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Intl. Trans. in Op. Res. 19 (2012) 581597DOI:
10.1111/j.1475-3995.2012.00844.x
INTERNATIONALTRANSACTIONSINOPERATIONAL
RESEARCH
Aggregate planning through the imprecise goal programmingmodel:
integration of the managers preferences
Mouna Mezghania, Taicir Loukila and Belad Aounib
aLogistique, Gestion Industrielle et de la Qualite, Institut
Superieur de Gestion Industrielle de Sfax, Universite de Sfax,Route
MHarza, km 1.5 P.B. 954, 3018 Sfax, Tunisie
bDecision Aid Research Group, School of Commerce and
Administration, Laurentian University, Sudbury, Ontario,P3E 2C6,
Canada
Email: [email protected] [Mezghani];
[email protected] [Loukil]; [email protected]
[Aouni]
Received 30 December 2010; received in revised form 9 November
2011; accepted 8 January 2012
Abstract
Aggregate planning involves planning the best quantity to be
produced during time periods in the medium-range horizon at the
lowest cost. Usually, the production manager seeks a plan that
simultaneously optimizesseveral incommensurable and conflicting
objectives, such as total cost, level of inventories, level of
customerservice, fluctuation inworkforce, and utilization level of
the physical facility and equipment. The goal program-ming
(GP)model is one of the best knownmulti-objective programmingmodels
that considers simultaneouslyseveral conflicting objectives to
select the most satisfactory solution among a set of feasible
solutions. In theproduction planning problem, the goals and the
technological parameters are naturally imprecise. Moreover,the
existing GP formulations developed in industrial engineering and
aggregate production planning do notexplicitly incorporate the
managers preferences. The aim of this paper is to develop a GP
formulation withinan imprecise environment where the concept of
satisfaction function will be utilized to explicitly introducethe
managers preferences into the aggregate planning model.
Keywords: aggregate production planning; imprecise goal
programming; satisfaction functions
1. Introduction
The main objective of aggregate planning is to provide a manager
with a production plan thatmeets the production requirements at the
lowest cost. The traditional aggregate planning analy-sis optimizes
the size of the workforce, production rate, and inventory levels.
This optimizationrequires adjusting the production capacity of the
company. A production plan should providethe portion of demand that
the company intends to meet, while not exceeding the
productioncapacity.An aggregate plan considers the overall or
aggregate level of the output and capacity that is
required to deliver such output. Dilworth (1989) considers two
basic approaches to estimate the
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capacity that will be required to produce an aggregation or
grouping of a companys products. Theseare top-down and bottom-up
approaches. The top-down approach to aggregate planning
involvesdevelopment of the entire plan by working only at the
highest level of consolidation. Under thisapproach, the products
are consolidated into an average product and then a single plan is
developedfor this aggregated product. Once the aggregate plan is
completed, it is disaggregated to allocatecapacity to individual
product families and individual products. This is the most
traditional ap-proach and is frequently observed in the literature
on aggregate planning. The top-down approachencompasses two major
categories: (a) methods that rely on subjective judgment to propose
alter-native arrangement of resources and to determine which plan
is best and (b) methods that attemptto mathematically model the
costs and systematically seek the best plan. The second category
canbe further subdivided into: (i) methods that mathematically seek
the optimal solution of the costfunction to set the best plan and
(ii) methods that perform a computer search for what appears tobe
the lowest cost plan.Linear programming (LP) is one of the widely
used optimization techniques. The objective of
aggregate planning is to find a production plan that leads to
the optimal use of the companysresources. Several methods of
aggregate planning have been proposed to express the cost of
theaggregate plan as amathematical expression and to seek the plan
thatminimizes this cost. Naturally,LP can be used if the cost of
the resources behaves as a linear function of the amount of the
resourcesused by the aggregate plan. Hansemann and Hess (1960) used
LP for production and employmentscheduling. The transportation form
of LP has also been proposed to solve aggregate planningproblems
(Bowman, 1956).The linear decision rule (LDR) is another
mathematical technique that is used in aggregate
production planning (APP). It contains linear equations that
recommend the best production rateand the best workforce size for
the upcomingmonths, based on the forecast demand for
themedium-range planning horizon. The data used to develop and test
the LDR are sometimes used to comparethe performance of other
aggregate planning techniques. Both LP and the LDR require a
specificmathematical structure for the functions to be optimized.
