Finalversionampmpinventorycontrolproblemunderinflationanddiscount.pdf

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/257140672

A multi-product multi-period inventory control problem under inflation and discount: A parameter-tuned particle swarm optimization algorithm Article  in  International Journal of Advanced Manufacturing Technology · February 2014 DOI: 10.1007/s00170-013-5378-y

CITATIONS

READS

35

463

4 authors: Seyed Mohsen Mousavi

Vahid Hajipour

University of Malaya

Islamic Azad University West Tehran Branch

37 PUBLICATIONS   586 CITATIONS   

49 PUBLICATIONS   449 CITATIONS   

SEE PROFILE

SEE PROFILE

Seyed Taghi Akhavan Niaki

Najmeh Alikar

Sharif University of Technology

University of Malaya

474 PUBLICATIONS   4,583 CITATIONS   

20 PUBLICATIONS   245 CITATIONS   

SEE PROFILE

SEE PROFILE

Some of the authors of this publication are also working on these related projects:

multi skilled project scheduling View project

A Mixed-Integer Model With Genetic Algorithm for Multi-Objective Assembly Line Balancing Problem in Fuzzy Manufacturing Environment View project

All content following this page was uploaded by Seyed Taghi Akhavan Niaki on 05 June 2014.

The user has requested enhancement of the downloaded file.

A Multi-Product Multi-Period Inventory Control Problem under Inflation and Discount: A Parameter-Tuned Particle Swarm Optimization Algorithm

Seyed Mohsen Mousavi, M.Sc. Faculty of Engineering, University of Malaya, Kuala Lumpur, Malaysia, Email: [email protected]

Vahid Hajipour, M.Sc. Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran, Phone: +98 281 3665275, Fax: +98 281 3665277, Email: [email protected] 

Seyed Taghi Akhavan Niaki1, Ph.D. Department of Industrial Engineering, Sharif University of Technology, P.O. Box 11155-0414 Azadi Ave, Tehran 1458889694 Iran, Email: [email protected]

 

Najmeh Aalikar, M.Sc. Department of Science and Technology, Universiti Kebangsaan Malaysia,43600, Bangi, Selangor, Malaysia Email: [email protected]

Abstract In this paper, a seasonal multi-product multi-period inventory control problem is modeled in which the inventory costs are obtained under inflation and all-unit discount policy. Furthermore, the products are delivered in boxes of known number of items and in case of shortage, a fraction of demand is considered backorder and a fraction lost sale. Besides, the total storage space and total available budget are limited. The objective is to find the optimal number of boxes of the products in different periods to minimize the total inventory cost (including ordering, holding, shortage, and purchasing costs). Since the integer nonlinear model of the problem is hard to solve using exact methods, a particle swarm optimization (PSO) algorithm is proposed to find a near optimal solution. Since there is no benchmark available in the literature to justify and validate the results, a genetic algorithm (GA) is presented as well. In order to compare the performances of the two algorithms in terms of the objective function and the required CPU time, they are first tuned using the Taguchi approach, in which a metric called “smaller is better” is used to model the response variable. Then, some numerical examples are provided to demonstrate the application and to validate the results obtained. The results show that while both algorithms have statistically similar performances, PSO tends to be the better algorithm in almost all problems. Keywords: Multiproduct inventory control; Inflation; Discounts; Shortages; Particle swarm optimization; Genetic algorithm; Taguchi method                                                              1

Corresponding author

1   

1. Introduction In inventory control problems, determining the ordering times and the order quantities of products are the two strategic decisions either to minimize total costs or to maximize total profits. In this regard, numerous research works were performed in the past decade. Some of these works are connected to the well-known economic order or production quantity models (EOQ and EPQ) to achieve maximum profit or minimum cost (see for example [1]-[7]). Inventory problems dealing with seasonal and fashion products are usually modeled in a known number of periods. To name a few research works on multi-periodic inventory control problems, Ahmed et al. [8] investigated a multi-period single-item inventory problem with linear cost, where the objective function was a coherent risk measure. Sepehri [9] formulated an integrated flow network and expanded it to a multi-period multi-product inventory control problem with the possibility of holding inventories in a multi-stage multi-member cooperative supply chain. Zhang et al. [10] presented some convex stochastic programming models for multiperiod inventory control problems where the market demand was random and order quantities needed to be decided before the demand was realized. Choi et al. [11] proposed a solution scheme for a periodic review multi-period inventory problem under a mean-variance (MV) framework. In addition to managing multiple items, real-world inventory control systems have many limitations in warehouse space, budget, shortage, and the like. Zhou [12] developed a deterministic replenishment model with multiple warehouses (one owned warehouse and others rented) possessing limited capacity, in which the replenishment rate was infinite and the demand rate was increasing at a decreasing rate with respect to time. Taleizadeh et al. [13] considered a

2   

multiproduct multi-constraint inventory control system with budget and space constraints where replenishment intervals were stochastic and the items were sold under discount. Nowadays, due to high volume financial turnover between or inside companies, inflation is an important factor to model inventory costs. The effects of inflation on inventory control decisions have been considered in many works since 1975 when Buzacott [14] proposed an EOQ model with inflation subject to different types of pricing policies. Dey et al. [15] considered two storage inventory problems with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money. Sarkar and Moon [16] proposed a production inventory model for stochastic demand with the effect of inflation where due to real-life constraints (labor problems, machine breakdown, etc.), a certain percentage of products are of imperfect quality. Recently, Mirzazadeh et al. [17] introduced an inventory model with stochastic inflation rate for multiple items with budget constraint. Many researchers investigated the multi-item multi-period inventory control problem in the recent decade. This part of inventory control problems includes seasonal or fashion products that are programmed to start ordering in a specific season or period and finish in another. These products are just demanded in some season with no demand at the end of the season. Mousavi et al. [18] investigated a multi-product multi-period seasonal inventory control problem with incremental discount and inflation, in which no shortages were allowed and items were ordered in specific boxes of certain capacities. They developed a genetic algorithm (GA) and a simulated annealing (SA) to find a near optimum solution of the mixed integer nonlinear programming problem. Li et al. [19] considered a multi-item multi-period capacitated dynamic lot-sizing problem where each item faces a series of dynamic demands, and in each period multiple items

3   

share limited production resources. In their research, shortages are allowed in the forms of backorder and lost sale. In real-world inventory control systems, managers are usually faced with several constraints that make their decision-making hard. Providing suitable and helpful mathematical formulations of the problems and developing solution methods are justified in these cases. The two common constraints in inventory control problems are the total storage space and the total available budget [18, 20, 21, 22]. Further, not only the order quantities are limited due to constrained storage capacity, but also the required budget to purchase the products is one of factors that is usually limited. In supply chain systems, most often vendors propose different prices per an item of different order quantities, where there are some discounts associated with large order quantities. These kinds of discounts are usually proposed based on two policies of all-unit and sometimes incremental-quantity discount [20, 23, 24, 25, 26]. Meta-heuristics are kind of near-optimal algorithms that were proposed in the two recent decades to integrate basic heuristic methods in higher-level structures in order to effectively and efficiently search a solution space. Nowadays, these algorithms have a large number of applications in optimization of different hard-to-solve problems. The particle swarm optimization (PSO) proposed by Kennedy and Eberhart [27] is a population-based stochastic meta-heuristic algorithm that was inspired by social behavior of bird flocking or fish schooling. PSO is a meta-heuristic that requires few or no assumptions on the problem being optimized and can search very large spaces of candidate solutions. However, meta-heuristics such as PSO do not guarantee an optimal solution is ever found. Nonetheless, since PSO does not use the gradient, it does not require the optimization problem to be differentiable; a prerequisite needed

