A parameter-tuned genetic algorithm to optimize two-echelon continuous review inventory systems

A parameter-tuned genetic algorithm to optimize two-echelon continuous review inventory systems

Seyed Hamid Reza Pasandideh, Assistant Professor Department of Industrial Engineering, Qazvin Islamic Azad University, Nokhbegan Ave., Qazvin, Iran Phone: +98 281 3665275, Fax: +98 281 3665277, E-mail: [email protected]

Seyed Taghi Akhavan Niaki, Professor Department of Industrial Engineering, Sharif University of Technology Phone: +98 21 66165740, Fax: +98 21 66022702, E-mail: [email protected]

Nafiseh Tokhmehchi, M.Sc. Student Department of Industrial Engineering, Qazvin Islamic Azad University, Nokhbegan Ave., Qazvin, Iran Phone: +98 281 3665275, Fax: +98 281 3665277, E-mail: [email protected]

Abstract This paper deals with a two-echelon inventory system for a non-repairable item where the system consists of one warehouse and m identical retailers and uses continuous-review (R,Q) ordering policy. To find an effective stocking policy for this system, a mathematical model with the objective of minimizing the total annual inventory investment subject to constraints on the average annual order frequency, expected number of backorders, and budget is formulated. The mathematical model of the problem at hand is shown to be nonlinear integer-programming and hence a parameter-tuned genetic algorithm is proposed to solve it efficiently. A numerical example is provided at the end to illustrate the applicability of the proposed methodology. Key words: Multi-echelon inventory; Continuous review policy; Nonlinear-integer programming; Meta-heuristic algorithms; Genetic algorithm

1  

1. Introduction Successful inventory management is recognized as a crucial activity to increase operational efficiency across competitive business, to improve customer service, and to reduce inventory costs at different locations of a supply network. One of the most important aspects of inventory management that has vital role in supply chain operations is the distribution of goods in multi-echelon inventory systems. The multi-echelon inventory systems are becoming more prevalent and have begun to draw more attentions from both practitioners and academicians in many industries and communication networks. For instance, the military's usage of multi-echelon knowledge achievements is quite well known and air force used a model for a multi-item, multi-echelon, multi-indenture inventory system as a method to compute recoverable spare stock level for the F-15 weapon system (Muckstadt 1973). Multi-echelon spare parts inventory systems have been discussed broadly in the literature. METRIC, one of the earliest models in this topic is a multi-echelon technique for recoverable item presented by Sherbrook (1968). The objective of this mathematical based-depot supply system model is minimizing expected backorders subject to budget constraints with compound Poisson demand. He also approximated outstanding orders at the retailers. According to this study, the appropriate policy for high-cost, low-demand items is (s – 1, s). Deuermeyer and Schwarz (1981) developed a model based on an exact, single facility (R,Q) model of Hadley and Whitin (1963) to analyze service level in a system consisting of one warehouse and a number of identical retailers. Svoronos and Zipkin (1988) proposed several refinements of this model in a multi-echelon inventory system. They approximated each facility as a single location and calculated the mean and variance of the warehouse and retailer lead-time demand. Graves (1985) presented a multi-echelon inventory model with the failures generated by the compound Poisson process and deterministic shipment time from the repair depot to each site for a repairable item with one-for-one replenishment. He also presented an approximation for the steady-state distribution of net inventory level with ample serves at the repair depot and determined the average and the variance of outstanding orders at the retailers.

