THE SINGLE ITEM LOT SIZING PROBLEM WITH FUZZY
PARAMETERS: A POSSIBILITY APPROACH
Suphattra Ketsarapong*, Sripatum University, Thailand
Email: suphattra.ke@spu.ac.th, *corresponding author
Varathorn Punyangarm, Srinakharinwirot University, Thailand
Punyangarm@gmail.com
Kongkiti Phusavat, Kasetsart University, Thailand
fengkkp@ku.ac.th
ABSTRACT
The main objective of this paper is to deal with a single item lot sizing problem with fuzzy
parameters, which is called the fuzzy single item lot sizing problem (F-SILSP). Since F-SILSP
does not meet the crisp deterministic assumption, it cannot be solved by traditional
mathematical programming. In this paper, the possibility approach is chosen to convert the
fuzzy model to be an equivalent crisp single item lot sizing problem (EC-SILSP). The
equivalent crisp model from this transformation procedure is in the form of mixed integer
linear programming (MILP). It can be solved by traditional solver to find an optimal solution
for each pre-specified possibility level. A numerical example with both trapezoidal and
triangular fuzzy parameters is illustrated to demonstrate that an equivalent crisp model can
be used. In addition, the equivalent crisp model is employed in inventory planning in a case
study of inventory planning for bituminous coal with trapezoidal fuzzy demand and
triangular fuzzy unit price in a power plant of an example petrochemical company.
Keywords: Modeling, Inventory Planning, Lot Sizing Problem, Fuzzy Set Theory,
Possibility Theory, Petrochemical Company
INTRODUCTION
Lot sizing problems are production planning problems with order quantity between
purchasing or production lots (Brahimi et al., 2006). Small lot sizes lead to many orders and
low inventory levels. On the other hand, big lot sizes lead to few orders and high inventory
levels (Lee et al., 1991). Thus, the consideration of lot sizes is an economic problem in that
the objective of inventory models is to minimize total inventory cost, which comprises unit
price, ordering cost, and inventory holding cost, while satisfying demand (Lee et al., 1991;
Brahimi et al., 2006). The first inventory planning model, namely Economic Order Quantity
(EOQ), was proposed by Harris (1913). It is used to find an optimal order quantity in the case
of an uncapacitated single stage and single item of inventory control with a well-defined
demand pattern. However, sometimes the demand pattern cannot be defined. Wagner and
Whitin (1958) proposed an inventory model with time-vary demand, namely dynamic lot
sizing, and used dynamic programming techniques to find an optimal order quantity.
Subsequently, other inventory models have been constructed, based on the above models,
such as the Economic Manufactured Quantity (EMQ) model, the EOQ with backorder, the
EOQ with price breaks, EOQ based period order quantity, the EOQ for multiple items with
capacity constraints, the lot sizing problem for lead time minimization, the run-out time
model, the lot sizing problem in the case of shortages causing lost sales, the order-level
system with a power demand pattern, the uncapacitated single item lot sizing problem, the
capacitated single item lot sizing problem, the uncapacitated multiple item lot sizing problem,
the capacitated multiple item lot sizing problem, and so on (Baracsi et al., 1990; Askin and
Goldberg, 2002; Brahimi et al., 2006).
USILSP is a type of inventory model with time-vary demand. Brahimi et al. (2006) stated
that there are four basic formulations of USILSP in the form of mixed integer linear
programming (MILP), i.e., aggregate formulation (AGG), formulation without inventory
variables (NIF), the shortest path formulation (SHP), and facility location-based formulation
without inventory variables (FAL). They gave three reasons that motivate us to study
USILSP, i.e., some industries can aggregate products to obtain a single product, USILSP is a
basic model and can be extended towards a more complex model, and many solution
approaches for solving complex lot sizing problems must use USILP as sub-problems. As a
result, the authors focus on the AGG model in USILSP which is shown in equations (1)-(4).
Notations
The following notations are used in the paper:
St,
xt ,
It ,
pt ,
the tth period in the planning horizon t = 1, 2,…, T,
the ordering (or setup) variable, which is 1 when order or setup occurs in period t, and
0 otherwise,
the ordering (or setup) cost in period t,
the purchasing (or producing) quantity in period t,
the inventory level at the end of period t,
the unit price (or production cost) in period t,
ht ,
dt ,
the inventory holding cost in period t,
the demand in period t,
d kT ,
the sufficiently large number, where d kT =  d t ; ∀t.
t,
yt ,
T
t 1
Focused on the AGG model in USILSP, assume that there are no beginning and ending
stocks of the planning horizon ( I 0  I T  0 ), and inventory parameters i.e., s t , p t , h t , and
d t are deterministic. The AGG of USILSP in a form of MILP is as follows:
T
(USILSP-AGG) Min TC =  (s t y t  p t x t  h t I t )
t 1
(1)
Subject to:
I t -1  x t  I t = d t ; ∀t,
(2)
y t d kT > x t ; ∀t,
(3)
y t ∈ binary variables, x t > 0, I t > 0; ∀t.
