European Journal of Operational Research 228 (2013) 282–289

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Innovative Applications of O.R.

Monitoring delivery chains using multivariate control charts

Alireza Faraz a,⇑, Cédric Heuchenne a, Erwin Saniga b, Earnest Foster c

a HEC Management School, University of Liège, Liège 4000, Belgiumb Department of Business Administration, University of Delaware, Newark, DE 19716, USAc Industrial Engineering Department, University of Washington, Seattle, USA

a r t i c l e i n f o

Article history:Received 11 July 2012Accepted 22 January 2013Available online 31 January 2013

Keywords:TransportationSupply chain managementQuality controlMultivariate control chartsEconomic statistical design and geneticalgorithms

0377-2217/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2013.01.038

⇑ Corresponding author. Tel.: +32 471200881.E-mail addresses: alirea.faraz@ulg.ac.be, alireza.

heuchenn@stat.ucl.ac.be (C. Heuchenne), saniga@udnest@gmail.com (E. Foster).

a b s t r a c t

Delivery chains are concerned with the delivery of goods and services to customers within a specific timeinterval; this time constraint is added to the usual consumer demand for product or service quality. Inthis context, we address the idea of using process control tools to monitor this key variable of deliverytime. In applications, there are usually several production and delivery sites and a variety of differentways to transport, treat and provide goods and services; that makes the problem multivariate in nature.We therefore propose to control the process using multivariate T2 control charts economically designedwith the addition of statistical constraints, a design method called economic-statistical design. We illus-trate the application in general through an illustrative example.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

One of the main objectives when managing a supply chain is toprovide quick, reliable, high quality and economic services and/orproducts to customers. Using Porter’s (1996) value chain approachin which the supply chain includes all activities required to providea product or service to the final customer, the delivery chain is con-sidered as a part of the flow of goods and services in the supplychain. A delivery chain describes all activities related to the deliv-ery of goods and services to customers in the supply chain. In fact,all activities related to the delivery of products to customers bytransportation in a supply chain are included in a delivery chain.

The delivery chain, as the ultimate step of a supply chain, playsan important role in a companies business. As an example, con-sider a simple business which produces and delivers pizza. Cus-tomers want tasty pizza but they want it delivered on time aswell. Thus, managers need to control both attributes of their ser-vice and/or product. Deming (1986) pointed out that in spite ofthe involved costs, the benefits of controlling a process can be highin many cases. Forker et al. (1997), Kuei and Madu (2001), Trent(2001), Flynn and Flynn (2005) and Matthews (2006) investigatedthe application of quality management to supply chains. Theyestablished that improving the supply chain effectivenessimproved customer satisfaction. Ramos et al. (2007) studied the

ll rights reserved.

faraz@gmail.com (A. Faraz),el.edu (E. Saniga), fosterear-

importance of using quality management activities in the supplychains of Motorola and Rolls-Royce Deutschland companies. Theyshowed that blending quality management activities and SCMwould result in higher benefits than their sole application.

In this paper we address the problem of delivery chain controlin a comprehensive way through the well-known methods of pro-cess monitoring with control charts. Since a delivery chain is char-acterized by many time variables corresponding to ways betweenproduction and delivery sites, the control charts we employ aremultivariate. Moreover, control can be costly; in particular, thecontrol costs include the cost of sampling, the cost of poor perfor-mance of delivery chains, the cost of adjusting and repairing thedelivery process and so on. Effective but expensive control chartsschemes result in costs being passed onto the customer. Therefore,methods to monitor delivery chains must be cost effective, andmust maintain good statistical properties such as detecting deliv-ery delinquencies as soon as possible and not signaling delinquen-cies when they do not exist.

The paper is organized as follows. In Section 2, the importanceand necessity of economic monitoring of delivery chains and somebasic considerations on the use of control charts to monitor deliv-ery chains in general are discussed. In Section 3, the problem ismodeled mathematically. An illustrative example is presented inSection 4 while some conclusions are drawn in Section 5.

2. Delivery chains and multivariate control charts

The mission of a delivery chain is to provide customers withproducts within a specified and agreed delivery time. Since there

Fig. 1. The delivery chain.