The mathematical complexity of theproblem increases when step
functions and equations of a higher order than quadratic are
involved.The bottom-up approach, or subplan-consolidation approach,
involves development of plans
for major products or product families at some lower level
within the product line. These sub-plans are then summed up to
arrive at the aggregate plan, which gives the overall outputand the
capacity required to produce it. This approach has come to be more
widely utilized,as massive computing capability has become more
economical and widely available in morecompanies.The earlier
mathematical models used to solve aggregate planning problems are
based on a single-
objective function, namely the total cost, to be minimized.
However, in practice, the productionenvironment requires more than
one objective. For example, frequent layoff and hiring not
onlyinvolves the direct cost of those activities, but they also
have an impact on the workforce moralethat is hard to measure in
monetary units. So, while it is desirable to have a low-cost
productionplan, it is also desirable to have a production plan that
has the lowest workforce variation. Theobjectives that may be
considered for aggregate planning are total cost of the plan, level
of in-ventories, customer service, fluctuation in workforce, and
utilization of the physical facility andequipment. These objectives
are conflicting and incommensurable. Goal programming (GP) is oneof
the multi-objective mathematical programming models that can be
applied to determine the
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best production plan. The GP model aggregates simultaneously
several conflicting objectives forchoosing the most satisfactory
solution within a set of feasible solutions (Aouni and Kettani,
2001).Compared with the other techniques introduced earlier to
handle APP, the GP model is a moreflexible technique that is
relatively easy to understand and utilize. The GP has been applied
inindustrial engineering and APP (Jaaskelainen, 1969; Lee and
Jaaskelainen, 1971; Charnes et al.,1976; Welam, 1976; Leung and Ng,
2007; Leung and Chan, 2009). However, these formulationswere
developed to deal with deterministic and precise decision-making
situations. Nevertheless,in practice, the goals and technological
parameters such as demand, available resources, and ca-pacities are
often imprecise or fuzzy. Zimmermann (1976) first introduced fuzzy
set theory intoconventional LP problems. This study considered LP
problems with a fuzzy goal and constraints.Following the fuzzy
decision-making method proposed by Bellman and Zadeh (1970) and
usinglinear membership functions, the above study confirmed the
existence of an equivalent LP problem.Thereafter, fuzzy
mathematical programming has been utilized to solve APP problems.
Lee (1993)presented an interactive fuzzy LP (FLP) model to solve
APP problems. Lees model considerssituations with soft constraints
and a single-objective function for total cost. Tang et al.
(2000)solved the multiproduct APP problem with fuzzy demands and
fuzzy capacities. For obtaining asolution, the fuzzy model was
converted into its crisp equivalent by using defuzzification of
softequations according to the satisfaction of membership functions
at a defined degree of truth. Daiet al. (2003) proposed a FLP model
for the single-objective APP problem in a manufacturing
en-vironment, where the workforce level and the demand are
imprecise. A trapezoidal form of fuzzynumbers is utilized and all
membership functions are formulated with linear forms.Wang and
Liang(2004) developed a fuzzy multi-objective LP model for solving
the multiproduct APP problem ina fuzzy environment. The handled
problem has the following three objectives: minimizing
totalproduction costs, minimizing carrying and backordering costs,
and minimizing rate of change inlabor levels; fuzzy aspiration
levels were defined for each objective. Piecewise linear
membershipfunctions are utilized to solve this problem. The model
can yield an efficient compromise solution.Jamalnia and Soukhakian
(2009) proposed a hybrid fuzzy multi-objective non-LP model with
dif-ferent objective priorities for a multiproduct multiperiod APP
problem in a fuzzy environment.In the literature, we notice
additional papers dealing with the APP problem, such as Gen et
al.(1992), Wang and Fung (2001), Wang and Liang (2005), Sakalli et
al. (2010), and Baykasoglu andGocken (2010).The developed
formulations of the APP do not explicitly integrate the managers
preferences. The
aim of this paper is to propose a GP formulation for the APP
where the concept of satisfactionfunctions will be used to
elucidate the managers preferences and for modeling the
imprecisionrelated to the goals associated with objectives and some
technological parameters (right-hand sidesof the constraints).In
Section 2, we will outline the general formulation of the GP within
an imprecise environment.