4   

in many classical optimization methods such as the gradient descent and the quasi-Newton methods. PSO can therefore be used on optimization problems that are partially irregular and noisy over time [28]. In recent years, PSO has been well applied to solve different problems. In inventory control problems, Taleizadeh et al. [29] solved an integer nonlinear programming type of inventory control problem using PSO. Baykasoglu and Gocken [30] utilized a PSO to solve an EOQ inventory control problem in fuzzy environments. Moreover, Taleizadeh et al. [31] developed a multi-product single-machine economic production quantity model with discrete delivery solved by a PSO. Since meta-heuristic algorithms are sensitive to their parameter settings, in order to improve the quality of solutions obtained, the Taguchi method is usually applied. Some recent works that employed this method are in location-allocation problems [33, 34], in scheduling problems [35, 36, 37], in statistical quality control problems [38], and in inventory control problems [18]. While in the past decade, a considerable amount of research works have been devoted to modeling, development, and solution procedures of the inventory control problem, the main contribution of this paper is threefold. First, the modeling area in which an integer nonlinear mathematical formulation for multi-product multi-period EOQ problem  with inflation, discount, and shortage under limitations on budget, storage space, number of orders, and available number of transportation boxes is proposed. Second, a PSO algorithm is developed in this paper to solve the problem. Third, a genetic algorithm (GA) is utilized as well to validate the results obtained and to access the performance of PSO in terms of the solution quality and the required CPU time. GA is also a common population-based stochastic meta-heuristic algorithm proposed by Holland [32] and can be a good competent with PSO. In addition, since meta-heuristic algorithms are

5   

sensitive to their parameters, a Taguchi procedure is employed to tune the parameters of both algorithms, where a metric called “smaller is better” is used to model the response variable. To be more specific, the present paper involves an integer nonlinear programming of multi-product multi-period EOQ problem with shortages in which the planning horizon is finite and the demand rates of the items are different in various periods. The problem is modeled under realistic constraints of storage space, budget, and truck-space, where the vendor sells the products in boxes of known number of items under all-unit discount and inflation plays an important role affecting the inventory costs. Further, three binary variables are used to model the shortage, the ordering, and the purchasing costs; a novel approach that distinguishes this work with others. Moreover, the order quantity of a product in a period must fill the shortage, i.e. the order quantity of a period is greater than or equal to the shortage quantity. The objective is to find the optimum order quantities of the products in different periods such that the total inventory cost including ordering, holding, shortage, and purchasing is minimized. The remainder of this paper is organized as follows: In the next section, the notations and assumptions are presented. In Section 3, the problem is formulated. Solution algorithms are introduced in Section 4. Section 5 provides experimental results along with discussions. Conclusion and future works are given in Section 6.

2. Notation and assumptions To extend the mathematical model of the inventory replenishment policy, the indices, parameters, and decision variables adopted in this paper are as follows: Indices:

i :

An index for a product; i  1, 2,..., m

6   

j :

An index for a replenishment cycle; j  1,..., N

k :

An index for a price break point; k  1, 2,..., K

Parameters: N :

Number of replenishment cycles in the planning horizon

m:

Number of products

K:

Number of price break points

Ai , j :

Replenishment cost per order of product i in period j ($/order)

Pi ,k :

Purchasing cost per unit of the i th product at the k th price break point ($/unit)

hi :

Inventory holding cost of the i th product per unit per unit time ($/unit/month)

 iB, j :

Backorder cost per unit of the i th product in period j ($/unit)

 iL, j :

Lost sale cost per unit of the i th product in period j ($/unit)

C :

Total available budget ($)

S :

Total storage space ( m 2 )

Ii ,j ( t ) :

Inventory position of i th product at time t in period j (unit)

Di , j :

Demand of i th product in period j (unit)

Bi :

The fixed batch size of i th product (unit)

q i ,k :

k th price discount point for i th product ( q i,1  0 ) (unit)

Si :

Required storage space per unit of i th product ( m 2 /unit)

Ti ,j :

Total time elapsed up to and including the j th replenishment cycle of the i th product (month)

T iO, j :

The time in period j at which the inventory of item i reaches zero 7 

 

i :

Percentage of unsatisfied demand of i th product that is back ordered

M1:

An upper bound for order quantity of all products in period j (unit)

M 2:

An upper bound for the available number of boxes (unit)

f :

The inflation rate

Decision variables: Vi ,j :

Number of the boxes for i th product order in period j

bi , j :

Shortage quantity for i th product in period j (unit) (in the first period there is no shortages i.e. bi ,1  0 )

Qi , j :

Ordering quantity of i th product in period j (unit)

X i ,j :

Beginning nonnegative inventory of the i th product in period j (unit) (in j  1 , the beginning inventory of all items is zero)

Ei ,j :

The remaining inventory of the previous period for i th product in period j (unit) ( E i ,1  0 )

U i , j ,k :

A binary variable; set to 1 if item i is purchased at price break point k in period j , and 0 otherwise

Wi ,j :

A binary variable; set to 1 if a purchase of product i is made in period j , and 0 otherwise

Li , j :

A binary variable; set to 1 if a shortage occurs for product i in period j , and 0 otherwise

Moreover, the assumptions involved in the inventory problem at hand are: 1. Replenishments are instantaneous. 8   

2. Demand rates of all products are independent from each other and are constant in a period. 3. In each period, at most one order can be placed for a product. 4. All products are delivered in special boxes, i.e. the order quantity of the products is a multiple of a fixed-sized batch. 5. When a shortage occurs for a product, a fraction of demand (  i ) is considered backorder and a fraction ( 1   i ) lost sale. 6. The initial inventory level of all products is zero (i.e. X i ,1  0 ). 7. The order quantity of each product in each period is at least equal to the shortage quantity of the

product

in

the

previous

period

(i.e.,

Q i , j  bi , j for

i  1 , 2 ,  , m ; j  1, 2 ,  , N 1 ). 8. Planning horizon is finite and known. In the planning horizon, there are N periods of equal length. 9. The total available budget to purchase the products is limited. 10. The total storage space is limited. 11. The order quantity of a product in a certain period is limited. 12. The number of available boxes to deliver products in different periods is limited.

3. Mathematical formulation A graphical representation of the inventory control problem at hand with six replenishment cycles for product i is shown in Fig. 1, where some possible inventory scenarios are given. Based on Fig. 1, the inventory planning starts in period T i ,0 and ends in period T i ,6 , where shortages occur in some periods in between. Moreover, an order that is at least equal to

9   

the shortage quantity of the period is placed in a period. In the beginning period T i ,1 an order quantity Q i ,1 of product i is received. Since this quantity may not be enough to satisfy demand

D i ,1 , a fraction of the demand is the shortage quantity bi ,2 . After receiving the order quantity Q i ,2 in the next period, the shortage quantity bi ,2 is first filled and then the rest is transferred to the next period to satisfy Di ,2 . The inventory X i ,3 of the previous period plus the order quantity

Q i ,3 are available to satisfy demand D i ,3 during the third period where a fraction bi ,3 is shortage. When the order quantity Q i ,4 receives, the quantity bi ,3 is first filled and then the remaining is assigned to demand D i ,4 . In the final period, there is no demand and therefore the order Q i ,5 just satisfies the shortage of bi ,5 since a shortage occurs in this period. Besides, there is no order in period 6.