A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289 283

may be several production sites (different cities or countries),many customer delivery sites (different cities, countries) and dif-ferent transportation modes (e.g. trains, planes, trucks, ships . . .),the solution is complex.

Establishing a control system on the delivery chain imposesoverhead costs to the organization but done efficiently, will reducecosts. Deming (1986) argues this convincingly; more specificallyDrucker (1998) argues that in order to compete successfully inthe increasingly competitive global marketplace, an organizationmust be aware of its supply chain and their activities cost and seekeconomic solutions. Establishing an economic monitoring schemefor the delivery chain will help organizations to reduce costs andoptimize customer satisfaction.

2.1. A process approach to delivery chains

In order to monitor delivery chains by control charts, the data ofdelivery chains must be recorded so that identifying unusual pat-terns and special causes of excessive delays in the routine deliverytime of services and goods to the customers can be done. This pro-cess of data collection and analysis, of course, is the first step incontrolling any process. In particular, some of the important deliv-ery variables include time spent in transporting and deliveringproducts to customers. Then, if the data indicate that the deliverychain process is not performing accordingly, the causes of this poorperformance are investigated and removed. For example, dataanalysis may show that there are predictable shipping delays inusing trucks to ship to a metropolitan area because of roadimprovements being done for a stated period of time. Managersthen might want to switch the delivery method to trains duringthe construction period. In fact, when the mean value shifts to avalue that is not the one specified the implication is that theprocess has changed. When monitoring, which is the focus of thispaper, one develops an intervention that accommodates this shift.If the shift is permanent (e.g. suppose a rail system is deemedunprofitable and not used in the future) the change is permanentand the mean value of the process changes. Thus, the processcontrol system must be adjusted. In this case one must estimatethe new parameters of the process and recalculate the design.For example, Silver and Rohleder (1999) address this problem indetails.

Conceptually the problem is quite simple; applying it to con-trolling a delivery chain, on the other hand, can be complex.

In the example we present here, near cities and destinations aremerged together to form a single node for simplification withoutloss of generality. Different delivery activities such as air, marineand ground transportations are shown in different branchesbecause companies often have different contracts for differentmethods of shipping. Fig. 1 illustrates a delivery chain with its timevariables. The lengths of the branches in Fig. 1 correspond to theaverage time needed to complete the job. As is clear, the branchesAB and BC are common in all routes and also are significantly long-er in time and geographical distance than the others. Thus, thereare correlations between the different time variables in the deliv-ery chain. Furthermore, we note that delivery chain performancecan be measured indirectly by customer satisfaction level deter-mined by customer survey or directly by on time delivery data.

Because of the above-mentioned correlations, controlling thedelivery chain requires the use of multivariate process control,or, in particular, multivariate control charts. Duncan (1956)showed that establishing control charts costs money, but, ifdesigned economically, can save money. So, in this paper, ourapproach to monitoring a delivery chain is through the use ofmultivariate control charts that are designed economically.Additionally, to ensure that problems are identified quickly andproblems that do not exist are not signaled, we impose constraints

on the statistical properties of the control charts, yielding whatSaniga (1989) called an economic statistical design.

2.2. The data collection system

When customers place orders, they expect on time deliveries.Furthermore, they expect to receive information regarding the sta-tus of the goods, such as loading and unloading times, some infor-mation on transits and delivery activities and the expected arrivaltimes and so on. Ideally, customers should be confident that theywill have the ordered goods on the agreed time. Customers alsorequire the time schedule of deliveries and need to be notified ofany changes in the delivery times. As an example, Fig. 2 illustratesa data collection system used by the TNT express company.

When monitoring delivery chains, determining a source time torecord the data is important because if the reference point is notknown, observations obtained according to different source pointscan be misleading and hence control charts lose their efficiency.For example, one might aggregate observations based on whenthe product is first sent, or when a specific segment of shippingis completed, or when another specific segment of shipping isstarted. In this paper, we set the source time as the time goodsare first sent without loss of generality.