(4)
The objective function of USILSP-AGG in equation (1) is to minimize the total cost over the
t period horizon. Constraints in equation (2) are inventory balance equations. Demand in
period t ( d t ) can be satisfied when it is equal to the beginning inventory of period t ( I t -1 ) plus
the purchasing or producing quantities in period t ( x t ) minus the available on-hand inventory
at the end of period t ( I t ). Constraints in equation (3) are used to control ordering variables.
Since d tT is a large number, then y t is treated as 1 when inventory is ordered in period t ( x t
> 0), and y t is treated as 0 when inventory is not ordered in period t ( x t = 0).
To answer how much inventory is required to minimize total cost and satisfy demand in each
period, the inventory models are a good choice of approach. When a classical inventory
model is used, a crisp deterministic assumption is required. However, sometimes, this
assumption does not hold. In real-world problems, there are two types of uncertain parameter
in the inventory model, i.e., randomness and fuzziness, which respectively lead to the fuzzy
inventory model and the stochastic inventory model. Most information stochastic inventory
models are described by statistical or probabilistic information, which deals with probability
distributions of inventory parameters. Since statistical information is collected from a lot of
sampling data, high cost and time lost for data collection must be considered. There are many
researchers who use verbal information based on the experiences of veterans, and model
these experiences to be fuzzy parameters (Mandal et. al., 2005; Yao et al., 2006; Dutta et. al.,
2007; Tutuncu et. al., 2008). For these reasons, this paper used fuzzy set theory and deals
with SILSP with fuzzy parameters.
FUZZY UNCAPACITATED SINGLE ITEM LOT SIZING PROBLEM (F-USILSP)
Fuzzy set theory was first proposed by Zadeh (1965). It is a mathematical tool to describe
imprecision in the fuzzy environment. Imprecision refers to the sense of vagueness rather
than the lack of knowledge about the value of parameters. The vagueness is due to the unique
experiences and judgments of decision makers. For example, demand is classified as two
verbal sets, medium and high. When a classical set is chosen to classify this demand as verbal
classes, the range of 10,000-15,000 units of demand is defined as medium, and 15,00120,000 units of demand is defined as high. The exact boundary of medium and high is 15,000
units of demand. It means that 15,000 units of demand is assigned to the set of medium, but
15,001 units of demand is assigned to the set of high. This classification shows that when an
event is classified by the judgment of decision maker(s), an exact set boundary is
inappropriate. On the other hand, the fuzzy set boundary is not an exact demarcation but an
area of boundary. Members in a fuzzy set must be linked with a degree of possibility in the
range of [0, 1], which is called a membership function.
The fuzzy set was designed for vagueness and uncertainty, which are found in real life. In
addition, there are many fields which are built based on fuzzy set theory, such as fuzzy logic
and control, fuzzy clustering, fuzzy mathematical programming , etc. (Zimmermann, 1996).
Fuzzy mathematical programming or fuzzy optimization, which was proposed by
Zimmermann (1976, 1978, and 1983), is one application of fuzzy set theory. It was extended
to many fields in production planning and control, such as transportation, scheduling, flexible
manufacturing system (FMS) , location, aggregate planning, inventory planning, and so on
(Zimmermann, 1996; Klir et al., 1997). There has been a lot of research which deals with
vagueness in inventory models. For example, Yao and Lee (1999) introduced a group of
computing schemata for the fuzzy EOQ of inventory with or without backorders. Hsieh
(2002) used fuzzy arithmetical operations of the function principle to find the fuzzy total
production inventory costs, and used Graded Mean Integration Representation and the
Extension of the Lagrangean method to defuzzify and find optimal solutions for the inventory
models. Pai (2003) applied fuzzy sets theory to solve capacitated lot size problems with fuzzy
capacity. Yao and Chiang (2003) compared the results obtained by centroid and signed
distance methods in the case of the total cost of inventory without backorders. Mandala et al.
(2005) focused on fuzzy cost parameters, objective functions, and constraints in a multi-item
multi-objective inventory model with shortages and demand dependent unit cost with storage
space, number of orders and production cost restrictions. Chang et al. (2006) used fuzzy
concepts to optimize total inventory cost in the case of a mixture inventory model involving
variable lead time with backorders and lost sales. Chen and Chang (2008) studied the fuzzy
EPQ in the case of unrepairable defective products with fuzzy opportunity cost and
trapezoidal fuzzy costs under crisp or fuzzy production quantities in order to extend the
traditional production inventory model to the fuzzy model. Halim et al. (2011) developed
fuzzy production planning models for an unreliable production system with fuzzy production
rate and stochastic/fuzzy demand rate.