2  

Axsäter (1990) presented a simple solution procedure for a two-echelon inventory system with one-for-one replenishment, constant lead-time, and independent Poisson demand at retailers. He used an inventory cost function and focused directly on evaluating the average costs. Furthermore, Axsäter and Zhang (1996) considered a two-echelon inventory system with one warehouse and a number of identical retailers with constant transportation time and compound Poisson demand at retailers. They provided a simple recursive procedure for the evaluation of holding and shortage costs at different control policies. DeBodt and Graves (1985) presented an approximate model to minimize the expected costs including a fixed ordering cost, an echelon inventory holding cost for each level, and a back order cost for end item in the multi-echelon inventory system controlled by continuous review policy. Svoronos and Zipkin (1991) described a simple technique to approximate steady state behavior in a multi-echelon with one-for-one replenishment and stochastic transit times of the parts between locations.   Hope et al. (1997) formulated a constrained optimization model in a single location controlled by (R,Q) policy and developed three heuristic to solve it. The objective of this model was minimizing total inventory investment subject to constraints on order frequency and customer service. Axsäter (2000) presented a method for exact evaluation of control policies that provide the complete probability distributions of the retailer inventory levels in a two-echelon inventory system consisting of one central warehouse and N retailers. This system is controlled by different continuous review (R,Q) policies with constant transportation times and independent compound Poisson demand processes at retailers. Seo et al. (2001) developed an optimal reorder policy to utilize centralized stock information for a two-echelon inventory system consisting of one warehouse and multiple retailers controlled by continuous review batch ordering policy. Axsäter (2001) evaluated a technique where a high-demand system is approximated by a low-demand system in a two-echelon inventory system with stochastic demand. Marklund (2002) investigated a two level distribution system consisting of one warehouse and a number of non-identical retailers. In order to control the replenishment process at the warehouse, he introduced a new policy by centralized information in which retailers implement continuous review (R,Q) control policies. He also presented a method for exact evaluation of the 3  

expected inventory holding and backorder costs for the system. Axsäter (2003) considered a twoechelon distribution inventory system consists of a central warehouse and a number of retailers controlled by continuous review installation stock (R,Q) policies. He presented a simple method that uses normal approximations for the retailer demand and the demand at the warehouse in order to approximate optimization of the reorder points. Kiesmüller et al. (2004) developed analytical approximations based on asymptotic result from renewal theory for performance characteristics of a divergent multi-echelon network controlled by continuous review (s,nQ) installation stock policies under compound renewal demand. Caglar et al. (2004) investigated a two-echelon, multi-item spare parts inventory system and presented a mathematical model with the objective of minimizing the system-wide inventory cost subject to constraint on response time at each field depot. They also used a heuristic algorithm to solve it efficiently. Axsäter (2005) determined warehouse backorder cost and provided a newly decentralized way with optimizing sum of expected holding and backorder costs to warehouse and retailers regarding their reorder point in a two-echelon distribution system with installation stock (R,Q) inventory control policy. Seifbarghi and Jokar (2006) developed an approximate cost function to find optimal reorder points of given batch sizes in a two-echelon inventory system consisting of a warehouse and many identical retailers with lost sales and independent Poisson demands controlled by continuous-review policy. Jokar and Zangeneh (2006) developed a model in a two-echelon inventory system consisting of one warehouse, several retailers, and two items with lost sale and demand substitution. They also presented a heuristic algorithm to find cost effective based stock policies. Al-Rifai and Rossetti (2007) investigated a two-echelon inventory system consisting of a central warehouse and a number of identical retailers controlled by (R,Q) inventory policy for nonrepairable items. The objective function of their model is minimizing the total annual inventory investment subject to constraint on average annual order frequency and expected number of backorder. They solved the model by decomposing the system by echelon and location. They also derived expressions for the inventory policy parameters and developed an iterative heuristic

4  

optimization algorithm. Haji et al. (2008) considered a two-echelon inventory system consisting of one warehouse and a number of non-identical retailers with Poisson demand in which warehouse is facing a uniform and deterministic demand ordered by each retailer and introduced a new ordering policy for inventory control. In this paper, an inventory system is considered in which there is a network of inventory holding facilities organized into two levels. More specifically, the problem that is considered consists of a two-echelon non-repairable item inventory system of one warehouse and m identical retailers. We consider the reorder policy for a continuous-review distribution system, utilizing the total inventory investment in both echelons subject to constraints on the average annual order frequency and expected number of backorder. The dominant model of this system in practical applications is based on the assumptions of unlimited warehouse and retailers' budget. These assumptions can lead to a serious underestimating of the spare parts requirements. To deal with this dilemma, warehouse and retailers' budget constraints are also considered. Considering a new objective function and several constraints makes the model more applicable to real-world inventory control systems. In addition, the solution method that is based on a parameter-tuned meta-heuristic algorithm seems more compact and simpler than the ones provided in earlier studies. The remainder of the paper is organized as follows. In section 2, the problem is defined precisely. Section 3 is dedicated to the mathematical formulation of the problem. Comprehensive explanation of the methodology proposed to solve the model is discussed in section 4. In section 5, parameter adjustment and numerical example are given. Finally, conclusions are provided and future research directions are proposed in section 6.

