Open Journal of Applied Sciences, 2012, 2, 188-192
doi:10.4236/ojapps.2012.23028 Published Online September 2012 (http :/ /www.SciRP.org/journal/ojapps)
Stability of Production Planning Problem with
Fuzzy Parameters
Samir Abdou Abass
Department of Mathematics and Theoretical Physics, Nuclear Research Center, Atomic Energy Authority, Cairo, Egypt
Received June 3, 2012; revised July 5, 2012; accepted July 16, 2012
ABSTRACT
The traditional production planning model based upon the famous linear programming formulation has been well
known in the literature. The consideration of uncertainty in manufacturing systems supposes a great advance. Models
for production planning which do not recognize the uncertainty can be expected to generate inferior planning decisions
as compared to models that explicitly account the uncertainty. This paper deals with production planning problem with
fuzzy parameters in both of the objective function and constraints. We have a planning problem to maximize revenues
net of the production inventory and lost sales cost. The existing results concerning the qualitative and quantitative
analysis of basic notions in parametric production planning problem with fuzzy parameters. These notions are the set of
feasible parameters, the solvability set and the stability set of the first kind.
Keywords: Production Planning; Stability; Linear Programmin g; Fuzzy Parameters
1. Introduction
The classical linear programming (LP) models for pro-
duction planning have been around for many years. A
typical formulation of the LP planning models has the
objective minimizing the total production-related costs,
such as variable production costs, inventory costs, and
shortage costs, over the fixed planning horizon [1,2]. The
usual constraints employed are: 1) inventory balance
equations for making the inventory and/or shortages bal-
anced with those from the previous period, production
quantity, and the demand quantity; 2) capacity con-
straints which ensure the total workload for each re-
source not exceed the capacity in each period [3].
The LP model considers the limited availability of the
resources (labor, machine, etc.) through the capacity
constraints. In a real production system, such capacity
constraints may not correct. Galbraith [4] defined uncer-
tainty as the difference between the amount of informa-
tion required to perform a task and the amount of infor-
mation already possessed. Mula et al. [5] presented an
exhaustive literature survey about models for production
planning under uncertainty. Abouzar Jamalnia and M.
Ali Soukhakian [6] introduced a hybrid fuzzy multi ob-
jective nonlinear prog ramming model with differen t goal
priorities. Zrinka et al. [7] introduced the production
planning problem as a bilevel programming problem. In
the real world, there are many forms of uncertainty that
affect production process. Ho [8] categorizes them into
two groups: 1) environmental uncertainty and 2) system
uncertainty. Environmental uncertainty includes uncer-
tainties beyond the production process, such as demand
uncertainty and supply uncertainty. System uncertainty is
related to uncertainties within the production process,
such as operation yield uncertainty, production lead time
uncertainty, quality uncertainty, failure of production
system and changes to product structure, to mention
some. In this paper, we will use the first category of un-
certainty. The literature in production planning under
uncertainty is vast. Different approaches have been pro-
posed to cope with different forms of uncertainty (see,
for example, [8-10]).
This paper is organized as follows. In next section, a
model of production planning problem to maximize
revenues net with fuzzy parameters is formulated. Sec-
tion 3 presents a qualitative analysis of some basic no-
tions for the problem of concern. An illustrative numeri-
cal example is provided in Section 4. Finally, Section 5
contains the concluding remarks.
2. Problem Formulation
For some production planning problems we have the
option of not meeting all demand in each time period.
Indeed, there might not be sufficient resources to meet all
demand. In this case, the optimization problem is to de-
cide what demand to meet and how. We assume that de-
mand that cannot be met in a period is lost, thus reducing

S. A. ABASS 189
revenue. First we gi ve t he notion:
T: number of time periods;
I: number of items;
K: number of resources;
kt : fuzzy parameters represents the amount of re-
source k available in time period t;
b

it : fuzzy parameters represent the demand for item i
in time period t;
d

it
r
cp
: unit revenue for item i in time period t;
it : unit variable cost of production for item i in time
period t;
it : unit cost of not meeting demand for item i in
time period t;
cu
it : unit inventory holding cost for item i in time pe-
riod t.
cq
The Decision Variables
it
p
q
: production of item i during time period t;
it
u
: inventory of item i at end of time period t;
it
The optimization model of production planning prob-
lem to maximize revenues net with fuzzy parameters is
as follows:
: unmet demand of item i during time period t.

