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This article was downloaded by: [University of Windsor]On: 27 September 2013, At: 02:16Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Production ResearchPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tprs20
Multi-objective production scheduling withcontrollable processing times and sequence-dependentsetups for deteriorating itemsM. Karimi-Nasab a & S.M.T. Fatemi Ghomi ba Department of Industrial Engineering, Iran University of Science and Technology, Narmak,Tehran, Iranb Department of Industrial Engineering, Amirkabir University of Technology, Tehran, IranPublished online: 27 Feb 2012.
To cite this article: M. Karimi-Nasab & S.M.T. Fatemi Ghomi (2012) Multi-objective production scheduling with controllableprocessing times and sequence-dependent setups for deteriorating items, International Journal of Production Research,50:24, 7378-7400, DOI: 10.1080/00207543.2011.649800
To link to this article: http://dx.doi.org/10.1080/00207543.2011.649800
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International Journal of Production ResearchVol. 50, No. 24, 15 December 2012, 7378–7400
Multi-objective production scheduling with controllable processing times and
sequence-dependent setups for deteriorating items
M. Karimi-Nasaba1 and S.M.T. Fatemi Ghomib*
aDepartment of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran;bDepartment of Industrial Engineering, Amirkabir University of Technology, Tehran, Iran
(Received 3 January 2011; final version received 6 December 2011)
The production scheduling problem is to find simultaneously the lot sizes and their sequence over a finite setof planning periods. This paper studies a single-stage production scheduling problem subject to controllableprocess times and sequence-dependent setups for deteriorating items. The paper formulates the problem byminimising two objectives of total costs and total variations in production volumes simultaneously.The problem is modelled and analysed as a mixed integer nonlinear program. Since it is proved that theproblem is NP-hard, a problem-specific heuristic is proposed to generate a set of Pareto-optimal solutions.The heuristic is investigated analytically and experimentally. Computational experiences of running theheuristic and non-dominated sorting genetic algorithm-I over a set of randomly generated test problems arereported. The heuristic possesses at least 56.5% (in the worst case) and at most 94.7% (in the best case) oftotal global Pareto-optimal solutions in ordinary-size instances.
Keywords: multi-objective; production scheduling; controllable processing time; sequence-dependent setups;deteriorating items; Pareto-optimal solutions
1. Introduction
Production scheduling is one of the major tasks of a production manager. It consists of simultaneous decision-making about production planning and scheduling of multiple items with dynamic demands over a finite planninghorizon. On the other hand, all desires and tasks/limitations of a production manager can be translated bymathematical modelling including a number of objective functions and a set of constraints. In previous decades, thetraditional objective of a production manager has been to minimise the total costs. However, some recent studiesindicate that production managers also wish to deal with a smoothed series of production volumes over planningperiods. For example, Yavuz and Akcali (2007) expressed that there should be an ideal production level, thatvariations of batch sizes are as small as possible around it. Then, Karimi-Nasab and Aryanezhad (2011) proposedan ideal production band around the ideal production volume, as depicted in Figure 1 (the forecast demand (Dt), andsmoothed production volume (x�t ) are plotted for periods t¼ 1, 2, . . . , 12). Numerous studies have differentcontributions to various aspects of the problem, including theoretical concepts, pragmatic approaches, efficientsolution methods owing to working conditions, and so forth. However, none of the studies have clearly determinedthe ideal level and/or borders of such an ideal production band. In this paper, we propose a model to determine theborders of the ideal production band for each item.
Production managers take different approaches to deal with their production scheduling problems. JIT-orientedplanning is one of the most popular. Yavuz and Tufekci (2006) believe that smoothing production volumes canresult in filling the gaps in a just-in-time (JIT) system. A few studies are aimed at forcing their developed models toobtain an ideal production band limited to the maximum production capacity (Yavuz et al. 2006). Furthermore,Matsui (2007) provided guidelines about reaching JIT in practice. According to Matsui, the smoothness of theproduction along with a low total cost is a characteristic of a JIT production plan. According to JIT concepts,several other researchers have extended classical batch sizing models with new objectives and/or constraints(Ehrhardt 1998, Mollick 2004, Aryanezhad et al. 2009). Each model is claimed to best fit the main problem settings.
It should be noticed that existing models are not mature enough in contrast to complicated realistic settings.This originates from the fact that usually developing problem formulations are involved in taking relaxed
*Corresponding author. Email: [email protected]
ISSN 0020–7543 print/ISSN 1366–588X online
� 2012 Taylor & Francis
http://dx.doi.org/10.1080/00207543.2011.649800
http://www.tandfonline.com
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assumptions in hand. A common assumption is that demand values in each period are given or forecast by a goodmethod. Another assumption mentioned in the literature is that the maximum production capacity of all periods isconstant. However, the current research assumes that a dynamic change is allowed in production capacity fromperiod to period. Inventory deterioration is a fact in many real cases dealing with limited-shelf-life goods such assauces, foods, blood, and oil. Numerous types of inventory deterioration can be considered in a problem based onthe corresponding working conditions. This paper assumes that a fixed percentage of inventories deteriorate after aperiod. This is true for some cases when an approximately constant percentage of inventories becomes useless ininternal transportation, inspection sampling, and so forth.
Determining the speed/rate at which the machine(s) should process each product is another practical task of aproduction manager. In most real situations, it is possible to change the working speed of the machinery by payingmore costs. This is dealt with as process compressibility in the literature. Karimi-Nasab and Pakgohar (2010)considered the case of dealing with compressible processing times in production scheduling problem with sequence-independent setups. They reported that appending extra features more than basic ones to simple models results in aneed for developing problem-specific solution methods.
According to the literature, various solution methods are reported in Table 1. Also, Yavuz and Akcali (2007)surveyed the different solution methods for a single-level batch smoothing problem. Clearly, for a given productionenvironment, production scheduling is harder than production planning and/or scheduling but yields more desirableand reliable solutions. Most of the time, the constructed model is so complex and full of many integer variables thateven finding a feasible solution becomes challenging (Chinneck 2008). We know that for a given environmentconfiguration, scheduling problems with sequence-dependent setups (SDS) are NP-hard. This comes from the factthat scheduling with an SDS assumption is equivalent to finding the optimal tour in the Travelling SalesmanProblem (TSP) (Pochet and Wolsey 2006). As Table 1 confirms, most of the related studies considered the case ofsequence-independent setups, but in this paper we consider the problem under SDS assumption.