2. Problem definition Consider a two-echelon inventory system of non-repairable items consisting of a warehouse and m identical retailers in which all installations use continuous review (R,Q) policy to replenish their inventories. The system is assumed to work in the following manner: At the beginning of a period, an outside supplier with unlimited capacity delivers the bulk of inventory directly to the warehouse. The warehouse allocates the stock to the retailers. The demand 5  

at a retailer level is either satisfied or backordered. Backorders, both at the warehouse and retailer levels, are filled according to the first in first out (FIFO) policy. At both echelon, (R,Q) ordering policy continuously monitors the inventory position (minus backorders plus on hand and on order inventory) for each item. It means that as soon as the stock level declines to the reorder point R , an order of batch size Q is placed. Note that the real lead-time at the retailer level consists of two components: retard and delay times. Retard time is the time between placement of an order by a retailer and the release of a batch by the warehouse (Svoronos and Zipkin, 1988). Delay time is due to the ordering and transportation times. As a result, the effective lead-time at the retailer level is the sum of the waiting time due to a lack of stock in the warehouse (retard) and the transportation and ordering times (delay) .In this paper, we assume that the retard time at the retailer level is zero. In other words, the real lead-time at the retailer level is equal to retailer's delay. Figure (1) provides a pictorial representation of the system under study.

Supplier

Central warehouse

Retailers

Figure (1): The two-echelon inventory control system

In order to define the problem precisely, a set of critical assumptions are required as follows. a. The demand process at each retailer is a Poisson process with an annual rate Dri . b. Warehouse and retailer's backorders are allowed. c. The supplier has unlimited capacity with a constant lead-time. d. Average annual ordering frequency at the warehouse and retailers are limited. 6  

e. Expected number of backorder at the warehouse and retailers are limited. f.

The warehouse budget for all items is limited.

g. The retailers' budget for all items is limited.

3. Problem formulation The problem at hand is formulated with the objective of minimizing the total annual inventory investment subject to constraint on average annual order frequency, expected number of backorder, and budget. To do this the following notations are required.

3.1 Notations, parameters, and variables The following notations, parameters, and variables are used for mathematical formulation of the problem:

k

Number of inventory items

m

Number of retailers

r

An index for the retailers

w

The warehouse index

i

An index for the items where i  1,..., k

TC

Total inventory investment of both echelons ($)

Ci

Unit cost of the ith item ($)

Dw i

Demand rate of the ith item at the warehouse level (in units of Qri )

Dri

Demand rate of the ith item at the rth retailer (unit per year), r  1, 2,..., m

I w i ( Rw i , Qwi ) Expected on-hand inventory at the warehouse for item i (in units of Qri ) I ri ( Rr i , Qri ) Expected on-hand inventory at retailer r for item i (units) B wi ( R ri , Q ri ) Expected number of backorder at the warehouse for item i (in units of Qri )

B ri ( R ri , Q ri )

Expected number of backorder at retailer r for item i (units)

7  

N wi

Average order frequency at the warehouse for item i (orders per year)

N ri

Average order frequency at retailer r for item i (orders per year)

Qwi

Warehouse batch size of item i (in units of Qri )

Rwi

Warehouse reorder point of item i (units)

Qri

Retailer r batch size of item i (units)

Rr i

Retailer r reorder point for item i (units)

Lwi

Warehouse delay for item i (years)

Lri

Retailer r delay for item i (years)

Nw

Target order frequency at the warehouse (orders per year)

Nr

Target order frequency at retailer r (orders per year)