Insert Fig. 1 about here

3.1. Objective function Total inventory cost consists of ordering, holding, shortage, and purchasing costs that are modeled as follows. Since at most one order can be placed for a product in a period, in order to model the ordering cost, the binary variable W i , j is used where it is 1 if a purchase of a unit of product i is made in period j , otherwise 0. As a result, the total ordering cost when inflation is m N-1

not present is obtained as

 A W i,j

i,j

. In case of an existing inflation with a rate of f , the total

i=1 j=1

ordering cost ( A ) under continuous compounding policy becomes [15]

10   

m N 1

A    Ai , jW i , j e

 fT i , j

(1)

i 1 j 1

The holding cost of a product in a period is equal to the area of either a trapezoid or a triangle above the horizontal line of Fig. 1. Therefore, for T i , j  t  T i , j 1 (1  Li , j 1 ) T iO, j Li , j 1 the holding cost under inflation ( H ) is

H  hi e

 fT i , j



T i , j 1 ( 1 Li , j 1 ) T iO, j Li , j 1

Ti , j

I i , j 1( t ) e  ft dt

(2)

For T i , j  t  T i , j 1 (1  Li , j 1 ) T iO, j Li , j 1 , since the inventory position of product i at time t in period j is I i , j 1 (t )  I i , j  D i , j (t T i , j 1 (1  L i , j 1 )  T iO, j L i , j 1 ) (see Fig. 1) and that at time T i , j 1 T i O, j 

bi , j 1 Di , j

in period j a shortage occurs , by letting T i,j+1 (1  Li,j+1 ) T i,jO Li,j+1  a , Eq.

(2) becomes (see Appendix 1) m N 1

H   hi e

 hi  fTi , j fT  e fa )  ( e i , j ( 1 fTi , j )  e fa ( 1 fa))Di , j )  (3)  2 ( f ( I i , j  Di , j a)( e f 

fTi , j

i 1 j 1

The shortage cost consists of two parts; backorder and lost sale. Since i percentage of the demand of product i is backorder and the (1  i ) percentage is lost sale, based on Fig. 1, the backorder ( BO ) and lost sale ( LS ) costs under inflation are obtained as m N 1

BO     e i 1 j 1

B i,j

i

T i , j 1



I i , j 1 (t )e -ft dt

(4)

T iO, j

m N 1

LS     e i 1 j 1

 fT i , j

L i,j

 fT i , j

T i , j 1

(1  i )



I i , j 1 (t )e  ft dt

(5)

T iO, j

where for T iO, j  t  T i , j 1 we have I i , j 1 (t )  D i , j (t T iO, j ) .

11   

So the total backorder (BO) and lost sale (LS) costs under inflation after simplification are calculated as (see Appendix 2)

m N 1

BO     iB, j  i e

 fT i , j

i 1 j 1

m N 1

LS     iL, j (1  i )e

 D i , j   f (T ) (1  fT i , j 1 )   b  (T i , j 1  i , j 1 )      e i , j 1 ( f D i , j   f       bi , j 1      bi , j 1    1  f (T i , j 1   D i , j   bi , j 1      f T i , j 1  Di , j     T i , j 1   e   f D i , j             

 fT i , j

i 1 j 1

D    (1  fT i , j 1 )  bi , j 1     i , j  e  f (T i , j 1 )    T i , j 1       f D i , j     f           bi , j 1   )  bi , j 1   1  f (T i , j 1  D i , j     f (T i , j 1  Di , j )   bi , j 1     T i , j 1    e   f D i , j             

(6)

(7)

To formulate the purchasing cost under the all-unit discount policy, let the price break points be

Pi , k

 Pi ,1 P  i ,2    Pi ,K

0  Q i , j  q i ,2 q i ,2  Q i , j  q i ,3 

; Pi ,1  Pi ,2  ...  Pi ,K

(8)

Q i , j  q i ,K

Then, the purchasing cost ( P ) becomes m N 1 K

P    Q i , j Pi ,k U i , j ,k e

 fT i , j

(9)

i 1 j 1 k 1

Note that, the vendor spots a discount policy for selling his/her items in K different prices (price break points). In this policy, if a buyer orders a quantity Q i , j in q i ,2  Q i , j  q i ,3 for example,

12   

then the vendor sells the product with price Pi ,2 (i.e. k  2 ), where in the all unit discount (AUD) policy we have Pi ,1  Pi ,2  ...  Pi ,K (see [1]).

The

objective

function

is

the

minimization

of

the

total

inventory

cost

Z  A  H  BO  LS  P obtained as Min Z  A  H  BO  LS  P

(10)

3.2. Constraints

The remaining inventory of a certain product i at the end of period j is obtained as

E i , j 1  E i , j  Q i , j  D i , j ( T i , j 1 T i , j )

(11)

where E i , j must be either nonnegative denoted by X i , j (the beginning inventory of period j ) or b i , j (shortage quantity in period j ). In other words, if E i , j  0 then X i , j  E i , j , otherwise

bi , j  E i , j . Moreover, the starting inventory of product i in period j is equal to its remaining

inventory in the previous period plus the order quantity. In other words

I i , j  E i , j  Qi , j

(12)

Since the order quantity of product i in period j , Q i , j , is delivered in V i , j boxes, each containing B i units, the next constraint becomes Q i , j  B iV i , j

(13)

Since S i square meters of storage space is allocated to each product, the total storage space constraint is formulated as m N 1

  (Q i 1 j 1

i ,j

 X i , j )S i  S

(14)

13   

Due to real-world limitations on the transportation methods (e.g. truck space), the order quantity of all products in all periods cannot be greater than a given fixed number M 1 . In other words, m N 1

 Q i 1 j 1

i,j

 M 1 ; j

(15)

Furthermore, the number of available boxes to deliver product i in period j is limited and we have Vi , j  M 2

(16)

The purchasing price per unit of product i at price break point k is Pi ,k , the order quantity of product i in period j is Q i , j and the total budget is C . As a result, the budget constraint is m N 1 K

  Q i 1 j 1 k 1

i,j

Pi ,k  C

(17)

Finally, since at most one order can be placed for a product in a period and that it can be purchased at one price break point, we have K

U k 1

i , j ,k

1 if Q i , j  0  0 if Q i , j  0

(18)

Also, the following constraints affirm the relations of W i , j with Q i , j . 1 if Q i , j  0 Wi ,j   0 if Q i , j  0

In short, the mathematical formulation of the bi-objective inventory control problem at hand becomes m N 1

Min Z    A i , jW i , j e i 1 j 1

 fT i , j

m N 1

   hi e

 fT i , j

T i , j 1 (1 Li , j 1 ) T iO, j Li , j 1



i 1 j 1

Ti , j

14   

I i , j 1 (t ) e  ft dt 

m N 1

  i 1 j 1

B i,j

e

 fT i , j

i

T i , j 1



I i , j 1 (t ) e dt     e i 1 j 1

T iO, j

m N 1 K

  Q i 1 j 1 k 1

i,j

m N 1

 ft

Pi ,k U i , j ,k e

L i,j

T i , j 1

 ft

(1   i )



I i , j 1 (t )e - ft dt 

T iO, j

 fT i , j

(19)

s.t.:

E i , j 1  E i , j  Q i , j  D i , j ( T i , j 1 T i , j ) X i , j  E i , j   bi , j  E i , j

if Ei , j  0 if Ei , j  0

I i , j  E i , j  Qi , j Q i , j  B iV i , j m N 1

 Q i 1 j 1

 M 1 ; j

i,j

Vi , j  M 2 m N 1 K

Q i 1 j 1 k 1 m N 1

  (Q i 1 j 1

i,j

K

U k 1

i , j ,k

i ,j

Pi ,k  C

 X i , j )S i  S

1 if Q i , j  0  0 if Q i , j  0

1 if Q i , j  0 Wi ,j   0 if Q i , j  0

i  1, 2,..., m , j  1,..., N , k  1, 2,..., K In the next section, two parameter-tuned meta-heuristic algorithms are proposed to obtain a near-optimum solution of the above constrained inventory problem. 15   

4. Solution algorithms

Although different exact methods such as Lagrangian relaxation [17] and branch and bound [39] have been developed in the literature to solve less complicated inventory control models, due to complexity, they cannot be employed to find exact optimal solutions of the problem at hand. As a result, in this work, a PSO algorithm is developed to find a near-optimum solution. Moreover, since there are no other easier ways to justify the accuracy of the PSO, a GA is proposed as well to validate the results obtained. The details of the algorithms are given in the following subsections.

