Optimization of Order-Picking Problems by Intelligent

Research ArticleOptimization of Order-Picking Problems by IntelligentOptimization Algorithm

Zhong-huan Wu 1 Hong-jie Chen 2 and Jia-jia Yang 3

1Department of Business Administration Huashang College Guangdong University of Finance amp EconomicsGuangzhou 510000 China2School of Business Administration South China University of Technology Guangzhou 510000 China3Department of Public Policy Kingrsquos College London London WC2R 2LS UK

Correspondence should be addressed to Hong-jie Chen raphel888gmailcom

Received 29 April 2020 Accepted 3 June 2020 Published 23 July 2020

Guest Editor Wen-Tsao Pan

Copyright copy 2020 Zhong-huan Wu et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

To improve the efficiency of warehouse operations reasonable optimization of picking operations has become an important taskof the modern supply chain For the purpose of optimization of order picking in warehouses a new fruit fly optimizationalgorithm particle swarm optimization random weight and weight decrease model are used to solve the mathematical modelFurther optimization is achieved through the analysis of the warehouse shelves and screening of the optimal solution of thepicking time In addition simulation experiments are conducted in the MATLAB environment through programming +eshortest picking time is found out and chosen as an optimized method by taking advantage of the effectiveness of these sixalgorithms in the picking optimization and comparing the data obtained under the simulation +e result shows that theoptimization capacity of RWFOA is better and the picking efficiency is the best

1 Introduction

As a part of the logistics the efficiency of the automatedwarehouse is largely dependent on the efficiency of orderpicking +erefore the picking plays an important role inthe automated warehouse for improving the efficiency ofpicking operation Although the automated stereoscopicwarehouse provides more orderly and standardized man-agement and the error rate is also small for small batchwarehouse frequent warehousing and warehouses withvarious products logistics storage becomes more stringentand requirements for the efficiency of the logistics arehigher and thus the efficiency of picking needs to beimproved

+ere are a lot of algorithms for optimization of orderpicking all of which have made minor or major contribu-tions to the optimization of order picking [1ndash6] At presentthe ant colony algorithm [7] genetic algorithm [8] andmultipopulation fruit fly optimization algorithm [9] which

have been used to solve the picking operation problemsyielded good results Based on the existing research we willuse the new fruit fly optimization algorithm and particleswarm optimization to solve the mathematical model

+e main structure of this paper is as follows Section 1introduces the motive and purpose of this study Section 2presents the literature review Section 3 introduces researchmethodsmdashoriginal FOA original particle swarm algorithm(PSA) random weight algorithm weight decrease and re-lated literature Section 4 introduces case description Sec-tion 5 presents results and discussion and Section 6 putsforward the research conclusions and suggestions

2 Literature Review

21 Order Picking Across the various operations in awarehouse order picking is the most time-consuming op-eration in general [10] and accounts for around 55ndash75percent of total warehousing costs [11] +erefore order

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 6352539 12 pageshttpsdoiorg10115520206352539

picking has the highest priority for productivity improve-ment [12]

Order picking is a particular case of the travelingsalesman problem (TSP) +is problem introduced byDantzig et al [13] is one of the most studied problems inoperations research Efficient algorithms have been designedfor the TSP [14] +erefore in order to improve the per-formance of order picking reducing travel time is criticalSince the travel distance is proportional to travel time forpicker-to-parts system [15] minimizing the travel distance(total or average) of a picking tour is often considered as animperative factor to reduce travel time and consequentlyimprove warehouse operation efficiency [12] +ere are fourmethods to reduce the travel distance of an order picker [12]storage location assignment warehouse zoning orderbatching and pick-routing methods And this paper focuseson the pick-routing methods

To most order-picking research studies optimizationalgorithms are still the center of routing studies [16] Topursue the optimal order-picking route in a typical rectan-gular the order-picking routing problem is considered as theSTSP (Steiner traveling salesman problem) [12 17] +ere aretwo general methods to solve the STSP the first method is toreformulate an STSP into the classic TSP by computing theshortest paths between every pair of required nodes (egRenaud and Ruiz [17]) and the second one for the solution ofa STSP is by using dedicated algorithms (eg Lucie Pansart[4]) +e latter method is preferred to the former

+e dedicated algorithms include dynamic programminginteger programming and branch and bound method Al-though this kind of algorithm can get the exact solution thecalculation time is long and it is seldom used in the practicalapplication [18] +e common approximation algorithms arethe insertion algorithm the r-opt algorithm and the nearestneighbour algorithm Although this kind of algorithm canquickly get a feasible solution to the optimal solution thedegree of its close to the optimal solution is not satisfactory[19] Intelligent optimization algorithm is a more effectivealgorithm to solve this problem in recent years [20] +esealgorithms are mainly genetic algorithms [8] ant colonyalgorithm [7] particle swarm optimization [21] andmodifiedFOA eg MSFOA [22] and IFOA4WSC [23]

22 Automated Stereoscopic Warehouse Model At presentthe shelves of the automatic stereoscopic warehouse aremainly fixed shelves and each row of shelves in the ware-house is equipped with a stacker which is responsible forpicking up a cargo on the shelf +is paper takes some of theshelves in the warehouse as the object of study In an au-tomated warehouse the stacker enters from the entranceperforms order picking and chooses goods according to theprogrammed procedure Assuming that there are k cargospaces on each shelf the stacker can only get one cargo spacefor each picking removes the cargo from the shelve andtransports it to the exit and then returns to the shelves topick and the above steps were repeated As the order ofpicking is not the same the time required for each picking isnot the same We need to set the optimal sorting order so as

to minimize the time on picking and to improve the effi-ciency of order picking

+is study refers to the high-level rack model designedby Professor Ning and Hu [9] and the formulas from (1) to(8) are also proposed by them [9] +e structure is shown inFigure 1

