Research ArticleA Multifactory Integrated Production and DistributionScheduling Problem with Parallel Machines andImmediate Shipments Solved by Improved WhaleOptimization Algorithm

Vahid Abdollahzadeh ,1 Isa Nakhaikamalabadi ,2 Seyyed Mohammad Hajimolana ,1

and Seyyed Hesamoddin Zegordi3

1Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran2Department of Industrial Engineering, University of Kurdistan, Sanandaj, Iran3Department of Industrial Engineering, Tarbiat Modares University (TMU), Tehran, Iran

Correspondence should be addressed to Isa Nakhaikamalabadi; nakhaikamalabadi.i@wtiau.ac.ir

Received 26 April 2018; Revised 30 August 2018; Accepted 13 September 2018; Published 2 December 2018

Academic Editor: Roberto Natella

Copyright © 2018 Vahid Abdollahzadeh et al. This 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.

This paper discusses the integrated scheduling of production and distribution operations in a multifactory supply chain with make-to-order production system. For the production side of the supply chain, we considered distributed parallel-established factorieswith identical parallel machines available at each factory. We assumed that the factories could produce all customers’ orderswith different production rates and costs. For the distribution side of the supply chain, we considered a limited number ofhomogeneous vehicles that immediately distribute the finalized orders to the customers. Then, a mixed-integer nonlinearprogramming model is developed to determine the detailed scheduling of production and distribution that minimizes the totalcosts of the supply chain including production, distribution, and late delivery costs. To solve the real-world scale problems, wedeveloped a new whale optimization algorithm (WOA). Moreover, we conducted computational experiments by generatingseveral test problems to evaluate the proposed algorithm. Statistical analysis showed that the proposed algorithm has betterperformance than traditional WOA for different scales of the problem. Moreover, it confirms the capability of the improvedwhale optimization algorithm (IWOA) to solve the medium-scale instances; however, the results indicate the better performanceof genetic algorithm (GA) for the large-scale instances.

1. Introduction

The increasing competition in different marketplaces andthe necessity of appropriate responses to the rapid changesin market requirements have made companies shift from asingle production plant to multiple factories adjacent to theircustomers in different geographical areas. Such strategiesreduce the transportation costs of the supply chain and allowcompanies to utilize economic privileges such as availabilityand low expense of raw materials, expert labors, energy, andtax allowances. The advent of new joint venture companies

in cooperation with global brands in developing countriesis an indication of the fact.

Multifactory supply chains are classified in different cate-gories such as ownership structure of factories, interactionmechanism between factories, and shop configuration. Insome multifactory supply chains, all plants belong to the samecompany, while in some others, each factory belongs to differ-ent companies. For the first type, all factories work together tofulfill the objectives of the main company while in the secondcategory, factories cooperate to maximize self-benefits. Differ-ent mechanisms of interactions in the factories throughout the

HindawiComplexityVolume 2018, Article ID 5120640, 21 pageshttps://doi.org/10.1155/2018/5120640

supply chain may lead to parallel, series, or network structures.Parallel factories can produce various types of products. Inseries structures, each production plant produces semifinishedproducts for the next factory. The combination of parallel andseries structures makes the network one [1]. Shop configura-tion in each factory can be considered the same as basic sched-uling problem including single machine, parallel machine,flow shop, job shop, and open shop [2].

Numerous applications of the multifactory supply chainsin the last two decades attracted the attention of researcherson multifactory scheduling problems in different industriessuch as electronics, food and chemical process industries,automotive companies, and electric power generating indus-tries. Furthermore, integrating distribution and productionscheduling operations as two costly functions of the supplychains made another subject of interest in applied operationsresearch named integrated production and distributionscheduling problem (IPDSP). Independent optimization ofthe two following operations, as a traditional solution, wouldnot necessarily lead to global optimality of the whole net-work. Thus, it is critical to consider interactions between pro-duction and distribution operations in the supply chain [3].

Besides, IPDSP in single factories was widely studied inthe last decade. The researchers and practitioners extensivelyconsider multifactory scheduling problem with differentshop configurations and objective functions as maximizingtotal tardiness of the orders. Nevertheless, there is a limitedamount of literature for the multifactory integrated produc-tion and distribution scheduling problem (MIPDSP) to min-imize the total costs of the supply chain.

This paper is aimed at integrating the production andimmediate distribution scheduling of various orders at paral-lel factories with identical parallel machines. The best alloca-tion of the facilities to the orders along with the bestscheduling of production and distribution pursue minimiz-ing the operational costs including production, transporta-tion, and late delivery. Therefore, the contributions of thispaper are as follows:

(i) Integrated mathematical formulation of productionand immediate distribution scheduling in a parallelmultifactory supply chain with a limited numberof vehicles

(ii) Considering trade-offs between late delivery, pro-duction, and transportation costs in make-to-orderproduction systems to make decisions on thedetailed scheduling of production and distribution

(iii) Considering different orders with different amountsand due dates for any of the customers compatiblewith their dynamic lot sizing

(iv) Considering the limited and equal fleet sizes at anyof the factories with homogeneous vehicles

(v) Improving newly proposed WOA to solve MIPDSP

(vi) Introducing a new procedure used in the meta-heuristic algorithms to ensure the feasibility of thesolutions.

The remainder of this paper is divided into the followingsections. Section (2) consists of a brief literature review. Sec-tion (3) defines the problem including assumptions and themathematical formulation. Section (4) presents the solutionapproach consisting of a new proposed procedure usedwith metaheuristic algorithms and improved whale optimi-zation algorithm (IWOA). Section (5) presents computa-tional experiments’ results. Conclusions and the trend offurther researches are given in Section (6).

2. Literature Review

The existing papers regarding the scheduling of productionand distribution operations can be grouped into three differentcategories: (i) IPDSP for single-factory supply chains, (ii) pro-duction scheduling problem in multifactory supply chains,and (iii) production and distribution scheduling problem inthe multifactory environment. In this section, we review therecent papers proposed for each category in brief.

Chen [4] proposed the latest review article on IPDSP.They presented a unified model representation scheme andclassified the existing models by the shipping and deliverymethods. The IPDSP in the multifactory environment hadbeen declared as a research gap by that time. In a more recentwork on deterministic approaches of IPDSP, Guo et al. [5]presented a MINLPmodel for integrated production and dis-tribution scheduling with parallel machines in a make-to-order supply chain. They developed a harmony search algo-rithm and a heuristic to solve the problem. Sawik [6] consid-ered two inconsistent objectives including costs and servicelevel in a three-echelon supply chain and addressed a bi-objective stochastic MIPmodel for joint selection of suppliersand scheduling of production and distribution for somenumerical examples using the CPLEX solver. Cheng et al.[7] considered integrated scheduling of production and dis-tribution problem to minimize the total cost of productionand distribution. In the production side, there are capaci-tated batch-processing machines. In the distribution side,there is a third party logistic (3PL) provider to deliverthe produce batches. They proposed a metaheuristic algo-rithm based on colony optimization method to solve theproduction scheduling and a heuristic algorithm for thedistribution part to minimize total costs of scheduling forproduction and distribution operations in a single factory.Behnamian and Ghomi [2] discussed the significance ofmultifactory supply chains and for the first time provideda comprehensive review of the present papers.

A limited number of papers are available for the multifac-tory scheduling problem with parallel machines. Cicirello andSmith [8] presented a dynamicmultifactory problemwith par-allel machines and applied wasp-like agents for distributedcoordination of agents with parallel machines to decide onthe acceptance of an arriving job on a machine queue. Hooker[9] combined MILP and constraint programming to solve themultifactory scheduling problem. They allocated the jobs tofactories using MILP and scheduled on facilities using con-straint programming (CP) and the two parts of the problemlinked using logic-based Benders decomposition. Kerkhoveand Vanhoucke [10] studied a real problem for the production

2 Complexity

scheduling of knitted fabrics in a multifactory environmentwith unrelated single machines in each factory. They pre-sented a hybrid metaheuristic, which combines simulatedannealing and genetic algorithm to solve the large-scale prob-lems. Behnamian and Ghomi [11] considered the multifactoryproblem with several parallel identical machines in each fac-tory. The objective function for the problem was to minimizethe maximum completion time of the jobs available at thebeginning of the time horizon. They presented a MILP modelfor the problem as well as a heuristic and genetic algorithm forthe solution. The presented that the modeling approach hasbeen severely criticized by Yazdani et al. [12]. They analyzedand showed that the proposed model and related algorithmare suffering from serious shortcomings. They introducedthree newmodels for minimizingmakespan and total comple-tion time in this paper. Furthermore, three efficient metaheur-istics are put forward based on an artificial bee colony.

A limited literature is available for the integration ofproduction and transportation scheduling with the serialmultifactory structure. As the latest research, Karimi andDavoudpour [1] proposed a branch and bound algorithmfor the scheduling of production and transportation problemin a serial multifactory production system. They also con-sider the same problem with batch delivery [13] so that theprocessed jobs should wait until all the jobs of the same batchare complete. Moreover, they developed their model andconsidered waiting time to start the process at the receivingfactory [14]. A time-indexed formulation with an LP relaxa-tion of binary variables is proposed; they considered dead-lines for the same problem and suggested a new approachin the imperialistic competitive algorithm (ICA) to solvethe large-scale problems. Terrazas-Moreno and Grossmann[15] considered a multiproduct integrated scheduling of pro-duction and distribution problem for the petrochemicalindustry. The partial production of demand is allowed indifferent complexes.