The AGG model is selected to show how the possibility approach can transform the fuzzy
inventory model into an equivalent crisp inventory model. Let the unit price, ordering cost,
inventory holding cost, and demand be fuzzy variables. Thus the F-USILSP can be modeled
as follows:
T
~
pt x t  h t I t )
(F-USILSP) Min TC =  (~st y t  ~
(5)
~
Subject to I t -1  x t  I t = dt ; ∀t,
(6)
~
y t dkT > x t ; ∀t,
(7)
y t ∈ binary variables, x t > 0, I t > 0; ∀t.
(8)
t 1
The F-USILSP in equations (5)-(8) is in a form of fuzzy MILP. Thus, this model cannot be
solved by classical mathematical methods.
EQUIVALENT CRISP UNCAPACITATED SINGLE ITEM LOT SIZING PROBLEM
(EC-USILSP)
The possibility approach in the context of fuzzy set theory was introduced by Zadeh (1978) to
deal with non- stochastic imprecision and vagueness. According to Dubois and Prade (1988)
and Dubois (2006), the possibility approach appropriately was used to model various kinds of
information, such as linguistic information and uncertain formulae, in logical settings. In this
section, the possibility approach will be used to convert the F-USILSP to the EC-USILSP.
~
~
~
Let ~
a and b be fuzzy variables, and ~
a  and b be the complements of ~
a and b ,
respectively. Operation * means any one of the operators >, >, =, <, <. The possibility and
~
necessity measure of fuzzy event ~
a * b are respectively defined by (Zimmermann, 1990).
~
π(~
a * b ) = sup{min( μ ~a (x), μ ~b (y))/x * y; x, y  }
(9)
~
~
Ness(~
a * b ) = 1  π(~
a  * b )
(10)
~ ) and Ness(
~ ) means possibility and necessity of fuzzy variable ~ .
where π(
~
Let ~
a i for i = 1,…, n be n fuzzy variables, the right hand side b becomes a crisp variable,
and let f : n   be a real-value function. The possibility of fuzzy event f( ~
a ) * b is
i
defined by (Zimmermann, 1996).
π(f( ~
ai ) * b) = sup {min {μ ~ai (x i )} / f(x i ) * b; x i , b  , i}
(11)
x i  1in
In this paper, Chance-Constrained Programming (CCP) (Charnes and Cooper, 1959), which
is normally used to confront stochastic linear programming ( SLP) , is adopted as a way to
convert the F-USILSP to the EC-USILSP. The concept of CCP guarantees that the
probability of stochastic constraints is greater than or equal to a pre-specified minimum
probability. Let ĉ j , â ij , and b̂i for j = 1, …, n and i = 1, …,m be continuous random
variables, and xj be decision variables. The relationship between standard SLP and its
n
n
j1
j1
probability SLP is given by Min Z =  ĉ j x j ; subject to constraints,  â ij x j < b̂i ; xj > 0 for
n

n

all xj if and only if Min Z = f; Pr  ĉ j x j  f  > 1 – ; Pr  â ij x j  b̂ i  > 1 – i; f and xj > 0
 j1

 j1

for all xj, where f is an artificial variable, which should be the minimum value when it is not
greater than the objective function of standard SLP, Pr means probability, 1 –  and 1 – i are
pre-specified minimum probabilities. The researchers who are interested in the field of SLP
can refer to Birge and Louveaux (1997) for further details. In the same way, the F-USILSP
becomes the following possibility USILSP:
(Poss-USILSP) Min TC = Θ
(12)
T
~
Subject to π  (~st y t  ~
pt x t  h t I t )  Θ  > α; ∀t,
 t 1

(13)
~
π(I t-1  x t  I t  d t ) > α; ∀t,
(14)
~
π(y t dkT  x t ) > α; ∀t,
(15)
y t ∈ binary variables, x t > 0, I t > 0, Θ > 0; ∀t.
(16)
Based on the possibility ranking in equation (11), Lertworasirikul et al. (2003) proved and
proposed the Lemma 1, as follows:
a i for i = 1,..., n be fuzzy variables with normal and convex membership
Lemma1. Let ~
a i are
functions and b be a crisp variable. The lower and upper bounds of the -level set of ~
a ) L and (~
a ) U , respectively. Then, for any given possibility levels 1, 2 and 3
denoted by (~
i α
with 0 < 1, 2, 3 < 1,
i α
(i)
π(~
a1   ~
a n  b)  α1 iff (~
a1 )αL1   (~
a n )αL1  b ,
(ii) π(~
a1   ~
a n  b)  α 2 iff (~
a1 ) αU2   (~
a n ) αU2  b ,
(iii) π(~
a1   ~
an  b)  α3 iff (~
a1 ) αL3   (~
a n ) αL3  b and (~
a1 )αU3   (~
a n )αU3  b .