We will also present the concept of satisfaction functions as a
tool for handling the managerspreferences within the imprecise GP
model. We present in Section 3 the mathematical formulationof our
model for the APP problem, where the aspiration levels are
considered as fuzzy values. InSection 4, we will illustrate our
formulation through the numerical example proposed by Dai etal.
(2003). The obtained results will be discussed in Section 5.
Finally, Section 6 provides someconcluding remarks.
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2. General formulation of fuzzy goal programming (FGP) model
Zimmermann (1978) has developed the first fuzzy multi-objective
programming formulationthrough the concept of membership functions.
Dhingra et al. (1992), Rao (1987), and Zimmer-mann (1978, 1988)
have developed a linear approximation procedure for the nonlinear
membershipfunctions. Since early 1980, fuzzy sets have been used in
the GP model to represent imprecise andfuzzy knowledge about some
parameters and to represent a satisfaction degree of the
decisionmaker(DM) with respect to his/her preference structure
(Narasimhan, 1980; Hannan, 1981a, 1981b; Ig-nizio, 1982; Tiwari et
al. 1987; Wang and Fu, 1997). The FGP formulation proposed by
Hannanis simple and more efficient comparatively to the one
proposed by Narasimhan. The FGP modelconsiders goals as fuzzy. This
fuzziness is expressed through a triangular membership
functiondefined on the interval [0; 1]. The mathematical
formulation of Hannan (1981a) model is as follows:
Maximize Z = Subject to: n
j=1ai jx j/i
+i + i = gi/i (for i = 1, 2, . . . , p),
+ i + +i 1 (for i = 1, 2, . . . , p),x X,xj , ,
i and
+i 0 (for i = 1, 2, . . . , p and j = 1, 2, . . . , n),
where:i: The constant of deviation of the aspiration levels
gi.ai j : Technological coefficients associated with goals.X : The
set of feasible solutions.+i , i : The positive and negative
deviations associated with the objective i.
Chang (2007) has proposed linearization strategies to convert a
binaryFGPmodel into a standardversion formulation of the
FGP.Despite the fact that the FGPmodel allows imprecisionmodeling
ofthe goals, thismodel seems to be rigid. Ignizio (1982) stresses
the fact thatNarasimhan andHannansformulations are limited to
specific cases where the DM is supposed to have membership
functionsof particular forms like the triangular one. The use of
such triangular membership functions wasmainly criticized by
Ignizio (1982) and Martel and Aouni (1998). These criticisms are
related tothe fact that the triangular form of the membership
functions does not reflect adequately the DMspreferences and are
not appropriate for modeling the goals fuzziness. Moreover, Martel
and Aouni(1998) mentioned that this type of functions has some bias
to the central values of the objectiveachievement levels. Wang and
Fu (1997), Pal and Moitra (2003), and Chen and Tsai (2001) havesome
concerns regarding the way to deal with the goals fuzziness through
the triangular form ofthe membership functions and indicate that in
some applications, this type of functions leads tono desired
results. The GP model formulation proposed by Martel and Aouni
(1998) explicitly
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Fig. 1. General form of the satisfaction function.
integrates the DMs preferences in an imprecise environment. Two
limits are fixed for the goals: thelower limit (gli) and the upper
limit(g
ui ). The fuzzy values (aspiration levels) specified by the
DM
(gi) can also be arbitrarily chosen from the interval (gi [gli,
gui ]). This formulation is as follows:
MaximizexX
Z =p
i=1
(w+i F
+i (
+i ) + wi Fi (i )
)
Subject to :nj=1
ai jx j+i + i = gi for i = 1, . . . , p;
x X ;gi
[gli, g
ui
](for i = 1, . . . , p);
0 +i +iv (for i = 1, . . . , p);0 i iv (for i = 1, . . . ,
p),
where Fi(i) are the satisfaction functions associated with
positive and negative deviations (+i ,
i )
as presented in Fig. 1, id is the indifference threshold, i0 is
the null satisfaction threshold, +iv
and iv are the positive and negative veto thresholds, w+i and wi
are the importance coefficientsassociated with positive and
negative deviations, respectively.The satisfaction function concept
will be utilized for modeling the imprecision related to the
goals
within the GP model where the managers preferences will be
explicitly introduced in the aggregateplanning model.
