergence of evolutionary algorithms for the problem (20) (see online version for colours) 3

10

2

10

1

10

Figure 3

0

200

400

600

800

1000

1200

1400

1600

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Conclusions

One of the most important problems that every organisation has to deal with is procuring lot-sizing. In the real life, due to some issues such as decreasing the risk of purchasing or providing a competitive environment, it is very common. Being different from previous studies, this paper presents GP approach in multi-period inventory lot sizing scenario,

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where there are single supplier and multi products and shortage is allowable. Minimising the shortage is considering one of the important goals for buyer. This paper has been done based on modelling and we have used of operation research method in achieving objectives. Whereas in operation research science, we first modelling and then solve them, it has been done in this research, too. Proposed model in this study has been inspired from mathematic model in Demirtas and Ustun (2009) and Julai et.al (2010). They have more constraint in their research than this study, like as capacity constraint of buyer, and demand constraint, the least amount of order constraint. But in these two researches, they did not consider the assumption of allowed shortage. Because shortage in supply chain is indispensible problem, in this research shortage is a goal in which buyer is trying to reduce it. We have used of different solving method. In fact, this model has determined lot-sizing by paying attention to constraint and for approaching to the fact world; we have examined it by multi-objective viewpoint. The purpose of study is presenting multi-objective model for determining the amount of lot sizing of multi-item by the aid of GP. On getting to these three objects simultaneously, we introduce them by the name of ideal or buyer preference. Thus model goals includes: budget ideal, quality ideal, shortage ideal. After solving model in small numerical example form by the aid of GAMS 23.6 software, it has been cleared provided model is capable to meet all limitation and target function. Because research software are not capable of solving models in large size, we use of DE algorithm. Then problem by different sizes has been proposed for measuring the efficiency of proposed algorithm. Here, solving software of GAMS on the problem by more than mediate size could not reach to determined answer in rational time. But according to the result on small size, DE algorithm, in rational time of computation, produce good answer. The result of comparing proposed algorithm by exact answer in different sizes has been shown on Table 1. Therefore, in problems by large size, we could use of proposed algorithm. In comparing the efficiency of proposed algorithm in the problems by large size, we compare them to GA and based on Table 2. It is clear, proposed algorithm have has more efficiency than GA

8

Recommendations

We have recommendation for future development in this research in which includes: 1

applying model in case study by real data and comparing the result to this study

2

in increasing efficiency, it is better to mix it to supplier selection

3

using feasible weighting method for determining the weight of each ideal

4

adding more hypothesis than supplier capacity, proposed discount, and possible parameter for approaching more to the reality

5

in this study, limitation in sale is removed; we could consider it in future study on the type of back rent or the mix of it.

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References Bergey, P. and Ragsdale, C. (2005) ‘Modified differential evolution: a greedy random strategy for genetic recombination’, Management Science, Vol. 33, No. 3, pp.255–265. Bilsel, R. and Ravindran, A. (2011) ‘A multiobjective chance constrained programming model for supplier selection under uncertainty’, Transportation Research, Vol. 45, No. 8, pp.1284–1300. Buffa, F.P. and Jackson, M. (1983) ‘A goal programming model for purchase planning’, Journal of Purchasing and Materials Management, Vol. 19, No. 11, pp.327–334. Choudhary, D. and Shankar, R. (2011) ‘Modeling and analysis of single item multi-period procurement lot-sizing problem considering rejections and late deliveries’, Computers & Industrial Engineering, Vol. 61, No. 5, pp.1318–1323. Choudhary, D. and Shankar, R. (2012) ‘An STEEP-fuzzy AHP-TOPSIS framework for evaluation and selection of thermal power plant location: a case study from India’, Energy, Vol. 42, No. 1, pp.510–521. Demirtas, E. and Ustun, O. (2009) ‘Analytic network process and multi-period goal programming integration in purchasing decisions’, Computers & Industrial Engineering, Vol. 56, No. 2, pp.677–690. Gao, Z.H. and Tang, L. (2003) ‘A multi-objective model for purchasing of bulk raw materials of a large scale integrated steel plan’, International Journal of Production Economics, Vol. 83, No. 3, pp.325–334. Gollner, P. (2008) Dell and HP See Laptop Battery Shortage, 26 March [online] http://www. reuters.com/article/idUSN2540920320080326S (accessed 01-09-2010). Jiang, Z., Xuanyuan, S., Li, L. and Li, Z. (2010) ‘Inventory-shortage driven optimization for product configuration variation’, International Journal of Production Research, Vol. 49, No. 4, pp.1045–1060. Jolai, F., Yazdian, S.A., Shahanaghi, K. and Azari Khojasteh, M. (2010) ‘Integrating fuzzy TOPSIS and multi-period goal programming for purchasing multiple products from multiple suppliers’, Journal of Purchasing & Supply Management, Vol. 17, No. 1, pp.42–53. Kannan, D., Khodaverdi, R., Olfat, L., Jafarian, A. and Diabat, A. (2013) ‘Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain’, Journal of Cleaner Production, Vol. 47, No. 2, pp.355–367. Karpak, B., Rammohan, K.G. and Kasuganti, R. (2001) ‘Purchasing materials in the supply chain: managing a multi-objective task’, European Journal of Purchasing & Supply Management, Vol. 7, No. 3, pp.209–216. Khalili-Damghani, K., Sadi-Nezhad, S and Tavana, M. (2013) ‘Solving multi-period project selection problems with fuzzy goal programming based on TOPSIS and a fuzzy preference relation’, Information Science, Vol. 37, No. 22, pp.9308–9323. Kumar, K.S. and Babu, A.S. (2012) ‘An integrated method using AHP, DEA and GP for evaluating supply sources’, International Journal of Service and Operations Management, Vol. 11, No. 2, pp.123–150. Kumar, M., Vrat, P. and Shankar, R. (2006) ‘A fuzzy programming approach for vendor selection problem in a supply chain’, International Journal of Production Economics, Vol. 101, No. 2, pp.273–285. Lee, A.H.I., Kang, H.Y., Lai, C.M. and Hong, W.Y. (2013) ‘An integrated model for lot sizing with supplier selection and quantity discounts’, Applied Mathematical Modelling, Vol. 37, No. 7, pp.4733–4746. Narasimhan, R., Talluri, S and Mahapatra, S.M. (2006) ‘Multiproduct, multicriteria model for supplier selection with product life-cycle considerations’, Decision Sciences, Vol. 37, No. 4, pp.577–603.