4.1. The PSO

PSO optimizes a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to simple mathematical formulae over the particle's position and velocity. Each particle's movement is influenced by its local best-known position and is also guided toward the best-known positions in the search-space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions [28]. The basic stages of a PSO algorithm are initializing and updating positions and velocities for particles. The structure of the developed PSO for the inventory control problem at hand comes next.

4.1.1. Initializing particles’ positions and exploration velocities

Assuming a d-dimensional search space, PSO is initialized with a group of random particles (solutions) and then searches for optima by updating generations. The type of particles is associated with the number of variables involved in a problem. The swarm size or the population 16   

is denoted by Pop. While in this research there are eight decision variables, the determination of the number of boxes V i , j will automatically determine the rest of the decision variables. Using the upper and the lower bounds on the design variables, x min and x max (here, x min  0 and x max  M 2 ), the positions, x ki , and the exploration velocities, v ki , of the initial swarm of particle i in time k can be first randomly generated. Equations (20) and (21) are applied to initial particles

and exploration velocities, in which “Rand” is a uniform random variable that takes values between 0 and 1 and  is the constant time increment and assumed 1. Fig.2 depicts a structure of a particle that is generated randomly in the closed interval [0, M 2 ].

Please insert Figure 2 around here

x 0i  x min  Rand (x max  x min )

v 0i 

(20)

x min  Rand (x max  x min ) Position   

(21)

An important note for generation and initializing step is that solutions must be feasible and satisfy the constraints. To decrease run time and to reduce the chance of selecting infeasible particles (particles that do not satisfy all the constraints,) the penalty function approach is applied in this research. To do this, consider a typical constraint given in Ineq. (22).

O (x )  R

(22)

If ‫׊‬x the inequality is satisfied, then the corresponding penalty is zero. Otherwise, the penalty is first determined based on Eq. (23) and then will be added to the objective function. Penalty (x )  (O (x )  R )10

(23)

In other words, if F(x) denotes the objective function proposed in Eq. (10), then 17   

x  Feasible region F(x); F(x)=   F(x)+Penalty(x); otherwise

(24)

As a result, if a solution vector does not satisfy a constraint, the related vector solution will be penalized by a big penalty on its objective function shown in Eq. (24).

4.1.2. Updating exploration velocity

Let Pb ki and Pg k be the best objective function value for particle i until time k and the best particle among all until time k, respectively. Then, the exploration velocity is updated as

v ki 1  w .v ki  C 1.rand 1.(Pb ki  x ki )  C 2 .rand 2 .(Pg k  x ki )

(25)

where rand 1 and rand 2 are uniform random numbers between 0 and 1. The coefficients C 1 and C 2 are the given acceleration constants towards Pb and Pg respectively and w is the inertia

weight. Note that introducing a linearly decreasing inertia weight into the original PSO significantly improves its performance through the parameter study of the inertia weight in [40, 41]. The linear distribution of the inertia weight is expressed as follows [40]:

w  w max 

w max w min .k NOG

(26)

where NOG is the maximum number of iterations and k is the current number of iteration. Eq. (26) presents how the inertia weight is updated, considering w max and w min are the initial and final weights, respectively. The related results of the two parameters w max  0.9 and w min  0.4 were investigated by Shi and Eberhart [41] and Naka et al. [40].

4.1.3. Position updating

The positions of the particles for the next objective function evaluation are calculated using the following equation: 18   

x ki 1  x ki  v ki 1. 

(27)

4.1.4. Stopping criterion

A stopping criterion can be defined based on different conditions such as achieving to a predetermined solution, average and standard deviations of the solution do not vary much in several consecutive generations, stop at a certain or a predetermined (CPU) time, or the algorithm is repeated until a maximum number of iterations is reached. In this research, the last criterion is used. The pseudo code of the developed PSO is shown in Fig. 3.

Insert Figure 3 around here

4.2. The GA

GA that was formally introduced in 1970s by John Holland [32] is one of the most commonly used population-base meta-heuristic to solve various complicated optimization problems. It is essentially a method of "breeding" computer programs and solutions to optimization or search problems by means of simulated evolution based on natural selection, crossover, and mutation that are repeatedly applied to a population of binary strings that represent potential solutions. In this research, a GA is utilized to find a near optimum solution of the developed integer nonlinear inventory problem. The steps involved in this algorithm follow.

4.2.1. Initializing and evaluating

A population of chromosomes is generated randomly in this step where each chromosome is initialized in range of [0, M 2 ]. A structure of chromosome is shown in Fig. 2. After generation of the chromosomes, each chromosome is evaluated using the objective 19   

function given in (19). Besides, infeasible chromosomes are fined by a big penalty defined in Eq. (23).

4.2.2. Roulette wheel selection operator

In order to select the chromosomes for the crossover operator, a roulette wheel selection operator is used in this research.

4.2.3. Crossover operator

A crossover operation is used to generate new solutions. For a crossover probability of Pc, a uniform random number between 0 and 1 is first selected for each chromosome. If Pc is less

than the random number, the chromosome is selected for the crossover operation. Let V 1 and V 2 be two different chromosomes that are selected for the crossover operation. Then, the crossover operator applied in this research is shown in Fig. 4 in which V 1' and V 2' are offspring.

Insert Figure 4 around here

4.2.4. Mutation operator

Those chromosomes that are not selected for the crossover operation have chance to be selected in the mutation operator. For a mutation probability of Pm , a uniform random number Ra between 0 and 1 is first selected for each chromosome. If Pm is less than Ra , the chromosome

is selected for the mutation operation. The way this operator works is shown in Fig. 5 where V is the chromosome and V ' is a child. In the mutation operation of this research, two sequential

20   

random numbers are created for a period and two other random numbers are generated for the range.

Insert Figure 5 around here

4.2.5. Elitism process

While some chromosomes are selected for the crossover operation and some others for the mutation operator, the rest are selected from the elitism process.

4.2.6. Stopping criterion

The stopping criterion of GA is similar to the one used for PSO (to reaching a specific number of generations.) Figure 6 depicts a pseudo code of the developed GA of this research.

Figure 6 should be around here

5. Experiments and discussion

In order to demonstrate the application of the proposed methodology, to validate the results obtained by PSO, and to evaluate and compare the performances of the two metaheuristic algorithms, the parameters of the algorithms are first tuned by the Taguchi method using two numerical illustrations in the next section.