Where the position of the column x and tier y can be setto (i j) the position at the entrance is set to (0 0) the lengthof the shelf is D and the height of the shelf is G Assuming ashelf has x columns and y tiers the goods allocation is (Y j)(m 1 2 3 k) and position is (α β) so

αm D middot i

Xminus

D

2X (1)

βm G(j minus 1)

Y (2)

Assuming that the velocity in the horizontal direction ofthe stacker is v1 and the velocity in the vertical direction is v2and the time spent by two adjacent cargo spaces m (i j) andm+ 1 (p q) in the process of picking up the goods in thehorizontal direction is t1 and t2 the operating equation usedis as follows

t1 αm minus αm+1

11138681113868111386811138681113868111386811138681113868

v1

|i minus p|D

X middot v1 (3)

t2 βm minus βm+1

11138681113868111386811138681113868111386811138681113868

v2

|j minus q|G

Y middot v2 (4)

Since the horizontal and vertical movements of thestacker occur at the same time the time of operation at theadjacent cargo space is tm and the maximum value forrunning speed t1 in the horizontal direction and the runningspeed t2 in the vertical direction is given by

tm max t1 t21113864 1113865 (5)

+en the k cargo positions are selected and the totalrunning time Tz used by the stacker is given by

Tz 1113944n

m0tm (6)

If the picking time spent by the stacker is the same eachtime that is ts then the running time of all the cargo spacesis Ts as shown in the following equation

Ts 1113944

n

m1ts kts (7)

+us the total time T of k cargo spaces is as follows

T Tz + Ts (8)

Under above circumstances we will ask for the total timeof operation of the automated warehouse stacker and theminimum value T Six intelligence algorithms includingoriginal particle swarm particle swarm weight decreaseparticle swarm randomweight original FOA fruit fly weightdecrease and fruit fly randomweight are used to evaluate theminimum value of T

2 Mathematical Problems in Engineering

3 Research Methods

Particle swarm optimization is a new algorithm in recentyears which solves the TSP problem and a good result isobtained [20] And the fruit fly optimization algorithm(FOA) is a newly developed bioinspired algorithm +econtinuous variant version of FOA has been proven to be apowerful evolutionary approach to determining the optimaof a numerical function on a continuous definition domain[24] +e FOA and PSO are also easy to program and can bemodified to other practical applications Due to these ad-vantages they have been used to solve a wide range ofoptimization problems including prediction and classifi-cation problems [25ndash27] However the FOA and PSO mustbe modified in order to effectively manage the discretevariables associated with optimization issues+erefore RWand WD were integrated into FOA and PSO to improve itsadvantage and to look for the better optimal order-pickingtime

31 Fruit Fly Optimization Algorithm (FOA) +e originalFOA was invented by Professor Pan [24] and the FOA ishighly accurate Many studies will use the FOA to solve theoptimization problem FOA can be used in any field such asmilitary engineering medical science management and fi-nancial and other fields It can also be combined with otheralgorithms complementing each other FOA is a new methodof global optimization derived from foraging behaviors of fruitflies Because a fruit fly itself is superior to other animals inperception it comes close to the food using its olfactory organknowing where the food and partners gathered and then fly tothe destination Following is the original fruit fly algorithm

(1) Set initial location of fruit flies at random (x and y aretwo coordinate axes initial position on coordinates)

InitX axis

InitY axis(9)

(2) Random directions and distance of fruit fliessearching for food relying on good sense of smellwhich is equivalent to the initial location of the fruitflies plus random flight distance

Xi X axis + RandomValue

Yi Y axis + RandomValue(10)

(3) As the location of food cannot be obtained estimatethe distance (Di) to the origin first and then calculatethe decision value of Smelli (Si) and this value is thereciprocal of Di

Di

X2i + Y2

i

1113969

Si 1

Di

(11)

(4) Substitute decision value of Smelli (Si) into the abovefunction to get the Smelli of location of fruit flies

Smelli Function(Si) (12)

(5) Locate the fruit fly with the best Smelli from fruit flies(max)

[bestSmellbestIndex] max(Smelli) (13)

(6) Retain the smell best and X-axis and Y-axis and thefruit flies will fly to this position

Smellbest bestSmell

X axis X(bestIndex)

Y axis Y(bestIndex)

(14)

(7) Enter into iterative optimization repeat steps 2ndash5and judge whether the Smelli is superior to the Smelliof the previous iteration if yes execute step 6

+e foraging process of a fruit fly group is illustrated inFigure 2 [25]

In view of the optimization of picking in this paper weknow that the range of search distance of the original fruit flyin the coordinates is limited which leads to the weak optimalperformance If the weight is added to the original FOA thesearch range of fruit flies will be enlarged which will greatlyenhance the optimization ability of fruit flies

32 Particle Swarm Optimization (PSO) Particle swarmalgorithm [28] is a kind of random search algorithm whichis a new intelligent optimization technique and can con-verge on the global optimal solution with larger probabilityPSO is derived from the study of predatory behavior of birdsa group of birds randomly search for food in a region allbirds know how far they are away from the food and thenthe simplest and most effective strategy is to search thesurrounding area of birds that is closest to the food Inspiredby this model it is applied to solve the optimizationproblem +e basic PSO is as follows

(1) Suppose in a D-dimensional target search space Nparticles form a community where the i-th particle isexpressed as a D-dimensional vector

hellip hellip hellip hellip hellip

G

(1 j) hellip hellip (i j) hellip






(1 2) hellip hellip (i 2) hellip


D(0 0)

Figure 1 Structural model of high-level rack

Mathematical Problems in Engineering 3

Xi (xi1 xi2 xiD) i 1 2 N (15)