To the best of our knowledge, the single paper for the inte-grated scheduling of production and distribution problem in amultifactory environment with a make-to-order productionsystem is proposed by Sun et al. [16]. They consideredMIPDSP for a particular supply chain with two different trans-portation modalities: inland and maritime transportation.They also proposed a two-level genetic algorithm to solvethe problem, one level for the assignment part of the problemand the other one for scheduling of the assigned jobs.

The MIPDSP studied in this research is strongly differentfrom previous ones in supply chain structure and assump-tions. The most appropriate production system for suchsupply chains is the make-to-order production system withimmediate shipments, without any temporary warehousesand intermediate inventory. The real-world instances areproject-based procurements in construction projects, auto-motive industry, knitting and clothing industry, dairyproducts, etc.

3. Problem Description

In this paper, we consider a supply chain with a set of parallelfactories F, F = 1,… , f in different geographical locations

owned by a holding company. The factories are controlledthrough a central planning system. There aremα, α ∈ F paral-lel identical machines in each factory. Factories’ productionspeeds are different due to technological differences. Thereare a set of geographically dispersed customers L, L = 1,… , l . Each customer issues several orders with differentsizes. The production cost and time of the orders are differentat any of the factories. There are a limited number ofcapacitated vehicles to deliver the orders immediately afterproduction. The order sizes are smaller than the vehicles’capacity. The orders’ delivery cost is proportional to the typeof products and traveling distance. An order can wait in thefactory until its transportation starts without any holdingcost. The vehicles deliver the orders to the customers on orlater than their due dates, and the company incurs a time-dependent penalty for the late orders. Since all the factoriesbelong to the same company, the most appropriate opti-mization objective is the minimization of operational costsincluding production, transportation, and late delivery costs.Figure 1 shows the supply chain configuration.

3.1. Assumptions

(i) All parameters and required data are deterministicand available when scheduling is undertaken.

(ii) There are orders from different customers availableat time zero.

(iii) The order sizes are smaller than the capacity of thevehicles.

(iv) An equal number of homogeneous vehicles areavailable at each factory.

(v) All factories can produce all the customers’ orders.

(vi) Each order can be processed at most on onemachine, and any machine can process at mostone order at any time.

(vii) Factories can be idle, but machines cannot be idlewhen there are unprocessed orders in the plant.

MachineFactoryCustomer

Transportation

Figure 1: Supply chain structure.

3Complexity

(viii) Different orders of a customer can be processed invarious plants.

(ix) Preemption is not allowed for production anddelivery operations.

(x) The vehicles cannot be idle when there are undeliv-ered orders in a factory.

3.2. Mathematical Formulation. This section presents themathematical formulation for MIPDSP which is the firststep to understanding the characteristic of these problems.Table 1 presents the notations used in the model. A solu-tion for MIPDSP includes (i) plants’ allocation to theorders, (ii) the orders’ production scheduling on machines,and (iii) transportation scheduling of orders from facto-ries. We propose two lemmas used for simplification ofthe problem modeling.

Lemma 1. The different numbers of available vehicles at var-ious factories influence the assignment of jobs to the plants.

Proof. Consider two factories f1 and f2 with v1 and v2 homo-geneous vehicles and assume that v1 > v2. Consider only twoorders i, j that belong to two different customers in differ-ent geographical positions with the same size and distancefrom any of the factories but with different due dates di,dj and di > dj. Consider that f1 processes order j and f2processes order i. It means that the factory with more vehi-cles processes the most critical order. Changing the alloca-tion of the factories will not improve the objective functionso that the allocation of factories to the orders affects theobjective function and rationally the factories with moreavailable vehicles should produce the most critical orders.

Lemma 2. The same numbers of available vehicles at eachfactory does not influence the assignment of the machinesand sequencing of orders.

Proof. Consider two factories f1, f2 and the orders i, jthat belong to two different customers in different geograph-ical positions with the same distance from any of the fac-tories. Both of the orders have the same due date, and the

Table 1: Definition of sets, indexes, parameters, and variables.

Sets Definition

L Set of customers

N Set of orders

Nj Order set for customer j

F Set of factories

Mα Machine set for factory α

Indexes Definition

r, i Order indexes

j, k Customer indexes

α Factory index

γ Machine index

Parameters Definition

ni/nj/nα/nγ Number of orders/batches/factories/machines

Qij The size for order i of customer j

dij The due date for order i of customer j

πijLate delivery cost for each unit of order

i of customer j

pijαProcessing time for order i of customer

j at factory α

l jα Distance between customer j and factory α

vs Mean vehicle speed in a unit of distance

ctijTransportation cost for order i of customer

j per unit of distance

cpijαProduction cost for orderi of customer j at factory α

Decision variables Definition

xij,αγ1 if the order i of customer j is processedat factory α on machine γ; 0 otherwise

yij,rk,αγ1 if the order i of customer j is processedbefore the order r of customer k at factory

α on machine γ; 0 otherwise

srk,αγ1 if the order r of customer k is scheduledas the first job at factory α on machine

γ; 0 otherwise

eij,αγ1 if the order i of customer j is processedas the last job at factory α on machine

γ; 0 otherwise

srk,α′1 if the order r of customer k is thefirst job delivered from factory α; 0

otherwise

erk,α′1 if the order r of customer k is thelast job delivered from factory α;

0 otherwise

zij,rk,α1 if the order i of customer j is deliveredbefore the order r of customer k at factory

α; 0 otherwise

psij Nonnegative variables for productionstart time of order i of customer j

tsij Nonnegative variables for transportationstart time of order i of customer j

Table 1: Continued.

Intermediatevariables

Definition

pijNonnegative variables for production

completion time of the order i of customer j

TijNonnegative variables for traveling time of the

order i of customer j

CijNonnegative variables for delivering time of

the order i of customer j

MVPij,rk

Maximum of production completion time andvehicle available time to transport order r

of customer k when it is transportedimmediately after order i of customer j

4 Complexity

production and their delivery times are the same. The num-ber of vehicles available at each factory is equal. Consider thatf1 processes order i and f2 processes order j then changingthe allocation of factories will not result in the worst objectivefunction value. Therefore, the factories have the same prior-ity in assigning the orders. Furthermore, if we allocate factoryf1 to process both of the orders and schedule order i before j,in this case, changing the sequence of the orders will notresult in a worse objective function value. Hence, it is con-cluded that the equal number of vehicles has no influenceon the sequencing and scheduling of orders to the machines.

Considering the lemmas, the optimal sequencing solu-tion of the problem with the limited and equal number ofhomogeneous vehicles available at each factory without theloss of generality is the same as the optimal sequencing solu-tion with single homogeneous vehicles available at eachplant. For the simplification of the proposed model, weassume that there are single vehicles available at each factoryand the vehicles at factories are homogeneous. The simplifiedMINLP model is as follows:

Min  〠j∈L

〠i∈N j

Qij〠α∈F

cpijα 〠γ∈Mα

xij,αγ

+〠j∈L

〠i∈N j

〠α∈F

l jαctij 〠γ∈Mα

xij,αγ

+〠j∈L

〠i∈N j

πijQij Cij − dij ,

1

subject to 〠α∈F

〠γ∈Mα

xij,αγ = 1, ∀j ∈ L, i ∈Nj, 2

〠j∈L

〠i∈N j

〠α∈F

〠γ∈Mα

yij,rk,αγ + srk,αγ = 1,

 ∀k ∈ L, r ∈Nk, i ≠ r or j ≠ k,3

〠k∈L

〠r∈Nk

〠α∈F

〠γ∈Mα

yij,rk,αγ + eij,αγ = 1,

 ∀j ∈ L, i ∈Nj, i ≠ r or j ≠ k,4

xij,αγxrk,αγ ≥ yij,rk,αγ, ∀i ≠ r or j ≠ k, 5

〠j∈L

〠i∈N j

sij,αγ ≤ 1, ∀α ∈ F, γ ∈Mα, 6

〠j∈L

〠i∈N j

eij,αγ ≤ 1, ∀α ∈ F, γ ∈Mα, 7

sij,f m ≤ xij,αγ, ∀j ∈ L, i ∈Nj, α ∈ F, γ ∈Mα, 8

eij,f m ≤ xij,αγ, ∀j ∈ L, i ∈Nj, α ∈ F, γ ∈Mα, 9

yij,rk,αγ + yrk,ij,αγ ≤ 1, ∀j, k ∈ L, i ∈Nj, r ∈Nk, i ≠ r or j ≠ k,

10

〠k∈L

〠r∈Nk

yij,rk,αγ + yrk,ij,αγ + sij,αγ ≤ 2,

 ∀j ∈ L, i ∈Nj, α ∈ F, γ ∈Mα, i ≠ r or j ≠ k,11

〠k∈L

〠r∈Nk

yij,rk,αγ + yrk,ij,αγ + eij,αγ ≤ 2,

 ∀j ∈ L, i ∈Nj, α ∈ F, γ ∈Mα, i ≠ r or j ≠ k,12

Pij = psij + 〠α∈F

〠γ∈Mα

Qijpijαxij,αγ, ∀j ∈ L, i ∈Nj, 13

psij +M 1 − 〠α∈F

〠γ∈Mα

sij,αγ ≥ 0,

 ∀j, k ∈ L, i ∈Nj, r ∈Nk, i ≠ r or j ≠ k,14

M 1 − 〠α∈F

〠γ∈Mα

sij,αγ − psij ≥ 0,

 ∀j, k ∈ L, i ∈Nj, r ∈Nk, i ≠ r or j ≠ k,15

psrk − psij ≥ 〠α∈F

〠γ∈Mα

Qijpijαyij,rk,αγ −M 1 − 〠α∈F

〠γ∈Mα

yij,rk,αγ ,

 ∀j, k ∈ L, i ∈Nj, r ∈Nk, i ≠ r or j ≠ k,16

psrk − psij ≤ 〠α∈F

〠γ∈Mα

Qijpijαyij,rk,αγ +M 1 − 〠α∈F

〠γ∈Mα

yij,rk,αγ ,

 ∀j, k ∈ L, i ∈Nj, r ∈Nk, i ≠ r or j ≠ k,17

〠j∈L

〠i∈N j

〠α∈F

zij,rk,α + srk,α′ = 1,

 ∀α ∈ F, k ∈ L, r ∈Nk, i ≠ r or j ≠ k,18

〠k∈L

〠r∈Nk

〠α∈F

zij,rk,α + erk,α′ = 1,

 ∀α ∈ F, j ∈ L, i ∈Nj, i ≠ r or j ≠ k,19

zij,rk,α + zrk,ij,α ≤ 1, ∀α ∈ L, k ∈ L, r ∈Nk, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,

20

〠k∈L

〠r∈Nk

zij,rk,α + zrk,ij,α + srk,α′ ≤ 2,

 ∀j ∈ L, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,21

〠k∈L

〠r∈Nk

zij,rk,α + zrk,ij,α + erk,α′ ≤ 2,

 ∀j ∈ L, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,22

〠j∈L

〠i∈N j

sij,α′ ≤ 1, ∀α ∈ F, 23

〠j∈L

〠i∈N j

eij,α′ ≤ 1, ∀α ∈ F, 24

〠k∈L

〠r∈Nk

zij,rk,α + zrk,ij,α ≥ sij,α′ ,

 ∀j ∈ L, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,25

5Complexity

〠k∈L

〠r∈Nk

zij,rk,α + zrk,ij,α ≥ eij,α′ ,

 ∀j ∈ L, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,26

〠γ∈Mα

xij,αγ 〠γ∈Mα

xrk,f m ≥ zij,rk,α,

 ∀j ∈ L, k ∈ L, r ∈Nk, i ∈Nj, α ∈ F, i ≠ r or j ≠ k,27

Cij ≥ tsij + Tij, ∀j ∈ L, i ∈Nj, 28

Tij = 〠α∈F

〠γ∈Mα

l jαvs xij,αγ, ∀j ∈ L, i ∈Nj, 29

Cij ≥ dij, ∀j ∈ L, i ∈Nj, 30

tsij ≥ Pij −M 1 − 〠α∈F

sij,α′ , ∀j ∈ L, i ∈Nj, 31

tsij ≤ Pij +M 1 − 〠α∈F

sij,α′ , ∀j ∈ L, i ∈Nj, 32

MVPij,rk =max Prk〠α∈F

zij,rk,α, tsij + 2〠α∈F

〠γ∈Mα

l jαvs xij,αγ ,

 ∀j ∈ L, k ∈ L, r ∈Nk, i ∈Nj, i ≠ r ≠ k,33

tsrk ≥MVPij,rk −M 1 − 〠α∈F

zij,rk,α ,

 ∀j ∈ L, k ∈ L, r ∈Nk, i ∈Nj, i ≠ r or j ≠ k,34

tsrk ≤MVPij,rk +M 1 − 〠α∈F

zij,rk,α ,

 ∀j ∈ L, k ∈ L, r ∈Nk, i ∈Nj, i ≠ r or j ≠ k,35

xij,αγ, yij,rk,αγ, zij,rk,α ∈ 0, 1 , 36

sij,αγ, eij,αγ, sij,α′ , eij,α′ ∈ 0, 1 , 37

psij, tsij, Tij, Pij, Cij, MVPij,rk ∈ℝ+ 38

The objective function (i) minimizes the total produc-tion, transportation, and late delivery costs. The unit produc-tion costs for any of the orders are different in factories.For making the problem more realistic, it is assumed thattransportation and late delivery costs of the product unitdepend on the type of goods. The distribution cost of ordersdepends on the type of products and the transportationcost per unit of product and distance. The late deliverycost also depends on product type and delivery delay time.The first statement of the objective function represents thetotal production cost of the orders. The second statementdefines total distribution cost, and the last statement showsthe total late delivery cost.

Since each order should be processed in one factory andon one machine, Constraints (2) are established to assure it.

Constraints (3) ensure that any of the orders is a successorto another order or the first scheduled order on machines,and Constraints (4) ensure each order is the predecessor ofanother order or the last scheduled order on machines. Non-linear constraints (5) establish interconnections betweenmachine allocation and orders scheduling subproblems.According to this constraint, if two different machines pro-cess two distinct orders, they should not be a successor or apredecessor to each other. Constraints (6) guarantee theselection of at most one order as first order on a machine,and Constraints (7) guarantee the same for the last order.Constraints (8) allocate a machine to the first scheduledjobs on the machines, and constraints (9) do the samefor the last scheduled job on machines. Constraints (10)state that an order cannot be both predecessor and successoron a machine at the same time. Constraints (11) control thefirst scheduled order on a machine not to be proceeded bythe last scheduled order, and Constraints (12) control the lastorder of a production scheduling on a machine not to beproceeded by the first scheduled order on the machine.Constraints (13) determine the production completion oforders, which is equal to the sum of the production start timeof the order and its processing duration. Constraints (14) and(15) set the first order’s production to start at the time zero.Constraints (16) and (17) ensure that the difference betweenproduction start times of two consecutive orders on the samemachine is equal to the processing time of the successor.Constraints (18) assure that any order should be a successoror the first order delivered from a factory, and constraints(19) guarantee that any order should be a predecessor orthe last order delivered from a factory. Constraints (20)ensure that any two orders transported from a plant not tobe a predecessor and a successor to each other at the sametime. Constraints (21) control the first scheduled order fortransportation in a factory not to be proceeded by the lastscheduled order, and Constraints (22) control the last orderof shipping schedule in a plant not to be proceeded by thefirst order. Constraints (23) guarantee that there is at mostone order at the beginning of transportation scheduling ateach factory and, Constraints (24) guarantee at most oneorder at the end of transportation scheduling. Constraints(25) and (26) ensure that the orders selected as first and lastorder for transportation in a factory are in the transportationscheduling of the factory. Constraints (27) establish the inter-connections between assignment and transportation sched-uling subproblems. Constraints (28) calculate the deliverytime of the orders that is the sum of transport start andtravel times. Constraints (29) determine the traveling timebetween factory and customer proportional to the distanceand vehicles’ mean speed. Constraints (30) guarantee thatthe orders’ delivery times are later than their due dates.Constraints (31) and (32) ensure the immediate transpor-tation of produced orders. Constraints (33) determine fea-sible transportation start time. Transportation of an ordershould not start earlier than its production completiontime and vehicle available time which is already transport-ing the previous order. Thus, these constraints determinethe maximum of production completion time and thevehicle available time for the consecutively transported

6 Complexity

orders. Constraints (34) and (35) set the traveling starttime of the orders. Constraints (36), (37), and (38) indi-cate the restrictions on binary and nonnegative variablesincluding decision and intermediate variables.

4. Solution Approach

Since the assignment and parallel machine scheduling prob-lems are both NP-hard problems, without loss of generality,the multifactory integrated scheduling of production and dis-tribution problem is NP-hard as well [17, 18]. As a result, wepropose a simulation-based intelligent procedure linked withmetaheuristic algorithms to solve the large-scale problems.It is worth mentioning that every metaheuristic algorithmcan be linked to the proposed procedure. In this research,recently proposed WOA is utilized to solve the schedulingproblems for the first time. The helix-shape movementinspired from social behavior of humpback whale makesWOA different with existing algorithms. This mechanismprovides a superior exploitation capability to the algorithmin solving the complex problems.

Furthermore, we improved the whale optimization algo-rithm which is called IWOA and evaluated its performancewithin the proposed procedure. We also evaluated the valid-ity of the presented model and procedure by solving severalrandom test problems in different sizes.

4.1. Solution Representation. The proposed problem includesseveral binary variables like the assignment of orders to facto-ries and assignment of orders to machines. In this research,we use the random key method to represent the solutions.Each solution is represented by a linear chromosome includ-ing a certain number of genes with uniform random values in[0,1]. The length of the chromosomes is variable based on thenumber of customers, number of orders, number of factories,and number of machines. The structure of chromosomes isillustrated in Figure 2.