The Poss-USILSP is transformed into the EC-USILSP by the Lemma 1, as follows:
T
~
(EC-USILSP) Min TC =   (~st ) L y t  (~
pt ) L x t  ( h t ) L I t 
α
α
α 
t 1
(17)
~
Subject to I t -1  x t  I t > (dt ) L ; ∀t,
(18)
~
I t -1  x t  I t < (d t ) U ; ∀t,
(19)
~
y t (dkT ) U > x t ; ∀t,
(20)
y t ∈ binary variables, x t > 0, I t > 0; ∀t.
(21)
α
α
α
The special case of EC-USILSP, which is the EC-USILSP with trapezoidal and triangular
fuzzy parameters, will be shown in next section.
EC-USILSP WITH TRAPEZOIDAL AND TRIANGULAR FUZZY NUMBERS
Since trapezoidal and triangular fuzzy numbers, which have linear membership functions, are
basically fuzzy numbers, the F-USILSP with trapezoidal and triangular parameters will be
~
examined in this paper. Let A be the trapezoidal fuzzy number with a lower and upper crisp
value at α = 0 and 1 at each corner point of trapezoidal membership function, which is
~
~
~
~
denoted by (A) L , (A) L , (A) U , and (A) U , respectively (See Figure 1a). The membership
0
1
1
0
~
function of A is given by
~ L
 r  (A
)
0

 ~ L ~ L
 (A)1  (A) 0

1
μ A~ (r) = 
~ U
 r  (A
)
0

~ U ~ U
 (A
)  (A)
 1
0

0
~
~
; (A) L  r  (A) L
0
1
~
~
; (A) L  r  (A) U
1
1
~
~
; (A) U  r  (A) U
1
; Otherwise
0
(22)
~
where μ A~ (r) ∈ [0, 1], and r ∈ ℝ. Thus, the lower and upper crisp value of A at each α-cut
~ U
~ U
~ L
~ L
~
are defined by r
~ = (A)1 α  (A) 0 (1  α) , and r
~ = (A)1 α  (A) 0 (1  α) . Let B be
Lower/A
Upper/A
~
~
~
~
a trapezoidal fuzzy number with (B)1L = (B)1U = (B)1 , then B is called a triangular fuzzy
~
number (see Figure 1b). The lower and upper crisp value of B at each α-cut are defined by
~
~ L
~
~ U
r
~ = (B)1 α  (B) 0 (1  α) , and r
~ = (B)1 α  (B) 0 (1  α) .
Lower/B
Upper/B
Figure 1: Membership Function of (a) Trapezoidal Fuzzy Number, and (b) Triangular Fuzzy
Number
~
~
If the fuzzy parameters of F-USILSP ~st , ~
pt , h t and dt are trapezoidal fuzzy numbers, then
the general model of EC-USILSP follows mixed integer linear programming.
T
(EC-USILSP-Trapezoidal) Min TC =    α( ~st ) L  (1  α)(~st ) L  y t 
1
0 
t 1 
~
~
 α(~
pt ) L  (1  α)(~
pt ) L x t   α( h t ) L  (1  α)(h t ) L I t 
1
0
1
0  



(23)
~
~
Subject to I t -1  x t  I t > α( dt ) L  (1  α)(dt ) L ; ∀t,
(24)
~
~
I t -1  x t  I t < α( dt ) U  (1  α)(dt ) U ; ∀t,
(25)
~
~
y t  α( dkT ) U  (1  α)(dtT ) U  > x t ; ∀t,
1
0 

(26)
y t ∈ binary variables, x t > 0, I t > 0; ∀t,
(27)
1
1
0
0
~) L = (
~ ) U . Thus,
Note that a triangular fuzzy number is a trapezoidal fuzzy number with (
1
1
EC-USILSP with triangular fuzzy parameters can be modeled as EC-USILSP with triangular
fuzzy parameters. The numerical example with triangular fuzzy parameters is illustrated to
show how to use this model in the next section.
NUMERICAL EXAMPLE
To show how to use the EC-USILSP model, a numerical example with trapezoidal fuzzy unit
price and demand and triangular fuzzy inventory holding and ordering cost is discussed in
this section. Assume that the beginning and ending stock of the planning horizon are both
zero. Let the trapezoidal fuzzy unit price be classified as three levels, i.e., cheap, normal and
expensive. It is respectively denoted by ~p c = (10, 15, 20, 25), ~
pn = (20, 25, 30, 35), and ~p e =
(30, 35, 40, 45). The trapezoidal fuzzy demand is classified as three levels, i.e., low, medium,
~
~
and high. It is respectively denoted by dl = (60, 80, 100, 120), d m = (100, 120, 150, 170),
~
and dh = (150, 170, 180, 200). The triangular fuzzy ordering cost is classified as three levels,
i.e., low, medium, and high. ~sl = (200, 300, 400), ~sm = (300, 400, 500), and ~sh = (500, 600,
700). The triangular fuzzy inventory holding cost is classified as three levels, i.e., low,
~
~
~
medium, and high. hl = (1, 2, 3), h m = (2, 3, 4), and h h = (3, 4, 5). Based on the experience
of the decision maker, the parameter levels of the F-USILSP in terms of the verbal
classification for five periods are shown in Table 1.