(2) +e ldquoflyingrdquo velocity of the i-th particle is also a D-dimensional vector denoted as follows

Vi (vi1 vi2 viD) i 1 2 N (16)

(3) +e optimal position of the i-th particle searched sofar is called the individual extremum denoted asfollows

Pbest (pi1 pi2 piD) i 1 2 N (17)

(4) +e optimal position of the whole particle swarmsearched so far is called the global extremumdenoted as follows

gbest (gi1 gi2 giD) (18)

When these two optimal values are found the particleswill update their speed and position according to the fol-lowing two formulas

Vij(t + 1) wlowast vij(t) + c1r1(t)[pij(t) minus xij(t)]

+ c2r2(t)[pgj(t) minus xij(t)]

Xij(t + 1) xij(t) + vij(t + 1)

(19)

where c1 and c2 are learning factors also known as accel-eration constants r1 and r2 are uniform random numberswithin the scope [0 1] i 1 2 D vij is the velocity of theparticle vij isin [minusvmax vmax] in which vmax is a constant andthe speed of the particle is set by the user r1 and r2 arerandom numbers between 0 and 1 which increases therandomness of particle flight W refers to the extent to retainthe original speed the greater of the w is the stronger abilityof global convergence and weak ability of local convergenceand the reverse is also true

+e foraging process of a particle swarm group is il-lustrated in Figure 3 [28]

33 Weight Decrease (WD) In this paper we refer to theweight decrease and randomweight algorithmmentioned byGao [29] and theWD is based on the original PSO and FOA

+e larger weighting factor is beneficial to jump out of thelocal minimum point and is convenient for global searchand the smaller inertia factor is beneficial to the accuratelocal search of the current search area which is better foralgorithm convergence +erefore for the phenomenon thatPSO and FOA are easy to get premature and the algorithmsare easy to oscillate near the global optimal solution at a laterstage the weight of linear change can be used to reduce theinertia weight linearly from the maximum ωmax to theminimum ωmin +e formula for the number of iterationswith the algorithm is ωωmax minus (tlowast (ωmax timesωmin))tmaxwhere ωmax ωmin respectively represent the maximum andminimum values of ω t indicates the current number ofiterations and tmax indicates the maximum number ofiterations

+e weight decrease method can adjust the global andlocal search capabilities of PSO and FOA but it still has twoshortcomings first the local search ability of early iterationsis relatively weak even if the initial particles are close to theglobal optimal point it will be missed and the global searchability will become weak at the later stage so the program iscaught in the local optimal value second the maximumnumber of iterations is difficult to predict which will affectthe adjustment function of the algorithm [30]

34 RandomWeight (RW) +e random weight algorithm isbased on the original PSO and FOA In this paper the RWrefers to taking ω value randomly so that the impact of thehistorical speed of particles on the current speed is randomIn order to accord with a random number that is randomlydistributed (N(μ σ^2)) the shortcomings of ω linear decreasecan be overcome from two aspects In addition we can applythe random direction and distance of fruit flies in FOA toincrease its global search ability If the evolution is close tothe most power consumption at the beginning of evolutionthe linearity of ω decreases so the algorithm will notconverge to the best point and the random generation of ωcan overcome this limitation ω is calculated as follows

ω μ + σ lowastN(0 1)

μ μmin +(μmax minus μmin)lowast rand(0 1)(20)

where N (0 1) represents the random number of thestandard normal distribution and rand (0 1) represents arandom number between 0 and 1 Researches show thatRW-based PSO and FOA algorithm can avoid the localoptimum to a certain extent

4 Case Description

Suppose the length of the shelf is 80m the height is 8m anda complete shelf has 40 rows and 5 tiers +e lateralmovement speed Va of the stacker is 1ms and longitudinalvelocity Vb is 02ms+e picking time of each cargo space isassumed to be 10 s According to the above optimizationalgorithms Popsize1 5 and Popsize210 that is thenumber of all populations is Popsize1times Popsize2 50 +elargest number of iterations of six algorithms is 1000 timesIn terms of FOA parameter the random initial position of a

Food

Fruit fly 1(X1 Y1) Fruit fly 2

(X2 Y2)Iterative

evolution

Dist 1

Dist 3Fruit fly 3(X3 Y3)

Dist 2S1 S2

S3

Fruit fly group(X Y)

(0 0)

S2 = 1Dist 2

Figure 2 Foraging process of a fruit fly group


fruit fly swarm is [minus5 5] fruit flies searching for foodrandomly and the distance interval is [minus50 50] in terms ofPSO parameter C1 and C2 are set to be 149445 Vmax andVmin are set to be 1 popmax is set to be 50 and popmin is set tobe minus50 six algorithms are run independently of 20 times

We apply the RW and WD mathematical model to FOAand PSO and take the individual position as the encodingobject and the length of the code is a randomly generatedcargo space number We then assume that the number ofsubpopulations is Popsize1 the number of individuals ineach population is Popsize2 and the number of individualsin all populations is Popsize1timesPopsize2 and then thepopulation quantity is Popsize1timesPopsize2 If m cargospaces are randomly generated then the coding scheme ofNo b fruit flies in No a subpopulation is shown in Table 1

In order to check the optimization capability of theproposed FOA and PSO two groups of 10 cargo spaces and20 cargo spaces are randomly generated as shown inTables 2ndash5

5 Results and Discussion

+e results (subfigures) are shown below in proper orderPSO (upper left) WDPSO (center left) RWPSO (lower left)FOA (upper right) WDFOA (center right) and RWFOA(lower right)