21   

5.1. Parameter tuning

As mentioned previously, PSO and GA are employed to find optimal solution of the proposed multi-product multi-period inventory control problem in (19), in which C1, C2, the population size (Pop), and the number of generations (NOG) are the input parameters of PSO and Pop, NOG, the crossover probability (Pc), and the mutation probability (Pm) are the input parameters of GA. Since meta-heuristic algorithms are severely sensitive to their parameters, a Taguchi procedure is utilized to calibrate the parameters of the two algorithms. Taguchi method is a fractional factorial experiment introduced by Taguchi as an efficient alternative for full factorial experiments [42]. Taguchi procedure uses orthogonal array for setting family of experiences to investigate a group of factors. In this procedure, factors are categorized into two groups consisted of controllable or signals factors and noise factors. Now, based on the concept of the robustness, the method seeks to minimize the effect of noise and to determine the optimal level of signal factors. To do so, the signal to noise ratio (S/N), which calculates the amount of variation involved in the response, is used. While there are different ways to obtain variation in Taguchi procedure, in this research, the “smaller is better” type of response has been utilized (since the goal is to minimize S/N), where S/N is given as

 S (Y 2 )  S N  10  Log    n 

(28)

In Eq. (28), Y and n are the response and the number of orthogonal arrays, respectively. To implement the Taguchi procedure, first, the levels of the factors are provided in Table 1. In both algorithms, three levels are considered for each factor. Then, by selecting the L9 design and using the Minitab Software version 15, the orthogonal arrays along with their responses for both PSO and GA are presented in Tables 2 and 3, respectively. 22   

Table 1 should be around here Table 2 should be around here Table 3 should be around here

To decide about the best combinations of the PSO and GA parameters in the Taguchi method, drawing the mean of S/N ratios is needed. Tables 4 and 5 depict the mean of S/N ratios obtained by different parameter combinations of the PSO and GA, respectively. Moreover, Figures 7 and 8 show the plot of the S/N mean for the parameter levels of the GA and PSO, respectively. According to Figs 7 and 8, the best parameter levels are the highest mean of S/N values. Table 6 shows the optimal levels of the parameters for both algorithms.

Table 4 should be around here Table 5 should be around here Figure 7 should be around here Figure 8 should be around here

5.2. Analysis of results

The computer coding of both algorithms are developed using MATLAB software and the experiments are performed on a computer with 2.50 GHz of core 2 CPU and 3.00 GB of RAM. In order to validate the results obtained by PSO and to investigate the performance of the two meta-heuristic algorithms in terms of the solution quality and the required CPU time, 20 problems with different number of products and periods are randomly generated. The general data of these 20 problems are given by Table 7, where the number of products and the number of

23   

periods are between 3 to 20 and 5 to 15, respectively. Moreover, for each problem M 1 , M 2 , S , C and f are given in Table 7. Table 8 contains the results obtained by PSO and GA in terms of

the objective function and CPU time. The last two columns of Table 8 contain the percentage differences of the objective function values and the CPU times of the two algorithms. The results in Table 8 show that on the average PSO works better than GA 0.94% and 8.78% in terms of objective function value and CPU, respectively. In order to show how the 20 problems are used for the comparison, problem no. 4 that consists of 8 products in 6 periods is chosen. The input data of this problem is shown in Table 9 where based on Table 8 the objective function value and the CPU time are (549401 and 38.20) and (553648 and 44.36), for PSO and GA, respectively. Figures 9 and 10 display the near optimal solutions of V i , j , Q i , j X i , j , and bi , j for both PSO and GA algorithms, respectively. Furthermore, the convergence path of the objective function value for PSO and GA are drawn in Figs 11 and 12, respectively. Besides, according to Figs 13 and 14, it seems that PSO has a better performance than GA in both objective function and CPU time in almost all problems.

Table 7 should be around here Table 8 should be around here Table 9 should be around here Figure 9 should be around here Figure 10 should be around here Figure 11 should be around here Figure 12 should be around here Figure 13 should be around here

24   

Figure 14 should be around here

To compare the performances of the employed GA and PSO statistically, the one-way analysis of variance (ANOVA) is utilized. This process is executed using the Minitab 15 software. The ANOVA output shown in Tables 10 and 11 indicates that at a confidence level of 95% the two algorithms do not have significant differences in the average objective function and average CPU times. Similar performances of both algorithms can also be observed in Figs 15 and 16.

6. Conclusion

In this paper, a closer to reality multi-product inventory control problem was investigated with the goal of minimizing the total inventory cost. In this problem, the number of replenishment cycles was limited, shortages were allowed, and the costs were obtained under inflation and discount. The problem was first formulated into an integer nonlinear model and then two meta-heuristic algorithms of PSO and GA were employed to a find near optimal solution of the problem. Moreover, Taguchi method was implemented to tune the parameters of both algorithms, in which ‘smaller is better’ type of response was used. Some numerical illustrations were next given to demonstrate the application of the proposed methodology and to evaluate and compare the performances of the two algorithms in terms of objective function value and the CPU time of reaching the near-optimum solution. The two algorithms were compared using the trend of the objective function and CPU time. A one-way analysis of variance was also used to compare the performances statistically. While the trends showed that

25   

PSO had better performances in comparison with GA in almost all problems, a statistical significant difference was not found. This shows that valid results were obtained using PSO. The model developed in this paper suits systems that produce fashion and seasonal products, i.e. items with different demands in a certain period. In these systems, the production managers make the orders based on marketing reports that give detailed information on the demand forecast. In some periods, however, it is quite possible the demand exceeds the current ordering quantity of some items; causing a shortage. When a shortage occurs for an item in a period an order more than or equal to this shortage is made to satisfy the demand in that period. Moreover, the model to be more applicable in real-world environments, an inflation rate is considered in the cost calculations. The inflation rate in some countries like Iran, due to the intense political and economic fluctuations that directly affect the purchasing power of the people, plays an important role in managerial decisions where most firms have to make their decisions based on inflation changes. As recommendations for future research, demands or inflation rate can be considered stochastic or fuzzy to make the model’s usage more realistic. Moreover, the selling prices of the products can be used and the problem can be modeled into the framework of a supply chain management.

Acknowledgment

The authors are thankful for constructive comments of reviewers. Taking care of the comments certainly improved the presentation of the manuscript.

26   

Appendix 1: The holding cost

The holding cost for product i between periods T i , j and T i,j (1  L i,j+1 ) T i,jO L i,j+1 is calculated as hi 

a

hi 

a

Ti , j

Ti , j

I i , j 1 (t )dt  

a

Ti , j

I

i,j

 D i , j (t  a )  e  ft dt 

(I i , j  D i , j a )e  ft dt  

a

Ti , j

D i , j te  ft dt 





hD hi  fT  fT (I i , j  D i , j a )(e i , j  e  fa )  i 2i , j e  fa (1  fa )  e i , j (1  fT i , j )  f f hi  fT i , j e  f (I i , j  D i , j a)  D i , j fT i , j  1  e fa  f (I i , j  D i , j a)  (1  fa)D i , j )   f 2 hi  fT  fT f (I i , j  D i , j a )(e i , j  e  fa )  e i , j (1  fT i , j )  e  fa (1  fa ) D i , j 2 f













where a = T i,j+1 (1  Li,j+1 ) T i,jO Li,j+1 .