3. Model formulation
In the APP problems, the input data or parameters, such as
demand, available resources, and capac-ities, are generally
imprecise because some information is incomplete or unobtainable or
collectingprecise data is very difficult. Sometimes, the arrival of
orders can be random; the information about
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available resources can be imprecise because of environmental
factors, parameters such as machinetime and available labor time
cannot have precise values. These imprecise parameters can be
de-fined as random numbers with probability distribution, but, a
great deal of knowledge about thestatistical distribution of the
uncertain parameters is required. Thus, in practice, the
utilization offuzzy numbers for imprecise parameters is more
efficient and/or practical. The model that we willpropose assumes
that a company produces N types of products to meet the market
demand over aplanning horizon T. In this model, the market demand,
maximum number of employees, and ma-chine capacity are considered
as imprecise over the planning horizon. The following two
objectivesare to be considered in the APP: (a) minimize the total
production costs (Z1) and (b) minimize thechanges in the workforce
level (Z2). These objectives are fuzzy with imprecise aspiration
levels. Thefollowing notations are used:
ParametersDnt: Forecast demand of product n in period t.
CPnt: Production cost per unit of regular time for product n in
period t.COnt: Production cost per unit of overtime for product n
in period t.CSnt: Cost to subcontract one unit of product n for one
period.CI+nt : Inventory cost per unit for product n in period
t.CInt : Backorder cost per unit for product n in period t.CHt:
Cost to hire one employee in period t.CFt: Cost to layoff one
employee in period t.int: Labor time per unit of product n in
period t (man hour/unit).rnt: Machine hours per unit of product n
in period t (machine hour/unit).
Wtmax: Maximum workforce available in period t.Mtmax: Maximum
machine capacity available in period t (machine-hour).
a: Regular working hours per employee.bt: Fraction of working
hours available for overtime production.
Decision variablesWt: Workforce level in period t.Pnt: Regular
time production of product n in period t.Ont: Overtime production
of product n in period t.Snt: Subcontracted production of product n
in period t.I+nt : Inventory of product n at the beginning of
period t.Int : Backorder of product n at the beginning of period
t.Ht: Number of employees hired in period t.Ft: Number of employees
laid off in period t.
The analytical forms of the two objectives are as follows:
Objective 1: minimizeN
n=1T
t=1CPnt Pnt (regular time production cost)+N
n=1T
t=1COnt Ont(overtime production cost)+Nn=1 Tt=1CSnt Snt
(subcontracting cost)+Nn=1 Tt=1CI+nt I+nt(inventory cost) +Nn=1
Tt=1CInt Int (backordering cost).
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Objective 2: minimizeT
t=1 (Ht + Ft )(hiring and firing levels).Subject to:
Inventory level constraints
I+nt1 Int1 + Pnt +Ont + Snt I+nt + Int = Dnt (n = 1, . . . ,N) ,
(t = 1, . . . ,T ) , (1)
Workforce constraints
Wt = Wt1 +Ht Ft (t = 1, . . . ,T ) , (2)
Wt W tmax (t = 1, . . . ,T ) , (3)
Nn=1
intPnt aWt (t = 1, . . . ,T ) , (4)
Nn=1
intOnt abtWt (t = 1, . . . ,T ) , (5)
Machine capacity constraints
Nn=1
rnt(Pnt +Ont
) Mtmax (t = 1, . . . ,T ) , (6) Nonnegativity constraints
Wt,Pnt,Ont,Snt, I+nt , I
nt ,Ht,Ft 0 (n = 1, 2, . . . ,N) , (t = 1, 2, . . . ,T ) .