Table 1 provides a taxonomy of the literature from 1989 to 2011. According to this table, a multi-objective singlemachine production scheduling problem with sequence-dependent setups and process compressibility fordeteriorating inventories has not been comprehensively studied in the literature. The paper is organised as follows:Section 2 formulates and analyses the problem; Section 3 introduces solution method; Section 4 presents a numericalexample; and finally, Section 5 concludes and provides future research directions.
2. Proposed model
This section presents the proposed model. The model determines the volume and sequence of products ineach planning period by considering controllable process times. Because of imperfect knowledge about the trade-off between cost and time, their relation is assumed to be linear. There are two critical points as (pn, C(pn)) and
Figure 1. An example of an ideal production band.
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Table 1. Taxonomy for the literature.
No. ofitems
No. ofobjectives
Planninghorizon Type of setup
Processtimes
Inventorydeterioration
Solutionmethod
Authors Single
Multiple
Single
Multiple
Finite
Infinite
Sequence
independent
Sequence
dependent
Inputdata
Variable
Notallowed
Fixed
rate
Others
Exact
Heuristic
Hybrid
Metaheuristic
Monma and Potts (1989)p p p p p p p
Azoza and Bonney (1990)p p p p p p p
Browne and Yechiali (1990)p p p p p p p
Coleman (1992)p p p p p p p
Laguna and Glover (1993)p p p p p p p
Ghosh (1994)p p p p p p p
Sundararaghavan and Kunnathur (1994)p p p p p p p
Gawiejnowicz and Pankowska (1995)p p p p p p p
Kim et al. (1995)p p p p p p p
Ozgur and Brown (1995)p p p p p p p
Rubin and Ragatz (1995)p p p p p p p
Asano and Ohta (1996)p p p p p p p
Feo et al. (1996)p p p p p p p
Chen (1997)p p p p p p p
Fleischmann and Meyer (1997)p p p p p p p
Lee et al. (1997)p p p p p p p
Tan and Narasimhan (1997a)p p p p p p p
Tan and Narasimhan (1997b)p p p p p p p
Unal et al. (1997)p p p p p p p p
Kolahan and Liang (1998)p p p p p p p
Laguna (1999)p p p p p p p p
Sun et al. (1999)p p p p p p p
Wee and Wang (1999)p p p p p p p
Haase and Kimms (2000)p p p p p p p
Tan et al. (2000)p p p p p p p p p
Alidaee et al. (2001)p p p p p p p
Bylka and Rempala (2001)p p p p p p p
Franca et al. (2001)p p p p p p p
Manna and Chaudhuri (2001)p p p p p p p
McMullen (2001)p p p p p p p
Yao and Elmaghraby (2001)p p p p p p p
Chen and Lin (2002)p p p p p p p
Roslof et al. (2002)p p p p p p p
Xie and Dong (2002)p p p p p p p
Balkhi (2003)p p p p p p p
Merce and Fontan (2003)p p p p p p p
Rabadi et al. (2003)p p p p p p p
Rossetti and Stanford (2003)p p p p p p p
Spina et al. (2003)p p p p p p p
Zhou et al. (2004)p p p p p p p
Gupta and Magnusson (2005)p p p p p p p
Gupta and Sivakumar (2005)p p p p p p p
Yang (2005)p p p p p p p
Fandel and Stammen-Hegene (2006)p p p p p p
Law and Wee (2006)p p p p p p p
Ouyang et al. (2006)p p p p p p p
Yavuz and Tufekci (2006)p p p p p p p
Yavuz et al. (2006)p p p p p p p p
Alamri and Balkhi (2007)p p p p p p p
Ho et al. (2007)p p p p p p p
Maity et al. (2007)p p p p p p p
(continued )
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(pc, C(pc)) corresponding to (normal process time, cost of normal process time) and (crash process time, cost of crashprocess time) respectively. Now, the proposed model is constructed according to the following assumptions:
. There are some independent products that should be produced and delivered to the customer during a finitenumber of planning periods.
. There is a single machine to process all products.
. There are two main objectives: (1) obtaining a production plan as smooth as possible and (2) minimising thetotal cost.
. There is a linear trade-off relation between unit production time and unit production cost. The optimalprocess time of a machine on each product type should be the same in all planning periods, as desired by themanager. The unit production cost by a given process time p is C(p)¼ k�CS� p, where
k¼CS� pnþC(pn) and CS ¼��Cð pcÞ�Cð pnÞ
pc�pn
��.. Because of preventive maintenance activities, the available time in each planning period can be different
from the others.. Each product type has its own demand in every planning period that is a finite, deterministic, and integer
value.. There is no initial inventory stock of any product type.. % fi of inventories of each product type deteriorates in a period. The discipline of inventory usage for each
product type is assumed to be first in first out (FIFO).. Backorder is allowed in every period except the last one. In other words, for each product, the sum of
delivered items over all periods should not be less than the sum of demands over all periods (i.e. lost salesare not allowed).
Table 1. Continued.
No. ofitems
No. ofobjectives
Planninghorizon Type of setup
Processtimes
Inventorydeterioration
Solutionmethod
Authors Single
Multiple
Single
Multiple
Finite
Infinite
Sequence
independent
Sequence
dependent
Inputdata
Variable
Notallowed
Fixed
rate
Others
Exact
Heuristic
Hybrid
Metaheuristic
Yavuz and Tufekci (2007)p p p p p p p
de Araujo et al. (2008)p p p p p p p
Liao (2008)p p p p p p p
Maity and Maiti (2008)p p p p p p p
Rong et al. (2008)p p p p p p p
Xu and Zhao (2008)p p p p p p p
Aryanezhad et al. (2009)p p p p p p p
Berrichi et al. (2009)p p p p p p p
Ferreira et al. (2009)p p p p p p p
Jaber et al. (2009)p p p p p p p
Kovacs et al. (2009)p p p p p p p
Lee and Hsu (2009)p p p p p p p
Marvelias (2009)p p p p p p p
Minner (2009)p p p p p p p
Weidenhiller and Jodlbauer (2009)p p p p p p p
Aryanezhad et al. (2010)p p p p p p p
He et al. (2010)p p p p p p p
Karimi-Nasab and Pakgohar (2010)p p p p p p p
Kim et al. (2010)p p p p p p p
Sun et al. (2010)p p p p p p p
Supithak et al. (2010)p p p p p p p
James and Almada-Lobo (2011)p p p p p p p
Karimi-Nasab and Aryanezhad (2011)p p p p p p p
Karimi-Nasab and Konstantaras (2011)p p p p p p p
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. Unit shortage and inventory holding costs are deterministic values that can be different from one period to
another.. A unique sequence should be used to process all product types in all periods.. For each product type, setup consists of some activities such as adjusting machine speed, changing tool,
brushing and cleaning the machine, etc. Setups depend only on the sequence of product types such that for
each product type, only one setup time is required, disregarding the amount of product type produced in a
period.. Pre-emption is not allowed, and the amount of each product type that should be produced in every
planning period has to be an integer value.
