The Effect of Price Discount on Time-Cost Trade-off Problem Using Genetic Algorithm

doi:10.4236/eng.2009.11005

Paper Menu >>

Journal Menu >>

Engineering, 2009, 1, 1-54

Published Online June 2009 in SciRes (http://www.SciRP.org/journal/eng/).

The Effect of Price Discount on T ime-Cost Trade-off

Problem Using Genetic Algorithm

Hadi Mokhtari1, Abdollah Aghaie2

Industrial Engineering Department, K. N. Toosi University of Technology, Tehran, Iran

Email: 1Mokhtari_IE@yahoo.com, 2AAghaie@kntu.ac.ir

Received April 21, 2009; revised May 13, 2009; accepted May 18, 2009

Abstract

Time-cost trade off problem (TCTP), known in the literature as project crashing problem (PCP) and project

speeding up problem (PSP) is a part of project management in planning phase. In this problem, determining

the optimal levels of activity durations and activity costs which satisfy the project goal(s), leads to a balance

between the project completion time and the project total cost. A large amount of literature has studied this

problem under various behavior of cost function. But, in all of them, influence of discount has not been in-

vestigated. Hence, in this paper, TCTP would be studied considering the influence of discount on the re-

source price, using genetic algorithm (GA). The performance of proposed idea has been teste d on a medium

scale test problem and several computational experiments have been conducted to investigate the appropriate

levels of proposed GA considering accuracy and computational time.

Keywords: Project Management, PERT Networks, Time Cost Trade off, Genetic Algorithm, Discount

1. Introduction

Time-cost trade off problem (TCTP) is an important part

of the project management in planning phase. It is a sub

problem of project scheduling when finding the most

cost effective way to finish a project within time limit is

desirable. In the TCTP, the objective is to determine the

duration of each activity in o rder to obtain the minimum

costs of the proj ect. All existing patterns of this problem

satisfy the objective function through expediting the

activity durations. In this problem, determining the opti-

mal levels of activity d ur ations and activ ity costs leads to

a balance between the project completion time and the

project total cost. There are some procedures in TCTP

(e.g. extra resource allocation, improvement in technol-

ogy level, increase in the quality of materials, hiring

highly skillful human resou rces, etc.) used for expeditin g

the activity durations, according to activity characteris-

tics. These procedures can be summarized to a unique

cost function corresp onding to each activity.

Herroelen and Leus [1] evaluated some stochastic

models of the TCTP under two main categories: “The

stochastic discrete time/cost trade-off problem” and

“Multi-mode trade-off problem in stochastic networks”.

In the stochastic TCTP, when objective function of the

model is related to the time, Bowman [2] presents a

heuristic algorithm based on simulation technique and

solves a general project compression problem using the

derivative estimators. Besides, the application of mathe-

matical programming for the stochastic project crashing

problem was suggested by Abbasi and Mukattash [3], in

which activities are assumed to have three time estimates.

Unfortunately, it seems that the proposed approach has

been developed for a network with a single path and is

not applicable for the ne tworks with slight differences in

the path durations. Arisawa and Elmaghraby [4] sug-

gested the use of fractional linear programming for the

optimal allocation of resources to the activities in order

to maximize the reduction in project mean duration per

unit investment in GERT type networks. A novel re-

source-constrained project scheduling problem has been

modeled by Golenko and Gonik [5] as a knapsack re-

34 H. MOKHTA RI ET AL.

source reallocation problem in which a heuristic algo-

rithm has been to determine the optimal amount of re-

sources allocated to the activities and their start times.

Some authors have investigated the stochastic TCTP

to minimize the project total costs [6-8]. Gutjahr et al. [9]

studied the discrete model of this problem by using the

stochastic branch and bound method to improve the

probability of meeting project deadline. In their work,

they assumed that the crashing of activity durations (by

using additional costs) leads to higher direct costs and

lower penalty costs. In another research, Mitchell and

Klastorin [10] have formulated the objective function

with the direct, indirect, and penalty costs. This study

presents a Stochastic Compression Project (SCOP) heu-

ristic based on decomposition of PERT networks into th e

serial and parallel subnets. Recently, some researchers

have considered the multi criteria/objective formulation

for the stochastic TCTP [11-14].