Appendix 2: The shortage cost

The shortage cost for product i between two periods T iO, j and T i , j 1 is obtained as



T i , j 1

T iO, j

I i , j 1 (t )e  ft dt  

T i , j 1

T iO, j

D i , j (t T iO, j )e  ft dt  

T i , j 1

T iO, j

D i , j te  ft dt  

T i , j 1

T iO, j

D i , jT iO, j e  ft dt 

 fT  e  fT i , j 1 D i , j D i , j O  fT i , j 1   e i , j D i , j D i , jT iO, j  fT iO, j O  e (1  fT )  T e  (1  fT )    i , j 1 i,j i,j 2 2    f f f f    O

D i , j   fT i , j 1  (1  fT i , j 1 ) T iO, j e   f  f  If T i , j 1 T iO, j 



T i , j 1

T iO, j

bi , j 1 Di , j

I i , j 1 (t )e  ft dt 

then T iO, j  T i , j 1 

 (1  fT iO, j ) T iO, j  f 

bi , j 1 Di , j

. Now, have

bi , j 1 Di , j

  bi , j 1  )  1  f (T i , j 1   ) D i , j   bi , j 1     T i , j 1     f D i , j          27 

 

   

D i , j   f (T i , j 1 )  (1  fT i , j 1 ) bi , j 1    (T i , j 1  e    f  f D i , j   

 f (T i , j 1 

e

  fT iO, j  e 

   

References [1]. Taleizadeh AA, Moghadasi H, Niaki STA, Eftekhari A. An EOQ-joint replenishment policy to supply expensive imported raw materials with payment in advance. Journal of

Applied Sciences 2008; 8: 4263-4273. [2]. Taleizadeh AA, Sadjadi SJ, Niaki STA. Multiproduct EPQ model with single machine, backordering, and immediate rework process. European Journal of Industrial Engineering 2011; 5: 388-411. [3]. Pasandideh SHR, Niaki STA, Roozbeh Nia A. An investigation of vendor managed inventory application in supply chain: The EOQ model with shortage. International

Journal of Advanced Manufacturing Technology 2010; 49: 329-339. [4]. Pasandideh SHR, Niaki STA, Mirhosseyni SS. A parameter-tuned genetic algorithm to solve multi-products EPQ model with defective items, rework, and constrained space.

International Journal of Advanced Manufacturing Technology 2010; 49: 827-837. [5]. Pasandideh SHR, Niaki STA. A genetic algorithm approach to optimize a multi-products EPQ model with discrete delivery orders and constrained space. Applied Mathematics and

Computation 2008; 195: 506-514. [6]. Pasandideh SHR, Niaki STA. Optimizing the economic production quantity model with discrete delivery orders. Journal of Economic Computation and Economic Cybernetics

Studies and Research 2010; 44: 49-62. [7]. Saha A, Roy A, Kar S, Maiti M. Inventory models for breakable items with stock dependent demand and imprecise constraints. Mathematical and Computer Modelling 2010; 52: 1771-1782.

28   

[8]. Ahmed S, Cakmak U, Shapiro A. Coherent risk measures in inventory problems. European

Journal of Operational Research 2007; 182: 226–238. [9]. Sepehri M. Cost and inventory benefits of cooperation in multi-period and multi-product supply. Scientia Iranica 2011; 18: 731–741. [10]. Zhang D, Xu H, Wu Y. Single and multi-period optimal inventory control models with risk-averse constraints. European Journal of Operational Research 2009; 199: 420-434. [11]. Chio TM, Chiu CH, Fu PL. Periodic review multiperiod inventory control under a mean– variance optimization objective. IEEE Transactions on Systems, Man and Cybernetics,

Part A: Systems and Humans 2011; 41: 678-682. [12]. Zhou YW. A multi-warehouse inventory model for items with time-varying demand and shortages. Computers and Operations research 2003; 30: 2115-2134. [13]. Taleizadeh AA, Niaki STA, Aryanezhad MB, Tafti AF. A genetic algorithm to optimize multiproduct multiconstraint inventory control systems with stochastic replenishment intervals and discount. The International Journal of Advanced Manufacturing Technology 2010; 51: 311-323. [14]. Buzacott JA. Economic order quantities with inflation. Operational Research Quarterly 1975; 26: 553-558. [15]. Dey JK, Mondal SK, Maiti M. Two storage inventory problems with dynamic demand and interval valued lead-time over finite time horizon under inflation and time-value of money.

European Journal of Operational Research 2008; 185: 170-194. [16]. Sarkar B, Moon L. An EPQ model with inflation in an imperfect production system.

Applied Mathematics and Computation 2011; 217: 6159-6167.

29   

[17]. Mirzazadeh A, Fatemi Ghomi SMT, Seyed Esfahani MM. A multiple items inventory model under uncertain external inflationary conditions. Trends in Applied Sciences

Research 2011; 6: 472-480. [18]. Mousavi SM, Hajipour V, Niaki STA, Alikar N. Optimizing Multi-item Multi-period Inventory Control System with Discounted Cash Flow and Inflation: Two Calibrated Metaheuristic Algorithms. Applied Mathematical Modelling 2012; 37: 2241-2256. [19]. Li Y, Tao Y, Wang F. An effective approach to multi-item capacitated dynamic lot-sizing problems. International Journal of Production Research 2011; 50: 5348-5362. [20]. Nayebi MA, Sharifi M, Shahriari MR, Zarabadipour O. Fuzzy-chance constrained multiobjective. Programming applications for inventory control model. Applied Mathematical

Sciences 2012; 6: 209-228. [21]. Kundu A, Chakrabarti T. A multi-product continuous review inventory system in stochastic environment with budget constraint. Optimization Letters 2012; 6: 299:313. [22]. Maity K. Possibility and necessity representations of fuzzy inequality and its application to two warehouse production-inventory problem. Applied Mathematical Modelling 2011; 35: 1252-1263. [23]. Chen S-P, Ho Y-H. Optimal inventory policy for the fuzzy newsboy problem with quantity discounts. Information Sciences: an International Journal 2013; 228: 75-89. [24]. Lee AH, Kang H-Y, Lai C-M, Hong W-Y. An integrated model for lot sizing with supplier selection and quantity discounts. Applied Mathematical Modelling 2012; 37: 4733-4746. [25]. Guchhait P, Kumar Maiti M, Maiti M. A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach. Engineering Applications of Artificial Intelligence 2012; 26: 766–778.

30   

[26]. Yang H-L, Chang C-T. A Two-warehouse Partial Backlogging Inventory Model for Deteriorating Items with Permissible Delay in Payment under inflation. Applied

Mathematical Modelling 2012; 37: 2717-2726. [27]. Kennedy J, Eberhart R. Particle swarm optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks: IEEE 1995: 1942-1948. [28]. Gigras Y, Gupta K. Artificial Intelligence in Robot Path Planning. International Journal of

Soft Computing 2012; 2: 471:474. [29]. Taleizadeh AA, Niaki STA, Shafii N, Meibodi RG, Jabbarzadeh A. A particle swarm optimization approach for constraint joint single buyer-single vendor inventory problem with changeable lead time and (r, Q) policy in supply chain. The International Journal of

Advanced Manufacturing Technology 2010, 51: 1209-1223. [30]. Baykasoglu A, Gocken T. Solving fully fuzzy mathematical programming model of EOQ problem with a direct approach based on fuzzy ranking and PSO. Journal of Intelligent and

Fuzzy Systems 2011; 22: 237-251. [31]. Taleizadeh AA, Widyadana GA, Wee HM, Biabani J. Multi products single machine economic production quantity model with multiple batch size. International Journal of

Industrial Engineering Computations 2011; 2: 213-224. [32]. Holland JH. Adaptive in natural and artificial systems. Ann Arbor, University of Michigan 1975. [33]. Mousavi SM, Niaki STA, Mehdizadeh E, Tavarroth MR. The capacitated multi-facility location–allocation problem with probabilistic customer location and demand: two hybrid meta-heuristic algorithms. International Journal of Systems Science. In press. DOI:10.1080/00207721.2012.670301.