Turning back again to Fig. 1, the quality time variables can be rep-resented by a multivariate quality vector, in which each componentrepresents a route delivery elapsed time from the origin to custom-ers. For example the total elapsed time to deliver goods from produc-tion site 1 to customers’ location 1 includes O1A, AB, BC and CD1

branches in which the four segment times add up to form the totaldelivery time. The quality vector then is monitored over time using

Fig. 2. The TNT express information system.

284 A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289

a multivariate control chart. If an assignable cause is signaled, effortsto analyze the signal should be followed to identify the root of thedelays. Assignable causes linked to an out-of-control signal mightbe a traffic jam that occurred on the main road, for example, or othercauses that simultaneously affect more than one branch in a deliverychain such as, for example, delays due to decreased speed limits.

2.3. The multivariate Hotelling’s T2 control chart

Consider a delivery process with p correlated routes, targetdelivery times vector l00 ¼ ðl01; . . . ;l0pÞ and variance–covariancematrix

P. Because of its generality, one of the most popular mul-

tivariate process control charts is the T2 control chart. It is assumedthat the joint probability distribution of the p process characteris-tics is a p-variate normal distribution. Now suppose that x1j, x2j,x3j, . . . , xnj are independent random delivery vectors, observed atsampling stage j; j = 1, 2, . . . . The procedure requires the user tocompute the sample delivery means and then the sample statistic

v2j ¼ nð�xj � l0Þ

0P�1ð�xj � l0Þ is plotted on a control chart in

sequential order, where �xj ¼ 1n

Pni¼1xij. The chart signals as soon

as v2j P k. If the sample value falls below the control limit k the

delivery process is considered in control, otherwise the deliveryprocess is said to be out of control for that period and the searchto find the assignable causes of poor delivery performance starts.Generally, the delivery chain parameters l0 and

Pare unknown

and it is necessary to estimate l0 andP

from m initial samplesby ��x ¼ 1

m

Pmj¼1�xj and S ¼ 1

m

Pmj¼1Sj respectively, where

Sj ¼ 1n�1

Pni¼1ðxij � �xjÞðxij � �xjÞ0. This stage is called phase I. The chart

statistics are calculated by T2j ¼ nðxj � ��xÞ0S�1ðxj � ��xÞ. If all the ini-

tial samples indicate that the process is in-control, the estimatedparameters shall be used for the monitoring purpose. Otherwise,the suspicious subgroups are eliminated and the parameters areestimated again.

In phase II which is the process monitoring phase, ifn > 1; mðn�1Þ�pþ1

ðmþ1Þðn�1Þp T2 is distributed as the F distribution with p andm(n � 1) � p + 1 degrees of freedom. Then each of the T2 values iscompared with

k ¼ pðmþ 1Þðn� 1Þmðn� 1Þ � pþ 1

Faðp;mðn� 1Þ � pþ 1;g ¼ 0Þ; ð1Þ

where Fa(v1,v2,g) is the upper a percentage point of the F distribu-tion with v1 and v2 degrees of freedom and non-central parameter g(see Alt, 1985).

In most cases it is difficult to obtain more than n = 1. In this casethe delivery chain parameters l0 and

Pare estimated by

�x ¼ 1m

Pmj¼1xj and S ¼ 1

2� V0Vðm�1Þ respectively, where

V ¼

x02 � x01x03 � x02

:x0m � x0m�1

2664

3775. Sullivan and Woodall (1996) showed that the

estimator S ¼ 1m�1

Pmj¼1ðxj � �xÞðxj � �xÞ0 is sensitive to outliers. The

chart statistic is T2j ¼ ðxj � �xÞ0S�1ðxj � �xÞ and the control limit is

calculated as follows:

k ¼ pðmþ 1Þðm� 1Þmðm� pÞ Faðp;m� p;g ¼ 0Þ: ð2Þ

In the next section we deal with the mathematical model andmodel assumptions regarding monitoring delivery chains.

3. Mathematical model

There are essentially two approaches to account for assignablecauses in economic control chart models. The Duncan (1956) andLorenzen and Vance (1986) models assume that there is only oneassignable cause while Duncan (1971) and Knappenberger andGrandage (1969) proposed models in which there are manydifferent assignable causes with different occurrence times. The

A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289 285

latter two papers concluded that an economic model with a singleassignable cause assumption can closely approximate the expectedminimal cost per hour of a multiple assignable cause model with aweighted average shift and weighted average time to out of control.Thus, in this paper the cost model proposed by Lorenzen and Vance(1986) is used. The assumptions summarized below are relativelystandard in most published work.