In all of the presented papers, TCTP have investig ated

under various behaviors of the cost function such as

discrete cost function [5,15-19], linear continuous cost

function [20-22], nonlinear convex cost function [23,24],

nonlinear concave cost function [25] and linear piece-

wise cost function [26,27]. But, in all of them, influence

of discount has not been investigated. In this study, we

develop a new TCTP based on discoun t in resource price

to maximize the project completion probability in a pre-

defined deadline using a limited available budget. In

order to construct the model we assume that activity

durations follow the normal distribution and the activity

mean durations represent the decision variables. In order

to solve this problem, we organize the genetic algorithm

(GA) technique. Recently, some researchers use the GA

in order to solv e the project scheduling problems such as

TCTP [11].

This paper is organized in the following way. The

mathematical model is presented in Section 2. Section 3

describes the structure of the proposed approach, which

is based on the GA algorithm. The characteristics of a

large scale numerical example and results of applying the

proposed approach to that example are presented in Sec-

tion 4. The appropriate levels of GA parameters are in-

vestigated in this section. Finally Section 5 consists of

concluding remarks.

2. Mathematical Model

Let be an acyclic Activity on Arrow ((,)GNA

OA )

graph with arrow set and node set, where the

source and sink node are denoted by

and

, respectively.

( ,,...,)

Nnn n



represents the set of events,

( ,,...,)

aa a



represents the set of activities in a

project PERT netw or k wi t h m nodes andn activities.

Considered notation s are presented as follows:

(, )ij

Activity with i head node and j tail node

Random variable of activity (i, j) duration



Mean duration of activity (i, j) (decisionvariable)



Standard deviation of activity (i, j)

Cost slope of activity (i, j)

Amount of allocated resource to activity (i, j)

Project completion deadline



Upper limit of mean of ith activity.



Lower limit of mean of ith activity.

Total number of paths.

Amount of available budget (i, j).

Number of activities which lie on path r.

Random variable of path r duration.



The mean duration of path r.



The standard deviation of path r.



Cumulative distribution function (CDF) ofnormal

standard distribution.

Generally, the TCTP is a nonlinear optimization prob-

lem. In order to investigate the effect of discount on this

problem, we extract the objective function (project com-

pletion probability) and constraints, separately. The ob-

jective function (OF) is concerned with the maximization

of project completion probability. To construct the OF,

we can write a single path completion probability ac-

cording to the central limit theo rem (CLT), in the fo llow-

ing way:





















































rij





rij ij

rij ijd











Pr ij



ZTT PrPr

(1)

Equation (1) is related to a single path and can’t im-

prove the total project completion probability. Thus,

considering all of the paths, the following objective fun c-

tion is suggested:

},r

rijij











2,1{

MinMax d

















,.. L.. (2)

Two type of restrictions should be modeled in a

mathematical formulation to ensure the obtained solu-

H. MOKHTARI ET AL. 35

Using (4) we can formulate the cost function for activ-

ity as a mathematical representation, in the following way:

),(ji

tions satisfy the budget limitation and real conditions.

Each activity can be planned between two maximum and

minimum limit of its mean duration. Following mathe-

matical representation show this type of constraint:

(4)

min 1

11 2

22 3

max

ijijij ij

ijij ijij

ijijij ij

ijij ijij

SRRR



























(3)

ijij





It is assumed that each activity needs a unique type of

resource. Using this assumption, in order to model the

budget limitation, we can define the effect of discount on

resource price in a mathematical representation. For this

purpose, cost slope (CS) due to various levels of ij



presented below:



max max

()

max11 1

()()()

() 1

min 1(mi

max1minmin 1min

()()( )

Sijijij ij

ij ij

nkkk nnnn

SSS

ij ijij

ijij ijijij ijijij ij

Cij ijk

kk k

ij ijij

ijij ijijijijijijij

 

 



 







 





 













 n) 1

 

 



 

 

(5)

Using (4) and (5), we can formulate the second con-

straint that satisfies budget limitation, in the following

way:

model help us to use the available extra budget more

efficiently, when we could find suppliers with discount

policies.