Bw

Target number of backorder at warehouse

Br

Target number of backorder at retailer r

Xw

Available warehouse budget for all items ($)

Xr

Available retailers' budget for all items ($)

H wi

Warehouse lead-time demand for item i

Hri

Retailer r lead-time demand for item i

F (x)

The cumulative distribution function of the standard normal distribution

f (x)

The probability density function of the standard normal distribution



The mean of the lead-time demand



The standard deviation of the lead-time demand

Now, the inventory control system under consideration can be formulated as the following constrained optimization model:

8  

Minimize total inventory investment Subject to: Average annual order frequency at the warehouse level  N w

(1)

Average annual order frequency at a retailer r level  N r , r  1, 2,..., m

(2)

Total expected number of backorders at the warehouse level  Bw

(3)

Total expected number of backorders at retailer r level  B r , r  1, 2,..., m

(4)

Total warehouse purchase  X w

(5)

Total retailers' purchase  X r

(6)

Thus, the mathematical formulation of the problem becomes:

Minimize: k

k

i 1

i 1

Min TC  m  C i I ri ( R r i ,Q ri )   C i Q ri I wi ( Rw i ,Qw i )

(7)

Subject to: k

1 k

Dwi

Q i 1

k

1 k

D ri

Q i 1

k

B

(8)

Nr

(9)

ri

wi

( Rw i ,Qwi )  B w

(10)

ri

( R r i ,Q r i )  B r

(11)

i 1 k

B i 1 k

C Q i

i 1

 Nw

wi

wi

Qr i  X w

(12)

k

m C i Q r i  X r

(13)

i 1

9  

R w i  Q w

i  1,..., k

(14)

R r i  Q ri

i  1,..., k

(15)

Q ri  1

i  1,..., k

(16)

i  1,..., k

(17)

i  1,..., k

(18)

i

Q wi  1 Q ri ,Qwi , R ri , and Rw Integers

Equation (7) gives the objective function of the model and refers to the total inventory investment of both echelons. Constraints (8) and (9) assure that average order frequencies at the warehouse and retailers are not greater than the target order frequencies at the warehouse and retailer levels, respectively. Constraints (10) and (11) assure that the total expected number of backorders at the warehouse and the retailers are not greater than the corresponding target numbers. Constraint (12) assures that total warehouse purchase is not greater than the available warehouse budget for all items. Constraint (13) assures that total retailers' purchases is not greater than the available retailers' budget for all items. Constraints (14) and (15) assure that when a replenishment order is received, the outstanding backorder are satisfied. Constraints (16) and (17) show that warehouse and retailer batch size are greater than zero. Finally, constraint (18) represents the ranges of the decision variables.

The expected on-hand inventory of the warehouse is calculated using the following equation developed by Hadley and Whitin (1963): 

Qw i  1  E[ H w i ] 2

I w i  B w i ( R w i , Qw i )  R w i 

(19)

Similar equation can be used to calculate the expected on-hand inventory of retailer r. Furthermore, Svoronos and Zipkin (1988) developed the mean and the variance of the lead-time demand at the warehouse as:

E[ H wi ] 

mDri Lwi Qri

V [ H wi ] 

mDri Lwi Qri

2

(20)



m Qri2

Qri 1

[1  exp( p Dri Lw i ) cos( p Dri Lwi )]

p 1

p



(21)

10  

Where

 p  1  cos(2 p / Q ri )

(22)

 p  sin ( 2  p / Q ri )

(23)

And N r i and Dw i are given as follows:

Nri 

Dr i Q ri

(24)

Dw i  mN ri 

mDr i Qri

(25)

Svoronos and Zipkin (1988) developed the mean and the variance of the lead-time demand at retailer r as:

E[ H ri ]  Dri Lri  V [ H ri ]

(26)

The warehouse backorder of the ith item is obtained via Equation (27). Similar equation can be used to calculate the retailers' backorder quantities of the ith item (Hopp and Spearman 2001).