(7)
The forecasted demand Dnt of product i in period t is imprecise.
In practice, the forecasted demandcannot be obtained precisely in a
dynamic market. The sum of regular and overtime
production,inventory levels, and subcontracting levels essentially
should be equal to the market demand, asin equation 1. Moreover,
the maximum workforce available 3 and the maximum available
machinecapacity 6 are imprecise. The formulation proposed by Martel
and Aouni (1998) neglected theintegration of fuzzy information in
the right-hand sides of the constraints and supposes that
allconstraints are precise. But, in our APP model we have flexible
constraints for which the right-handsides are imprecise and
expressed through intervals. These constraints may be violated and
can beconsidered as goals. Constraints 1, 3, and 6 are flexible and
constraints 2, 4, and 5 are consideredto be crisp.The proposed
mathematical model considers only one item (N = 1) during a horizon
of T
periods. The objective is to elaborate an aggregate production
plan where the managers preferencesare explicitly integrated in an
imprecise environment. The concept of satisfaction functions will
be
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used for modeling both the imprecise nature of the information
and the managers preferences. Themathematical model is as
follows:
Z =2i=1
w+i F+i
(+i)+ T
j=1
(w+j F
+j
(+j)+ wj Fj (j ))+
Tj=1
w+j F+j
( +j)+ T
j=1w+j F
+j
(+j),
Subject to:Tt=1
(CPt Pt +COt Ot +CSt St +CI+t I+t +CIt It
) +1 + 1 = Z1,Tt=1
(Ht + Ft
) +2 + 2 = Z2,I+t1 It1 + Pt +Ot + St I+t + I+t +j + j = Dt (for
j = 1, . . . ,T ),
Wt +j + j = Wtmax (for j = 1, . . . ,T ),
rt(Pt +Ot
) +j + j = Mtmax (for j = 1, . . . ,T ),Wt = Wt1 +Ht Ft,
itPt aWt,
itOt abtWt,
0 +i +iv (for i = 1, 2),
0 +j +jv (for j = 1, . . . ,T ),
0 j jv (for j = 1, . . . ,T ),
0 +j +jv (for j = 1, . . . ,T ),
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where:+j and
j ( j = 1, . . . ,T ) are the positive and negative deviations
for the inventory level goal,
respectively. +j and
j ( j = 1, . . . ,T ) are the positive and negative deviations
for the workforce goal, respec
tively.+j and
j ( j = 1, . . . ,T ) are the positive and negative deviations
for the machine capacity goal,
respectively.
4. Model simulation
In this section, we illustrate the developed APP model through
the same numerical example pre-sented by Dai et al. (2003). The
objective function Zcorresponds to the total costs, given by
theproduction costs and the changes in workforce levels costs. The
maximum number of employeesand machine capacity are precise over
the planning horizon. Table 1 summarizes the intervals forthe
market demand. Other relevant data are as follows:
(1) The initial inventory (I0) is 0. The inventory carrying cost
is $2 per unit per period and thebackorder cost is null. The
subcontracting cost is $67 per unit.
(2) The initial workforce (W0) is 100 employees/day. The costs
associated with the regular payroll,hiring, and firing are $64,
$30, and $40 per employee, respectively.
(3) The production cost is $20 per unit and 3 h of labor are
needed for each unit. The regular timeper employee is 8 h/day.
(4) Overtime production is limited to no more than 30% of
regular time production. The overtimecost is $15/h.
(5) Regular capacity of the machines is as follows: 800, 700,
820, 650, 750, and 720 h. The fractionof regular machine capacity
available in overtime is ct = 0.5, 0.6, 0.5, 0.6, 0.4, and 0.4 for
allperiods. The machine time is 2.5 machine h/unit.
In this case, the total cost is imprecise and the permissible
limit is 83,00091,000. For modelingthe imprecise goals, we use two
satisfaction functions types for interval goals: type II and type
III(Martel and Aouni, 1990). The shapes of these satisfaction
functions are presented below.