3.1. The model assumptions

1. It is assumed that the joint probability distribution of the p pro-cess characteristics is a p-variate normal distribution and so theT2 statistic follows the Fisher distribution. Since the T2 is a uni-variate statistic with a univariate distribution, one could per-form a goodness-of-fit test to evaluate the assumption orappeal to the Central Limit Theorem since a delivery time is asum of intermediate delivery times which are independentand positive. Also, observations taken from a multivariate nor-mal can be transformed to a T2 statistic having a F distribution(see Mason and Young, 2002).

2. The process is characterized by an in-control state l = l0.3. An assignable cause produces ‘‘step changes’’ in the agreed deliv-

ery times from l = l0 to l = l1. This results in a known value ofthe Mahalanobis distance by using the following equation:

d ¼ ðl1 � l0Þ0R�1ðl1 � l0Þ: ð3Þ

Data Analysis could identify these step changes quickly in prac-tice. As stated before, here we assumed only a single assignablecause model which closely approximates a multiple assignablecause model (Duncan, 1971; Knappenberger and Grandage,1969). Saniga (1992) has shown that the control charts are ro-bust to changes in the design parameters for constrained designssuch as the ones we use in this paper. Another issue here is esti-mating the parameter d using past data. The mean value of thedelivery process can be shifted to any other value and thus pastdata might not come from the same population, see Celano et al.(2011), who discuss the problem of shift size estimation for theunivariate Shewhart charts. As Saniga (1992) mentioned, the de-sign of a control chart is robust to errors in the specification ofthe out of control process parameter in the sense that misspeci-fication of the parameter does not have a large impact on theperformance of the design. Of course, if a new shift occurs pastdata analysis will not be accurate but the control chart will stillbe effective if not optimal at that point. And since the controlchart is used continuously, the data stream that is collected thatis the basis of the control chart will reflect the new value of theassignable cause immediately and, if necessary, the control chartdesign may be adjusted, although, as we note above, the robust-ness of the control chart makes this continuous adjustmentunnecessary. The same argument is true if the in control meanshifts, which can occur as a result of an intervention or learning.If this shift is permanent, the in control mean shift will requirethe design be changed over time. We wish to emphasize thatcontemporary literature such as that referenced above is basedupon the concept that the control chart is a dynamic tool subjectto change rather than a static tool since dynamic processes arethe rule rather than the exception since the advent of modernquality improvement methods. Finally, in estimating the param-eter d sometimes we do not know the status (in or out of controlor more specific states) of the past data at all. It is possible to usedata driven clustering techniques to detect out of control data(‘‘the highest times’’) in the past data set. That could be achieved,for example, according to a threshold on the aggregation level ofthose methods that would determine what should be consideredas out of control data (see for example Everitt, 1993).

4. We assume the covariance matrix R is constant over time.

5. The assignable cause is assumed to occur according to a Poissonprocess with intensity k occurrences per day, which of coursehas the memoryless property. That is, assuming that the deliv-ery process begins in the in-control state, the time interval thatthe process remains in control is an exponential random vari-able with mean 1/k days. Again, k could be estimated easilyfrom past data. This assumption is made without loss of gener-ality. Many authors have investigated design when there ismemory; see, e.g. Saniga (1979)

6. The quality control cycle starts with the in-control state andcontinues until the delivery process is repaired after an out-of-control signal. When the process goes out of control at a cer-tain level it stays out of control at that level unless the processshift has been detected and corrected. It is assumed that qualitycontrol cycle follows a renewal reward process.

3.2. The cost of monitoring delivery chains

Duncan (1956) observed that a quality control cycle is dividedinto the following four periods which are illustrated in Fig. 3.

1. In-control period,2. Time to detect the assignable cause (time to signal),3. Time to take a sample and interpret the results,4. Time to find and repair an assignable cause.

These four periods form the basis of the renewal reward processand can be used to calculate the quality control cycle cost per dayfor a specified set of design parameters.