Table 1: Verbal Estimation of F-USILSP parameters for Five Periods.
Period t
Unit Price
Demand
Ordering Cost
1
Cheap
Medium
Medium
2
Expensive
High
High
3
Cheap
Low
Low
4
Normal
Medium
Low
5
Expensive
High
High
Holding Cost
High
Medium
High
Medium
Low
From Table 1, the F-USILSP of this example is modeled as follows:
(F-USILSP) Min TC = (Medium)y 1  (High)y 2  (Low)y 3  (Low)y 4  (High)y
 (Cheap)x 1  (Expensive )x 2  (Cheap)x 3  (Normal)x 4  (Expensive )x 5
 (High)I 1  (Medium)I 2  (High)I 3  (Medium)I 4  (Low)I 5
5
(28)
Subject to
x1  I1 = Medium,
(29)
I1  x 2  I 2 = High,
(30)
I 2  x 3  I3 = Low,
(31)
(32)
I 4  x 5 = High,
(33)
I 3  x 4  I 4 = Medium,
~
y1d15 > x 1 ,
~
y 2 d25 > x 2 ,
(35)
~
y3 d35 > x t ,
(36)
~
y 4 d45 > x 4 ,
(37)
~
y5 d55 > x 5 ,
(38)
y t ∈ binary variables, x t > 0, I t > 0 for t = 1, 2, …, 5
(34)
(39)
~
In accordance with the extension principle, fuzzy numbers d tT for all t = 1, …, T are given
5 ~
5 ~
5 ~
5 ~
5 ~
~
 5 ~

by d15 =   (d t ) L ,  (d t ) U  =  α  (d t ) L  (1  α)  (d t ) L ,  α  (d t ) U  (1  α)  (d t ) U  ,
t 1
1
0
1
0
α t 1
α 
t 1
t 1
  t 1


 t 1
~
~ U
~ U
then the crisp interval of d15 is (d15 ) = 760α  860(1  α) . So, (d25 ) = 610α  690(1  α) ,
α
α
~
~
~
(d35 ) U = 430α  490(1  α) , (d45 ) U = 330α  370(1  α) , and (d55 ) U = 180α  200(1  α) .
α
α
α
Thus, the EC-USILSP with triangular and trapezoidal parameters is a crisp MILP as follows:
(EC-USILSP) Min TC = (400α  300(1  α))y1  (500α  400(1  α))y 2  (300α  200(1  α))y 3
 (300α  200(1  α))y 4  (500α  400(1  α))y 5  (15α  10(1  α))x1  (35α  30(1  α))x 2
 (15α  10(1  α))x 3  (25α  20(1  α))x 4  (35α  30(1  α))x 5  (4α  3(1  α))I1
 (3α  2(1  α))I 2  (4α  3(1  α))I 3  (3α  2(1  α))I 4  (2α  (1  α))I5
(40)
Subject to
x1  I1 > 120α  100(1  α) ,
(41)
x1  I1 < 150α  170(1  α) ,
(42)
I1  x 2  I 2 > 170α  150(1  α) ,
(43)
I1  x 2  I 2 < 180α  200(1  α) ,
(44)
I 2  x 3  I3 > 80α  60(1  α) ,
(45)
I 2  x 3  I3 < 100α  120(1  α) ,
(46)
I 3  x 4  I 4 > 120α  100(1  α) ,
(47)
I 3  x 4  I 4 < 150α  170(1  α) ,
(48)
I 4  x 5 > 170α  150(1  α) ,
(49)
I 4  x 5 < 180α  200(1  α) ,
(50)
y1 (760α  860(1  α)) > x 1 ,
(51)
y 2 (610α  690(1  α)) > x 2 ,
(52)
y3 (430α  490(1  α)) > x 3 ,
(53)
y 4 (330α  370(1  α)) > x 4 ,
(54)
y5 (180α  200(1  α)) > x 5 ,
(55)
y t ∈ binary variables, x t > 0, I t > 0 for t = 1, 2, …, 5
(56)
This above crisp mixed integer linear programming is solved with α = 0, 0.1, 0.2, …, 1. Then
the total cost, ordering variable in period t, buying quantity in period t, and inventory level at
the end of period t for t = 1, 2, …, 5 are shown in Table 2.