The whole problem consists of two complexity aspects: (1) the capacitated lot sizing and scheduling problem
(also, referred to as CLSP) is NP-hard (Pochet and Wolsey 2006); and (2) the single-stage scheduling of product
types with SDS assumption is also NP-hard (Haase and Kimms 2000).The input parameters and decision variables are as follows:
Input parameters:
T total number of planning periods;t index of planning period;n number of product types;i index of product type;
di,t demand of product type i in period t;ATt available time in period t;pni normal process time on product type i;pci crashed process time on product i;
Si,0 setup time of machine to produce product type i in the first order;Si,f setup time of machine to produce product type i after product type f;
SCi,0 setup cost of machine to produce product type i in the first order;SCi,f setup cost of machine to produce product type i after product type f;�i,t unit shortage cost;hi,t unit inventory holding cost;CSi cost slope of processing product i on machine;ki fixed cost of processing product i on machine;fi rate of deterioration for product type i.
Decision variables:
Bi,t amount of slackness for product type i at the end of period t;Ii,t amount of inventory for product i at the end of period t;
ri,t1 if product i is produced in period t0 else;
�Oi,0
1 if product type i is the first produced in all periods0 else;
�
Oi,f1 if product type i is produced after product type f in all periods i 6¼ fð Þ
0 else;
�pi process time of product type i;
xi,t amount of product type i produced in period t;xmini minimum production (lower limit of ideal production band) of product type i over all periods;
xmaxi maximum production (upper limit of ideal production band) of product type i over all periods.
The current research considers the following issues further than the paper of Karimi-Nasab and Aryanezhad
(2011):
. sequence-dependent setups;
. inventory deterioration;
. a linear formulation for the smoothing objective (15) instead of the nonlinear objective presented in
Equation (1).
7382 M. Karimi-Nasab and S.M.T. Fatemi Ghomi
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The following model translates the problem assumptions into the mathematical formulation:
Model 1
min Z1 ¼Xni¼1
XT�1t¼1
ðxi,tþ1 � xi,tÞ2
ð1Þ
min Z2 ¼XTt¼1
Xni¼1
SCi,0Oi,0 þXf 6¼i
SCi,fOi,f
!ri,t þ
XTt¼1
Xni¼1
ðki � CSipiÞxi,t
þXTt¼1
Xni¼1
�i,tBi,t þXTt¼1
Xni¼1
hi,tIi,t ð2Þ
s.t.
Xni¼1
ðSi,0Oi,0 þXf 6¼i
Si,fOi,fÞri,t þ xi,tpi
!� ATt ð8 tÞ ð3Þ
XT
t¼1xi,t ¼
XT
t¼1di,t ð4Þ
pci � pi � pni ð8iÞ ð5Þ
xi,t �M:ri,t ð8i, tÞ ð6Þ
Oi,0 þXf 6¼i
Oi,f ¼ 1 ð8iÞ ð7Þ
Oi,f þOf,i � 1 ð8i 6¼ f Þ ð8Þ
Ii,0 ¼ 0 ð8 iÞ ð9Þ
Bi,0 ¼ 0 ð8 iÞ ð10Þ
Ii,tBi,t ¼ 0 ð8i, tÞ ð11Þ
Ii,t � Bi,t ¼ xi,t þ ð1� fiÞIi,t�1� �
� Bi,t�1 � di,t ð8i, tÞ ð12Þ
Oi, 0,Oi,f, ri,t 2 0, 1f g ð8i, tÞ ð13Þ
xi,t,Bi,t, Ii,t 2 Zþ [ 0f g ð8i, tÞ: ð14Þ
Objective (1) aims to obtain a smoothed production plan, while objective (2) minimises the sum of setup,
production, shortage, and inventory holding costs. Constraint (3) is satisfied when, in every period t, the sum of
setup and production times becomes less than the available time. Constraint (4) ensures that in every feasible
production plan, no lost sales could occur, but a backorder could. Constraint (5) states that the unit production time
of each product type should be in its permitted interval. Constraint (6) determines whether, in period t, product type
i requires a setup or not. Constraints (7) and (8) determine the process sequence of product types. Constraints (9)–
(12) determine the amount of inventory and shortage for product type i in period t. In this paper, rb c means the
maximum integer value that is not larger than a real value r. Constraints (13)–(14) state the type of decision
variables except pi that should satisfy Equation (5).
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The model can be more simplified as follows. First of all, we believe that objective (1) can be replaced by a linear
relation using xmini and xmax
i , where xmini � xi,t � xmax
i for all t. Then, objective (1) can be replaced by Equation (15)
along with appending constraint (16):
min Z1 ¼Xni¼1
xmaxi � xmin
i
� �ð15Þ
xmini � xi,t � xmax
i ð8i, tÞ ð16Þ
Also, the inventory balance is shown with the recursive constraint (12). However, Equation (12) can be replaced by
Equation (17), which is not recursive:
Ii;t � Bi;t ¼Xtq¼1
ð1� fiÞt�qðxi;q � di;tÞ
� �ð8i; tÞ ð17Þ
Relation (17) could be inducted in case there is no initial inventory and/or backorder. Furthermore, the
multiplicative terms of Oi,0ri,t and Oi,fri,t could be linearised by introducing new decision variables yi,0,t and yi,f,talong with a set of the following constraints (this is true because objective (2) is of minimisation):
Oi,0 þ ri,t � 1 � yi,0,t ð8i, tÞ ð18Þ
Oi,f þ ri,t � 1 � yi,f,t ð8i 6¼ f, tÞ ð19Þ
According to the relations presented above, Model 2 is as follows:
Model 2
min Z1 ¼Xni¼1
xmaxi � xmin
i
� �ð15Þ
min Z2 ¼XTt¼1
Xni¼1
SCi,0yi,0,t þXf 6¼i
SCi,fyi,f,t
!þXTt¼1
Xni¼1
ðki � CSipiÞxi,t
þXTt¼1
Xni¼1
�i,tBi,t þXTt¼1
Xni¼1
hi,tIi,t ð2Þ
s.t.