51 Iteration Verification of 10 Cargo Spaces in Group 1According to the data of Figure 4 the optimal search time ofPSOWDPSO and RWPSO is 243 s 235 s and 234 s and theoptimal search time of FOAWDFOA and RWFOA is 236 s228 s and 226 s

According to the data of Table 6 the optimal averagesearch time of PSO is 237 s the optimal search time of FOAis 230 s and the optimization of FOA is better +e averageoptimal search time of the original WD and RW is 2345 s232 s and 231 s respectively and the optimization of RW isbetter +us RWFOA is the best

From the standard deviation in Table 7 RWFOA is thesmallest better than the other five+erefore PSO algorithmis featured with good accuracy and speed but its optimi-zation performance is worse than FOA For six differentalgorithms the optimization of RWFOA is relatively good

+e optimal picking time of 10 cargo spaces is 226 s andthe corresponding picking order is as follows8ndash5ndash6ndash9ndash2ndash1ndash10ndash7ndash4ndash3

52 Iteration of 10 Cargo Spaces in Group 2 Accordingto the data of Figure 5 the optimal search time of PSOWDPSO and RWPSO is 216 s 214 s and 212 s and theoptimal search time of FOAWDFOA and RWFOA is 209 s208 s and 207 s

According to the data of Table 8 the optimal averagesearch time of PSO is 214 s the optimal search time of FOAis 208 s and the optimization of FOA is better +e averageoptimal search time of the original WD and RW is 2125 s211 s and 2095 s respectively and the optimization of RWis better +us RWFOA is the best



Pbest

Gbest

Individual

pij (t)

pij (t) ndash xij (t) pgj (t) ndash xij (t)

xij (t)

pgj (t)

vij (t)

vij (t + 1)

Y

X

Figure 3 Process of a Particle swarm group

Table 1 Coding scheme of fruit flies randomly generated

Cargo space 1 2 7 8 mminus 1 mTier xab1 xab2 xab7 xab8 xab (nminus 1) xabnRow yab1 Yab2 yab7 yab8 yab (nminus 1) yabnSmell Sab1 Sab2 Sab7 Sab8 Sab (nminus 1) Sabn

Table 2 10 cargo spaces in Group 1

Tier 24 32 40 26 17 12 38 15 7 29Row 4 1 4 5 3 2 2 3 5 1


Tier 22 32 12 28 40 12 25 34 17 27Row 2 1 3 3 1 5 3 2 2 4


Tier 20 32 42 35 22 6 19 43 18 38Row 3 2 3 5 1 3 2 1 4 5Tier 25 10 44 16 41 17 28 3 7 15Row 1 5 2 4 4 2 1 3 5 4


Tier 8 20 22 6 12 13 28 14 34 4Row 4 3 2 4 2 1 3 6 3 1Tier 33 11 32 3 36 27 40 4 22 25Row 4 2 2 5 3 1 3 4 2 6


53 Iteration of 20 Cargo Spaces in Group 1 According to thedata of Figure 6 the optimal search time of PSO WDPSOand RWPSO is 553 s 550 s and 549 s and the optimal searchtime of FOAWDFOA and RWFOA is 545 s 543 s and 541 s

According to the data of Table 10 the optimal averagesearch time of PSO is 550 s the optimal search time of FOAis 543 s and the optimization of FOA is better +e averageoptimal search time of the original WD and RW is 549 s

240

245

250

255

260

265

270

275

280

Y

Optimization process

100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280

290Optimization process

100 200 300 400 500 600 700 800 900 10000Iteration number

(b)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

240

250

260

270

280



(d)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

240

250

260

270

280



(f )

Figure 4 Iterative changes of 10 cargo spaces in Group 1 (a) PSO (b) FOA (c) WDPSO (d)WDFOA (e) RWPSO (f ) RWFOA Note X isthe total time of iteration and Y is 1000 iterations


Table 6 Average of picking time of 10 cargo spaces in Group 1

Algorithm Original (s) WD (s) RW (s) Average (s)PSO 243 235 234 237FOA 236 228 226 230Average 2345 232 231

Table 7 Standard deviation of picking time of 10 cargo spaces in Group 1

Algorithm SDAlgorithm Original WD RWPSO 66 64 62FOA 49 38 37


210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

220

230

240

250

260

Fitn

ess

(b)


210

220

230

240

250

260

270

Fitn

ess


(c)



205

210

215

220

225

230

235

240

245

Fitn

ess

(d)

Figure 5 Continued



210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

220

230

240

250

260

Fitn

ess

(f )




Table 9 Standard deviation of picking time of 10 Cargo spaces in Group2



100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

640

660

680

700

(b)

Figure 6 Continued


5465 s and 545 s respectively and the optimization of RWis better +us RWFOA is the best


+e optimal picking time of 20 cargo spaces is 541 s andthe corresponding picking order is as follows9ndash12ndash19ndash13ndash20ndash15ndash4ndash17ndash8ndash1ndash10ndash2ndash16ndash5ndash14ndash3

54 Iteration of 20Cargo Spaces inGroup 2 According to thedata of Figure 7 the optimal search time of PSO WDPSOand RWPSO is 544 s 542 s and 540 s and the optimal search


100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)

Optimization processFi

tnes

s


540

560

580

600

620

640

660

680

(f )




Table 11 Standard deviation of picking time of 20 cargo spaces inGroup 1




100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )



time of FOA WDFOA and RWFOA is 531 s 528 s and514 s



+e optimal picking time of 20 cargo spaces is 514 s andthe corresponding picking order is as follows 8ndash18ndash19ndash4ndash5ndash1ndash12ndash2ndash10ndash6ndash16ndash15ndash20ndash14ndash11ndash7ndash9ndash3ndash13ndash17