4.2. Simulation-Based Intelligent Procedure. Constraints sat-isfaction is a critical issue in metaheuristic algorithms for cre-ating and updating the populations. MIPDSP is a complexproblem that includes several interrelated subproblems andconstraints that need to be feasible during the executionof metaheuristic algorithms. For example, a vehicle shouldship an order from a factory that is produced in, an order’sproduction should be scheduled in a plant that is assignedto, and the shipment of an order should be planned inthe available time of vehicles. The penalty function is a tra-ditional approach to deal with these kinds of constraints.Using penalty functions for the violating constraints doesnot necessarily lead to a final feasible solution. Thus, we pro-posed a four-step intelligent procedure based on random

simulation of operational situation and metaheuristic algo-rithms. In the proposed procedure, simulation of operationsguarantees always feasible solutions and metaheuristic algo-rithms ensure intelligent search within the feasible solutions.Figure 3 illustrates the steps of the procedure.

Step 1. Initial random populationThe proposed procedure is initiated with a random pop-

ulation in which the solutions are represented in random keyalphabet. Each random solution in initial population isdenoted by u i, j, α, γ ∈ 0, 1 .

Step 2. Generate priority vectorsAt this step, the procedure decodes the random variable

u i, j, α, γ to a feasible solution x i, j, α, γ ∈ 0, 1 by deter-mining the priority of orders to assign factories andmachines. Algorithm 1 represents the pseudocode of decod-ing mechanism.

Step 3. Operation simulationAt this step, all the operational conditions including

order assignment to factories, production scheduling onmachines, and transportation scheduling are simulatedaccording to priority vectors determined at Step 2. Interme-diate variables like production completion times, deliverytimes, and delays are calculated accordingly.

Step 4. Fitness evaluationThe objective function is calculated at this step based

on the main and intermediate variables. Moreover, the bestsolution within the population is determined if the stoppingcriterion was not satisfied.

Step 5. Evolutionary processesAt this step, evolutionary processes are implemented on

the population to generate a new population to continuethe procedure in Step 2.

To clarify the decoding mechanism, we proposed anillustrative example with two customers and two orders foreach. We used Algorithm 1 to generate a random feasiblesolution to produce the orders at two factories with two par-allel machines i, j, α, γ = 1, 2 . Consider the initial randomsolution which is represented in Figure 4. The algorithmreshapes the chromosome to u i, j, α, γ which is illustratedin Table 2. According to the algorithm, when i = 1 the sumshould be 3 and when i = 2 the sum should be 3.6; on theother hand, when j = 1 the sum should be 2.7 and when j =2 the sum should be 3.6. Thus, the algorithm adjusts the pri-

ority vectors I = 2, 1 and J = 2, 1 Then, the algorithmassigns random machines at random factories to the orderswith the sequence (1) second order of customer two, (2) sec-ond order of the customer one, (3) then the first order of thecustomer two, and (4) first order of the customer one. Wealso used the same concept to determine the productionand distribution sequencing of the orders.

4.3. Improved Whale Optimization Algorithm. The whaleoptimization algorithm is one of the newly nature-inspiredmetaheuristic algorithms, which is classified as swarm-

⋯

ni × nj × n�훼 × n�훾

Figure 2: Chromosome structure.

7Complexity

based algorithms [19]. This algorithm mimics the socialbehavior of whales and their bubble-net hunting strategy.This algorithm assumes that the current best position is thetarget pray or is close to the optimal solution, so the othersearch agents will try to update their position towards thebest search agent. This algorithm includes a new exploitationmechanism known as “bubble-net attacking method” andexploration mechanism called “search for prey.” The algo-rithm starts with a set of random solutions. At each iteration,search agents (whales) update their position towards eitherthe best search agent or a randomly chosen agent. Search

agents update their position towards the best position usingthe following equations:

D t = C X∗ t − X t , 39

X t + 1 = X t + A t ⋅ D t , 40

where t indicates the iteration index, A and D are coefficient

vectors, X∗ is best position vector, and A · D is an element-by-element multiplication of two vectors. While updating theposition of search agents, the algorithm makes a randomselection between two approaches designed for the exploita-tion phase: (i) shrinking encircling mechanism and (ii) spiral

Step 1: Generate priority vectors

Step 2: Operation simulation

Step 3: Fitness evaluation

(i) Fitness evaluation of solutions(ii) Determining the best solution

(iii) Evaluating the stopping criterion

Step 4: Evolutionary processes

Step 0: Initialrandom population (i) Orders’ priority vector

(ii) Customers’ priority vector

(i) Exploration and exploitation

(ii) Updated population based onoperators

metaheuristic algorithm

(i) Orders’ production scheduling on

(ii) Orders’ distribution scheduling(iii) Main and intermediate variables’

machines

calculation

Figure 3: Intelligent procedure based on simulation and metaheuristic algorithms.

Input datai, j, α, γ: indexes and number of orders, customers, factories, machines;npop: number of population;sort: sort descendingun i, j, α, γ : random variable ∈ 0, 1procedurei, sum j, α, γ =∑j∑α∑γun i, j, α, γ ;

I → i of sorted i, sum j, α, γ by sum j, α, γ as the priority vector of orders;j, sum i, α, γ =∑i∑α∑γun i, j, α, γ ;;

j → j of sorted j, sum α, γ by sum α, γ as the priority vector of customers;

for i ∈ I

for j ∈ Jselect a random factory α to be assigned to order i of customer jselect a random machineγ at factory α α, γ to be assigned to order i of customer j i, jset xn i, j, α, γ = 1;if order i, j is proceeding the order i′, j′ on α, γ then: yn i, j, i′, j′, α, γ → 1, zn i, j, i′, j′, α → 1

end;end;

Algorithm 1: Decoding Mechanism P1 .

0.80.20.50.30.40.70.10.60.70.40.10.30.60.10.20.3

Figure 4: An example chromosome.

8 Complexity

updating position. The following equations are designed toperform a random search by updating all search agentstowards a random solution in the exploration phase.

D′ t = C Xrand t − X t , 41

X t + 1 = Xrand t − A t ⋅ D t 42

While conducting computational experiments usingbasic WOA, we observed some shortcomings in exploringsolutions. Early convergence of WOA resulted in weak solu-tions compared with the other algorithms like GA and PSO.It seemed that the functions (41) and (42) designed for therandom search of the algorithm are not able to ensure agood exploration of the solution region. To resolve theproblem, we designed two perturbation mechanisms forboth exploration and exploitation phases of the algorithmwithout disturbing the rationale behind the basic algorithmas follows:

(a) On the exploitation phase, while using the shrink-ing mechanism, we considered adding a randomvector to the search agents updated towards the bestsolution. We inspired this mechanism from theswarm-based firefly algorithm to increase explora-tion capability of WOA and diversification of solu-tions. We used (44) instead of (40) while updatingthe search agents using the shrinking mechanism inthe improved algorithm.

D t = C X∗ t − X t , 43

X t + 1 = X t − A t ⋅ D t − α ϵ t 44

(b) On the exploration phase, besides using a randomsearch agent for updating the current population,we considered a partial random movement consid-ering the direction of the best search agent. Theproposed improvement mechanism is designed toprovide the algorithm with maximum explorationwhile updating the population. We used the fol-lowing equations instead of (42) in our algorithm.

Dd t = X∗ t − X t ,

X t + 1 = Xrand t − A t ⋅ D′ t

+ ϵ t ⋅ Dd t ,

45

where ϵ t in the third term of both equations is avector of random numbers drawn from Gaussianor uniform distribution or any random vectoraccording to the domain of the decision variables,

and α is a constant number in 0, 1 and Dd t isthe vector from the current population towards thebest search agent.

For detailed information on this algorithm and the equa-

tions for updating coefficients A and D , you can refer to therecently published paper [19]. Algorithm 2 proposes thepseudocode of the IWOA using P1.

4.4. Solution Algorithm. The integrated nature of the problemwith several interrelated subproblems creates a complexstructure to MIPDSP, which results in too many constraintgroups and decision variables. To confront this complexity,a new algorithm is proposed to solve the large-scale problemswith respect to the proposed procedure. The steps of the algo-rithm are presented in Figure 5.

5. Computational Experiments

This section represents the results of computational exper-iments designed to examine the validity of the proposedsolution approach using IWOA and other metaheuristicalgorithms. We performed statistical analysis using one-way analysis of variance (ANOVA) technique to comparethe performance of IWOA with WOA as well as differentversions of two prevalent metaheuristic algorithms, GAand PSO.

5.1. Genetic Algorithm (GA). GA is an intelligent searchalgorithm that seeks optimal solutions for the NP-hardoptimization problems. Furthermore, a standard and mod-ified genetic algorithm has been used extensively to solvescheduling and routing problems [20–23]. Thus, standardGA with uniform crossover and mutation operators as wellas a new version of GA with multiparent crossover operator(GA-MPC) [24] are utilized as measures to evaluate theefficiency of the proposed IWOA.