Bwi (Rw i ,Qw i ) 

1 [  (Rw i )   (Rw i  Qwi )] Qw i

(27)

Where

 (x ) 

2 2

{(

(x   ) 2



2

 1)[1  F (

(x   )



)] 

(x   )



f (

(x   )



)}

(28)

As hinted before the demand process of item i at retailer r is Poisson with an annual rate of Dri . However, the demand at the warehouse level does not follow a Poisson process. This is due to the fact that when the stock-out is backordered in the retailers and the demand process at each retailer is Poisson, the arrival process of the orders at the warehouse is a superposition of the order processes from the retailers, specifically, a superposition of independent renewal processes, each with Erlang inter-renewal time with Qri stages and rate per state Dri (see Svoronos and Zipkin 1988). In the next section, a search-based algorithm is developed the inventory model of the problem at hand.

11  

4. A solution procedure The model that was formulated and introduced in the previous section belongs to nonlinear integer-programming (NIP) problems and involves a complex shape of search space. These characteristics lead us to use a meta-heuristic algorithm to solve such a hard problem reliably. Genetic algorithm (GA) is one of the most efficient intelligent randomized search procedures for solving optimization problem based on the principle of survival of the fittest in biological evaluation and genetics, introduced first by Holland (1975). Although GA was successfully applied to some inventory control problems, the application of GA to multi-echelon inventory systems is still rare.

4.1 The proposed GA method The underlying GA's idea is to generate an initial set of random solution called population. Each individual in the population is called a chromosome that is represented as a candidate solution to a problem and consists of a number of genes .The chromosomes must go through a successive set of solution called generation and the fitness function is used to evaluate all individuals. Crossover and mutation operators are used to create new chromosomes called offspring. A new population is created with respect to fitness value of chromosomes. The population of solution become better and better from generation to generation until satisfying solution is obtained. Generally, the proposed GA woks according to the scheme that is described and given in Algorithm (1) as follows.

1. Input initial requirements 2. Set generation = 0 3. Create initial population of solutions randomly 4. Evaluate all individuals of the current generation via the fitness function 5. Repeat the following until a satisfying solution is obtained -Select parents using roulette-wheel method -Apply crossover on parents 12  

-Make random mutation -Evaluate each chromosome using fitness function -Preserve the best chromosomes among parents and offspring -Set generation = generation + 1 -Select new population 6. Output

Algorithm (1): The general steps involved in the proposed GA

4.2 Chromosome One of the most important factors for successful implementation of GA is designing a more suitable chromosomal structure. In this article, the chromosomal solution consists of a matrix with four rows and k columns. The first and the second rows show replenishment batch sizes and reorder points of all items at the warehouse level, respectively. The third and the fourth rows show replenishment batch sizes and reorder points of all items at the retailers, in turns. Furthermore, each column shows the replenishment batch sizes and the reorder points at the warehouse and the replenishment batch sizes and the reorder points of the retailers for each item. These lead to a 4×k matrix as the chromosomal structure of all items depicted in Figure (2). Note that due to the constraint described in (18), each matrix component (gene) must be integer.

Q wi R wi Q ri R ri

Qw1 Rw1 Qr 1 Rr 1

Qw 2 Rw 2 Qr 2 Rr 2

... ... ... ...

... ... ... ...

... ... ... ...

.... ...... .... .... Qwk .... ..... .... .... Rwk .... .... .... .... Qrk .... .... .... .... Rrk

Figure (2): The chromosomal structure

13  

4.3. Initial population As noted previously, the initial population of solutions is generated randomly. Although random generation covers almost all of the search space, a high-quality solution, obtained from another heuristic technique as an initial population, might help a GA to find better solution.

4.4 Evaluation A fitness function is required to evaluate the chromosomes of each generation. In most GA applications, the objective function of the optimization model at hand is considered a fitness function. However, as explained before, the inventory model of this research has 10 constraints on the average annual order frequencies, the expected number of backorders, and the budget. These characteristics make the probability of a generated chromosome being feasible very low. In order to promote this chance, a penalty function is defined to be a positive and known sum of squared violation of each constraint. As a result, the penalty and the fitness function in which violation per constraint is denoted by Ei is defined as follows:



10

Penalty function=

i 1

The chromosome  feasible region

Ei2

The chromosome  feasible region

0

(29) Fitness function = Penalty function + objective function

4.5 Initial requirement The initial requirement described in step 1 of Algorithm (1) consists of two parts: 1. Model

data:

consists

of

proper

constants

to

calculate

required

parameter

such

as

D r i , Lw i , L ri , and C i . These values are determined by the user and depend on the system ability and economic criteria.