11
max TI
itititititit itit it
ti
rducpp cqq cuu


 

 

t
d


(1)
subject to
1
,
I
ik itkt
i
ap bkt


 (2)
,1 , i titititit
qpqudi

 (3)
,,0 ,
it it it
pqu it (4)
The  -level set of the fuzzy numbers are
defined as the ordinary set and respec-
tively for which the degree of their membership function
exceeds the level
  [0,1]. This definition is introdu ced
by Dubois and Prade [11,12]. For a certain degree
 ,
problems (1)-(4) can be understood as the following non-
fuzzy production planning problem:
and
kt it
b


Ld



Lb



11
max TI
itititititit itit it
ti
rducpp cqq cuu

  

 (5)
subject to
1 ,
I
ik itkt
i
ap bkt


 (6)
,1 , i titititit
qpqudi
 t
(7)

,
it it
dLd it


 (8)

,
kt kt
bLbkt


 (9)
,,0 ,
it it it
pqu it (10)
The nonfuzzy production planning problem can be re-
written in the following equivalent form:

11
max TI
itititititit itit it
ti
rducpp cqq cuu


  


 (11)
subject to
1
,
I
ik itkt
i
ap bkt


 (12)
,1 , i titititit
qpqudi
t

 (13)
,
it itit
hd Hit

 (14)
,
kt ktkt
lbL kt

 (15)
,,0 ,
it it it
pqu it (16)
where are lower and upper bounds on it
respectively and are lower and upper bounds
on respectively.
and
it it
hH d
and
it it
lL
it
b
3. Qualitative Analysis of Basic Notions for
the Problems (11)-(16)
Let ,, and ,,
itit ktkt
hHlL ikt

are assumed to be pa-
rameters rather than constants. The decision space of
problem (11)-(16) can be defined as follows:



3
,,,
,,,satisfies the constraints (12)-(16)
IT
it itit
XhHlL
pqu Rit
In what follows we are give the definitions of some
basic notions for the problem (11)-(16). Such notions are
the set of feasible parameters, the solvability set and the
stability set of the first kind (see [13,l4]).
3.1. The Set of Feasible Parameters
The set of feasible parameters of the prob lems (11)-( 16),
which is denot e d by U, is defined by:





2
,,,,,, is not emp ty
TI K
UhHlLRX hHlL


3.2. The Solvability Set
The solvability set of problems (11)-(16), which is de-
noted by V, is defined by




,,,
p
roblem (11)-(16) has -optimal solution.
VhHlL
U 


3.3. The Stability Set of the First Kind
Suppose that with a corresponding
α-optimal solution
****
,,,hHlL V


***
,,
it it it
pqu for problems (11)-(16)
together with the α-level optimal parameters


**
,
kt it
bd .

S. A. ABASS 190
The stability set of the first kind of problems (11)-(16) that is denoted by


***
,,
it it it
Spqu is defined by





**
of problems (1
,,, ,
it kt
db ik
***
***,,,,, is -optimal solution1)-(16)
,, with corresponding -level optimal parameters
it it it
it it it
hHlLVpqu
Sp qu
t










3.4. Determination of the Stability Set of the
First Kind
The Lagrange function of problems (11)-(16) can be
written as follows:



1
,1
++
++
I
ktik itkt
i
ititit it itit
kt ktktkt ktkt
itititit itit
ititititit it
LFZa pb
qpqud
bH hb
dL ld
pqu









 




 




where

11
max TI
itititititit itit it
ti
Z
rducpp cqq cuu




 .
The Kuhn-Tucker necessary optimality conditions for
problems (11)-(16) are as follows:
0, 0, 0 ,
it itit
LFLFLF it
up q
 
 
 

t
t
t
it
1
0 ,
I
ik itkt
i
ap bkt



,1 0 , i titititit
qpqud i

0, 0 ,
it ititit
hdd Hit 
0, 0 ,
kt ktktkt
lb bLkt
,,0 ,
it it it
pqu it
10 ,
I
ktik itkt
i
ap bkt 










,1 0 ,, iti titititit
qpqudik 
 

0 ,
kt ktkt
bH k  

0 ,
kt ktkt
hb kt  

0 ,
it itit
dL it  

0 ,
it itit
ld it  
0, 0, 0 ,
ititit itit it
pqu

 
,,,,,,,,0 ,,
ktit ktktit it it ititikt

  
where all the relations of the above system are evaluated
at the α-optimal solution with the corre-
sponding α-level optimal parameters