6 Conclusion

With the increasing pursuit of efficiency in logistics ware-housing order picking has also become an important re-search and it is constantly proposed to apply a variety ofdifferent algorithms to optimize picking time +is paperassumes a model of automated warehouse shelves By re-ferring to previous studies the study is designed to set thepicking route to get the optimal picking time so as to im-prove the efficiency of order picking It has been widely usedin various industries including electronic appliancespharmaceutical logistics tobacco logistics machinery au-tomation and food industry

A new FOA PSO RW and WD are used to improveFOA and PSO and to look for the optimal order pickingtime +e result shows that the optimization capacity ofRWFOA is better and the picking efficiency is the best+erefore it can be applied to the order picking in auto-mated warehouses thereby improving warehouse operationefficiency and reducing the time cost of order picking

RWFOA is a more effective local search method whichcan be used in future work +e proposed RWFOA could beapplied to other variations of the TSP for example fixededges are listed that are required to appear in each solutionto the problem path problem or vehicle routing problemetc +erefore future work could focus on the developmentof adaptive algorithms with the implementation of other

problem-specific features that could improve the perfor-mance of the RWFOA

+is study also has certain limitations For example thepaper assumes that the stacker is moving at a constant speedbut the speed in the actual operating conditions is uncertainSecondly this paper takes part of the shelves as the object ofstudy instead of shelf-to-shelf which means it is the localoptimal in the warehouse rather than the global optimal

Data Availability

+e data used to test the algorithm are randomly generatedreaders need to pay more attention to intelligent algorithmsAnyway the data used to support the findings of this studyare available from all the authors upon request

Conflicts of Interest

+e authors declare there are no conflicts of interest re-garding the publication of this paper

Acknowledgments

+is research was financially supported by the 2018 SocialScience Planning Project of Guangzhou ldquoResearch on theConstruction and Development of Guangzhou Smart In-ternational Shipping Center Based on the One Belt OneRoad Strategyrdquo (Grant no 2018GZGJ169) and 2016 Hu-manities and Social Sciences Research Projects of Univer-sities in Guangdong Province ldquoConstruction of keydisciplines in business administrationrdquo (Grant no2015WTSCX126)

References

[1] R L Daniels J L Rummel and R Schantz L J and R SchantzldquoA model for warehouse order pickingrdquo European Journal ofOperational Research vol 105 no 1 pp 1ndash17 1998

[2] H-I Jeong J Park and R C Leachman ldquoA batch splittingmethod for a job shop scheduling problem in an MRP en-vironmentrdquo International Journal of Production Researchvol 37 no 15 pp 3583ndash3598 1999

[3] W Lu McF Duncan V Giannikas and Q Zhang ldquoAn al-gorithm for dynamic order-picking in warehouse operationsrdquoEuropean Journal of Operational Research vol 248 no 1pp 107ndash122 2016

[4] L Pansart N Catusse and H Cambazard ldquoExact algorithmsfor the order picking problemrdquo Computers amp OperationsResearch vol 100 pp 117ndash127 2018

[5] J Bolantildeos Zuntildeiga J A Saucedo Martınez T E Salais Fierroand J A Marmolejo Saucedo ldquoOptimization of the storagelocation assignment and the picker-routing problem by usingmathematical programmingrdquo Applied Sciences vol 10 no 22020

[6] H Hwang W J Baek and M-K Lee ldquoClustering algorithmsfor order picking in an automated storage and retrievalsystemrdquo International Journal of Production Research vol 26no 2 pp 189ndash201 1988

[7] M J LI X B Chen and C Q Liu ldquoSolution of order pickingoptimization problem based on improved ant colony algo-rithmrdquo Computer Engineering vol 35 no 3 pp 219ndash2212009






[8] C-M Hsu K-Y Chen and M-C Chen ldquoBatching orders inwarehouses by minimizing travel distance with genetic al-gorithmsrdquo Computers in Industry vol 56 no 2 pp 169ndash1782005

[9] X Ning and H Hu ldquoMultiple-population fruit fly optimi-zation algorithm for scheduling problem of order pickingoperation in automatic warehouserdquo Journal of LanzhouJiaotong University vol 33 no 3 pp 108ndash113 2014

[10] K J Roodbergen and R Koster ldquoRouting methods forwarehouses with multiple cross aislesrdquo International Journalof Production Research vol 39 no 9 pp 1865ndash1883 2001

[11] D M-H Chiang C-P Lin and M-C Chen ldquo+e adaptiveapproach for storage assignment by mining data of warehousemanagement system for distribution centresrdquo Enterprise In-formation Systems vol 5 no 2 pp 219ndash234 2011

[12] R De Koster T Le-Duc and K J Roodbergen ldquoDesign andcontrol of warehouse order picking a literature reviewrdquoEuropean Journal of Operational Research vol 182 no 2pp 481ndash501 2007

[13] G Dantzig R Fulkerson and S Johnson ldquoSolution of a large-scale traveling-salesman problemrdquo Journal of the OperationsResearch Society of America vol 2 no 4 pp 393ndash410 1954

[14] M Hahsler and K Hornik ldquoTSP infrastructure for thetraveling salesperson problemrdquo Journal of Statistical Softwarevol 23 no 1 pp 1ndash21 2007

[15] C G Petersen and G Aase ldquoA comparison of pickingstorage and routing policies in manual order pickingrdquo In-ternational Journal of Production Economics vol 92 no 1pp 11ndash19 2004

[16] C +eys O Braysy W Dullaert and B Raa ldquoUsing a TSPheuristic for routing order pickers in warehousesrdquo EuropeanJournal of Operational Research vol 200 no 3 pp 755ndash7632010

[17] J Renaud and A Ruiz ldquoImproving product location andorder picking activities in a distribution centrerdquo Journal of theOperational Research Society vol 59 no 12 pp 1603ndash16132007