(6) ()C

ij ij





3. Applying Genetic Algorithm

So, the proposed TCTP can be summarized as a

nonlinear optimization model, as follows:

Genetic Algorithm (GA) is one of the popular metaheu-

ristic algorithms, which can explore the feasible space

using computational intelligence inspired from natural

genetics and evolutionary concepts. It is a search method

based on random selection policy can be used to solve

the nonlinear mathematical programs. In this technique,

an initial populatio n containing several feasib le solutions

(chromosome) is generated randomly. Interesting fact is

that GA does not need a good initial solution. In this

procedure, algorithm starts from an initial solution and

then it would be improved through an evolutionary proc-

ess. After production of initial randomly generated popu -

lation, parents are selected from this primary population

and produce offspring. Second population is a combina-

tion of parent and offspring. The generation procedure

has been done using two effective operators, cross over

and mutation. The crossover is a genetic operator used to

vary the programming of a chromosome or chromo-

somes from one generation to the next and the mutation

operator can prevent th e premature convergence of a new

{1,2,...., }

Tdij

ij r

ax MinrL

ij r

























 (7)

Subject to:

() ,C

ij ij





M (8)

ctivitiy ijr

ijijij



  (9)

This is a new formulation of TCTP in PERT networks.

In order to shorten each activity, some amount of extra

resources is needed. This amount of resources may be

provided by external su ppliers, who may determine some

incentive policy such as price discount. Our proposed

TCTP, models this situation. Using this model, project

manager can increase confidence level of on-time project

completion and prevent project delay. Also, proposed

36 H. MOKHTA RI ET AL.

generation [28]. Desirability of obtained offspring is

investigated using fitness function which is concerned

with objective function and feasibility. The most com-

mon steps for GA follow:

 Generate the initial population

 Evaluate the fitness of all solutions in popula-

tion

 Repeat:

Select parents

Apply the cross over and mutation operators to

produce children

Evaluate the fitness of children

Establish new population using a combination of

parents and children

 Until a satisfactory solution has been obtained

About the proposed GA for the discount based TCTP, we

develop the chromosome structure as a simple string of

decision variables. Indeed, each chromosome repre-

sents ij



for all of the activities. Proposed GA algorithm

steps are presented in the fo llowing way:

 Initial random population is generated using

uniform random number generator.

 Fitness of individuals is evaluated.

 Population is ranked in terms of computed fit ness.

 Parents are selected randomly.

 Cross over and mutation are applied to produce

children.

 Produced offspring are evaluated using fitness

function.

 Form the new population using half of the fit-

test parent and half of the fittest children .

About the fitness function some points should be men-

tioned. In order to evaluate a solution (chromosome),

two parameter should be investigated, goodness of solu-

tion and meaningfulness of solution. The goodness is

concerned with objective function (project completion

probability) and the meaningfulness indicates the feasi-

bility. According to these parameters the fitness function

can be represented as follows:

1, 2,...,

Tdij

ij r

itness FunctionMinrL

ij ij

ij projectij r





























 

























 (10)

In which, the first term represents feasibility and sec-

ond one is concerned with project completion probability.



would be equal to one, ifl

ijij iju



 &

and otherwise, indicates zero.

()C

ij ij



M



4. Numerical Example

In order to investigate the performance of the presented

GA approach for the proposed discount based model of

stochastic TCTP, a numerical example are described in

this section. We conducted the proposed approach for the

large scale example with 15 nodes, 20 independent ac-

tivities and 44 interrelated paths to show that how the

presented approach optimally improves the project com-

pletion probability. Characteristics of this network are

presented in Table 2. The objective is to obtain optimal

allocated budget to the activities in order to improve the

all path completion probability from a risky value to a

maximum confident one. It is assumed that the time unit

is in weeks and the co st is in thousan d dollars. Accord ing

to the presented characteristics, initial probability of the

project completion time at Td =150 which can be com-

uted by using the Central Limit Theorem (CLT) is equal

to 55%. This value is related to a situ ation that all activi-

ties are planned in their upper bound of



. The pro-

posed approach attempts to improve this value to a

maximum level using the limited available budget in

order to decrease the risk of project tardiness. It is also

assumed that the available amount of additional budget

for this purpose is equal to 10,000 thousand dollars. We

conducted our analysis of the presented example in two

steps. In first step we run the developed procedure with

the random initial values for parameters of the proposed

GA. Then the obtained results are considered to deter-

mine the efficient values of the parameters in terms of

the accuracy and computation time. For this purpose we

developed a computer simulation program based on the

proposed approach and randomly initial values of GA

parameters have been considered in primary computa-

tional effort. The obtained simulatio n results for the con-

sidered ex ample are organized in Table 1.