3.2.1. Expected cycle timeThe process failure mechanism (PFM) is assumed Markovian,

which implies the distribution function is exponential. Generally,the delivery process is a continuous process, i.e. the process isnot shut down when the chart signals. Hence, the average timeuntil an occurrence of an assignable cause is 1/k. The out-of-controltime is given as the expected time of the following events:

� The adjusted average time to signal or the average time fromwhen the process shifts untill the chart gives an out-of-controlsignal which is composed of the following two components:

The time between an occurrence of an assignable cause andthe next sample, which is given by h � s where h is sampling

interval (in hours) and s ¼R ðjþ1Þh

jhkðt�jhÞe�kt dtR ðjþ1Þh

jhke�kt dt

¼ 1�ð1þkhÞe�kh

kð1�e�khÞ .

The expected time from the first sample after the processmean shifts until an out-of-control signal which is givenby h(ARL-1) where ARL is the average run length when theprocess has shifted to an out-of-control state. Note that theexpected number of samples required to detect that the pro-cess is out of control follows a geometric distribution withmean ARL = 1/(1 � b) where b is the usual probability of aType II error or the probability of failing in detecting out-of-control states.

� The expected time to take a sample of n items and interpret theresults; this depends on the number of items and is assumed tobe nE where E is the expected time to sample and interpret theresult for one item.� The expected time to find and repair the assignable cause,

which is given as T.

Therefore, the expected cycle time is given as the sum of thein-control and out-of -control periods as follows:

EðTÞ ¼ 1k� sþ nEþ hARLþ T ð4Þ

Fig. 3. A quality control cycle.

286 A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289

3.2.2. Expected cycle costMonitoring costs of the delivery chains can be divided into four

main components: the cost of delivery delays while the process isin control, the cost of delays while the process is out of control, thecost of evaluating alarms – both false and true alarms-, the cost offinding assignable causes and repairing the process and the cost ofsampling.

The expected cost of delays while the process is in and out ofcontrol is given by the following:

C0

kþ C1ðhARL� sþ nEþ TÞ ð5Þ

where C0 and C1 stand for the costs per hour for delivery delayswhen the process is in control and out of control respectively. Thedifference C1 � C0 can be considered the opportunity cost per hourof not operating under the in control state of the process; i.e. theextra costs incurred.

Considering a as the occurrence of an assignable cause, theexpected number of samples taken before a shift is given by:X1j¼0

jP½a 2 ðjh; ðjþ 1ÞhÞ� ¼X1j¼0

j½e�kjh � e�kðjþ1Þh� ¼ e�kh

ð1� e�khÞ

Hence, the expected cost of evaluating false alarms and repairingthe process is given by:

We�kh

ð1� e�khÞaþ Y ð6Þ

where a is the probability of having a Type I error, W is the cost ofinvestigating false alarms and Y stands for the cost of locating andrepairing an assignable cause.

The expected sampling cost per cycle is given by:

Sh� 1

k� sþ nEþ hARLþ T

� �ð7Þ

where S is the sampling and testing cost.Now the expected total cost per cycle, E(C), is given by the sum

of the above terms.Due to the renewal reward assumption, the expected cost per

hour is:

EðAÞ ¼ EðCÞEðTÞ ð8Þ

3.3. The optimization problem and genetic algorithm

One design method for the T2 chart is a heuristic design inwhich samples of size n are drawn every h hours, and the valueof the T2 statistic is plotted on a control chart at the level of typeI error a = 0.005. A much more preferred method is the economic

statistical design (ESD) method proposed by Saniga (1989) wherethe goal is to find the three chart parameters (k,n,h) which mini-mize the corporation’s expected cost (8) while maintaining tightstatistical control. Tight statistical control means that problemswith the delivery chain will be detected quickly and furthermore,non-existent problems will not be incorrectly detected. Therefore,the general optimization problem is defined as follows:

min EðAÞðk;n;hÞs:t :