Table 2: The result of numerical example.
Alpha Level
Total Cost
Period t
0
7,600.00
0.1
8,082.80
0.2
8,577.20
0.3
9,083.20
0.4
9,600.80
0.5
10,130.00
0.6
10,670.80
0.7
11,223.20
0.8
11,787.20
0.9
12,362.80
1.0
12,950.00
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
yt
xt
It
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
0
250
0
310
0
0
254
0
316
0
0
258
0
322
0
0
262
0
328
0
0
266
0
334
0
0
270
0
340
0
0
274
0
346
0
0
278
0
352
0
0
282
0
358
0
0
286
0
364
0
0
290
0
370
0
0
150
0
250
150
0
152
0
254
152
0
154
0
258
154
0
156
0
262
156
0
158
0
266
158
0
160
0
270
160
0
162
0
274
162
0
164
0
278
164
0
166
0
282
166
0
168
0
286
168
0
170
0
290
170
0
From Table 2, at the lowest possible level (α = 0), the inventory plan orders 250 units in
period one to cover the first two periods and 310 units to cover periods three, four, and five.
The total cost is 7,600.00 THB. At the highest possible level (α = 1), the inventory plan
orders 290 units in period one to cover the first two periods and 370 units to cover periods
three, four, and five. The total cost is 12,950.00 THB. The trend of total cost from ECUSILSP, solving with α = 0, 0.1, 0.2,…, 1, is an increasing linear trend, which is illustrated
by plotting TC and α in x and y axis, respectively. It means that when higher possible levels
are required, the TC will increase.
CASE STUDY
The case study of this paper focuses on the inventory planning of bituminous coal in a power
plant of a petrochemical company. The main purpose of this power plant is to produce
electrical power for the company’s petrochemical plants, which is the core business of this
company. Bituminous coal is the main raw material for the generation of the electrical power.
Demand for bituminous coal measured in metric tons is dependent on the demand for
electrical power, which is required for petrochemical production. In this study, the demand
for bituminous coal is classified as 5 trapezoidal fuzzy numbers, i.e., lowest, low, medium,
high, and highest (see the membership function of fuzzy demand in Figure 2). Based on the
experience of the production manager and the fuzzy information in Figure 2, the demand for
bituminous coal in the planning horizon (12 periods) is predicted in the form of verbal
demand, which is shown in Table 3. The price of bituminous coal per metric ton is classified
as 7 triangular fuzzy numbers, i.e., the cheapest, very cheap, cheap, normal, expensive, very
expensive, the most expensive (see the membership function of fuzzy unit prices in Figure 3).
Based on the experience of the purchasing manager and the fuzzy information in Figure 3, the
unit price of bituminous coal in the planning horizon is predicted in the form of verbal unit
prices, which are shown in Table 3. The company has two outsourcers, namely companies A
and B, which transport bituminous coal from the distributor to the inventory area. Focusing
on the inventory area, there is a large area for the storage of 80,000 metric tons of bituminous
coal. According to the company’s policies, 20,000 metric tons of bituminous coal must be
stored as a safety stock. It cannot be used up. Thus, the net inventory space is equal to 60,000
metric tons. The inventory level at the end of period 0, or the beginning stock of period 1, is
equal to 15,000 tons. So the net demand of period 1 is equal to (20,000, 25,000, 30,000,
35,000) metric tons minus 15,000 metric tons, that is (5,000, 10,000, 15,000 20,000) metric
tons. Company A charges 1 million Thai Baht (THB) per order, and the maximum
transportation capacity of company A is 50,000 metric tons per period. Company B charges
1.5 million THB per order, and the maximum transportation capacity of company B is equal
to that of company A. Note that although the quantity of each order is less than the maximum
transportation capacity of each company, companies A and B cannot discount the charge rate
per order. Based on the policies of the company, the holding cost is fixed as a crisp value. It
is equal to 200 THB per metric ton per period.
Table 3: Demand (x 1,000 Metric Tons) and Unit Price (x 100 THB per Metric Ton).
Demand
Unit Price
Period
Verbal
Trapezoidal
Verbal
Triangular
t
Prediction
Fuzzy Number
Prediction
Fuzzy Number
1
Low
(20, 25, 30, 35)
Normal
(20, 25, 30)
2
Low
(20, 25, 30, 35)
Cheap
(15, 20, 25)
3
Medium
(30, 35, 40, 45)
Cheap
(15, 20, 25)
4
Medium
(30, 35, 40, 45)
Very Cheap
(10, 15, 20)
5
Medium
(30, 35, 40, 45)
Cheap
(15, 20, 25)
6
Medium
(30, 35, 40, 45)
Cheap
(15, 20, 25)
7
High
(40, 45, 50, 55)
Normal
(20, 25, 30)
8
High
(40, 45, 50, 55)
Expensive
(25, 30, 35)
9
Medium
(30, 35, 40, 45)
Normal
(20, 25, 30)
10
Low
(20, 25, 30, 35)
Expensive
(25, 30, 35)
11
Lowest
(15, 15, 20, 25)
Normal
(20, 25, 30)
12
Lowest
(15, 15, 20, 25)
Expensive
(25, 30, 35)
Figure 2: Membership Function of Fuzzy Demand.