To investigate efficiency of the proposed GA ap-

proach, best obtained project completion probability and

CPU time have been considered. In order to make trade

off between the accuracy and computation time, we also

solve the example with the different values of popula-

tion size (k), mutation and cross over probabilities, then

H. MOKHTARI ET AL. 37

Table 1. Characteristics of considered example.

compute thecomputation time and efficiency measure.

The appropriate levels of these parameters can conduct

the GA toward optimal solution, directly and therefore,

the analysis of these parameters is an important aspect of

the proposed GA approach. Tables 2, 3 and 4 provide the

behavior of the model results according to different lev-

els of parameters, for presented example.

According to the simulation results provided by Tables

2, 3 and 4 the best prob ability of cross over and mutation

rates are 0.8 and 0.2, respectively and best values of

considered criteria (accuracy and CPU time) obtained by

setting population size at 25-30. In this example the

project completion probability has been improved from

55% to 0.79%.

5. Model Validation

The appropriate levels of the GA parameters have been

investigated in section 4 to reach the accurate results

with reasonable computation time. By using this proce-

dure we reach the maximum capabilities of the proposed

GA approach in terms of accuracy and computational

time, but these maximum capabilities should be compared

Table 2. Model results (objective function) for different

levels of (k = 25).

P,2.0

P Run 1* Run 2 Run 3 Run 4 Best

Obj.

0.1 0.73900.71420.6952 0.7202 0.7390

0.2 0.69660.64140.7157 0.7377 0.7377

0.4 0.72220.61970.7585 0.6726 0.7585

0.6 0.72470.62530.7822 0.6792 0.7822

0.8 0.70090.69570.7951 0.7823 0.7951

* Each runs containing 200 iterations

Table 3. Model results (objective function) for different

levels of (k = 25).

P,7.0

PRun 1 Run 2 Run 3 Run 4 Best

Obj.

0.10.71080.75610.6970 0.7896 0.7896

0.20.68760.78570.7952 0.7396 0.7952

0.40.71850.69690.7355 0.7291 0.7355

0.60.68270.68730.6269 0.7565 0.7565

0.80.70070.70670.7477 0.6900 0.7477

Activity l







S 1

S 

S ij



1 8.51 12.32 13.29 14.26 201.59 250.00 265.95 0.25

2 11.60 - 14.56 16.70 - 195.25 225.45 0.60

3 9.56 - 10.05 15.09 - 302.70 321.28 0.45

4 12.95 15.29 16.09 17.77 259.30 295.52 305.04 0.15

5 11.72 12.56 14.05 16.79 252.27 265.33 294.37 0.51

6 11.14 13.00 15.25 16.34 257.84 301.04 326.65 0.67

7 10.43 - 12.05 15.77 - 254.05 295.73 0.65

8 13.43 14.59 17.05 18.15 194.51 201.36 254.00 0.41

9 13.03 15.05 16.32 17.83 241.08 259.32 294.21 0.62

10 12.61 - 15.02 17.50 - 201.51 285.54 0.57

11 9.46 10.25 12.98 15.01 199.65 209.21 225.31 0.14

12 13.22 - 16.02 17.99 - 295.84 309.74 0.03

13 12.41 - 17.02 17.34 - 229.74 276.48 0.24

14 11.32 - 15.25 16.48 - 207.97 257.54 0.10

15 10.17 14.52 15.02 15.57 154.95 174.04 198.74 0.58

16 8.58 10.21 14.09 14.32 174.00 198.32 254.13

0.36

17 9.00 - 12.06 14.65 - 205.27 211.37 0.50

18 10.50 11.65 14.21 15.84 298.57 314.45 365.36 0.25

19 9.31 - 14.09 14.89 - 302.12 341.91 0.17

20 9.41 11.64 13.92 14.97 157.37 164.18 187.47 0.63

38 H. MOKHTA RI ET AL.

Table 4. Model results (computational time, objective function) for different levels of k ().

0.7,P

c0.2Pm

with other standard solutions which are available in the

literature to validate the solution. Since, this paper is

the first combination of price discount with the TCTP,

there isn’t any standard test problem for the proposed

model, in the published literature. We use the global

optimum solution obtain ed by LINGO software to evalu-

ate the performance of proposed approach. Indeed,

LINGO can handle nonlinear programming problems

involving bo th continuous and binary variables and solve

such problem by generalized reduced gradient (GRG).