***
,,
it it it
pqu



**
,.
it kt
db
,,,, ,
ktit ktktititit,,, and it it it

  
and
it
 are the La-
grange multipliers.
4. Illustrative Example
Let us consider the following production planning prob-
lem to maximize net revenues with fuzzy parameters.
Consider I = 3, K = 3 and T = 2. Table 1 contains the
values of triangle fuzzy parameters kt, db
 1, 2,3,i

1, 2,1, 2, 3tk .


,, and 1,2,cp cq rcuk
Table 2 contains the values of aik,
1,2,3,i t1, 2, 3.


itit itit
We a ss ume th at th e me mb er sh ip functions to the trian-
gle fuzzy numbers is take the following form:
and
it kt
db



12
dd

32
d
1
dd d




  

and


121
bb

2
b
3
bb b




  

let α = 0.5, problems (11)-(16) can be written as follows:
1121 21
31 12
22 32
max 29
2
4 91
Zd d
d
d
11 11
31 31
22 22
8 4
378
116
u q
duq
duq
21
12
32
8
7
u
u
u
12
32
6
89
q
q
q



 

21 2,5,6d


 


11 1, 3, 6d



 

3,4,6
Table 1. Values of triangle fuzzy parameters.
31 d


12 2,3,4d


22 1,5, 6d


32 4,6,9d


11 2,5,6b


21 3,4,7b


22 3,6,8b


31 1, 4,6b


12 2,5,7b


32 2,4,5b

Table 2. Values of aik, cpit, cqit, rit and cuit . ,,itk
11 3a

21 1a 31 2a

12 4a

22 6a 32 5a

13 4a

23 5a 33 6a

11 2cp

21 5cp  31 4cp

12 3cp

22 4cp  32 6cp

11 2cq

21 3cq  31 4cq

12 3cq

22 2cq  32 3cq

11 6cu

21 8cu  31 4cu

12 5cu

22 7cu  32 9cu

11 2r

21 1r 31 3r

12 2r

22 4r 32 2r


S. A. ABASS 191
subject to
11213111
32pp pb , ,
112131 21
465pppb
122131 31
456pppb,
122232 12
32pp pb , ,
122232 22
465pppb
122232 32
456pppb,
.5,
.5
1,0 11111111
0,qpqud 
2,0212121210,qpqud 
3,031313131 0,qpqud
1,112121212 0,qpqud
2,122222222
0,qpqud
3,132323232
0,qpqud 
11
3.5 5.5,b
21
3.5 5.5,b 31 2.5 5,b
12
3.5 6,b
22
4.5 7,b 32 34b
11
24d.5,21 3.5 5.5,d
31
3.5 5,d
12
2.5 3.5,d
22
35d.5,
32
57d
So, we get the following results by using any software
package for solving linear pr ogram ming problem:
1132 12 22 11
21 311121
1.375,0.5, 1.5,0.5, 0.625,
3.5, 2,3.5,
ppqqu
uud d

 
31 12223211
21 31 1222
3.5,5.5,7.5, 4.125,
5.5,2.5, 3.5,4.5.
dd dd b
bbbb
 

32 and all other variables are equal to zero. Objec-
tive function value is equal to 41.5. The sets of feasible
parameters, solvability set and the stab ility set of the first
kind are calculated as follows:
3b


112 233112233
,,,0,,1,2, 3,1,
2, ,,,,
itit ktkt
UhHlL ikt
hHh HhHlLlLl L

 
5. Conclusion
In this paper, we have discussed the concept of stability
for the planning problem to maximize revenues net of the
production inventory and lost sales cost. We used fuzzy
parameters to represent both of amount of available re-
sources and demand for item in period. We have defined
and characterized some basic notions for the problem of
concern. These notions are the set of feasible parameters,
the solvability set and the stability set of the first kind.
Although an extensive literature on models for produc-
further research is identified as the development of new
models that contain additional sources and types of un-
certainty, such as supply lead times, transport times,
quality uncertainty, failure of production system and
changes to product structure. Also as a point of further
research, an investigation of incorporation all types of
uncertainty is needed .
tion planning under uncertainty was reviewed, a need for
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