31   

[34]. Mousavi SM, Niaki STA. Capacitated location allocation problem with stochastic location and fuzzy demand: A hybrid algorithm. Applied Mathematical Modelling 2012; 37: 5109– 5119. [35]. Bagher M, Zandieh M, Farsijani H. Balancing of stochastic U-type assembly lines: an imperialist competitive algorithm. The International Journal of Advanced Manufacturing

Technology 2011; 54: 271–285. [36]. Forouharfard S, Zandieh M. An imperialist competitive algorithm to schedule of receiving and shipping trucks in cross-docking systems. The International Journal of Advanced

Manufacturing Technology 2010; 51: 1179–1193. [37]. Ayough A, Zandieh M, Farsijani H. GA and ICA approaches to job rotation scheduling problem: considering employee’s boredom. The International Journal of Advanced

Manufacturing Technology 2012; 60: 651-666. [38]. Niaki STA, and Ershadi MJ. A parameter-tuned genetic algorithm for statistically constrained economic design of multivariate CUSUM control charts: a Taguchi loss approach.

International

Journal

of

Systems

Science.

In

press.

DOI:10.1080/00207721.2011.570878. [39]. As’ad R, Demirli K. A bilinear programming model and a modified branch-and-bound algorithm for production planning in steel rolling mills with substitutable demand.

International Journal of Production Research 2011; 49: 3731-3749. [40]. Naka S, Genji T, Yura T, Fukuyama Y. Practical distribution state estimation using hybrid

particle swarm optimization. In Proceedings of the IEEE Power Engineering Society Winter Meeting, 2001.

32   

[41]. Shi Y, Eberhart RC. Empirical study of particle swarm optimization. In Proceedings of the 1999 Congress on Evolutionary Computation 1999; 1945–1950. [42]. Peace GS. Taguchi Methods. Addison-Wesley Publishing Company, 1993. [43]. Montgomery DC. Design and analysis of experiments. Wiley, 2008.

33   

Fig. 1 Some possible scenarios for an inventory problem with 6 periods

 V 1,1 V 1,2 ... V 1,N 1    V 2 ,1 V 2 ,2 ... V 2 ,N 1   . . . .  V   . . .   .  . . . .    V V ... V 1 2  1 m , m , m ,N   Fig. 2 An article and a chromosome representation

For i=1:Pop Generate position x 0i and exploration velocity v 0i randomly in between [0, M 2 ] End For k=1: NOG For i=1:Pop Evaluate objective functions of the positions ( F (x ki ) ) Penalize infeasible solutions by function penalty (x ) and add it to its objective function End Update positions and exploration velocities Check stopping criteria (if k= NOG then stop, otherwise go next generation) End Fig. 3 Pseudo code of PSO

34   

1 1  r11 2 r V 1   21 3  r31  4  r41 1 1  a11 2  a 21 V2  3  a31  4  a 41

3

2 r1 2 r22

r13 r23

r32 r42

r33 r43

2 a12

3 a13

a 22 a32 a 42

a 23 a33 a 43

4 r14  r24  r34   r44 

' 1

V

4 a14  a 24  a34   a 44 

V

' 2

1 1  a11 2  a   21 3  r31  4  r41

2 a12 a 22

3 a13 a 23

r32 r42

r33 r4 3

1 1  r11 2  r21  3  a31  4  a 41

2 r1 2

3 r13

r22 a32 a 42

r23 a33 a 43

4 r14  r24  r34   r44  4 a14  a 24  a34   a 44 

Fig. 4 The uniform crossover operation

 220  47 V  123   220

335 251 101 4

91 183 4 364

88 3 76 366

     

V

Fig. 5 The mutation operation

35   

'

 220  47  123   220

335 129 86 4

91 183 4 364

88 3 76 366

     

Begin; Define ( Pc ; Pm ; Pop ; and NOG ) For k=1: NOG

Chromosomes=Generate between [0, M 1 ] randomly Objective function value for each chromosome= Evaluate (chromosomes) R=Best chromosome with minimum objective function value Selection process (based on Roulette wheel method) Generate r1 between (0,1) for each chromosome IF Pc  r1 Do crossover operator on each chromosome Else Generate r1 between (0,1) IF Pm  r2 Do mutation operator on each chromosome End End Elitism process Updating (Objective function value and R) EndFor Return (R)

Fig. 6 Pseudo-code of GA

36   

Main Effects Plot for SN ratios Data Means A

B

-116.6 -116.7

Mean of SN ratios

-116.8 -116.9 -117.0 1

2 C

3

1

2 D

3

1

2

3

1

2

3

-116.6 -116.7 -116.8 -116.9 -117.0

Signal-to-noise: Smaller is better

 

Fig. 7 Taguchi S/N ratio plot for PSO

Main Effects Plot for SN ratios Data Means A

B

-116.6 -116.7

Mean of SN ratios

-116.8 -116.9 1

2 C

3

1

2 D

3

1

2

3

1

2

3

-116.6 -116.7 -116.8 -116.9

Signal-to-noise: Smaller is better

 

Fig. 8 Taguchi S/N ratio plot for GA

37   

 79 35 324 349  54 44 326 272   78 218 65 243  118 54 368 96   154 262 328 67   116 27 164 10  14 175 30 77   2 159 41 52

V PSO

X PSO

0 0  0  0  0  0 0  0

0 158 192 44 1186 556 0 0

201  11  15   201  ; 183  135  217   23 

Q PSO

2152 4624  0 1842 3086  1974 2229 3771   86 2620 3228  ; 3274 5796 5819   628 1422 1242  720 140 162   572 450 266 

632 378  702  944  1386  696 84  16

0

b PSO

2592 2792 1608  308 2282 1904 77  1962 585 2187 135   432 2944 768 1608  2358 2952 603 1647   162 984 60 810  1050 180 462 1302   1272 328 416 184  280

0 0  0  0  0  0 0  0

168

0

0

0 0

374 0

0 0

0 0

0 0

0 0

0 46

0 0

0 0

734

0

0

0 0  0  0 0  0 0  0 

Fig. 9 The near optimal solution obtained by PSO

V GA

X GA

17  40  118  29   22  316  26   219

0 0  0  0  0  0 0  0

233 37

11 49

53 106

208 35 80 120

387 196 163 212

199 345 121 27

82 183

168 170

1 18

0 60 552 0 0 1756 26 1002

84  2  229   11  ; 9   354  55   359 

1584 1232 1336  0 0 82  2244 5397 6543   0 1158 3758  ; 450 1487 1996   2386 3468 3390   188 436 2  1766 2676 2220 

QGA

136  280  1062  232  198  1896 156  1752

bGA

0 0  0  0  0  0 0  0

1864 259 1872 280 720 720 492 1464

672  14  3483 1791 2061   1568 2760 88  1467 1089 81   1272 162 2124  1008 6 330   1360 144 2872  88 343

664 0

0 521

0 97

0 668 2

0 110 0

0 0 0

0 0 0

0 0 0

0 0 0

Fig. 10 The near optimal solution obtained by GA

38   

424 742

0 0  0  0 0  0 0  0 

Fig. 11 The convergence path of objective function for problem No. 4 by PSO

Fig. 12 The convergence path of objective function of problem No. 4 by GA

39   

Fig. 13 The trend of objective function values of the generated problems for the proposed algorithms

Fig. 14 The trend of CPU times of solving the generated problems by the proposed algorithms

40   

Individual Value Plot of Fitness vs Solving Methodologies 800000 750000

Fitness

700000 650000 600000 550000 500000 450000 GA

PSO Solving Methodologies

Fig. 15 Individual value plot of the objective function values

Individual Value Plot of CPU vs Solving Methodologies 110 100 90

CPU

80 70 60 50 40 30 GA

PSO Solving Methodologies

Fig. 16 Individual value Plot of CPU times

41   

Table 1: PSO and GA parameter ranges and levels

 

Algorithm Parameters

Parameters Range

Low (1)

Medium (2)