14  

2. GA data: consists of the probability of performing crossover called crossover rate denoted by pc and the probability of performing mutation called mutation rate denoted by p m . The number of chromosomes that is called population size and is denoted by Pop_size plays the main role in the run time of the algorithm to reach the near-optimal solution.

4.6. Parent selection At this stage, parents are selected using one of the most popular selection methods called roulette-wheel. In this method, the parents are chosen based on the probability distribution of their fitness value and copied into the mating pool. Thus, the chance of selecting the best individuals becomes higher.

4.7. Crossover After parent selections, several pairs of chromosomes are selected randomly from mating pool by predetermined crossover rate ( pc ) and are mixed to produce offspring. In a crossover operation, some of the genes in the first selected parent are replaced with the corresponding genes in the other parent. In the proposed GA, at first a binary chromosome is created for the parents under consideration. Then, the matrix components (genes) of the two selected parents that correspond to zero values of the binary chromosome are replaced with each other. Those genes that correspond to value one do not change. As an example, the crossover operation of six items is performed as given in Figure (3).

4.8. Mutation To explore new solutions, mutation operator performs random alteration in chromosome genes by a predetermined mutation rate of p m . For the mutation operation of this research, first a random chromosome whose components are between 0 and 1 is created and applied to the selected parent. Then, the parent genes that correspond to values less than p m are mutated within the

15  

boundaries of their corresponding variable. The other genes of the offspring are exactly the same as its parent. Figure (4) illustrates a mutation operation in which p m is set at 0.25.

15 21 62 93

43 382 31 28

5 61 191 341

83 12 23 44

9 7 82 10

30 5 9 14

7 41 3 1

355 82 90 10

121 33 48 51

Parent1

18 121 44 37

21 41 371 45

62 74 11 23

Parent2 1 0 0 1

0 0 0 0

0 1 0 0

1 0 1 0

1 1 1 1

0 0 0 1

Binary chromosome

7 21 62 1

43 382 31 28

5 33 191 341

18 12 44 44

21 41 371 45

30 5 9 23

15

355

121

83

9

62

41 3 93

82 90 10

61 48 51

121 23 37

7 82 10

74 11 14

Offspring 1

Offspring 2

Figure (3): An illustration of the crossover operation

0 . 36

0 . 38

0 . 61

0 . 64

0 .5

0 . 84

0 . 11

0 . 77

0 . 93

0 . 31

0 . 49

0 . 24

0 . 83 0 . 51

0 . 29 0 . 65

0 . 28 0 . 41

0 . 01 1

0 . 38 0 . 59

0 . 71 0 . 27

Randomly generated chromosome 15

43

5

83

9

30

15

43

5

83

9

30

21 62

382 31

61 191

12 23

7 82

5 9

93

28

341

44

10

14

33 62 93

382 31 28

61 191 341

12 1 44

7 82 10

168 9 14

Parent

Offspring

Figure (4): An illustration of the mutation operation

16  

4.9 New population During this phase, the fitness function value of all members consisting parents and offspring are evaluated. After evaluation, the chromosomes with better fitness scores are selected to create new population. In order to achieve better solution, the fittest chromosomes must be preserved at the end of this phase. Note that the number of selected chromosomes must be equal to Pop_size.

4.10. Termination The population of solutions improves from generation to generation based on the theory of survival of the fittest. GA obtains a near optimal solution when the fittest member of the population satisfies a termination condition. In this research, the algorithm stops when the fitness function values for several consecutive generations do not improve. In the next section, a numerical example is given to illustrate the application of the proposed methodology in real-world environments.