Xni¼1
Si,0yi,0,t þXf 6¼i
Si,fyi,f,t þ xi,tpi
!� ATt ð8tÞ ð3Þ
constraints (4) to (11)
xmini � xi,t � xmax
i ð8i, tÞ ð16Þ
Ii,t � Bi,t ¼Xtq¼1
ð1� fiÞt�qðxi,q � di,tÞ
� �ð8i, tÞ ð17Þ
Oi,0 þ ri,t � 1 � yi,0,t ð8i, tÞ ð18Þ
Oi, f þ ri,t � 1 � yi,f,t ð8i 6¼ f, tÞ ð19Þ
yi,0,t, yi,f,tOi, 0,Oi,t, ri,t 2 0, 1f g ð8i, tÞ ð13Þ
xi,t,Bi,t, Ii,t 2 Zþ [ 0f g ð8i, tÞ ð14Þ
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According to nonlinear constraint (17) along with nonlinear terms of pi�xi,t both in objective (2) and inconstraint (3), Model 2 is a mixed integer nonlinear program. Hence, in the next section, a simple heuristic isdeveloped based on the problem characteristics to obtain Pareto-optimal solutions.
3. Solution method
In this section, a heuristic method is proposed to solve the problem, and then several properties are given as lemmasand theorems to analyse the model and its Pareto-optimal solutions. First, the following new notations are used inthe heuristic:
iter, si counter variables;� intermediate variable to determine pi (where 05 �5 1);
Z1temp intermediate variable to store the last best value of Z1 during the search;Z2temp intermediate variable to store the last best value of Z2 during the search;
Zcomtemp intermediate variable to store the last best combined value of Z1 and Z2 during the search(calculated according to relation (27));
Feasibility a logical variable to determine whether or not the constructed solution is feasible;SS jump length of change in � (where 05SS� 0.05);
w1, w2 weights of Z1 and Z2 respectively;RTt idle time in period t;R a random uniform value in (0, 1);
tmax, tmin intermediate indices of periods to change the production volume in these periods.
The proposed heuristic is organised in the following steps:
Step 0. Let iter¼ 0, si¼ 0, �¼ 0, Z1temp¼M, Z2temp¼M, Zcomtemp¼M and Feasibility¼ true. Request from thedecision-maker the values of SS, w1, and w2 such that 05SS� 0.05, w1þw2¼ 1 and w1, w2� 0.Step 1. Generate a random sequence for processing n product typesfor i¼ 1 to n
if i is sequenced in the first position, let si¼Si,0 and Oi,0¼ 1else if i is sequenced after another product type f, let si¼Si,f and Oi,f¼ 1
Step 2. For i¼ 1 to n
pi ¼ pci þ �ð pni � pci Þ ð20Þ
for t¼ 1 to T
xi,t ¼ATt �
Pnj¼1 sj
� PTk¼1 di,k
� Pn
j¼1 pj
� Pnj¼1
PTk¼1 dj,k
� 6664
7775 8i, t:
ri, t ¼ 1 ð21Þ
Step 3. For i¼ 1 to nIi,0¼ 0, Bi,0¼ 0ifPT
t¼1 ðxi,t � di,tÞ5 0 thenfor t¼ 1 to T
RTt ¼ ATt �Xn
j¼1ðsjrj,t þ pjxj,tÞ 8 t ð22Þ
xi,t ¼ xi,t þmin RTt=Pi
� �, T� 1þ
XT
k¼1ðdi,k � xi,kÞ
� =T
j kn o8i, t ð23Þ
Ii,t ¼ maxf0,xi,t þ ð1� fiÞIi,t�1� �
� di,t � Bi,t� 1g
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Bi,t ¼ maxf0,Bi,t� 1þ di,t � ð1� fiÞIi,t�1� �
00A0� xi,tg
go to step 4elsego to step 7.
Step 4. For t¼ 1 to T
RTt ¼ ATt �Xnj¼1
ðsjrj,t þ pjxj,tÞ 8t ð24Þ
if RTt� 0 thenFeasibility¼ false
for i¼ 1 to nif Bi,T4 0 thenFeasibility¼ false
if Feasibility¼ true thenfor i¼ 1 to n:generate a random number R (i.e. R� uniform(0, 1))if R4 0.5 then
find periods tmax and tmin with maximum and minimum values of xi,t–di,t respectively,else
find periods tmax and tmin with maximum and minimum values of xi,t,update xi,tmax and xi,tmin by (25) and (26),
xi,tmax ¼ xi,tmax
� �� 1 ð25Þ
xi,tmin ¼ xi,tmin
� �þmin
�RTtmin=pi� �
, 1
ð26Þ
elseiter¼ iterþ 1; si¼ 0,go to step 2.
Step 5. Compute the corresponding values of Z1, Z2, and Zcom, where Zcom is the combined (maximisation) objective
function,
Zcom ¼ w1Z1NIS � Z1
Z1NIS � Z1PISþ w2
Z2NIS � Z2
Z2NIS � Z2PISð27Þ
Step 6. If Z1�Z1temp and Z2�Z2temp simultaneously, thenZ1temp¼Z1; Z2temp¼Z2 and Zcomtemp¼Zcom,add the current solution to the list of Pareto-optimal solutions
elseif Zcom�Zcomtemp thenZ1temp¼Z1, Z2temp¼Z2 and Zcomtemp¼Zcom,add the current solution to the list of Pareto-optimal solutions
si¼ siþ 1,iter¼ iterþ 1.
Step 7. If iter¼ itermax thenstop and report the results,
elseif si¼ simin then�¼min{�þSS, 1}si¼ 0,go back to step 1,
elsego back to step 4.
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It should be noted that because of the multi-objectivity of the proposed model, the solution method is to explorethe feasible solution space and find some efficient solutions during a few iterations. Now, some of the theorems andlemmas are provided with detailed proofs to facilitate applying the proposed solution method and to show some ofthe mathematical features of both the model and the solution method. Lemmas 1 and 2 provide the positive andnegative ideal values of Z1 and Z2. PIS and NIS are solution vectors corresponding to positive and negative idealvalues respectively. The meaning of positivity and negativity in PIS and NIS comes from the originality of theobjective functions. In other words, when the objective function is to find a minimum value over the feasiblesolution space, its minimum value or every value less than it would be considered as its positive ideal value and viceversa. On the contrary, the maximum of the above-mentioned objective function or every value higher than it wouldbe considered as its negative ideal value.
Lemma 1: Possible values of Z1 are between Z1PIS and Z1NIS (i.e. Z1PIS�Z1�Z1NIS):
Z1PIS ¼ 0 ð28Þ
Z1NIS ¼ ðAT1Þ2þ 2
XT�1t¼2
ðATtÞ2þ ðATTÞ
2
! Xn
i¼1
1
pci
� �2 !