For this purpose a mathematical model based on a sin-

gle path of PERT network has been considered. The

results of comparing the proposed GA’s best objective

function (obtained by appropriate levels of GA parame-

ters based on the procedure described in section 4) and

LINGO’s global optimu m for a single path are presented

in Ta ble 5.

According to the Table 5, the proposed GA approach

shows a difference with LINGO’s global optimum with

the average of 1.84%. Such a little difference can be

reliable and shows the good performance of proposed

approach.

6. Conclusions

In this paper, we proposed a new approach based on

price discount for TCTP in PERT networks in which

activity durations follow the normal distribution. The

main objective o f the developed model is to improve the

Table 5. Comparing the proposed GA best objective function and LINGO’s global optimum.

Run 1 Run 2 Run 3 Run 4

k CPU

time Obj.

Func. CPU

Time Obj. Func.CPU

Time Obj. Func.

CPU T ime Best

Obj.

5 10.40 0.692 11.24 0.642 9.80 0.716 15.590.701 10.40 0.716

10 16.10 0.722 15.92 0.707 25.21 0.757 20.650.740 15.92 0.757

12 17.41 0.710 16.10 0.735 29.51 0.740 23.620.727 16.10 0.740

18 30.60 0.731 29.25 0.765 39.74 0.788 45.410.751 29.25 0.788

25 40.25 0.758 45.62 0.761 54.30 0.780 42.200.796 40.25 0.796

30 51.50 0.793 47.26 0.757 59.20 0.798 65.200.795 47.26 0.798

Problem Limited Budget (M) Proposed GALINGODifferences (LINGO-GA) % Differences

1 10,000 0.7425 0.7489 0.0064 0.8620

2 15,000 0.7689 0.7909 0.0220 2.8612

3 20,000 0.7849 0.7981 0.0132 1.6817

4 25,000 0.8001 0.8214 0.0213 2.6622

5 30,000 0.8149 0.8236 0.0087 1.0676

6 35,000 0.8311 0.8544 0.0233 2.8035

7 40,000 0.864 0.8764 0.0124 1.4352

8 60,000 0.9049 0.9171 0.0122 1.3482

Average 1.8402 %

H. MOKHTARI ET AL. 39

project completion probability from a risky amount to a

maximum value using limited additional budget. To our

knowledge, this is the first combination of price discount

as a supplier’s policy, with the TCTP, in the published

literature. The presented model determines the optimal

additional resources should be allocated to activities

using the genetic algorithm. The GA algorithm has or-

ganized to allocate the available budget to activities. In

order to illustrate the model efficiency, a computer pro-

gram using MATLAB 7.6.0 was developed and the

model was tested on a medium scale PERT network. Best

amount of objective function and CPU time have been

recorded and efficient levels of GA parameters have been

investigated through the several computational experi-

ments. In order to validate the GA approach, proposed

model have been compared with the LINGO’s global

optimum solution and obtained results showed the good

performance of the proposed solution approach.

7. References

[1] W. Herroelen and R. Leus, “Project scheduling under

uncertainty: Survey and research potentials,” European

Journal of Operational Research, Vol. 165, pp. 289–306,

2005.

[2] R. A. Bowman, “Stochastic gradient-based time-cost

tradeoffs in PERT networks using simulation,” Annals of

Operations Research, Vol. 53, pp. 533–551, 1994.

[3] G. Abbasi and A. M. Mukattash, “Crashing PERT networks

using mathematical programming,” International Journal o f

Project Management, Vol. 19, pp. 181–188, 2001.

[4] S. Arisawa and S. E. Elmaghraby, “Optimal time-cost

trade-offs in GERT networks,” Management Science, Vol.

18, pp. 589–599, 1972.

[5] L. V. Tavares, “A multi stage non-deterministic model for

a project scheduling under resource consideration,” Euro-

pean Journal of Operational Research, Vol. 49, pp. 92–

101, 1990.

[6] R. L. Bergman, “A heuristic procedure for solving the

dynamic probabilistic project expediting problem,” Euro-

pean Journal of Operational Research, Vol. 192, pp. 125–

137, 2009.

[7] S. Foldes and F. Soumis, “PERT and crashing revisited:

Mathematical generalization,” European Journal of Op-

erational Research, Vol. 64, pp. 286–294, 1993.