aðk;n;hÞ 6 a0

bðk;n;hÞ 6 b0

h > nE

k;h > 0

n 2 Zþ

ð9Þ

The statistical constraints a 6 a0 and b 6 b0 are added to havegood protection against Type I and II error probability rates,respectively. The constraint h > nE ensures that the sampling inter-val is larger than the time needed to get a statistic from a samplehaving size equal to n and interpreting the results. The optimiza-tion problem (9) has both continuous and discrete decision vari-ables and a discontinuous and non-convex solution space. Thisproblem can be solved with the genetic algorithm approach whichhas successfully been applied to economic control chart designs(see, e.g. Celano and Fichera, 1999; Faraz et al., 2012; Faraz andSaniga, 2012). The genetic algorithm approach (GA) (see, e.g. Hauptand Haupt, 2002) is a method for solving both constrained andunconstrained optimization problems which is based on naturalselection, the process that drives biological evolution. GA repeat-edly modifies a population of individual solutions. At each step,GA selects individuals at random from the current population tobe parents and uses them produce the children for the next gener-ation. Over successive generations, the population evolves towardan optimal solution. GA can be applied to solve a variety of optimi-zation problems that are not well suited for standard optimizationalgorithms, including problems in which the objective function isdiscontinuous, non-differentiable, stochastic, or highly nonlinear.GA has received a great deal of attention in the recent literaturedue to the following facts:

1. GA does not rely on analytical properties and derivative infor-mation of the function to be optimized which make it well sui-ted to a wide class of optimization problems.

2. GA considers many points in the search space simultaneously,rather than a single point;

3. GA works directly with strings of characters representing theparameter set, not the parameters themselves;

A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289 287

4. GA uses probabilistic rules, not deterministic rules to guide thesearch direction of finding the optimal solution. Hence, it can beapplied for many kinds of optimization problems.

5. GA can lead to a global optimum by mutation and crossovertechniques to refrain from trapping in a local optimum.

6. GA is able to search for many possible solutions (or chromo-somes) at the same time. Hence, it can obtain the global optimalsolution efficiently.

Using GA requires one to determine the values of the most sig-nificant GA parameters, i.e. the crossover rate (rC), number of elites(Nelit), the initial population size (Npop) and the mutation rate (rM).Selecting the GA parameters optimal set is often done through trialand error experiments and is very difficult due to the many possi-ble combinations of GA parameters. Traditionally large populationshave been used to thoroughly explore complicated cost surfaces.Crossover is then the operator of choice to exploit promisingregions of phase space by combining information from promisingsolutions. The role of the mutation is somewhat nebulous. As sta-ted by Back and Schutz (1996), mutation is typically consideredas a secondary operator of little performance. He found that largervalues than typically used are best for the early stages of GA run. Inone sense, greater exploration is achieved if the mutation rate isgreat enough to take the gene into a different region of solutionspace.

In general, the parameters settings are sensitive to the costfunction, options in GA and bounds on the parameters. Conse-quently different studies result in different conclusions about theoptimum values. For example, Grefenstette (1986) proposedNpop = 30, rC = 0.95 and rM = 0.01 while Haupt and Haupt (2002)suggested Npop 6 16 and 0.05 6 rM 6 0.2. Faraz et al. (2010) deter-mined the optimal values of the GA parameters through a 23 facto-rial design with five iterations and parameter ranges25 6 NPOP 6 100, 1 6 Nelit 6 5 and 0.05 6 rC 6 0.95 to perform thestudy. The ANOVA results indicated that the optimal values ofthe three GA parameters are NPOP = 100, Nelit = 5 and rC = 0.05 thatare used here. These optimal values also are applied by Faraz andSaniga (2011) to study the economic statistical design of T2 controlcharts with variable sampling ratio schemes.

Our actual procedure is as follows:

Step 1: For satisfying the Type I error rate a 6 a0, the lower boundof the parameter k is set to the 100(1 � a0)th percentile ofthe F distribution and then a pool of chromosomes (k,n,h)having size 100 are randomly generated to get the initialpopulation. For each chromosome from the initial popula-tion which is an arbitrary subset of design parameters theType II error rate b is evaluated. The E(A) of each chromo-some that satisfies the constraint b 6 b0 is evaluated to getthe fitness function. The other chromosomes get a largepenalty of E(A) say infinity.