: ð29Þ
Proof: Relations (28) and (29) are true for both objective function (1) and its linear equivalent (15). However, if themodel were linear, a more efficient bound could be obtained for Z1. See Karimi-Nasab and Aryanezhad (2011) for acomplete proof. œ
Lemma 2: The lower and upper bounds of Z2 are Equations (30) and (31) respectively, shown as Z2PIS and Z2NIS. Inother words, Z2PIS�Z2�Z2NIS is always true for every feasible solution.
Z2PIS ¼Xni¼1
min SCi,0, minf 6¼i
SCi,f
�� �þXni¼1
ðki � CSipni ÞXT
t¼1di,t
� ð30Þ
Z2NIS ¼ TXni¼1
max SCi,0, maxf 6¼i
SCi,f
�� �þ
XTt¼1
ATt
! Xni¼1
ki � CSipci
pci
!
þXni¼1
XT�1t¼1
�i,tð1� fiÞt�1
Xtq¼1
di,q
! !
þXTt¼1
Xni¼1
hi,tð1� fiÞt�1
Xtq¼1
ATq
pci� di,q
� �����������
!: ð31Þ
Proof: See Appendix 1. œ
Lemma 3: For a given sequence of processing n products, the solution method has the following properties:
(I) if there exists any feasible set of batch sizes, the algorithm is certainly capable to find it,(II) if (I) holds true, all the succeeding solutions will be feasible until a new sequence is generated.
Proof: See Appendix 2. œ
Theorem 1: Possible values of Zcom are between zero and one.
Proof: See Appendix 3. œ
Theorem 2: The positive ideal value of Zcom is one, and its negative ideal value is zero.
Proof: See Appendix 4. œ
4. Computational experiments
There are many differences between a single-objective problem and a multi-objective counterpart in some features,mainly CPU times and solution types. In a single-objective problem, we are searching for a unique optimal solution.
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However, in a multi-objective problem, we are searching for a set of non-dominated solutions. Thus, it is not
unusual to find some Pareto-solutions in a few milliseconds. However, most multi-objective algorithms, such as
NSGA-I, will find local Pareto-solutions at their starting times such that many of them will be dominated afterward
by newly found global Pareto-solutions. Most multi-objective algorithms, such as NSGA-I and our heuristic, will
not find any new (global/local) Pareto-solution(s) after a time interval. The length of this time interval depends on
many factors such as: (1) the input data for the problem; (2) the tuning of the algorithm; (3) the type of constraints;
(4) internal relations between the decision variables of the problem; (5) search strategies of the algorithm in the
search space; (6) computing power of the used computer; (7) language of the programming; (8) level of specialisation
of the programmer about coding the algorithm; and so forth. In this section, computational experiments are
reported in two subsections: ordinary-sized problems and extremely large problems. In both subsections, first a set
of test problems are generated randomly according to the data in Table 2.Also, all computations are performed in MATLAB 7.10 on a Pentium IV, 2.53GHz CPU, with 512MB of
RAM. For each problem size (i.e. each pair of n and T), three instances are generated randomly. Then, the proposed
heuristic is run on each data set with w1¼w2¼ 0.5 along with NSGA-I independently. Srinivas and Deb (1994)
proposed a non-dominated sorting genetic algorithm (NSGA) to obtain a Pareto-frontier in hard multi-objective
problems. To start the NSGA-I, it is necessary to code the solutions with chromosomes. Also, there are a number of
decision variables in the problem. However, only the independent variables are considered in the chromosome
structure. Therefore, we have used a three-parts chromosome as shown in Figure 2. The first part shows the position
of product type i in the sequence (i.e. Oi values); the second part shows the production volume of product type i in
period t (i.e. xi,t values); and the third part of the chromosome shows the unit production time of product type i
(i.e. pi values).Genetic algorithm operators such as crossover and mutation are to be performed for each part of the
chromosomes independently, as they have different dimensions. NSGA-I is run on all test data with a population
size of 25, and the obtained sets of the Pareto-optimal solutions of each algorithm are combined altogether in order
to screen the global Pareto-optimal solutions (i.e. totally non-dominated solutions) among them.
Table 2. Range of input data to generate random test problems.
Input parameter Notation Range of variations
Demand di,t [0, 100]Available time ATt [500, 600]Unit shortage cost �i,t [5, 20]Unit inventory holding cost hi,t [5, 20]Normal process time pni [10, 20]Cost of normal process time Cið p
ni Þ [50, 100]
Crash process time pci [1, 10]Cost of crash process time Cið p
ni Þ [200, 600]
Percentage of periodically deterioration fi [0, 0.5]Setup time Si,f [1, 20]Setup cost SCi,f [100, 1000]
O1 x1,1 … x1,t … x1,T p1
: : : : :Oi xi,1 … xi,t … xi,T pi
: : : : :On xn,1 … xn,t … xn,T pn
Figure 2. Three parts’ chromosome structure in NSGA-I.
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4.1 Ordinary-size problems
Scheduling decisions are usually made for short-term horizons, and based on this point, it is assumed that the
production manager tends to apply the model in a rolling horizon framework with newly corrected forecast demand
data. Thus, sizes of ordinary problems are limited. Tables 3–5 report the percentage of global Pareto-optimal
solutions for the proposed heuristic and NSGA-I under different CPU time limits as the stopping criterion. This
stopping criterion was selected in order to have a fair comparison between the two algorithms.In Figure 3, the horizontal and vertical axes of the figures are the CPU time limit and 100GP % index
respectively.According to Tables 3–5 and Figure 3, the following points can be stated:
. The proposed heuristic poses at least 56.5% (in the worst case) and at most 94.7% (in the best case) of total
global Pareto-optimal solutions.. For a fixed n, with an increase in T, the average GP index of the heuristic approximately decreases, but that
of NSGA-I gradually increases. This issue is also true when T is constant, and n increases.. For small and medium-sized problems, the heuristic is superior to NSGA-I, but for large sizes, they have
the same performance in the GP index. The heuristic seems suitable for shorter CPU time limits and/or
problem instances with small to medium sizes. The following reasons might justify the improving trend in
the GP index of NSGA-I: (1) NSGA-I has several mechanisms to generate new neighbourhoods by
Table 3. Comparison of the heuristic versus NSGA-I on different problem sizes for CPU time limit¼ 1 (s).
No. of planning periods (T)
6 12 18
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
3 88.8 91.6 94.7 21.7 27.1 33.3 77.7 86.1 91.6 22.2 29.6 33.3 66.7 71.2 76.9 46.2 48.7 508 81.4 86.5 90.3 23 30.9 34.5 75.8 82 88.4 24.9 33.3 38.5 63.5 70 75.8 48.3 49.5 52.413 80.1 85.9 89.2 24.6 31.5 35.8 75.2 81.5 87.8 25.1 34.4 38 63.1 69.4 74.3 48.6 50.2 52.818 79.7 84.3 88.8 24.2 31.1 34.4 75 81.3 87.9 25.7 34.2 38.8 62.6 68 74.2 48.9 50.1 53
Note: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of global Pareto-optimal solution obtained by NSGA-I.