Figure 3: Membership Function of Fuzzy Unit Prices.
This case study is different from the numerical example. Since there are two transporters, and
transportation capacity of them is limited, sometimes only company A can be selected (the
service charge of company A is less than that of company B), but sometimes both companies
A and B must be selected (when the buying quantity exceeds the transportation capacity of
company A). Let decision variables x t A and x tB for t = 1,…, 12 be the buying quantities
which are transported by companies A and B in period t, respectively. Thus, the equation of
the bituminous coal price can be formulated as follows:
Unit price = (2,500α  2,000(1  α))(x1A  x1B )  (2,000α  1,500(1  α))(x 2A  x 2B )
 (2,000α  1,500(1  α))(x 3A  x 3B )  (1,500α  1,000(1  α))(x 4A  x 4B )
 (2,000α  1,500(1  α))(x 5A  x 5B )  (2,000α  1,500(1  α))(x 5A  x 5B )
 (2,500α  2,000(1  α))(x 7A  x 7B )  (3,000α  2,500(1  α))(x 8A  x 8B )
 (2,500α  2,000(1  α))(x 9A  x 9B )  (3,000α  2,500(1  α))(x10A  x10B)
 (2,500α  2,000(1  α))(x11A  x11B)  (3,000α  2,500(1  α))(x12A  x12B) .
(57)
Let binary variables y tA and y tB for t = 1,…, 12 be ordering variable, which is one when
companies A and B receive orders, respectively, and 0 otherwise. The equation of the
ordering cost can be formulated as follows:
12
12
t 1
t 1
Ordering cost = 1,000,000y tA  1,500,000y tB .
(58)
Let decision variable I t be the inventory level at the end of period t. The equation of the
inventory holding cost can be formulated as follows:
12
Inventory holding cost =  200I t .
t 1
(59)
Because the demand for bituminous coal for t = 1,…, 12 was based on the judgment of
decision maker, it is a fuzzy demand. Therefore, there are 24 equivalent crisp inventory
balance constraints of 12 periods, as follows:
x1A  x1B  I1 > 10,000α  5,000(1  α) ,
(60)
x1A  x1B  I1 < 15,000α  20,000(1  α) ,
(61)
I1  x 2A  x 2B  I 2 > 25,000α  20,000(1  α) ,
(62)
I1  x 2A  x 2B  I 2 < 30,000α  35,000(1  α) ,
(63)
I 2  x 3A  x 3B  I3 > 35,000α  30,000(1  α) ,
(64)
I 2  x 3A  x 3B  I3 < 40,000α  45,000(1  α) ,
(65)
I3  x 4A  x 4B  I 4 > 35,000α  30,000(1  α) ,
(66)
I3  x 4A  x 4B  I 4 < 40,000α  45,000(1  α) ,
(67)
I 4  x 5A  x 5B  I 5 > 35,000α  30,000(1  α) ,
(68)
I 4  x 5A  x 5B  I 5 < 40,000α  45,000(1  α) ,
(69)
I5  x 6A  x 6B  I6 > 35,000α  30,000(1  α) ,
(70)
I5  x 6A  x 6B  I6 < 40,000α  45,000(1  α) ,
(71)
I6  x 7A  x 7B  I7 > 45,000α  40,000(1  α) ,
(72)
I6  x 7A  x 7B  I7 < 50,000α  55,000(1  α) ,
(73)
I7  x 8A  x 8B  I8 > 45,000α  40,000(1  α) ,
(74)
I7  x 8A  x 8B  I8 < 50,000α  55,000(1  α) ,
(75)
I8  x 9A  x 9B  I9 > 35,000α  30,000(1  α) ,
(76)
I8  x 9A  x 9B  I9 < 40,000α  45,000(1  α) ,
(77)
I9  x10A  x10B  I10 > 25,000α  20,000(1  α) ,
(78)
I9  x10A  x10B  I10 < 30,000α  35,000(1  α) ,
(79)
I10  x11A  x11B  I11 > 15,000,
(80)
I10  x11A  x11B  I11 < 20,000α  25,000(1  α) ,
(81)
I11  x12A  x12B > 15,000,
(82)
I11  x12A  x12B < 20,000α  25,000(1  α) .