[18] C G Petersen II ldquo+e impact of routing and storage policieson warehouse efficiencyrdquo International Journal of Operationsamp Production Management vol 19 no 10 pp 1053ndash10641999

[19] X Yu X Liao W Li X Liu and T Zhang ldquoLogistics au-tomation control based on machine learning algorithmrdquoCluster Computing vol 22 no 6 pp 14003ndash14011 2019

[20] X H Shi Y C Liang H P Lee C Lu and Q X WangldquoParticle swarm optimization-based algorithms for TSP andgeneralized TSPrdquo Information Processing Letters vol 103no 5 pp 169ndash176 2007

[21] R C Eberhart and J Kennedy ldquoA new optimizer usingparticle swarm theoryrdquo in Proceedings of the Sixth Interna-tional Symposium on Micro Machine and Human Sciencepp 39ndash43 Nagoya Japan October 1995

[22] Y Zhang G Cui J Wu W-T Pan and Q He ldquoA novelmulti-scale cooperative mutation Fruit Fly OptimizationAlgorithmrdquo Knowledge-Based Systems vol 114 pp 24ndash352016

[23] Y Zhang G Cui S Zhao and J Tang ldquoIFOA4WSC a quickand effective algorithm for QoS-aware servicecompositionrdquoInternational Journal of Web and Grid Services vol 12 no 1pp 81ndash108 2016

[24] W-T Pan ldquoA new fruit fly optimization algorithm taking thefinancial distress model as an examplerdquo Knowledge-BasedSystems vol 26 pp 69ndash74 2012

[25] H-z Li S Guo C-j Li and J-q Sun ldquoA hybrid annual powerload forecasting model based on generalized regression neuralnetwork with fruit fly optimization algorithmrdquo Knowledge-Based Systems vol 37 pp 378ndash387 2013

[26] S-M Lin ldquoAnalysis of service satisfaction in web auctionlogistics service using a combination of fruit fly optimizationalgorithm and general regression neural networkrdquo NeuralComputing and Applications vol 22 no 3-4 pp 783ndash7912013

[27] D Shan G Cao and H Dong ldquoLGMS-FOA an improvedfruit fly optimization algorithm for solving optimizationproblemsrdquo Mathematical Problems in Engineering vol 2013pp 1ndash9 2013

[28] T A M Abdel M B Abdelhalim and S E-D HabibldquoEfficient multi-feature pso for fast gray level object-trackingrdquoApplied Soft Computing vol 14 pp 317ndash337 2014

[29] F Gao Matlab Intelligent Algorithm Super Learning ManualPost amp Telecom Press Beijing China 2014

[30] Y Shi and R C Eberhart ldquoA modified particle swarm op-timizerrdquo in Proceedings of the Congress on Evolu-TionaryComputation pp 79ndash73 Washington DC USA July 1998


picking has the highest priority for productivity improve-ment [12]

Order picking is a particular case of the travelingsalesman problem (TSP) +is problem introduced byDantzig et al [13] is one of the most studied problems inoperations research Efficient algorithms have been designedfor the TSP [14] +erefore in order to improve the per-formance of order picking reducing travel time is criticalSince the travel distance is proportional to travel time forpicker-to-parts system [15] minimizing the travel distance(total or average) of a picking tour is often considered as animperative factor to reduce travel time and consequentlyimprove warehouse operation efficiency [12] +ere are fourmethods to reduce the travel distance of an order picker [12]storage location assignment warehouse zoning orderbatching and pick-routing methods And this paper focuseson the pick-routing methods

To most order-picking research studies optimizationalgorithms are still the center of routing studies [16] Topursue the optimal order-picking route in a typical rectan-gular the order-picking routing problem is considered as theSTSP (Steiner traveling salesman problem) [12 17] +ere aretwo general methods to solve the STSP the first method is toreformulate an STSP into the classic TSP by computing theshortest paths between every pair of required nodes (egRenaud and Ruiz [17]) and the second one for the solution ofa STSP is by using dedicated algorithms (eg Lucie Pansart[4]) +e latter method is preferred to the former

+e dedicated algorithms include dynamic programminginteger programming and branch and bound method Al-though this kind of algorithm can get the exact solution thecalculation time is long and it is seldom used in the practicalapplication [18] +e common approximation algorithms arethe insertion algorithm the r-opt algorithm and the nearestneighbour algorithm Although this kind of algorithm canquickly get a feasible solution to the optimal solution thedegree of its close to the optimal solution is not satisfactory[19] Intelligent optimization algorithm is a more effectivealgorithm to solve this problem in recent years [20] +esealgorithms are mainly genetic algorithms [8] ant colonyalgorithm [7] particle swarm optimization [21] andmodifiedFOA eg MSFOA [22] and IFOA4WSC [23]

22 Automated Stereoscopic Warehouse Model At presentthe shelves of the automatic stereoscopic warehouse aremainly fixed shelves and each row of shelves in the ware-house is equipped with a stacker which is responsible forpicking up a cargo on the shelf +is paper takes some of theshelves in the warehouse as the object of study In an au-tomated warehouse the stacker enters from the entranceperforms order picking and chooses goods according to theprogrammed procedure Assuming that there are k cargospaces on each shelf the stacker can only get one cargo spacefor each picking removes the cargo from the shelve andtransports it to the exit and then returns to the shelves topick and the above steps were repeated As the order ofpicking is not the same the time required for each picking isnot the same We need to set the optimal sorting order so as

to minimize the time on picking and to improve the effi-ciency of order picking

+is study refers to the high-level rack model designedby Professor Ning and Hu [9] and the formulas from (1) to(8) are also proposed by them [9] +e structure is shown inFigure 1