Table 1Imprecise intervals of the market demand (Dt,Dt )
Demand\period Dt (lower limit) Dt (upper limit)1 250 3002 285
3753 400 5004 260 3405 250 3506 200 340
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Fig. 2. Total production cost satisfaction function.
4.1. Satisfaction function for Z
The main objective for any manufacturing company is to minimize
the total cost over the planninghorizon. So, we have a tendency
toward the lower bound of the target interval ($83,000).
Thepreferences decrease as the values tend toward the upper bound
($91,000). We have utilized thesatisfaction functions with linear
preference type (type III, Fig. 2). The managers satisfaction
willbe higher with smaller values of the deviation from the goal.
The satisfaction is decreasing whendeviations are within the
following interval [0; 7,000]. Solutions with deviations greater
than $8,000will be rejected.The satisfaction function F+1 (
+1 ) can be written as follows:
F+1 (+1 ) =
{f1(
+1 ) = 1 0.000143+1 if 0 +1 7,000;
f2(+1 ) = 0 if 7,000 +1 8,000.
The equivalent representation of this function requires the
introduction of two binary variables11 and 12. These variables are
defined as follows:
11 ={1 if 0 +1 7,000;0 otherwise.
and 12 ={1 if 7,000 +1 8,000;0 otherwise.
Thus, the function can have the following equivalent form:
F+1 (+1 ) = 11 f1(+1 ) + 12 f2(+1 );
= 11(1 0.000143+1 ) + (0)12= 11 0.00014311+1
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Fig. 3. Market demand satisfaction functions.
The objective is to maximize the satisfaction function F+1 (+1
). The mathematical program can
be written as follows:
Maximize Z = 11 0.00014311+1 ,Subject to:
7,00012 +1 0;
+1 7,00011 8,00012 0; (8)
11 + 12 = 1;
11, 12 {0; 1}; +1 0,
4.2. Satisfaction function for Dt
The market demand of the products is considered imprecise and
expressed through the intervals[Dt, Dt]. However, the manager
cannot provide an exact value of the demand for each period.
Theparameter Dt can be any point within this interval. We have
defined the value of this parameteras: Dt = Dt = (Dt +Dt )/2. Two
types of satisfaction functions are used as shown in Fig. 3.
Thesatisfaction function thresholds for the positive and the
negative deviations are available in Tables2 and 3.
Table 2Negative deviations
1 2
3
4
5
6
j0 25 40 45 40 50 70 jv 30 45 50 45 55 75
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Table 3Positive deviations
+1 +2
+3
+4
+5
+6
jd 25 40 45 40 50 70 jv 30 45 50 45 55 75
According to these figures, the satisfaction is at its maximum
when the deviation from the centralvalue of the demand interval for
each period is within the interval [0, jd ]. The satisfaction
functiondrops to zero as soon as the positive deviation goes beyond
the indifference threshold jd (Fig. 3)and decreases for negative
deviation in the following interval: [0, j0] (Fig. 3). If the
positive andnegative deviations from the goals are within the
interval [ jd , jv], [ j0, jv], respectively. Hencethe manager is
not satisfied but still he/she is willing to accept solutions.
However, if the deviationsare larger than the veto threshold jv,
then the solution will be rejected.The satisfaction functions Fj
(
j )( j = 1, 2, . . . , 6) can be written as follows:
Fj (j ) =
f1(j ) = 1 1
/ j0 if 0 j j0;
f2(j ) = 0 if j0 j jv.
The equivalent representation of this function requires the
introduction of two binary variables j3 and j4. These variables are
defined as follows:
j3 =1 if 0 j j0;0 otherwise.
and j4 =1 if j0 j jv;0 otherwise.
Thus, the function can have the following equivalent form:
Fj (j ) = j3 f1(j ) + j4 f2(j ) = j3 1
/ j0 j3
j .
The satisfaction functions F+j (+j ) ( j = 1, 2, . . . , 6) can
be written as follows:
F+j (+j ) =
f1(+j ) = 1 if 0 +j jd ;
f2(+j ) = 0 if jd +j jv.
The equivalent representation of this function requires the
introduction of two binary variables j5 and j6. These variables are
defined as follows:
j5 =1 if 0 +j jd ;0 otherwise.
and j6 =1 if jd +j jv;0 otherwise.