5. Numerical illustration Consider an inventory control system of this research that consists of six items with the general data given in Table (1).

Table (1): General data item

Dr i

L wi

Lr i

Ci

1

9.00

0.01

0.59

23.00

2

2.00

0.40

0.21

143.00

3

71.00

0.50

0.11

176.00

4

14.00

0.10

1.00

831.00

5

15.00

0.05

0.21

16.00

6

40.00

0.06

0.45

123.00

m =6, N w =12, N r =24, Bw =4, Br =2, X w = 2003545, X r =1500433 Qwi  [1, 50], Qri  [1, 30], Rwi  [0, 45], Rri [0, 25] 17  

In order to determine the best value for GA parameters (the crossover & mutation rates and the population size) that result in a better solution, a fine-tuning procedure is followed. In this procedure the proposed GA runs 150 times with different values of pc , p m , and Pop_size. Then the fitness function is recorded in each run. Table (2) shows the results of the first 25 runs. To investigate the relationship between the fitness value and the parameters, a regression analysis using the SPSS software is then employed. This relationship is obtained as Fitness  101515.529  37488.559 Pc  10919.906 Pm  885.129 Pop _ size  54555.486 Pc 2  49565.787 Pm2  2.746( Pop _ size ) 2  8526.294 Pc Pm  103.750( Pop _ size ) Pc  135.701( Pop _ size ) Pm

Next, the LINGO software is employed to the following optimization problem in which the best GA parameter values are found. M in Fitness  101515.529  37488.559 Pc  10919.906 Pm  885.129 Pop _ size  54555.486 Pc 2  49565.787 Pm2  2.746( Pop _ size ) 2  8526.294 Pc Pm 

(30)

103.750( Pop _ size ) Pc  135.701( Pop _ size ) Pm

Subject to:

0  pc  1

(31)

0  pm  1

(32)

Pop _ size  0

(33)

As a result, the combination of the crossover rate of 0.52, the mutation rate of 0.18, and the population size of 175 results in the best solution. Employing the proposed GA with the obtained values of the parameters and after one hundred generations, the algorithm converges to the following solution with a total inventory investment of $9016.42: 11

9

48

6

14

35

5 6

2 1

40 4

2 1

18 2

7 1

6

0

1

21

6

10

Furthermore, the graph of the convergence path of the GA is presented in Figure ( 5).

18  

Table (2): Numerical results of 25 runs No. of problem

Pc

Pm

Pop_ size

1

0.10

0.01

50.00

68703.84

2

0.10

0.01

200.00

20257.72

3

0.10

0.01

100.00

38467.87

4

0.10

0.40

100.00

30670.78

5

0.10

0.40

200.00

37633.25

6

0.10

0.95

50.00

119752.33

7

0.10

0.95

100.00

78870.47

8

0.10

0.95

200.00

47556.75

9

0.50

0.01

200.00

19707.38

10

0.50

0.01

100.00

24761.40

11

0.50

0.40

100.00

36219.62

12

0.50

0.40

150.00

41641.57

13

0.50

0.40

200.00

32147.98

14

0.50

0.40

50.00

38157.20

15

0.50

0.95

150.00

55337.74

16

0.50

0.95

50.00

108021.08

17

0.50

0.95

100.00

49865.37

18

1.00

0.01

150.00

20066.67

19

1.00

0.01

50.00

90170.75

20

1.00

0.40

100.00

40397.76

21

1.00

0.40

150.00

48007.82

22

1.00

0.40

200.00

18482.64

23

1.00

0.95

150.00

70515.29

24

1.00

0.95

50.00

115671.24

25

1.00

0.95

200.00

44777.06

Fitness

19  

Fitness value

Number of Generations

 

Figure (5): The graph of the convergence path

7. Conclusion and future research directions In this research, a new model for a two-echelon inventory system was developed to find the continuous review inventory control parameters that minimize the total annual inventory investment subject to constraints on the average annual order frequency, the expected number of backorder, and budgets. This model is closer and hence more applicable to real-world inventory problems than the other existing ones. We also explored the use of a GA to solve this problem efficiently. Future research may extend the model by considering different assumptions or different solving methods .In addition, many variations in GA operators, initialization method, fitness definitions, and replacement strategies are obviously possible and might lead to more efficient solutions.