Table 2: Reshaped chromosome to un i, j, α, γ .

i j α γ u

1 1 1 1 0.3

1 1 1 2 0.2

1 1 2 1 0.1

1 1 2 2 0.6

1 2 1 1 0.3

1 2 1 2 0.1

1 2 2 1 0.4

1 2 2 2 0.7

2 1 1 1 0.6

2 1 1 2 0.1

2 1 2 1 0.7

2 1 2 2 0.4

2 2 1 1 0.3

2 2 1 2 0.5

2 2 2 1 0.2

2 2 2 2 0.8

9Complexity

5.2. Particle Swarm Optimization (PSO). Since PSO is aswarm-based intelligent algorithm, it can be considered as auseful measure to evaluate the effectiveness of the proposedIWOA. Moreover, PSO is applied to solve several discreteoptimization problems like machine scheduling, routing,and location-allocation problems [25–27]. Thus, canonicalPSO and standard PSO 2011 called SPSO [28] are usedwithin the proposed simulation-based algorithm (Figure 2)to evaluate IWOA for randomly generated instances.

5.3. Experiment Setup. Several experiments in different sizesincluding (i) small-scale, (ii) medium-scale, and (iii) large-scale instances are conducted to cover all the configurationsof the problem. For our experiments, metaheuristic algo-rithms including GA, GA-MPC, PSO, SPSO, WOA, and

IWOAwere coded inMATLAB (R2015b). Moreover, the lin-earized MINLP model was coded in the General AlgebraicModeling System (GAMS). Then, we generated several ran-dom instances for each size of the problem. Each experimentin this paper includes a random instance which is solved 30times using each metaheuristic algorithm. Moreover, smalland medium-scale instances are solved once with GAMS,CPLEX solver 12.2, restricting the run time to 500 seconds.

The experiments were performed on a Notebook PC IntelCore i7 (2.93GH and 4GB of RAM). We applied a design ofexperiments to tune the parameters of IWOA, PSO, GA, andGA-MPC which are represented in Table 3. Furthermore, wetuned the parameters of SPSO based on [28].

To evaluate the performance of the metaheuristic algo-rithms, we define three metrics: (i) percentage of deviation

Input datainput l, b, αInitializefor n = 1 to npop;generate un i, j, f ,m ∈ 0, 1 for all i, j, f , m;

decode the continuous vector u n to binary variable X n ∈ 0, 1 using P1 ;evaluate the fitness of the initial population;end;save the best solution→ X∗;Main stepswhile t≤ maximume number of iterations;update (A, C);update p as random variable ∈ 0, 1 ;for n = 1 to npopif 2 p ≤ 0 5if 2 A ≤ 1;update the current search agent un t using improved shrinking encircling mechanism;

D t = C u∗ t − u t

u t + 1 = u t − A t D t + α ε tend if 2if 2 A > 1select a random search agent from urand t ;update the current search agent using improved equations

D′ t = C urand t − u t

Dd t = u∗ t − u t

u t + 1 = urand t − A t D′ t + ε t Dd t ;end if 2

end if 1if 3 p > 0 5update the current search agent un t using spiral updating mechanism;end if 3

decode the continuous vector u n t + 1 ∈ 0 1 to the binary variable X n t + 1 ∈ 0, 1 using P1 ;implement the production and delivery scheduling considering the constraints;

evaluate the fitness of the X n t + 1 ;end;update the X∗ if better;t = t + 1;end while;return X∗;

Algorithm 2: Pseudocode for Improved WOA.

10 Complexity

(PD) as represented by (46), (ii) standard deviation (SD), andaverage of CPU times (ACT).

PDa =Fita − Fit∗

Fit∗ × 100 46

PDa is the percentage of deviation of algorithm a, Fita isthe fitness of algorithm a, and Fit∗ is the best solution everfound for a random instance. For small-size instances, Fit∗is the optimal solution of the GAMS CPLEX solver, and formedium- and large-scale problems, it is the best solution everfound by metaheuristic algorithms or the GAMS solver.Since each experiment includes 30 observations, we usedthe mean value of PDs (MPD) to evaluate the metaheuristicalgorithms’ performances. Equation (47) representsMPDa as

MPDa =130〠

30

i=1PDai

, 47

Yes

No

Yes

No

Yes

No

Start

Generate I , J using P1

Select the first i ∈ I

Select the first j ∈ J

Assign a random machine at random factory for order i of customer J

Schedule production operations

J = J − {j}

J = Ø

Schedule distribution operations

Generate random keys for initial population or updated random keys

Calculate fitness of solution

Solutions < Npop

Implement the steps of the metaheuristic algorithm (IWOA/WOA/GA/GA-MPC/PSO/SPSO)

End

→ →

→

→

→

→

I = �휙→

→

I = I − {i}→ →

Figure 5: Simulation-based algorithm.

Table 3: Parameters of metaheuristic algorithms.

Algorithm Parameters Small-scaleMedium- andlarge-scale

IWOA, WOA

Npop 40 80

Iterations 300 500

b 6 10

GA (GA-MPC)

Npop 40 80

Iterations 200 300

Pc 0.6 (0.5) 0.6 (0.5)

Pm 0.4 (0.5) 0.4 (0.5)

PSO

Npop 40 80

Iterations 200 300

W 0.3 0.3

C1 3 3

C2 3 3

11Complexity

Table4:Average

ofobjectivevalues

formetaheuristicalgorithmsandmeanPD

s(%)forsm

all-scaleinstances.

i/j/f/m

Num

ber

ofvariables

GAMSCPU

time(s)

GAMS’