Table 4. Comparison of the heuristic versus NSGA-I on different problem sizes for CPU time limit¼ 3 (s).
No. of planning periods (T)
6 12 18
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
3 86.9 88.4 89.4 34.8 40 50 69.2 76.9 84.6 38.4 46.1 53.8 57.1 63.5 66.7 53.3 58.7 64.38 79.3 85.8 89.1 37.6 45.4 52.2 67.7 74.3 88 39.9 50.2 54.3 59.2 62.8 65.4 54 59.9 65.613 78.2 84.7 88.5 38.3 46.7 52.5 67.5 73.8 84.1 40 51.6 54.5 56.9 61.6 64.3 54.1 60.2 65.818 77.8 84 88.6 38.1 46.1 52.9 67.1 73.6 83.9 40.3 52 54.7 56.5 61.2 64.9 54.3 60.5 64.9
Note: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of global Pareto-optimal solution obtained by NSGA-I.
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combining the elements of some solutions, (2) NSGA-I works with a population of solutions instead of asingle solution (like the heuristic) in each iteration.
Also, Figures 4–6 demonstrate Pareto-optimal solutions for three instances of three problem sizes of n¼ 3, andT¼ 6, 12, 18. Furthermore, the other sizes have the following issues in common with what Figures 4–6 confirm:
. For smaller sizes, a larger number of global Pareto-optimal solutions are obtained for a fixed CPU timelimit, on average. In other words, as the problem size increases, one would expect to observe fewer globalPareto-optimal solutions for a fixed CPU time limit. This observation comes back to the growth of problemcomplexity by augmenting the problem size.
. For a specific size and a given instance, by increasing the CPU time limit, the average number of globalPareto-optimal solutions increases.
. There are several dominated solutions among those of the heuristic in a given instance. It seems that thesesolutions are obtained because there is no constant effort in the heuristic to update the obtained set ofPareto-optimal solutions. Also, it might be for steps 5 and 6 of the heuristic in accepting a solution as aPareto-optimal with relation (27). Surely, this acceptance depends on w1 (and w2¼ 1�w1), and tightness ofthe bounds for Z1 and Z2 (i.e. Z1NIS�Z1PIS and Z2NIS�Z2PIS).
. For a given instance and fixed CPU time limit, the heuristic has always found the Pareto-optimal solutioncorresponding to the best value of Z1. This is true for Z2 in small- and medium-sized instances.
. For a given instance, the heuristic and NSGA-I have more solutions in common, irrespective of whether ornot they are global Pareto-optimal solutions.
In Figures 4–6, the horizontal and vertical axes of the figures are the normalised values of Z1 and Z2.
4.2 Extremely large problems
Here, the computational results on extremely large instances are reported for the test data generated according toTable 2. We do not expect to observe a real case dealing with 65 independent product types for 36 months, but theexperiments are mainly for observing the behaviour of the two competitive algorithms on very large data sets.Tables 6–9 report the GP index for CPU time limits¼ {30, 35, 40, 45} s.
We believe that the obtained results can be interpreted as follows:
. a rough pattern is observed for the GP index of the two algorithms when comparing the new results inTables 6–9 with the results of ordinary-size instances. This similarity can be for (1) randomness of decisionsin the two algorithms or (2) multi-objectivity of the problem (i.e. we do not search for a unique globaloptimal solution).
. the rule ‘the larger the problem sizes, the bigger the CPU time limits’ should be taken into account torun the algorithms. However, suitable CPU time limits should be estimated by trial and error. For example,
Table 5. Comparison of the heuristic versus NSGA-I on different problem sizes for CPU time limit¼ 5 (s).
No. of items (n)
6 12 18
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
3 72 75.3 80 40 53.6 60.9 62.5 63.4 64.7 52.6 59.9 64.7 58.8 61.2 64.7 58.8 65.4 70.68 71.6 74.9 79.2 43.5 58.1 61.7 60.3 61.5 63.8 53.2 60 65.1 58.1 61.1 62.6 56.9 62.5 66.313 70.8 73 78.4 44.3 58 61.4 61.2 61.4 63.3 54.5 60.4 65.3 58 60.5 61.9 57.2 60.7 66.118 70.2 72.7 78.1 44.2 57.5 61.6 60.1 61.1 63.2 54.8 60.7 64.5 57.4 60.5 61.2 57.3 61.6 66.2
Note: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of global Pareto-optimal solution obtained by NSGA-I.
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Figure 3. Variation in 100 GP % index of the heuristic and NSGA-I for different n and T values over the CPU timelimit¼ 1, 3, 5.
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Figure 4. Pareto-optimal solutions for three instances of size (n¼ 3,T¼ 6) with CPU time limit¼ 1, 3, 5 (s).
Figure 5. Pareto-optimal solutions for three instances of size (n¼ 3,T¼ 12) with CPU time limit¼ 1, 3, 5 (s).
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if the CPU time limit is taken below 14 s for the current extremely large sizes, very poor results will be
obtained by NSGA-I, but the heuristic is moderate in this sense. This may come back to the fact that
NSGA-I makes all decisions in a stochastic manner.. it seems that since the heuristic is a problem-specific algorithm, it can yield good results in short CPU time
limits. However, in general-purpose algorithms such as NSGA-I, less number of problem characteristics are
used in the algorithm. In other words, special features of a problem could help in devising more efficient
algorithms.
Figure 6. Pareto-optimal solutions for three instances of size (n¼ 3,T¼ 18) with CPU time limit¼ 1, 3, 5 (s).
Table 6. Comparison of the heuristic versus NSGA-I on extremely large problems for CPU time limit¼ 30 (s).
No. of planning periods (T)
12 24 36
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
55 81.5 86.5 90.2 23.1 31.1 34.3 75.9 81.9 88.3 25.1 33.5 38.6 63.3 70.2 75.8 48.4 49.3 52.360 80.0 86.1 89.4 24.6 31.6 35.8 75.2 81.7 87.8 25.2 34.4 37.8 63.1 69.5 74.3 48.8 50.2 52.665 79.8 84.2 88.7 24.0 31.3 34.3 75.1 81.5 88.1 25.9 34.4 38.6 62.6 68.2 74.0 48.8 50.2 52.9
Notes: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of globalPareto-optimal solution obtained by NSGA-I.