[8] L. Sunde and S. Lichtenberg, “Net-present value cost/

time trade off,” International Journal of Project Manage-

ment, Vol. 13, pp. 45–49, 1995.

[9] W. J. Gutjahr, C. Strauss and E. Wagner, “A stochastic

branch-and-bound approach to activity crashing in project

management,” INFORMS Journal on Computing, Vol. 12,

pp. 125–135, 2000.

[10] G. Mitchell and T. Klastorin, “An effective methodology

for the stochastic project compression problem,” IIE

Transaction, Vol. 39, pp. 957–969, 2007.

[11] A. Azaron, C. Perkgoz, and M. Sakawa, “A genetic algo-

rithm approach for the time-cost trade-off in PERT net-

works,” Applied Mathematics and Computation, Vol. 168,

pp. 1317–1339, 2005.

[12] A. Azaron and R. Tavakkoli-Moghaddam, “A multi

objective resource allocation problem in dynamic PERT

networks,” Applied Mathematics and Computation, Vol.

18, pp. 163–174, 2006.

[13] A. Azaron, H. Katagiri, and M. Sakawa, “Time-cost

trade-off via optimal control theory in Markov PERT net-

works,” Annals of Operations Research, Vol. 150, pp. 47–

64, 2007.

[14] P. C. Godinho and J. P. Costa, “A stochastic multimode

model for time cost tradeoffs under management flexibil-

ity,” OR Spectrum, Vol. 29, pp. 311–334, 2007.

[15] W. Crowston and G. L. Thompson, “Decision CPM: A

method for simultaneous planning, scheduling, and con-

trol of projects,” Operations Research, Vol. 15, pp. 407–

426, 1967.

[16] E. Demeulemeester, S. E. Elmaghraby, and W. Herroelen,

“Optimal procedures for the discrete time/cost trade-off

problem in project networks,” European Journal of Op-

erational Research, Vol. 88, pp. 50–68, 1996.

[17] E. Demeulemeester, B. De Reyck, B. Foubert, W. Herroe-

len, and M. Vanhoucke, “New computational results on

the discrete time/cost trade-off problem in project net-

works,” Journal of the Operational Research Society, Vol.

49, pp. 1153–1163, 1998.

[18] D. R. Robinson, “A dynamic programming solution to

cost-time tradeoff for CPM,” Management Science, Vol.

22, pp. 158–166, 1975.

[19] M. Vanhoucke and D. Debels, “The discrete time/cost

trade-off problem: Extensions and heuristic procedures,”

Journal of Scheduling, Vol. 10, pp. 311–326, 2007.

[20] I. Cohen, B. Golany, and A. Shtub, “The stochastic time–

cost tradeoff problem: a robust optimization approach,”

Networks, Vol. 49, pp. 175–188, 2007.

[21] D. R. Fulkerson, “A network flow computation for pro-

ject cost curves,” Management Science, Vol. 7, pp. 167–

178, 1961.

[22] P. S. Pulat and S. J. Horn, “Time-resource tradeoff prob-

lem,” IEEE Transactions on Engineering Management,

40 H. MOKHTA RI ET AL.

Vol. 43, pp. 411–417, 1996.

[23] E. B. Berman, “Resource allocation in PERT network

under activity continuous time-cost functions,” Manage-

ment Science, Vol. 10, pp. 734–745, 1964.

[24] R. Lamberson and R. R. Hocking, “Optimum time com-

pression in project scheduling,” Management Science,

Vol. 16, pp. B597–B606, 1970.

[25] J. Falk and J. Horowitz, “Critical path problems with

concave cost-time curves,” Management Science, Vol. 19,

pp. 446–455, 1972.

[26] R. Kelley, “Critical-pathplanning and scheduling: Mathe-

matical basis,” Operations Research, Vol. 9, pp. 296–320,

1961.

[27] P. Vrat and C. Kriengkrairut, “A goal programming model

for project crashing with piecewise linear time-cost trade-

off,” Engineering Costs and Production Economics, Vol.

10, pp. 161–172, 1986.

[28] I. Kaya, “A genetic algorithm approach to determine the

sample size for control charts with variables and attrib-

utes,” Expert Systems with Applications, Vol. 36, pp.

8719–8734, 2009.