High (3)

C1 (A)

1.5-2.5

1.5

2

2.5

C2 (B)

1.5-2.5

1.5

2

2.5

Pop (C)

50-200

50

100

200

 

NOG (D)

100-500

200

300

500

 

Pop (A)

80-150

80

100

150

Pc (B)

0.6-0.99

0.8

0.7

0.6

Pm (C)

0.01-0.4

0. 1

0.2

0.3

NOG (D)

100-300

200

500

700

             

  PSO

GA

 

    Table 2: Calibration process of PSO

A

B

C

D

R1

R2

R3

R4

R5

1

1

1

1

697197

627230

724903

685920

711451

-116.779 689340.2

1

2

2

2

718150

634103

709161

629965

605145

-116.402 659304.8

1

3

3

3

680746

691909

696805

713670

717192

-116.904 700064.4

2

1

2

3

682430

657763

689410

633324

712359

-116.594 675057.2

2

2

3

1

735230

635773

674105

693663

673845

-116.692 682523.2

2

3

1

2

674599

645180

729375

672780

-116.748

3

1

3

2

744490

727583

743793

695152

731876

-117.252 728578.8

3

2

1

3

649565

685730

736719

735309

719560

-116.978 705376.6

3

3

2

1

717920

639135

726114

640898

667055

-116.641 678224.4

713046

                42   

S/N ratio

Mean

686996

      Table 3: Calibration process of GA

A

B

C

D

R1

R2

R3

R4

R5

S/N ratio

Mean

1

1

1

1

569730

638250

710570

728010

698350

-116.541

668982

1

2

2

2

635450

692930

694700

695290

726807

-116.773 689035.4

1

3

3

3

681300

734210

736830

755835

641410

-117.04

2

1

2

3

673950

678670

571550

711838

687765

-116.476 664754.6

2

2

3

1

654950

672460

678810

634806

766715

-116.689 681548.2

2

3

1

2

665810

689190

712930

716615

644020

-116.73

3

1

3

2

703930

709140

700920

728570

724024

-117.067 713316.8

3

2

1

3

690310

662210

655250

639230

699112

-116.516 669222.4

3

3

2

1

684680

674950

727600

662815

672490

-116.712

 

Table 4: The S/N mean for the factor levels of PSO Factors

Level A 1 -116.7 2 -116.7 3 -117.0 Delta (max-min) 0.3 Rank 2

B -116.9 -116.7 -116.8 0.2 3

C -116.8 -116.5 -116.9 0.4 1

D -116.7 -116.8 -116.8 0.1 4

   

Table 5: The S/N mean for the factor levels of GA Factors Level

A

B

C

D

1

-116.8

-116.7

-116.6

-116.6

2

-116.6

-116.7

-116.7

-116.9

3

-116.8

-116.8

-116.9

-116.7

Delta (max-min)

0.2

0.2

0.3

0.2

Rank

4

3

1

2

  43   

709917

685713

684507

Table 6: Optimal values of the parameters

 

Algorithms

Parameters

Optimal Value

 

C1

2

C2

2

PS

100

 

NOG

100

 

PS

100

Pc

0.7

Pm

0.1

NOG

200

PSO

 

GA

 

 

Table 7: The general data of the generated problems

Problem No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

m 3 4 6 8 10 15 8 10 15 10 13 15 5 8 10 15 10 15 15 20

N 5 5 5 6 6 6 7 7 7 8 8 8 10 10 10 10 12 12 15 15

M1

M2

400 400 400 400 400 400 500 500 500 500 600 600 700 700 700 900 900 1000 1000 1000

30000 34000 39000 50000 53000 56000 60000 69000 78000 85000 95000 110000 125000 140000 148000 157000 169000 180000 200000 230000

44   

S 150000 280000 360000 500000 550000 600000 650000 700000 780000 830000 900000 950000 1100000 1150000 1240000 1300000 1400000 1500000 1700000 1950000

C 700000 780000 850000 1050000 1090000 1110000 1200000 1290000 1400000 1480000 1560000 1630000 1700000 1780000 1880000 1950000 2150000 2250000 2400000 2600000

f 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

Table 8: Objective function and CPU time of the generated problems GA

PSO

Problem No. Objective CPU (sec) Objective

CPU (sec)

1

465320

41.22

465320

35.91

0

14.79

2

491232

42.87

477321

37.22

2.91

15.18

3

531091

43.21

525211

37.89

1.12

14.04

4

553648

44.36

549401

38.20

0.77

16.13

5

582311

46.20

579932

42.18

0.41

09.53

6

632807

51.13

627441

46.32

0.85

10.38

7

593205

48.81

590112

44.29

0.52

10.20

8

605821

50.33

596730

47.19

1.52

06.65

9

628090

54.41

620085

49.98

1.29

08.86

10

621988

51.52

615720

48.23

1.02

06.82

11

643672

57.34

638902

53.59

0.75

07.00

12

668732

65.43

661298

60.73

1.12

07.74

13

653201

59.45

647903

55.11

0.82

07.87

14

662543

63.22

658490

59.34

0.61

06.54

15

678321

70.69

674329

67.33

0.59

04.99

16

698740

79.18

693592

74.21

0.74

06.70

17

690324

76.09

685309

72.48

0.73

04.98

18

720564

89.24

713987

85.11

0.92

04.85

19

748734

96.43

740082

90.32

1.17

06.76

20

771209

104.15

764315

98.60

0.90

05.63

Avg.

632078

61.76

626274

57.21

0.94

8.78

45   

% % CPU Objective difference function difference

Table 9: Input data for problem no. 4 i D i ,1

1

2

3

4

5

6

7

8

200

220

170

300

100

140

130

150

Di ,2

140

280

60

130

270

90

110

140

D i ,3

220

110

110

205

215

190

190

90

Di ,4

160

220

215

80

290

240

220

120

Bi

8

7

9

8

9

6

6

8

hi

9

7

8

8

10

7

9

10

Ai ,1

20

18

20

18

13

13

21

14

Ai ,2

16

21

16

21

17

17

16

16

A i ,3

11

15

12

14

15

16

18

17

Ai ,4

18

17

21

13

18

17

19

19

A i ,5

20

18

23

15

20

19

21

21

Si

8

11

12

10

11

9

12

9

o

π i,1

8

7

7

7

7

8

6

4

π i,2

o

6

8

4

8

4

7

7

9

π i,3

o

3

4

5

6

7

4

4

6

π i,4

o

5

5

4

9

6

5

9

7

π

9

5

9

5

9

9

10

8

4

9

4

9

7

4

4

5

6

5

4

7

8

6

8

9

7

8

5

4

6

8

7

8

βi

0.4

0.5

0.5

0.6

0.3

0.4

0.3

0.45

T

0

0

0

0

0

0

0

0

T

4

1

3

3

2

1

1

5

T

6

4

6

6

3

2

4

10

T

8

8

9

8

5

3

8

15

T

10

11

12

10

7

4

10

20

T

12

13

13

12

8

5

11

22

π π π

i,1 i,2 i,3 i,4

i,1

i,2 i,3

i,4 i,5 i,6

46   

Table 10 Analysis of variance results to compare the algorithms in terms of mean objective function value

Source

DF

SS

MS

F

pvalue

Solving Methodologies

1

336823533

336823533

0.05

0.820

Error

38

2.44029E+11

6421804067

Total

39

2.44029E+11

Table 11 Analysis of variance results to compare the algorithms in terms of CPU time

Source

DF

SS

MS

F

pvalue

Solving Methodologies

1

207

207

0.60

0.445

Error

38

13201

347

Total

39

13408

47   

View publication stats