Aggregate Planning Through the Imprecise Goal Programming

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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M. Mezghani et al. / Intl. Trans. in Op. Res. 19 (2012) 581597 583

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.



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



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



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).



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



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 ),

0 +j +jv (for j = 1, . . . ,T ),C 2012 The Authors.International Transactions in Operational Research C 2012 International Federation of Operational Research Societies


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



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



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



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.


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.


F+j (+j ) = j5 f1(+j ) + j6 f2(+j ).



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.



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.



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.



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Aggregate Planning Through the Imprecise Goal Programming

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