Thus, the function can have the following equivalent form:
F+j (+j ) = j5 f1(+j ) + j6 f2(+j ).
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The objective is tomaximize Fj (j ) and F
+j (
+j ) ( j = 1, 2, . . . , 6). Themathematical program
can be written as follows:
Maximize Z = j3 1/ j0 j3
j + j5
Subject to:
j0 j4 j 0 (for j = 1, 2, . . . , 6),
j j0 j3 jv j4 0 (for j = 1, 2, . . . , 6),
jd j6 +j 0 (for j = 1, 2, . . . , 6), (9)
+j jd j5 jv j6 0 (for j = 1, 2, . . . , 6),
j3 + j4 + j5 + j6 = 1 (for j = 1, 2, . . . , 6)
j3, j4, j5, j6 = {0; 1}; +j , j 0 (for j = 1, 2, . . . , 6).For
the purpose of illustration, we have assumed that the two
objectives have the same importance.
The objectives of the same category have the same weight (wj =
0.5/6 for j = 1, 2, . . . , 6). We haveconsidered the same
constraints used by Dai et al. (2003); constraint (6) is replaced
by rPt Mtand rPt ctMt. Because the developed model contains
nonlinear terms 9 and 8, we have utilizedthe linearization
procedure developed by Oral and Kettani (1992). The software Lindo
6.1 is usedto solve the mathematical program.
5. Discussion of the results
The corresponding aggregate production plan is generated and the
results are summarized inTable 4. The global satisfaction level is
79%. As can be seen from Table 5, the best total costobtained by
our model is $83,000 as compared to $87,114 obtained by Dai et al.
(2003). Thesatisfaction for this goal reached the level
100%.According to the Figs 5 and 6, the production level is almost
stationary and varies around the
average. Moreover, demand movement is seasonal and reaches a
peak during the third period. Theproduction plan obtained by Dai et
al. (2003) shows a big variation for workforce level to meetthe
optimistic forecasted demand as presented by Fig. 4. Whereas, a
minimum cost is obtainedwith lower variation of the workforce level
by using the proposed GP model and the satisfactionfunction. It is
obvious that the solution is very satisfactory.
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Fig. 4. Workforce level movement.
Table 4Best production plan
Period 1 2 3 4 5 6
Regular time production 280 280 280 260 272 270Overtime
production 0 0 115 40 0 0Inventory level 5 5 0 0 0 0Subcontracted
production 0 0 0 0 0 0Workforce level 105 105 105 102 102
102Employee hired 5 0 0 0 0 0Employee layoff 0 0 0 3 0 0
Table 5Results comparison
FLP FGP
Production total cost $87,114 $83,000
Fig. 5. Results of (Dai et al.) model.
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595
Fig. 6. Proposed FGP model results.
Table 6Comparative results
FLP FGP
Total demand 1,848 1,861Total production 1,616 1,642Overtime
production 187 219Inventory level 29 5Subcontracted production 45
Employee hired 9 5Employee layoff 8 3
Our model allows to satisfy more demand with lower variation of
the workforce level with adeterministic total cost objective
$87,114. The comparative results are reported in Table 6.
6. Conclusion
To deal with imprecision and the fuzziness in APP, we have
developed a GP model on the basisof the concept of satisfaction
functions. The proposed model integrates the managers
preferenceswhere the goals and technological parameters are
imprecise and expressed through intervals. Thismodel has been
utilized to generate a production planwith higher satisfaction
level for the numericalexample presented byDai et al. (2003).
Aminimum total cost is obtained with lower variation of
theworkforce level. One of the advantages of this formulation is
its flexibility and it explicitly integratesthe managers
preferences in an easy manner. The main shortage is that the
obtained results dependon the form of the satisfaction function
used. Real data from companies can be used to validate theproposed
model. The manager should generate the appropriate objectives,
input data, and type ofsatisfaction functions on the basis of
subjective judgment and/or historical resources. However, thenumber
of constraints and variables varies according to the practical APP
application.
C 2012 The Authors.International Transactions in Operational
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581597
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