Where the position of the column x and tier y can be setto (i j) the position at the entrance is set to (0 0) the lengthof the shelf is D and the height of the shelf is G Assuming ashelf has x columns and y tiers the goods allocation is (Y j)(m 1 2 3 k) and position is (α β) so

αm D middot i

Xminus

D

2X (1)

βm G(j minus 1)

Y (2)

Assuming that the velocity in the horizontal direction ofthe stacker is v1 and the velocity in the vertical direction is v2and the time spent by two adjacent cargo spaces m (i j) andm+ 1 (p q) in the process of picking up the goods in thehorizontal direction is t1 and t2 the operating equation usedis as follows

t1 αm minus αm+1

11138681113868111386811138681113868111386811138681113868

v1

|i minus p|D

X middot v1 (3)

t2 βm minus βm+1

11138681113868111386811138681113868111386811138681113868

v2

|j minus q|G

Y middot v2 (4)

Since the horizontal and vertical movements of thestacker occur at the same time the time of operation at theadjacent cargo space is tm and the maximum value forrunning speed t1 in the horizontal direction and the runningspeed t2 in the vertical direction is given by

tm max t1 t21113864 1113865 (5)

+en the k cargo positions are selected and the totalrunning time Tz used by the stacker is given by

Tz 1113944n

m0tm (6)

If the picking time spent by the stacker is the same eachtime that is ts then the running time of all the cargo spacesis Ts as shown in the following equation

Ts 1113944

n

m1ts kts (7)

+us the total time T of k cargo spaces is as follows

T Tz + Ts (8)

Under above circumstances we will ask for the total timeof operation of the automated warehouse stacker and theminimum value T Six intelligence algorithms includingoriginal particle swarm particle swarm weight decreaseparticle swarm randomweight original FOA fruit fly weightdecrease and fruit fly randomweight are used to evaluate theminimum value of T


3 Research Methods




InitX axis

InitY axis(9)





Di

X2i + Y2

i

1113969

Si 1

Di

(11)






Smellbest bestSmell

X axis X(bestIndex)

Y axis Y(bestIndex)

(14)







G









D(0 0)



Xi (xi1 xi2 xiD) i 1 2 N (15)


Vi (vi1 vi2 viD) i 1 2 N (16)









(19)










4 Case Description


Food


(X2 Y2)Iterative

evolution

Dist 1


Dist 2S1 S2

S3


(0 0)

S2 = 1Dist 2
















Pbest

Gbest

Individual

pij (t)


xij (t)

pgj (t)

vij (t)

vij (t + 1)

Y

X





Tier 24 32 40 26 17 12 38 15 7 29Row 4 1 4 5 3 2 2 3 5 1


Tier 22 32 12 28 40 12 25 34 17 27Row 2 1 3 3 1 5 3 2 2 4


Tier 20 32 42 35 22 6 19 43 18 38Row 3 2 3 5 1 3 2 1 4 5Tier 25 10 44 16 41 17 28 3 7 15Row 1 5 2 4 4 2 1 3 5 4


Tier 8 20 22 6 12 13 28 14 34 4Row 4 3 2 4 2 1 3 6 3 1Tier 33 11 32 3 36 27 40 4 22 25Row 4 2 2 5 3 1 3 4 2 6




240

245

250

255

260

265

270

275

280

Y


100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280



(b)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

240

250

260

270

280



(d)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

240

250

260

270

280



(f )








210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

220

230

240

250

260

Fitn

ess

(b)


210

220

230

240

250

260

270

Fitn

ess


(c)



205

210

215

220

225

230

235

240

245

Fitn

ess

(d)

Figure 5 Continued



210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

220

230

240

250

260

Fitn

ess

(f )







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

640

660

680

700

(b)

Figure 6 Continued







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)


tnes

s


540

560

580

600

620

640

660

680

(f )








100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )







6 Conclusion






Data Availability




Acknowledgments


References





































3 Research Methods




InitX axis

InitY axis(9)





Di

X2i + Y2

i

1113969

Si 1

Di

(11)






Smellbest bestSmell

X axis X(bestIndex)

Y axis Y(bestIndex)

(14)







G









D(0 0)



Xi (xi1 xi2 xiD) i 1 2 N (15)


Vi (vi1 vi2 viD) i 1 2 N (16)









(19)










4 Case Description


Food


(X2 Y2)Iterative

evolution

Dist 1


Dist 2S1 S2

S3


(0 0)

S2 = 1Dist 2
















Pbest

Gbest

Individual

pij (t)


xij (t)

pgj (t)

vij (t)

vij (t + 1)

Y

X





Tier 24 32 40 26 17 12 38 15 7 29Row 4 1 4 5 3 2 2 3 5 1


Tier 22 32 12 28 40 12 25 34 17 27Row 2 1 3 3 1 5 3 2 2 4


Tier 20 32 42 35 22 6 19 43 18 38Row 3 2 3 5 1 3 2 1 4 5Tier 25 10 44 16 41 17 28 3 7 15Row 1 5 2 4 4 2 1 3 5 4


Tier 8 20 22 6 12 13 28 14 34 4Row 4 3 2 4 2 1 3 6 3 1Tier 33 11 32 3 36 27 40 4 22 25Row 4 2 2 5 3 1 3 4 2 6




240

245

250

255

260

265

270

275

280

Y


100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280



(b)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

240

250

260

270

280



(d)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

240

250

260

270

280



(f )








210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

220

230

240

250

260

Fitn

ess

(b)


210

220

230

240

250

260

270

Fitn

ess


(c)



205

210

215

220

225

230

235

240

245

Fitn

ess

(d)

Figure 5 Continued



210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

220

230

240

250

260

Fitn

ess

(f )







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

640

660

680

700

(b)