objectivevalue

IWOA

WOA

GA

GA-M

PC

PSO

SPSO

Average

ofobj.values

MPD

Average

ofobj.values

MPD

Average

ofobj.values

MPD

Average

ofobj.values

MPD

Average

ofobj.values

MPD

Average

ofobj.values

MPD

2/2/2/2

112

0.353

4.61E+08

4.61E+08

04.61E+08

04.61E+08

04.61E+08

04.61E+08

04.61E+08

0

2/2/2/2

112

0.326

1.36E+09

1.36E+09

01.36E+09

01.36E+09

01.36E+09

01.36E+09

01.36E+09

0

2/2/2/2

112

0.312

7.24E+08

7.24E+08

07.24E+08

07.24E+08

07.24E+08

07.24E+08

07.24E+08

0

2/2/2/2

112

0.308

5.72E+08

5.72E+08

05.72E+08

05.72E+08

05.72E+08

05.72E+08

05.72E+08

0

2/2/2/2

112

0423

2.43E+09

2.43E+09

02.43E+09

02.43E+09

02.43E+09

02.43E+09

02.43E+09

0

2/2/2/2

112

0.321

3.16E+08

3.16E+08

03.16E+08

03.16E+08

03.16E+08

03.16E+08

03.16E+08

0

2/2/2/2

112

0.354

1.95E+09

1.95E+09

01.95E+09

01.95E+09

01.95E+09

01.95E+09

01.95E+09

0

2/2/2/2

112

0.236

8.63E+08

8.63E+08

08.63E+08

08.63E+08

08.63E+08

08.63E+08

08.63E+08

0

2/2/2/2

112

0.325

7.49E+08

7.49E+08

07.49E+08

07.49E+08

07.49E+08

07.49E+08

07.49E+08

0

2/2/2/2

112

0.247

1.38E+09

1.38E+09

01.38E+09

01.38E+09

01.38E+09

01.38E+09

01.38E+09

0

3/2/2/2

240

67.4

1.77E+11

1.79E+11

1.6

1.79E+11

1.6

1.79E+11

1.6

1.79E+11

1.6

1.79E+11

1.6

1.79E+11

1.6

3/2/2/2

240

196.3

1.79E+10

1.79E+10

01.79E+10

01.79E+10

01.79E+10

01.79E+10

01.79E+10

0

3/2/2/2

240

20.59

2.83E+11

2.83E+11

02.83E+11

02.83E+11

02.83E+11

02.83E+11

02.83E+11

0

3/2/2/2

240

171.03

8.82E+11

8.82E+11

08.82E+11

08.82E+11

08.82E+11

08.82E+11

08.82E+11

0

3/2/2/2

240

393.95

1.32E+09

1.36E+09

3.5

1.36E+09

3.5

1.36E+09

3.5

1.36E+09

3.5

1.36E+09

3.5

1.36E+09

3.5

3/2/2/2

240

58.43

4.80E+09

4.92E+09

2.3

4.92E+09

2.3

4.92E+09

2.3

4.92E+09

2.3

4.92E+09

2.3

4.92E+09

2.3

3/2/2/2

240

110.52

7.31E+10

7.31E+10

07.31E+10

07.31E+10

07.31E+10

07.31E+10

07.31E+10

0

3/2/2/2

240

93.44

6.41E+09

6.41E+09

06.41E+09

06.41E+09

06.41E+09

06.41E+09

06.41E+09

0

3/2/2/2

240

213.67

9.32E+09

9.32E+09

09.32E+09

09.32E+09

09.32E+09

09.32E+09

09.32E+09

0

3/2/2/2

240

43.72

1.42E+09

1.44E+09

1.23

1.44E+09

1.23

1.44E+09

1.23

1.44E+09

1.23

1.44E+09

1.23

1.44E+09

1.23

2/3/2/2

240

68.56

4.81E+10

4.81E+10

04.81E+10

04.81E+10

04.81E+10

04.81E+10

04.81E+10

0

2/3/2/2

240

234.79

4.59E+09

4.64E+09

1.04

4.64E+09

1.04

4.64E+09

1.04

4.66E+09

1.34

4.65E+09

1.21

4.65E+09

1.21

2/3/2/2

240

62.99

5.30E+09

5.32E+09

0.4

5.32E+09

0.9

5.42E+09

2.1

5.46E+09

2.81

5.46E+09

2.1

5.46E+09

2.1

2/3/2/2

240

26.5

1.52E+10

1.67E+10

8.9

1.69E+10

9.8

1.69E+10

9.6

1.70E+10

10.3

1.67E+10

8.9

1.67E+10

8.9

2/3/2/2

240

229.23

3.38E+10

3.46E+10

2.4

3.46E+10

2.4

3.46E+10

2.4

3.46E+10

2.4

3.46E+10

2.4

3.46E+10

2.4

2/3/2/2

240

279.3

2.37E+09

2.46E+09

3.4

2.46E+09

3.4

2.46E+09

3.4

2.48E+09

4.23

2.46E+09

3.4

2.46E+09

3.4

2/3/2/2

240

95.34

1.69E+09

1.71E+09

1.03

1.73E+09

2.34

1.71E+09

1.03

1.74E+09

2.65

1.71E+09

1.03

1.71E+09

1.03

2/3/2/2

240

162.49

3.47E+10

3.52E+10

1.45

3.52E+10

1.45

3.52E+10

1.45

3.52E+10

1.45

3.52E+10

1.45

3.52E+10

1.45

2/3/2/2

240

348.26

2.04E+11

2.04E+11

02.04E+11

02.04E+11

02.04E+11

02.04E+11

02.04E+11

0

2/3/2/2

240

279.36

1.60E+09

1.64E+09

2.12

1.64E+09

2.35

1.64E+09

2.12

1.64E+09

2.12

1.64E+09

2.12

1.64E+09

2.12

12 Complexity

Table5:MeanPD

%,stand

arddeviation

SD×10

8,average

ofCPUtime(s),andgapof

GAMS(%

)formedium-scaleinstances.

i/j/f/m

Num

berof

variables

IWOA

WOA

GA

GA-M

PC

PSO

SPSO

GAMS

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

PD

GAP

2/4/2/2

416

0.2

0.02

160.80

0.07

160.30

0.14

141.43

0.52

130.30

0.03

83.25

0.63

1111.2

57.6

2/5/2/2

640

4.1

0.19

186.30

0.24

184.80

0.19

166.23

0.17

145.30

0.24

99.12

0.62

1425.7

54.8

2/6/2/2

912

3.3

0.54

163.60

0.87

207.70

1.14

149.36

0.97

153.90

1.10

87.12

3.82

1450.1

58.2

2/7/2/2

1232

3.8

0.83

235.01

1.40

224.40

0.84

197.84

0.73

185.40

1.24

119.70

4.85

1860.4

61.3

2/8/2/2

1600

5.6

1.50

198.20

1.43

205.80

1.20

197.34

1.01

193.50

0.90

156.56

1.31

2480.5

62.7

2/9/2/2

2016

4.4

1.20

218.70

4.10

227.20

1.70

2211.5

2.45

194.20

2.15

138.22

5.32

2380.7

63.5

2/10/2/2

2480

7.6

2.40

2713.4

3.24

258.90

1.90

2113.6

4.51

197.30

2.30

1211.9

3.39

2180.6

59.1

2/5/3/2

960

6.5

0.29

299.46

0.90

228.20

0.57

2711.9

0.43

266.80

5.40

199.73

0.53

2522.9

57.3

2/6/3/2

1368

6.2

0.67

2212.1

0.89

239.20

0.89

2013.2

1.73

205.70

0.99

149.34

1.82

2339.1

58.3

2/7/3/2

1848

8.5

0.80

1813.3

1.34

199.10

0.80

2216.1

1.06

1911.2

1.87

1213.5

0.58

2128.3

61.4

2/8/3/2

2400

6.8

2.20

2611.7

3.8

2610.9

2.40

2516.3

4.23

2310.3

2.20

2215.5

5.36

2783.5

62.7

2/9/3/2

3024

11.5

1.40

2813.2

2.15

2311.6

2.55

2817.5

3.44

2612.1

2.86

1321.3

3.58

30138

64.2

2/10/3/2

3720

12.8

1.80

3121.5

3.16

2511.5

1.66

2514.1

3.49

189.50

2.26

1415.3

7.84

26158.3

63.6

3/3/2/2

522

1.97

0.31

182.99

0.35

192.36

0.39

207.34

0.67

192.45

0.28

106.13

0.56

1710.8

59.4

3/4/2/2

912

5.10

0.86

205.75

0.95

195.81

1.01

1910.6

1.14

184.69

1.02

117.84

1.23

1354.3

56.7

3/5/2/2

1410

2.9

0.57

196.40

2.30

175.30

1.80

229.92

2.03

195.60

1.90

1210.5

2.01

2341.1

59.5

3/6/2/2

2016

4.3

0.85

206.13

1.90

203.90

1.42

1911.1

4.48

173.20

1.45

107.88

4.35

1950.5

60.2

3/7/2/2

2730

4.1

2.05

269.21

3.30

245.70

2.61

2012.2

4.43

183.90

2.06

148.23

4.36

2559.4

56.1

3/8/2/2

3552

5.7

3.70

2511.1

4.36

247.50

2.90

2312.7

4.84

206.30

4.00

1212.9

5.71

2881.3

57.4

3/9/2/2

4482

4.3

2.47

276.61

4.95

223.90

4.90

218.42

4.11

214.70

6.01

158.19

4.43

3455.2

67.6

3/10/2/2

5520

106.72

4519.5

11.2

5710.7

9.33

3914.1

9.60

3010.5

8.13

1618.4

13.9

3078.5

62.3

3/10/3/2

8280

9.8

4.80

4718.1

5.60

559.10

5.70

5111.8

4.80

559.20

5.50

1918.1

5.88

3697.9

74.3

4/2/2/2

416

0.33

0.20

191.70

0.20

150.41

0.16

163.45

0.24

140.40

0.10

94.23

0.12

1413.4

55.9

4/3/2/2

912

6.1

0.86

1811.7

1.43

1612.0

1.32

1919.2

1.74

197.04

1.10

1212.8

1.78

2351.6

57.5

4/4/2/2

1600

4.4

0.86

279.70

1.86

227.40

1.43

2415.5

2.31

214.10

1.10

1111.9

1.68

2059.4

55.4

4/5/2/2

2480

2.8

1.38

298.20

3.34

256.20

2.15

2012.6

5.32

184.50

2.15

109.82

4.39

3041.1

61.5

4/6/2/2

3522

4.1

1.25

277.50

2.32

266.40

3.55

254.71

12.2

203.40

1.33

145.32

2.27

1967.7

62.8

4/7/2/2

4816

5.6

1.88

317.81

4.08

298.20

4.69

288.85

4.06

225.20

4.12

189.38

5.48

3360.4

66.9

4/8/2/2

6272

8.04

4.11

4712.7

8.51

416.10

4.20

3211.2

6.94

276.70

7.70

913.4

10.2

2480.2

68.4

4/9/2/2

7920

8.91

7.7

5211.5

12.6

497.17

10.8

3013.3

15.5

475.90

13.8

1810.5

11.3

3480.11

72.3

4/10/2/2

9760

9.40

10.6

5811.9

16.3

4511.8

14.0

4913.4

15.8

478.20

20.7

1710.2

11.4

31106.8

89.1

4/5/3/2

3720

11.1

1.74

2815.3

2.69

2715.3

4.14

2220.6

3.60

2011.1

4.08

1217.6

3.04

29135.9

58.0

4/6/3/2

5328

11.3

3.58

4412.7

4.09

3911.7

4.09

3210.9

3.88

288.56

3.01

1519.2

3.02

40180.8

69.3

5/2/2/2

640

1.60

0.23

155.05

0.33

164.93

0.34

167.7

0.65

174.97

0.48

95.47

7.08

1714.89

57.3

5/4/2/2

2480

7.58

2.35

229.48

3.98

2212.5

4.24

2015.7

1.97

216.19

2.77

1215.6

4.49

19102.5

58.1

13Complexity

Table5:Con

tinu

ed.

i/j/f/m

Num

berof

variables

IWOA

WOA

GA

GA-M

PC

PSO

SPSO

GAMS

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

MPD

SDACT

PD

GAP

5/6/2/2

5520

5.79

3.6

3910.3

5.1

338.59

8.56

2911.4

4.79

238.41

9.29

1713.1

6.91

3878.8

66.0

6/4/2/2

3552

8.46

3.00

2615.1

4.64

247.71

2.09

2215.2

2.88

195.97

3.03

1314.3

3.37

3298.7

61.2

14 Complexity

where PDaiis the percentage of deviation for observation i of

algorithm a.In addition to graphical analysis of the results, one-way

ANOVA test is conducted on the observations’ MPDs forthe statistical analysis of the results.

5.4. Results and Discussions. To evaluate the complexity ofthe problem, a single experiment is designed to solve amedium-scale random instance with 416 decision variables(2/4/2/2) using the GAMS CPLEX solver relaxing runtime

constraint. The best feasible solution obtained after a weekruntime was just 9% better than the best solution found bymetaheuristic algorithms in less than a minute. The resultsreveal the weakness of exact algorithms implemented by con-ventional software and the necessity of the computationalmethods to solve medium- and large-scale problems.