Step 2: Chromosomes are scaled on the basis of their fitness; theprobability to be selected from one of the genetic opera-tors is directly proportional to their fitness value: thus, achromosome with a lower E(A) (i.e. a higher fitness value)has a higher chance to be selected as a parent ormutated.

Step 3: Five chromosomes having the highest fitness values in thecurrent generation are directly copied to the nextgeneration.

Step 4: From the mating pool of 95 chromosomes, four ones areselected as parents and coupled together to get the nextgeneration offspring: the crossover selection is randombut biased by the fitness values of chromosomes withinthe current generation.

Step 5: Each couple of parents is recombined by means of thecrossover and the fitness of the two offspring are com-puted. Repeat steps 5 and 6 until four children are bornto form the new generation.

Step 6: The mutation operator is applied to the remaining chro-mosomes (96) of the current population to obtain as manyoffspring chromosomes as the parents.

Step 7: repeat the steps from 2 to 6 until the termination condi-tion is met, i.e., when the fitness function E (A) has notbeen improved in the last 1000 successive generations.

The Genetic Algorithm has been coded within the ‘‘MATLAB Ge-netic Algorithm Toolbox’’ environment.

4. An illustrative example

Suppose an Industrial Group has two main storage facilities (2origins) and three wholesalers (3 destinations, see Fig. 1). Aftercrossing the local routes with trucks (branches O1A and O2A), theproducts are transited to B (branch AB) and then to C (branchBC) by train. The goods are then delivered to three customers lo-cated in D1 (destination 1), D2 (destination 2) and D3 (destination3) centers by trucks (branches CD1, CD2 and CD3, respectively).The goods are then distributed to retail stores by the wholesalers.Fig. 1) illustrates the company delivery chain. Accordingly, theCompany is responsible for the on times deliveries of all the sixroutes and unusual delays result in fines being imposed. As is clear,all the delivery routes have AB and BC branches in common. Eachbranch consists of inevitable stops and waiting, loading, transit andunloading activities. This process has been running twice a day foralmost 3 years (One run corresponds to one travel of a train from Ato C, its content being a combination of products from O1 and O2 tobe distributed to D1, D2 and D3). Fig. 2 shows the on-line TNTexpress company information system. Fig. 4 shows the designedExcel worksheet which is helpful for calculating delivery data.

Consider the delivery chain process with p = 6 correlated deliv-ery times characteristics. Based on the latest 60 initial samples(m = 60) when the deliveries were satisfactory, the in-controldelivery process mean vector and covariance matrix are estimatedas follows (These simulated data are available upon request):

��x ¼

525049514948

2666666664

3777777775

; S ¼

2:98 2:52 1:63 4:45 3:88 2:722:37 4:18 3:73 1:85 2:31

2:13 3:38 2:68 1:982:10 2:15 3:41

1:72 4:062:21

2666666664

3777777775

For example, ��x01 is the average delivery time from origin 1 todestination 1, estimated as 52 hours and ��x02 ¼ 50 hours is the aver-age delivery time from origin 1 to destination 2, and so on. Theaverage branches times are also indicated on Fig. 1. We note thatit is not important for the Company whether or not the branchesmeet their time windows but rather overall on time delivery isthe objective. The Corporation wants to monitor the delivery pro-cess at least one time a week and the process runs twice a day,3 days a week and 10 months a year. So the sampling interval his set to fixed sampling intervals 0.5, 1, 1.5, . . . , 6.5, 7 days. Oneprocedure is to apply a heuristic design for the T2 chart. The Heu-ristic design is a design method used by practitioners where littleprior knowledge of the specific system is utilized in the designmethod. The usual argument is that heuristic design works wellfor controlling many processes but many authors have showncan yield far from optimal solutions (see e.g. Saniga, 1989). In this

Fig. 4. The data sheet in Excel.

Table 1The TNT delivery chain parameters.

S = $10 k = 0.003 E = 0.23 day C0 = $3150 C1 = $29637T = 0.62 day W = $250 Y = $10375 d = 1.5 m = 60; p = 6

288 A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289

paper the heuristic Hotelling T2 chart is chosen by the supply chainmanager in which samples of size n = 1 are drawn every day (h = 1),and the value of the T2 statistic is plotted on a control chart at thelevel of type I error a = 0.005 (resulting costs are indicated in thelast column of Table 2).