Figure 6 Continued







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)


tnes

s


540

560

580

600

620

640

660

680

(f )








100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )







6 Conclusion






Data Availability




Acknowledgments


References





































Xi (xi1 xi2 xiD) i 1 2 N (15)


Vi (vi1 vi2 viD) i 1 2 N (16)









(19)










4 Case Description


Food


(X2 Y2)Iterative

evolution

Dist 1


Dist 2S1 S2

S3


(0 0)

S2 = 1Dist 2
















Pbest

Gbest

Individual

pij (t)


xij (t)

pgj (t)

vij (t)

vij (t + 1)

Y

X





Tier 24 32 40 26 17 12 38 15 7 29Row 4 1 4 5 3 2 2 3 5 1


Tier 22 32 12 28 40 12 25 34 17 27Row 2 1 3 3 1 5 3 2 2 4


Tier 20 32 42 35 22 6 19 43 18 38Row 3 2 3 5 1 3 2 1 4 5Tier 25 10 44 16 41 17 28 3 7 15Row 1 5 2 4 4 2 1 3 5 4


Tier 8 20 22 6 12 13 28 14 34 4Row 4 3 2 4 2 1 3 6 3 1Tier 33 11 32 3 36 27 40 4 22 25Row 4 2 2 5 3 1 3 4 2 6




240

245

250

255

260

265

270

275

280

Y


100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280



(b)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

240

250

260

270

280



(d)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

240

250

260

270

280



(f )








210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

220

230

240

250

260

Fitn

ess

(b)


210

220

230

240

250

260

270

Fitn

ess


(c)



205

210

215

220

225

230

235

240

245

Fitn

ess

(d)

Figure 5 Continued



210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

220

230

240

250

260

Fitn

ess

(f )







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

640

660

680

700

(b)

Figure 6 Continued







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)


tnes

s


540

560

580

600

620

640

660

680

(f )








100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )







6 Conclusion






Data Availability




Acknowledgments


References


















































Pbest

Gbest

Individual

pij (t)


xij (t)

pgj (t)

vij (t)

vij (t + 1)

Y

X





Tier 24 32 40 26 17 12 38 15 7 29Row 4 1 4 5 3 2 2 3 5 1


Tier 22 32 12 28 40 12 25 34 17 27Row 2 1 3 3 1 5 3 2 2 4


Tier 20 32 42 35 22 6 19 43 18 38Row 3 2 3 5 1 3 2 1 4 5Tier 25 10 44 16 41 17 28 3 7 15Row 1 5 2 4 4 2 1 3 5 4


Tier 8 20 22 6 12 13 28 14 34 4Row 4 3 2 4 2 1 3 6 3 1Tier 33 11 32 3 36 27 40 4 22 25Row 4 2 2 5 3 1 3 4 2 6




240

245

250

255

260

265

270

275

280

Y


100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280



(b)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

240

250

260

270

280



(d)

230

240

250

260

270

280

290

Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

240

250

260

270

280



(f )








210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

220

230

240

250

260

Fitn

ess

(b)


210

220

230

240

250

260

270

Fitn

ess


(c)



205

210

215

220

225

230

235

240

245

Fitn

ess

(d)

Figure 5 Continued



210

220

230

240

250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

220

230

240

250

260

Fitn

ess

(f )







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

640

660

680

700

(b)

Figure 6 Continued







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)


tnes

s


540

560

580

600

620

640

660

680

(f )








100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )







6 Conclusion






Data Availability




Acknowledgments


References







































240

245

250

255

260

265

270

275

280

Y


100 200 300 400 500 600 700 800 900 10000X

(a)

Fitn

ess

230

240

250

260

270

280



(b)

230

240

250

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Y


100 200 300 400 500 600 700 800 900 10000X

(c)

Fitn

ess

220

230

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260

270

280



(d)

230

240

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Y


100 200 300 400 500 600 700 800 900 10000X

(e)

Fitn

ess

220

230

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280



(f )








210

220

230

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270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

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260

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ess

(b)


210

220

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250

260

270

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ess


(c)



205

210

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240

245

Fitn

ess

(d)

Figure 5 Continued



210

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250

260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(e)



200

210

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230

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250

260

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ess

(f )







100 200 300 400 500 600 700 800 900 10000X

540

560

580

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620

640

660

680

700

Y

(a)


Fitn

ess


540

560

580

600

620

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660

680

700

(b)

Figure 6 Continued







100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

Y

(c)


Fitn

ess


540

560

580

600

620

640

660

680

(d)


100 200 300 400 500 600 700 800 900 10000X

500

550

600

650

700

Y

(e)


tnes

s


540

560

580

600

620

640

660

680

(f )








100 200 300 400 500 600 700 800 900 10000X

540

560

580

600

620

640

660

680

700

Y

(a)


Fitn

ess


520

540

560

580

600

620

640

660

(b)


Fitn

ess


540

560

580

600

620

640

660

(c)


Fitn

ess


520

530

540

550

560

570

580

590

600

(d)


Fitn

ess


540

550

560

570

580

590

600

(e)


Fitn

ess


500

550

600

650

(f )







6 Conclusion






Data Availability




Acknowledgments


References










































210

220

230

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260

270

Y

100 200 300 400 500 600 700 800 900 10000X

(a)



200

210

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260

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ess

(b)


210

220

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ess


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Figure 5 Continued



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6 Conclusion






Data Availability




Acknowledgments


References






































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6 Conclusion






Data Availability




Acknowledgments


References










































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(f )







6 Conclusion






Data Availability




Acknowledgments


References






































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6 Conclusion






Data Availability




Acknowledgments


References









































6 Conclusion






Data Availability




Acknowledgments


References





























































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Optimization of Order-Picking Problems by Intelligent