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Table 8. Comparison of the heuristic versus NSGA-I on extremely large problems for CPU time limit¼ 40 (s).
No. of planning periods (T)
12 24 36
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
55 71.7 74.8 79.3 43.6 58.0 61.7 60.4 61.4 63.9 53.2 60.0 65.1 58.0 61.0 62.7 57.0 62.4 66.260 70.8 73.1 78.5 44.3 58.0 61.5 61.2 61.4 63.2 54.5 60.3 65.3 57.9 60.5 61.9 57.2 60.8 66.265 70.1 72.6 78.2 44.3 57.6 61.5 60.1 61.0 63.1 54.8 60.8 64.5 57.5 60.4 61.1 57.2 61.6 66.2
Notes: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of globalPareto-optimal solution obtained by NSGA-I.
Table 7. Comparison of the heuristic versus NSGA-I on extremely large problems for CPU time limit¼ 35 (s).
No. of planning periods (T)
12 24 36
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
55 79.2 85.7 89.0 37.8 45.5 52.0 67.8 74.4 88.0 40.1 50.1 54.3 59.0 62.7 65.6 54.1 59.7 65.660 78.4 84.6 88.6 38.2 46.6 52.5 67.4 73.8 84.1 40.1 51.4 54.5 56.9 61.6 64.5 54.0 60.3 65.765 78.0 84.2 88.4 38.2 46.1 52.9 67.2 73.6 83.8 40.5 51.9 54.9 56.5 61.2 64.9 54.3 60.4 64.7
Notes: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of globalPareto-optimal solution obtained by NSGA-I.
Table 9. Comparison of the heuristic versus NSGA-I on extremely large problems for CPU time limit¼ 45 (s).
No. of planning periods (T)
12 24 36
% GPH % GPNSGA-I % GPH % GPNSGA-I % GPH % GPNSGA-I
No. of items (n) Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
Minim
um
Mean
Maxim
um
55 71.7 74.9 79.1 43.4 58.1 61.8 60.3 61.5 63.8 53.3 60.0 65.2 58.1 61.1 62.7 57.0 62.5 66.360 70.9 73.0 78.3 44.4 58.0 61.3 61.3 61.4 63.2 54.6 60.4 65.2 57.9 60.5 61.8 57.2 60.7 66.065 70.1 72.7 78.1 44.2 57.5 61.6 60.2 61.1 63.2 54.8 60.6 64.6 57.3 60.4 61.2 57.2 61.6 66.2
Notes: % GPH, percentage of global Pareto-optimal solution obtained by the heuristic; % GPNSGA-I, percentage of globalPareto-optimal solution obtained by NSGA-I.
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5. Conclusions and future research
This paper has modelled and solved the multi-objective production scheduling problem subject to specific workingconditions such as process compressibility, inventory deterioration, and sequence-dependency of setups. As theproblem is NP-hard, and there are natural conflicts between minimising the total costs and obtaining a smoothedproduction plan, a simple heuristic was proposed, based on the problem structure. The heuristic was shown to beefficient, especially for short CPU time limits and small to medium-sized problems.
It seems that the model could be developed for many other environments such as parallel machine, flow shop,job shop, open shop, cellular manufacturing systems, etc. Furthermore, the proposed solution method has somepotential to be developed for any new environment. The algorithm could be developed to behave more intelligent.Also, the current paper could be extended to deal with stochastic demands and indefinite cost coefficients. It wasassumed that the available time in each period is determined by subtracting the maintenance time from potentialworking time. However, in the real world, there are cases involving uncertain maintenance time. Another potentialarea for future research is when there are several dependent products among products under consideration.Alternatively, the paper could be extended to consider an infinite planning horizon. On the other hand, the problemcan be solved with other solution methods such as VEGA, HLGA, NSGA-II, NPGA, AMGA-I, AMGA-II, and soforth, in the future.
Note
1. M. Karimi-Nasab. Email: [email protected], [email protected].
Acknowledgements
The authors thank the anonymous referees for their insightful comments on previous versions of the current paper. Also, theythank V. Aminian for his kind help in double-checking grammatical and/or spelling issues in the text.
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Appendix 1. Proof of Lemma 2
Z2 is the sum of several non-negative variables with non-negative coefficients. Thus, it could be considered as the total sum ofsetup cost, production cost, shortage and inventory holding costs. Then, it is clear that the term of setup cost will be maximum ifthere will be one setup for each product in every planning period (i.e. ri,t¼ 1) as:
maxXTt¼1
Xni¼1
SCi,0Oi,0 þXf 6¼i
SCi,fOi,f
!ri,t �
XTt¼1
Xni¼1
max SCi,0, maxf 6¼i
SCi,f
�� �
¼ TXni¼1
max SCi,0, maxf 6¼i
SCi,f
�� �: ð1aÞ
On the contrary, this term will be minimised when there will be only one setup during all of planning periods as:
minXTt¼1
Xni¼1
SCi,0Oi,0 þXf 6¼i
SCi,fOi,f
!ri,t �
Xni¼1
min SCi,0, minf 6¼i
SCi,f
�� �: ð2aÞ
The term of production cost will be maximised if all products are processed in crash time. Hence, each xi,t should bemaximised as possible and (3a) is obtained from (3) as an upper bound for xi,t:
xi,t �ATt
pci
8i, t: ð3aÞ
Consequently, the following holds true for the term of production cost in Z2:
maxXTt¼1
Xni¼1
ðki � csipiÞxi,t �XTt¼1
Xni¼1
ðki � csipci ÞATt
pci¼
XTt¼1
ATt
! Xni¼1
ki � csipci
pci
!: ð4aÞ
Contrarily the term of production cost will be minimum if each xi,t is minimised such that (4) is satisfied and all of productsare produced in normal time as:
minXTt¼1
Xni¼1
ðki � csipiÞxi,t ¼Xni¼1
ðki � csipiÞXTt¼1
xi,t �Xni¼1
ðki � csipni ÞXTt¼1
di,t
!: ð5aÞ
The term of shortage cost in Z2 is maximised if there are T� 1 periods of shortage that do not correspond to the minimum of�i,t� di,t for each product type. Another fact obtained from Equation (3) is that only backorder is allowed, not lost sales for each
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product type. Consequently, there should be no shortage in the last planning period. Therefore, Equation (6a) will be gained byconsidering Equations (4) and (17) simultaneously:
maxXni¼1
XTt¼1
�i,tBi,t �Xni¼1
XT�1t¼1
�i,tð1� fiÞt�q
Xtq¼1
ðxi,q � di,qÞ
����������
!