Modelling and solving of a lot sizing problem for multiple items

117

Nazari-Shirkouhi, S., Shakouri, H., Javadi, B. and Keramati, A. (2013) ‘Supplier selection and order allocation problem using a two-phase fuzzy multi-objective linear programming’, Applied Mathematical Modelling, Vol. 37, No. 22, pp.9308–9323. Raut, R.D., Bhasin, H.V., Kamble, S.S. and Banerjee,, S. (2011) ‘An integrated fuzzy-AHP-LP (FALP) approach for supplier selection and purchasing decision’, International Journal of Service and Operations Management, Vol. 10, No. 4, pp.400–425. Rezaei, J. and Davoodi, M. (2011) ‘Multi-objective models for lot-sizing with supplier selection’, Int. J. Production Economics, Vol. 130, No. 1, pp.76–86. Seifbarghy, M. and Esfandiari, N. (2011) ‘Modeling and solving a multi-objective supplier quota allocation problem considering transaction costs’, Journal of Intelligent Manufacturing, Vol. 24, No. 1, pp.201–209. Sencer Erdem, A. and Gocen, E. (2012) ‘Development of a decision support system for supplier evaluation and order allocation’, Expert Systems with Applications, Vol. 39, No. 5, pp.4927–4937. Senyigit, E. and Soylemez, I. (2012) ‘The analysis of heuristics for lot sizing with supplier selection problem’, World Conference on Business, Economics and Management, Antalya, Turkey, pp.672–676. Sharma, D., Benton, W.C. and Srivastava, R. (1989) ‘Competitive strategy and purchasing decisions’, in Proceedings of the 1989 Annual Conference of the Decision Sciences Institute, pp.1088–1090. Storn, R. and Price, k.V. (1996) ‘Minimizing the real functions of the ICEC’96 contest by differential evolution’, in IEEE International Conference on Evolutionary Computation, pp.842–844. Walsh, L. (2010) ‘Dell hit by server inventory shortages’, Channelinsider, 14 January [online] http://www.channelinsider.com/c/a/Dell/Dell-Hit-by-Server-Inventory-Shortages-100399/S (accessed 1 September 2010). Wang, G., Huang, S.H. and Dismukes, J.P. (2004) ‘Product-driven supply chain selection using integrated multi-criteria decision-making methodology’, International Journal of Production Economics, Vol. 91, No. 1, pp.1–15. Weber, C.A., Current, J. and Desai, A. (2000) ‘An optimization approach to determining the number of vendors to employ’, Supply Chain Management: An International Journal, Vol. 5, No. 2, pp.90–98. Wu, S.H. and Hwang, J. (2011) ‘Manufacturing and operational management, marketing and services’, International Journal of Services and Operations Management, Vol. 9, No. 1, pp.32–51. Xu, H. (2010) ‘Inventory policies for a supplier facing mixed periodic demand in a single-supplier multi-retailer supply chain’, International Journal of Production Economics, Vol. 126, No. 2, pp.306–313.