References Al-Rifai

MH, Rossetti MD, 2007. An efficient heuristic optimization algorithm for a two echelon (R,

Q) inventory system. International Journal of Production Economics 109: 195–213. Axsäter S, 1990. Simple solution procedures for a class of two echelon inventory problems. Operations Research 38: 64–69.

20  

Axsäter S, Zhang W, 1996. Recursive evaluation of order-up-to-S policies for two-echelon inventory systems with compound Poisson demand. Naval Research Logistics 43: 15 1-1 57. Axsäter S, 2000. Exact analysis of continuous review (R,Q) policies in two-echelon inventory systems with compound Poisson demand. Operations Research 48: 686-696. Axsäter S, 2001. Scaling down multi-echelon inventory problems. International Journal of Production Economics71: 255-261. Axsäter S, 2003. An approximate optimization of a two-level distribution inventory system. International Journal of Production Economics 81–82: 545–553. Axsäter S, 2005. A simple decisions rule for decentralized two-echelon inventory control. International Journal of Production Economics 93–94: 53–59. Caglar D, Chung-Lun L, Simchi-Levi D, 2004. Two-echelon spare parts inventory system subject to a service constraint. IIE Transactions 36: 655-666. Debodt MA, Graves SC, 1985. Continuous review policies for a multi-echelon inventory problem with stochastic demand. Management Science 31: 1286-1299. Deuermeyer BL, Schwarz LB, 1981. A model for the analysis of system service level in warehouseretailer distribution Systems: The identical retailer case. TIMS Studies in the Management Sciences 16: 163–193. Graves SC, 1985. A multi-echelon inventory model for a repairable item with one-for-one replenishment. Management Science 31: 1247–1256. Hadley G, Whitin TM, 1963. Analysis of inventory Systems. Prentice-Hall, Inc., Englewood Cliffs, NJ. Haji R, Neghab MP, Baboli A, 2008. Introducing a new ordering policy in a two-echelon inventory system with Poisson demand. International Journal of Production Economics 117: 212–218. Holland JH, 1975. Adaption in natural and artificial systems. University of Michigan Press, Ann Arbor, Michigan; re-issued by MIT Press (1992). Hopp WJ, Spearman ML, Zhang RQ, 1997. Easily implementable inventory control policies. Operations Research 45: 327–340. Hopp WJ, Spearman ML, 2001. Factory Physics, second ed., McGraw-Hill, New York. 21  

Jokar MA, Zangeneh S, 2006. Developing a model for a two-echelon two-item inventory system with lost sale and demand substitution. Management of Innovation and Technology 2: 926-930. Kiesmüller GP, de Kok TG, Smits SR, van Laarhoven PJM, 2004 .Evaluation of divergent N-echelon (s, nQ)-policies under compound renewal demand. OR Spectrum 26: 547-577. Marklund J, 2002. Centralized inventory control in a two-level distribution system with Poisson demand. Naval Research Logistics 49: 798–822. Muckstadt JA, 1973. A model for a multi-item, multi-echelon, multi-indenture inventory system. Management Science 20: 472–481. Seifbarghi M, Jokar MR, 2006. Cost evaluation of a two-echelon inventory system with lost sales and approximately Poisson demand. International Journal of Production Economics 102: 244–254. Seo Y, Jung S, Hahm J, 2001.Optimal reorder decision utilizing centralized stock information in a two-echelon distribution system. Computers & Operations Research 29: 171-193. Sherbrooke CC, 1968. METRIC: A multi-echelon technique for recoverable item control. Operations Research 16: 122-141. Svoronos A, Zipkin P, 1988. Estimating the performance of multi-level inventory systems. Operations Research 36: 57–72. Svoronos A, Zipkin P, 1991. Evaluation of one-for-one replenishment policies for multi-echelon inventory systems. Management Science 37: 68-83.

 

22