�Xni¼1
XT�1t¼1
�i,tð1� fiÞt�q
Xtq¼1
di,q
! !: ð6aÞ
On the other hand, the term of shortage cost in Z2 is minimised when there is no shortage in every planning period for eachproduct type (i.e. yi,t¼ 0 for each i, t). So, (7a) holds true:
minXTt¼1
Xni¼1
�i,tBi,t � 0: ð7aÞ
Similarly the term of inventory holding cost in Z2 is maximised if there is more product than is needed in each planningperiod for each product type. Consequently, there should be no shortage in all of periods for each product type (i.e. Bi,t¼ 0 foreach i, t). On the other hand by considering (17) and (3a), an upper bound is obtained as (8a):
maxXTt¼1
Xni¼1
hi,tIi,t �XTt¼1
Xni¼1
hi,tð1� fiÞt�qXtq¼1
max ðATq
pci� di,qÞ, 0
� �( ): ð8aÞ
In the same manner as with (7a), relation (9a) always holds true:
minXTt¼1
Xni¼1
hi,tIi,t � 0: ð9aÞ
Finally, the positive ideal value of Z2, shown with Z2PIS, could be calculated as Equation (30) by considering (2a), (5a), (7a),and (9a) simultaneously. And the negative ideal value of Z2, shown with Z2NIS, could be elaborated as Equation (31) byconsidering Equations (1a), (4a), (6a), and (8a) altogether. Now, the proof is complete. œ
Appendix 2. Proof of Lemma 3
A feasible solution should satisfy all of constraints simultaneously, especially constraints (3) to (6). Thus, constraint (3) could berewritten as:
ðx1,1p1 þ � � � þ xi,1pi þ � � � þ xn,1pnÞ � AT1 � ðS1,0y1,0,1 þ S1,2y1,2,1 þ � � � þ Si,fyi,f,1þ � � � þ S1,ny1,n,1Þ
..
.
ðx1,tp1 þ � � � þ xi,tpi þ � � � þ xn,tpnÞ � ATt � ðS1,0y1,0,1 þ S1,2y1,2,t þ � � � þ Si,fyi,f,t þ � � � þ S1,ny1,n,tÞ
..
.
ðx1,Tp1 þ � � � þ xi,Tpi þ � � � þ xn,TpnÞ � ATT � ðS1,0y1,0,T þ S1,2y1,2,T þ � � � þ Si,fyi,f,T þ � � � þ S1,ny1,n,TÞ:
It is clear that every feasible solution vector should be such that the sum of the corresponding setup times is much lower than theavailable time. In other words, the right-hand side of all inequalities is to be strongly non-negative. Furthermore, it is clear thatthe coefficients of all xi,t in all inequalities are strongly positive, too. As both sides of the above inequalities are strongly non-negative, it is straightforward to show that Equation (10a) constructs a solution vector satisfying all inequalities of constraint (3)simultaneously:
xi,t ¼ATt �
Pnj¼1 sjPn
j¼1 pj
$ %8i, t, ð10aÞ
where, si¼Si,f if Oi,f¼ 1; or si¼Si,0 if Oi,0¼ 1, and Equation (21) is preferred to Equation (10a), since first, it not only satisfies allinequalities of constraint (3) but also considers the total demand of each product type, and second, the ratioPT
k¼1 di,k=Pn
j¼1
PTk¼1 dj,k is clearly greater than 0 and less than 1.
xi,t ¼ATt �
Pnj¼1 sj
� PTk¼1 di,k
� Pn
j¼1 pj
� Pnj¼1
PTk¼1 dj,k
� 6664
7775 8i, t: ð21Þ
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It should be noted that Equation (20) controls the value of pi to vary between its permitted values by 0� �� 1 as:
pi ¼ pci þ �ð pni � pci Þ: 8i: ð20Þ
For �¼ 0, constraints (3) and (5) are satisfied, but Equation (4) may not be satisfied. So, in such casesPT
t¼1 ðxi,t � di,tÞ5 0holds in step 3, and Equation (23) will update xi,t such that Equation (4) will be satisfied without denying Equation (3). RTt
measures the idle time of period t.
RTt ¼ ATt �Xni¼1
ðsiri,t þ pixi,tÞ 8t ð22Þ
xi,t ¼ xi,t þmin RTt=Pi
� �, T� 1þ
XTk¼1
ðdi,k � xi,kÞ
!B
�T
$ %( )8i, t: ð23Þ
Also, Equation (23) is still feasible, since RTt=pi� �
states that the potential capacity of producing product type i in the periodt is less than or equal to (RTt /pi). For �¼ 0 and a given sequence of product types, there should exist at least one vector ofXt¼ (x1,t, . . . , xn,t) for every planning period t, if and only if the feasible solution space is not null for that sequence. Furthermore,if there is no � that could satisfy Equations (3) and (5), the feasible solution space does not exist. In this way, property (I) holds inthe algorithm. Furthermore, all changes in the values of xi,t, as independent decision variables, are made by considering relations(20) to (23), until a feasible set of batch sizes is reached. On the other hand, if a solution is feasible, new feasible solutions (if doexist) will be generated by Equations (25) and (26). Thus, property (II) holds, too. Then, the proof is complete. œ
Appendix 3. Proof of Theorem 1
It is clear that Z1 and Z2 are all convex functions and the minimum values of Z1 and Z2 (shown with Z1PIS and Z2PIS) are smallerthan the maximum value of Z1 and Z2 respectively (shown with Z1NIS and Z2NIS). Also, it is clear that the corresponding valuesof Z1 and Z2 for all feasible solutions are less than or equal to Z1PIS and Z2PIS respectively. For all the above reasons, thefollowing relations hold true:
0 � U1 ¼Z1NIS � Z1
Z1NIS � Z1PIS� 1 ð11aÞ
0 � U2 ¼Z2NIS � Z2
Z2NIS � Z2PIS� 1: ð12aÞ
On the other hand, w1 and w2 are to be determined such that 0�w1, w2� 1 and w1þw2¼ 1. This means that Zcom is a positivecombination of Ui, and it could be re-written as Zcom¼w1U1þw2U2. This relation completes the proof. œ
Appendix 4. Proof of Theorem 2
From Theorem 1, it was proved that the domain of variation of Zcom is the interval of [0, 1]. So, because of the definition of Ui asin relations (11a) and (12a), it is clear that for the minimum seeking characteristic of Z1 and Z2, the positive ideal value of each Ui
is one, and on the contrary, the negative ideal value of each Ui is zero. As Zcom is the positive combination of Ui, the positive idealvalue of Zcom is one, and its negative ideal value is zero. œ
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