The economic statistical design (ESD) procedure aims to findthe three chart parameters (k,n,h) which minimize the corporationexpected cost (8). This method, in comparison uses GA to solve theconstrained optimization problem given by (9) and is thus optimalin finding the minimum cost solution that meets statistical con-straints. Among the chart parameters, due to the nature of the pro-cess, the sample size n is set to 1 and the parameter k iscontinuous. Therefore, the general optimization problem tailoredto the imperative of this application is:

min EðAÞs:t :

a 6 0:005b 6 0:99h > E

k > 0h ¼ 0:5ð0:5Þ7

ð10Þ

The statistical constraints a 6 0.005 and b 6 0.99 are added tohave a good protection against false alarms and seasonal changes(Type II error rate).

Table 1 gives the estimated cost parameters of the deliverychain process. Table 2 provides the ESD of the T2 control chartfor monitoring the delivery chain using different values of theparameter d as well as providing a cost comparison to the heuristicdesign. The Corporation is willing to have a good protection againstthe estimated process mean shift d = 1.5. The optimal chart param-eters are (k = 23.59,n = 1,h = 0.5), i.e., a sample of size 1 should be

taken every 12 hours (twice a day). In fact, the process should bemonitored and inspected every run. This scheme costs $4717.38a day while the corresponding heuristic design costs $6002.40 dai-ly (compare both of these to a scheme where the process is notmonitored or controlled at all. Here the opportunity cost wouldbe C1 � C0 = $29637 � $3150 = $26487 per day which quantifiesDeming (1986)s statements on the economic value of process con-trol). Comparing the heuristic method versus the optimal methodwe propose shows that the optimal method saves almost $1285per day. On an annual basis, since the delivery process is running3 days a week and 10 month a year the company can save$154200 annually. The benefits of optimal economic statistical de-sign of the control process are apparent but there are other advan-tages as well.

When the control chart indicates that unacceptable delays haveoccurred, the manager may implement previously constructed re-routing plans that may involve temporarily using alternate rail andtruck segments to reduce delays. Additionally, the logistics man-ager will not over-react to long shipment times unless they are sig-nificantly large. This will avoid costs associated with expeditingorders, as well as the added variability in delivery times comingfrom over-correction, a well-known problem associated with theuse of control charts that has been discussed at length by Deming(1986). The decision rule presented here is especially helpful fordelivery chain managers in high demand seasons in which a meanshift may occur from a shortage of railcar availability. Anothercause of such a mean shift could be the winter seasonal influence

Table 2The ESD of the T2 control chart for different values of d.

d ESD Heuristic

K n h a ARL E (A) E (A)

1.00 23.59 1 0.5 0.005 84.15 6205.97 8542.451.25 23.59 1 0.5 0.005 57.92 5358.27 7128.541.5 23.59 1 0.5 0.005 39.26 4717.38 6002.401.75 23.59 1 0.5 0.005 26.59 4262.50 5171.122.00 23.59 1 0.5 0.005 18.18 3951.01 4585.772.25 23.59 1 0.5 0.005 12.63 3741.31 4184.042.50 23.59 1 0.5 0.005 8.96 3600.74 3911.242.75 23.59 1 0.5 0.005 6.51 3506.15 3726.033.00 23.59 1 0.5 0.005 4.85 3441.91 3599.49

A. Faraz et al. / European Journal of Operational Research 228 (2013) 282–289 289

on transit time. On a national or international delivery chain net-work, driving conditions are significantly impacted by cold weath-er. At the very least, the control chart can be used to alert vehicledealers of the anticipated increase in delivery times.

5. Concluding remarks

In this paper, a new approach to manage delivery chains is pro-posed by applying an economic-statistically designed T2 controlchart for monitoring the process. The study illustrates the advanta-ges of this new approach, not only in economic benefits that can bequantified, but in other less quantifiable benefits that most cer-tainly are associated with customer satisfaction.

Acknowledgement

We thank the three anonymous referees, the Editor and prof.Giovanni Celano for their valuable comments, criticisms and valu-able suggestions.

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