5.4.1. Comparison of Small-Scale Instances. For small-scaleproblems, 30 random numerical experiments are conductedconsidering all of the available configurations of the problem.

Table 6: Results of one-way ANOVA test for medium-scale instances.

DF SS MS F-Value P-Value Test results

IWOA-WOA 1 270.3 270.34 16.21 0.00 Means are significantly different

Error 72 1200.9 16.68

Total 73 1471.2

IWOA-GA 1 40.89 40.89 3.51 0.065 Means are equal

Error 72 839.14 11.65

Total 73 880.04

IWOA-PSO 1 0.396 0.396 0.04 0.839 Means are equal

Error 72 686.03 9.528

Total 73 686.43

GA-PSO 1 33.25 33.25 3.20 0.078 Means are equal

Error 72 747.25 10.38

Total 73 780.49

DF: degree of freedom; SS: sum of squares; MS: mean squares.

0

5

10

15

20

2541

652

264

091

296

013

6816

0018

4820

1624

8024

8030

2435

5237

2044

8253

2855

2079

2097

60

MPD

Number of variables

IWOA and WOA

IWOAWOA

0

5

10

15

20

416

522

640

912

960

1368

1600

1848

2016

2480

2480

3024

3552

3720

4482

5328

5520

7920

9760

MPD

Number of variables

IWOA and GA

IWOAGA

02468

101214

416

522

640

912

960

1368

1600

1848

2016

2480

2480

3024

3552

3720

4482

5328

5520

7920

9760

MPD

Number of variables

IWOA and PSO

IWOAPSO

0

50

100

150

200

416

522

640

912

960

1368

1600

1848

2016

2480

2480

3024

3552

3720

4482

5328

5520

7920

9760

MPD

Number of variables

IWOA and GAMS

IWOAGAMS

Figure 6: MPD comparison of IWOA with WOA, GA, PSO, and GAMS for medium-scale instances.

15Complexity

MPDs of these experiments are calculated using the optimalsolutions obtained from the GAMS CPLEX solver in lessthan 500 seconds. The reasonable optimal solutions for thesmall-scale problems confirm the validity of the proposedMINLP model. Table 4 represents the comparative per-formance of different algorithms. Zeros in the MPD columnof this table indicate the conformity of metaheuristicalgorithms’ solutions with the optimal solutions and insistthe validity of metaheuristic algorithms as well. For smallinstances, while MPD of an instance is not zero, it meansthat there is at least one run among multiple runs of the algo-rithm that the solution of the metaheuristic algorithm wasnot equal to the optimal solution. Moreover, there was noexperiment for IWOA with worse performance than othermetaheuristic algorithms while there are two experimentsfor basic WOA that their MPDs are worse than other algo-rithms. Thus, we can conclude that the competence of IWOA

is at least the same as GA, GA-MPC, PSO, and SPSO forsmall-scale problems.

5.4.2. Results’ Comparison for Medium-Scale Instances. Toevaluate the performance of IWOA, 37 experiments areconducted for the medium-scale problems. The experi-ments’ MPD, SD, and ACT are investigated in Table 5 formedium-scale instances. The last column of this table repre-sents the deviation from best feasible solution and the gapsfor solutions obtained from the GAMS CPLEX solver. Com-parative analysis of GA and PSO with GA-MPC and SPSOindicated that MPDs of all the experiments for GA andPSO were better than their new versions. However, ACT ofGA-MPC is better than GA and ACT of PSO is better thanSPSO. Due to the significance of the solutions’ quality in thisresearch, traditional GA and PSO are used to evaluate IWOAfor the small-scale problems.

WOAIWOA

20

15

10

5

0

MPD

IWOAGA

16141210

86420

MPD

PSOIWOA

14

12

10

8

6

4

2

0

MPD

PSOGA

16141210

86420

MPC

Figure 7: Boxplot diagrams for medium-scale instances.

0

5

10

15

20

416

522

640

912

960

1368

1600

1848

2016

2480

2480

3024

3552

3720

4482

5328

5520

7920

9760

SD

Number of variables

IWOA and WOA

IWOAWOA

02468

10121416

416

522

640

912

960

1368

1600

1848

2016

2480

2480

3024

3552

3720

4482

5328

5520

7920

9760

SD

Number of variables

IWOA and GA

IWOAGA

Figure 8: SD comparison of IWOA with WOA and GA for medium-scale instances.

16 Complexity

The graphical analysis of the results, as illustrated inFigure 6, implies that the performance of IWOA is betterthan WOA for all the instances considering the MPDmatrix.On the other hand, the performance of IWOA is better thanGA for 89.2% of the experiments, and also it is better thanPSO for most of the instances.

Furthermore, one-way ANOVA test is conducted tocompare IWOA with WOA as well as GA and PSO whichhad better performances. Although the graphical analysisillustrates a better performance for IWOA compared withGA and PSO, the results of ANOVA test on MPDs of thesealgorithms as illustrated in Table 6 and Figure 7 indicate

Table 7: Mean PD % , standard deviation SD × 108 , and average of CPU time (s) for large-scale instances

i/j/f /m Number ofvariables

IWOA WOA GA GA-MPC PSO SPSOMPD SD ACT MPD SD ACT MPD SD ACT MPD SD ACT MPD SD ACT MPD SD ACT

3/10/3/3 11070 10.4 3.10 59 20.3 3.52 50 8.04 3.73 48 11.2 3.63 40 17.1 4.50 20 10.7 3.52 38

3/10/3/4 13860 14.3 4.4 61 19.9 7.63 54 14.7 4.61 48 15.3 3.19 39 25.2 6.90 22 13.4 5.35 39

3/10/3/5 16650 17.7 3.79 65 24.3 3.77 60 17.2 1.75 51 10.1 1.89 44 27.2 4.42 23 14.0 3.73 45

4/7/3/2 7224 9.50 3.00 43 10.8 3.89 41 7.72 1.76 37 12.1 4.38 32 8.73 4.24 18 13.6 5.65 39

4/8/3/2 9408 11.4 5.07 40 15.2 4.60 37 8.62 2.85 35 10.2 5.63 34 9.71 9.31 15 11.8 3.95 34

4/10/2/3 13040 12.5 3.06 58 14.2 6.63 44 13.3 6.55 43 12.6 5.28 39 13.4 6.70 21 14.5 6.59 40

4/10/2/4 16320 13.1 3.79 63 17.3 5.64 58 12.4 13.4 46 11.6 9.00 35 21.1 7.20 25 14.1 9.76 43

4/10/2/5 19600 15.3 4.12 68 18.9 10.9 51 13.6 11.7 49 11.6 5.95 43 25.8 11.5 24 11.9 8.74 46

4/10/3/5 29400 24.2 4.49 80 26.2 7.03 74 17.6 5.22 63 16.4 5.17 54 38.2 7.61 30 22.9 5.79 64

4/10/4/5 39200 19.6 6.11 87 21.9 5.69 83 18.8 5.11 68 12.5 3.77 61 40.3 7.20 37 22.9 4.85 67

4/10/5/5 49000 26.1 6.39 107 29.1 5.32 99 14.6 4.71 79 10.6 4.18 63 42.5 5.32 42 32.8 5.17 101

5/7/3/2 11235 17.2 6.09 54 21.4 8.03 45 11.5 7.15 41 15.6 3.78 35 18.2 5.10 18 25.2 7.82 36

5/8/3/2 14640 12.1 9.70 57 17.3 7.39 49 11.6 8.62 43 12.1 6.27 38 15.3 9.05 21 12.1 8.17 37

5/10/4/2 30400 10.4 10.6 85 16.1 11.9 84 4.48 8.00 71 7.61 8.58 59 17.2 15.4 34 12.8 7.32 63

5/10/5/2 38000 16.7 4.54 91 18.6 11.2 83 15.8 12.6 76 10.2 6.00 68 20.9 17.4 36 18.8 9.62 69

5/10/5/3 50750 21.7 5.90 125 21.8 7.11 119 13.8 13.5 84 13.1 12.3 73 32.7 11.6 47 26.8 9.76 103

6/4/4/2 7014 15.8 2.90 39 21.1 2.90 44 11.7 1.38 38 28.3 1.19 30 12.3 2.42 17 27.1 2.84 40

6/8/2/2 14016 4.85 11.9 62 12.6 13.8 53 5.86 12.9 49 5.75 12.3 39 6.67 32.3 24 7.49 10.3 42

6/10/2/2 21840 9.01 13.2 71 10.5 21.1 64 6.64 16.2 50 7.11 27.1 44 6.68 19.9 26 10.5 23.2 55

6/10/5/2 54600 16.5 12.2 132 21.3 18.3 129 10.1 8.68 121 8.82 7.27 97 16.6 16.7 49 13.4 14.9 98

7/5/5/2 18725 12.6 5.94 67 13.2 4.83 52 15.2 37.8 45 11.4 5.60 42 19.5 8.80 23 15.3 6.88 48

7/8/4/2 38080 20.5 19.6 76 27.2 13.9 69 16.0 19.6 54 14.9 11.9 47 23.3 15.2 27 20.7 15.9 61

7/10/5/2 74200 21.1 18.7 153 26.2 25.1 149 8.81 23.6 131 8.12 21.5 104 26.1 17.7 57 20.8 21.2 107