(83)
Based on transportation capacity limitations, the maximum transportation capacity of
companies A and B is 50,000 metric tons per period, the constraint for the control ordering
variable can be formulated as follows:
50,000y tA > x tA and 50,000y tB > x tB for t = 1,…, 12.
(84)
Focusing on the inventory space limitations, the bituminous coal cannot be stored at a
quantity greater than 60,000 metric tons. Thus the warehouse capacity constraints are added
as follows:
I t -1  x tA  x tB < 60,000 for t = 1,…, 12.
(85)
The minimum total cost, binary variables, order quantity, and inventory level for each α-level
set can found by solving MILP, min (57) + (58) + (59), subject to (60)-(85), where y tA and
y tB ∈ binary variables, x t A , x tB , and I t > 0 for t = 1, …, 12. The results at α = 0 and 1 are
shown in Table 4.
Table 4: Inventory plan at α = 0 and 1.
α = 0 and TC = 540,000,000
Period
Company
t
y
x
I
A
1
5,000
1
0
B
0
0
A
1
20,000
2
0
B
0
0
A
1
30,000
3
0
B
0
0
A
1
50,000
4
30,000
B
1
10,000
A
0
0
5
0
B
0
0
A
1
50,000
6
30,000
B
1
10,000
A
1
30,000
7
20,000
B
0
0
A
1
20,000
8
0
B
0
0
A
1
50,000
9
20,000
B
0
0
A
0
0
10
0
B
0
0
A
1
30,000
11
15,000
B
0
0
A
0
0
12
0
B
0
0
α = 1 and TC = 813,000,000
Y
x
I
1
10,000
0
0
0
1
25,000
0
0
0
1
35,000
0
0
0
1
50,000
25,000
1
10,000
1
10,000
0
0
0
1
50,000
25,000
1
10,000
1
35,000
15,000
0
0
1
30,000
0
0
0
1
50,000
25,000
1
10,000
0
0
0
0
0
1
30,000
15,000
0
0
0
0
0
0
0
If a decision maker wants to plan for the lowest total cost, the lowest of possible level (α = 0)
is chosen. The inventory plan orders 5,000, 20,000, and 30,000 metric tons of bituminous
coal, which is transported by company A, in periods one, two, and three, respectively. To
serve the demand of periods four and five, 60,000 metric tons must be refilled. Since the
maximum transportation capacity of company A is less than 60,000 metric tons, an excess
demand (10,000 metric tons) must be transported by company B. In period six, 60,000 metric
tons of bituminous coal is ordered. An aggregate total of 30,000 metric tons, which is ordered
in period seven, and 30,000 metric tons, which was transferred from period six, must be used
to serve the demand of periods seven and eight. In period nine, 50,000 metric tons is ordered
to cover periods nine and ten, and 30,000 metric tons is ordered in period eleven to cover the
demand of the last two periods. The total cost is 540 million THB. On the other hand, when
the highest possible level (α = 1) is required, the total cost increases to 813 million THB. The
inventory plan of this scenario orders 10,000, 25,000, and 35,000 metric tons in periods one,
two, and three, respectively. To serve the demand of periods six, seven, and eight, 60,000 and
35,000 metric tons is ordered respectively, in periods six and seven. In period nine, 60,000
metric tons is ordered to cover periods nine and ten, and 30,000 metric tons is ordered in
period eleven to cover the last two periods.
CONCLUSIONS
This paper used Possibility Approach to convert the fuzzy inventory model (F-USILSP) to an
equivalent crisp inventory model (EC-USILSP). Consequently, the EC-USILSP can be
solved by traditional mathematical programming methods. In addition, this EC-USILSP was
adopted for inventory planning in the case study of inventory planning for bituminous coal
with trapezoidal fuzzy demand and triangular fuzzy unit price in a power plant of a
petrochemical company. The EC-USILSP is very flexible to solve problems, for example,
before using EC-USILSP in the case study, the EC-USILSP could be modified to support
additional restrictions on two issues; transportation capacity limitations and inventory space
limitations. The results found higher possible levels are required, and the total cost will
increase accordingly.
FUTURE STUDIES
In this article, SILSP with fuzzy parameters, which is the smallest problem, was selected as
the sample to study. Future work could focus on, firstly, using other more sophisticated
inventory models, such as a multiple item lot sizing problem in a single-stage and multiplestage production; and secondly, a study of SILSP with several periods, where these problems
would be more complex. Consequently, solving these problems requires a lot of time.
Therefore, for future work, comparisons should be made between ‘optimization methods and
Meta Heuristics’ and ‘optimization methods or Meta Heuristics’ to test which is more
appropriate in these cases. Lastly, this article focuses only on inventory models with fuzzy
parameters, but some parameters of inventory models may have variations in other
characteristics, such as random variables or random fuzzy variables. Therefore, in future
study, models should be developed to support other parameters.
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