8/9/3/2 47088 12.2 21.5 103 14.2 31.3 96 11.4 32.9 82 7.69 20.6 74 11.7 21.3 44 11.7 30.4 90

8/10/5/2 96800 19.6 16.9 192 20.4 19.0 183 11.4 25.6 139 10.3 14.4 112 21.3 18.8 66 21.2 13.7 122

10/5/2/3 20300 10.5 10.31 67 15.3 14.8 59 7.61 14.2 53 5.23 9.45 42 9.50 16.2 24 6.49 15.0 53

10/5/2/4 25400 12.2 9.67 72 12.7 6.29 71 11.1 7.37 66 5.84 10.6 55 14.4 8.85 28 11.3 11.3 69

10/5/2/5 30500 7.69 5.28 79 9.93 4.82 78 7.33 1.14 64 5.23 7.71 52 18.4 9.48 24 11.4 7.93 64

10/5/3/5 45750 17.1 4.63 99 19.1 5.83 91 9.70 8.3 82 7.81 6.99 74 36.1 9.39 34 20.7 6.92 61

10/5/4/5 61000 21.4 4.11 158 24.2 6.96 154 19.2 9.9 129 9.61 6.90 97 43.9 6.62 54 18.9 8.60 99

10/5/5/5 76250 23.1 6.50 167 24.1 7.78 159 12.5 5.6 136 10.1 5.28 109 43.3 9.43 56 20.6 7.84 104

10/10/2/2 60400 9.03 65.3 137 9.34 52.2 138 5.43 44.3 116 4.16 32.2 95 9.10 95.6 39 6.67 46.6 99

10/10/4/5 242000 17.8 12.9 382 22.5 19.5 413 8.33 20.4 341 4.48 18.4 316 35.2 22.7 90 31.3 19.4 189

10/10/5/5 302500 11.4 21.5 403 15.2 17.6 396 6.40 20.6 361 5.66 10.1 328 28.9 17.2 99 25.6 18.6 183

15/10/2/2 135600 7.67 120 241 9.83 112 285 6.17 102 237 5.32 96.2 216 8.16 125 75 6.84 120 153

20/10/2/2 240800 5.68 141 372 6.82 115 385 4.06 180 363 3.18 100 331 8.32 146 84 6.68 153 158

25/10/2/2 376000 4.59 272 397 6.25 353 410 2.73 250 382 3.02 207 370 5.10 229 98 4.92 190 176

30/10/2/2 541200 6.08 162 440 9.13 204 463 3.34 324 424 2.72 220 419 6.05 254 126 5.05 305 213

35/10/2/2 736400 4.95 433 315 7.96 630 341 2.28 377 294 2.27 237 275 5.11 477 98 4.21 442 195

40/10/2/2 961600 5.38 530 410 5.40 511 439 2.01 466 396 2.06 355 314 6.47 547 93 5.06 442 162

45/10/2/2 1216800 4.04 402 469 8.81 587 482 1.75 692 451 1.06 321 412 4.18 456 112 4.14 371 239

50/10/2/2 1502000 4.75 863 472 8.23 595 458 1.94 825 462 1.43 401 464 5.88 975 87 4.75 667 235

17Complexity

similar competence for all three algorithms. Furthermore,the ANOVA test’s results imply that MPD of IWOA ismeaningfully better thanWOA which confirms the graphicalanalysis results.

On the other hand, MPD of IWOA and nonzero PDs ofthe results obtained from the GAMS CPLEX solver which isillustrated in Figure 4 reveal the worse performance of

standard software and deterioration of solutions when thenumber of variables increases.

Figure 8 represents the comparison of SD for IWOA,WOA, and GA. Smaller SD of the experiments for IWOAconfirms its improved robustness and stability of thesolutions compared with WOA, GA, and PSO for themedium instances.

05

1015202530

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

MPD

Number of variables

IWOA and WOA

IWOAWOA

05

1015202530

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

MPD

Number of variables

IWOA and GA-MPC

IWOAGA

05

101520253035

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

MPD

Number of variables

IWOA and SPSO

IWOAPSO

05

101520253035

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

MPD

Number of variables

GA-MPC and SPSO

GAPSO

Figure 9: MPD comparison of IWOA with WOA, GA-MPC, SPSO, and GA-MPC with SPSO for large-scale instances.

Table 8: Results of the one-way ANOVA test for large-scale instances

DF SS MS F value P value Test results

IWOA-WOA 1 313.2 313.2 9.39 0.003 Means are significantly different

Error 82 2737.2 33.34

Total 83 3047.4

IWOA-GAMPC 1 400.2 400.16 12.62 0.001 Means are significantly different

Error 82 2599.3 31.70

Total 83 2999.4

IWOA-SPSO 1 47.82 47.82 0.98 0.325 Means are equal

Error 82 4006.18 48.86

Total 83 4054.00

GAMPC-SPSO 1 724.7 724.65 16.56 0.00 Means are significantly different

Error 82 3587.2 43.75

Total 83 4311.8

DF: degree of freedom; SS: sum of squares; MS: mean squares

18 Complexity

5.4.3. Results’ Comparison for the Large-Scale Instances.Table 7 represents the results of the experiments on 42large-scale instances to compare the performance of differ-ent metaheuristic algorithms. In our experiments, GA-MPC had better performance than GA considering bothMPD and ACT metrics. Comparing MPDs of GA with GA-MPC indicated better performance of GA-MPC for 76.2%of the experiments. Furthermore, comparing PSO with SPSOwith respect to MPDmetrics revealed the better performanceof SPSO for 88.1% of the test problems. However, ACT ofPSO is better than SPSO. Due to the importance of solutions’quality, we have evaluated IWOA using GA-MPC and SPSOfor the large-scale problems.

The graphical analysis of the results as illustrated inFigure 9, indicates that the performance of IWOA is bet-ter than WOA for all of the instances. Moreover, IWOAhas better or same performance in 61.2% of the experi-ments compared with SPSO. However, the performanceof GA-MPC is better than IWOA in 83.3% of the exper-iments. Furthermore, the comparative graphical analysisof IWOA and GA-MPC also reveals that the performanceof IWOA has deteriorated as the number of decision var-iables increased.

On the other hand, one-way ANOVA test is conducted tocompare the performance of IWOA with WOA as wellas GA-MPC and SPSO which had better performances.The ANOVA test’s results as investigated in Table 8 andFigure 10 indicate that the performance of IWOA is stronglybetter than that of WOA which confirms the results of thegraphical analysis. Moreover, the ANOVA test’s results showthat IWOA has almost similar competence with SPSOwhile GA-MPC is more competent than other algorithmsfor the large-scale problems. However, IWOA reveals a

1

10

100

1000

SD

Number of variables

IWOA and WOA

IWOAWOA

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

Figure 11: SD comparison of IWOA with WOA for large-scaleinstances.

1

10

100

1000

SD

IWOA and GA-MPC

IWOAGA

Number of variables

7014

9408

1123

513

860

1464

016

650

1960

021

840

2940

030

500

3808

045

750

4900

054

600

6100

076

250

1356

0024

2000

3760

0073

6400

1216

800

Figure 12: SD comparison of IWOA with GA-MPC for large-scaleinstances.

WOAIWOA

30

25

20

15

10

5

MPD

IWOAGA-MPC

30

25

20

15

10

5

0

MPD

SPSOIWOA

35

30

25

20

15

10

5

0

MPD

SPSOGA-MPC

35

30

25

20

15

10

5

0M

PD

⁎

⁎

Figure 10: Boxplot diagrams for large-scale instances.

19Complexity

similar performance compared with SPSO considering MPDof the observations.

Figures 11 and 12 represent almost the same perfor-mance for IWOA compared with WOA and GA-MPC withrespect to solutions’ SD.

6. Conclusion and Future Studies

In this study, we proposed a formulation for the integratedscheduling of production and distribution operations in amultifactory supply chain. Considering the make-to-orderproduction system and direct shipment of finalized ordersas well as the limited number of transportation vehiclesincreased the complexity of the problem. The integratedsimulation-based solution approach that considers the inter-actions of subproblems also provides the best trade-offamong the individual costs of the objective function. Theadvantage of our work is that it optimizes the scheduling oftransportation integrated with production in the operationallevel of multifactory production systems. We used a conven-tional software to solve the small-size problems to evaluatethe validity of the proposed model. We improved the newlyintroduced WOA to solve the medium-scale problems andcomputationally confirmed the efficiency of the modifiedalgorithm (IWOA) compared to basic WOA for differentscales of the problem. To evaluate IWOA, we compared theresults of computational experiments with two versions ofGA and PSO. The results of experiments confirmed a signif-icant improvement in the mean value of the percentage ofdeviations while using IWOA rather than WOA for all sizesof the problem. However, unlike the improvements madeto WOA and the better performance of IWOA for themedium-scale instances, it is better to use GA-MPC for thelarge-scale problems.

We considered parallel machines and immediate directshipment of the orders in the problem of this research; otherconfigurations of machines and shipping methods could bethe trend of future studies. On the other hand, an integratedsupply chain with a single objective function is considered inthis paper. So, independent factories and multiobjectiveproblem can be presented in the future. Furthermore, onecan consider the application of the MIPDSP for the loadingand distribution operations, and the general evaluation ofIWOA as new directs of research.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

References

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