Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 20-29 
 

CONTACT:  

E. Susanti,               eka_susanti@mipa.unsri.ac.id             Departement of  Mathematics, Universitas Sriwijaya 

Palembang, Sumatera Selatan 30662, Indonesia 

 

       The article can be accessed here. https://doi.org/10.15642/mantik.2021.7.2.124-131   

 Optimization of Inventory Level Using Fuzzy Probabilistic 

Exponential Two Parameters Model 
 

 

Eka Susanti1, Indrawati2, Robinson Sitepu3, Karita Ondhiana4,  

Widya Dwi Wulandari5 

1,2,3,4,5 Department of Mathematics, Universitas Sriwijaya, Palembang, Indonesia 
 

 

Article history: 

Received Nov 30, 2020 

Revised, May 30, 2021 

Accepted, Oct 31, 2021 

 

Kata Kunci:  

Inventori fuzzy 

probabilistik, 

Distribusi 

eksponensial, 

Distribusi Pareto 

 

Abstrak. Pengendalian persediaan adalah faktor penting dalam 

kegiatan perdagangan. Pengendalian persediaan bertujuan untuk 

menjamin ketersediaan produk. Terdapat beberapa faktor yang 

mempengaruhi tingkat persediaan diantaranya faktor tingkat 

permintaan, persediaan maksimal produk dan tingkat kerusakan. Jika 

faktor yang mempengaruhi tidak dapat didefinisikan dengan pasti dan 

mengikuti distribusi statisik tertentu maka pendekatan fuzzy 

probabilistik dapat diterapkan. Pada penelitian ini dibahas masalah 

optimasi persediaan cabai merah pada tingkat pengecer. Tingkat 

kerusakan diasumsikan mengikuti distribusi eksponensial dan 

permintaan mengikuti distribusi Pareto. Parameter statistik diestimasi 

dengan metode Maksimum likelihood dan parameter biaya 

dinyatakan dengan bilangan fuzzy segitiga. Berdasarkan hasil 

perhitungan untuk beberapa nilai beta diperoleh total biaya tertinggi 

sebesar Rp 405143.6 dengan tingkat persediaan maksimum sebanyak 
15 kg dan waktu siklus pemesanan selama 0.923 hari. 

Keywords: 

Fuzzy probabilistic 

inventory, Exponential 

distribution, Pareto 

distribution    

 

 

Abstract. Inventory control is an important factor in trading 

activities. Inventory control aims to ensure product availability. 

Several factors affect the level of inventory including the level of 

demand factor, maximum inventory, and the level of deterioration. If 

the influencing factors cannot be defined with certainty and follow a 

certain statistic distribution then the fuzzy probabilistic approach can 

be applied. This research discusses the problem of optimizing the 

inventory of red chillies at the retail level. The level of deterioration 

is assumed to follow an exponential distribution and demand follows 

a Pareto distribution. Statistical parameters are estimated using the 

Maximum likelihood method and cost parameters are expressed by 

triangular fuzzy numbers. Based on the calculation results for 

several beta values, the highest total cost is 405143.6 rupiah, a 

maximum inventory level of 15 kg, and an order cycle time of 0.923 

days. 

 

How to cite: 

E. Susanti et al, “Optimization of Inventory Level Using Fuzzy Probabilistic Exponential Two 

Parameters Model, J. Mat. Mantik, vol. 7, no. 2, pp. 124-131, October 2021. 

  

Jurnal Matematika MANTIK 

Vol. 7, No. 2, October 2021, pp. 124-131 

 

ISSN: 2527-3159 (print) 2527-3167 (online) 

mailto:eka_susanti@mipa.unsri.ac.id
http://u.lipi.go.id/1458103791


 
 

 

Jurnal Matematika MANTIK  

Vol 7, No 2, October 2021, pp. 124-131 
 

125 

1. Introduction 

Inventory planning is an important stage in production, distribution and trading 

activities. Planning is carried out to ensure product availability in fulfilling consumer 

demand activities. The inventory model can be applied to inventory planning activities. 
The inventory system is given operational policies related to product storage control, such 

as how much is the maximum inventory, when to order and stock out time to minimize the 

total cost of ordering. Research related to inventory methods has been developed and 

applied in various fields. Inventory and supply chain problems to determine the optimal 

location, optimal route for distribution activities are discussed by [1]. The problem of 

inventory optimization with various payment systems is discussed by [2]. EOQ model with 

nonlinear constraints in inventory problem discussed by [3]. The solution method used by 

[4] for the inventory model is the algebraic method. The fruit fly algorithm modification 

method was introduced by [5] to solve the problem of inventory and allocation 

optimization. The problem of inventory for perishable products is discussed by [6]. the 

concept of inventory to the health sector introduced by [7]. 

The research that has been mentioned is an implementation of the deterministic 

inventory concept with demand and the parameters that affect it can be stated with 

certainty. In some cases, the parameters cannot be stated with certainty, for example, data 

obtained from the forecasting process, unequal and unpredictable demand data in the next 

period. Fuzzy probabilistic and stochastic approaches can be used to solve inventory 

problems with uncertainty. 

The following is research related to inventory problems with uncertainty. Stochastic 

inventory model to the problem of perishable product inventory introduced by [8]. The 

queuing theory is applied by [9] to solve the stochastic inventory problems. A Tri-Level 

Optimization model and a stochastic model with demand and lead time in uncertainty were 

introduced by [10]. In [11] discussed the problem of stochastic inventory with demand 

depending on price. If the parameters are considered following a certain statistical 

probability distribution and contain an element of uncertainty, then the probabilistic 

inventory model is appropriate to use, either the deterministic approach or the fuzzy 

approach. The probabilistic inventory model was introduced by [12]. A review of some 

literature related to inventory problems with a fuzzy approach is given by [13]. In [14] 

discussed probabilistic inventory with time-dependent storage costs and considering the 

extent of the damage. A two-parameter probabilistic inventory model was introduced by 

[15]  and applied to chemical inventory. In [16] discussed a probabilistic inventory model 

for waiting times. Fuzzy concepts can be applied to inventory problems with uncertainty. 

The following is research related to the fuzzy concept. The fuzzy concept to the forecasted 

data is discussed by [17]. In [18] studied the relationship between fuzzy and probabilistic 

concepts.  The method for solving nonlinear problems with fuzzy intervals was introduced 

by [19]. The concept of zero set in fuzzy optimization problems is used by [20]. The fuzzy 

concept is also discussed by [21] and applied to fuzzy transportation problems. 

In this study, a two-parameter probabilistic inventory model introduced by [15] was 

implemented in the red chilli inventory planning problem where the cost and time 

parameters are represented by triangular fuzzy numbers. The tolerance value to the left and 

right of the triangular fuzzy number is taken from the deviation value of the data. The 

defuzzification stage uses the concept of α-cut. Statistical parameters were estimated using 

the Maximum Likelihood. This research discusses optimizing the inventory level of red 

chillies and the optimal ordering cycle at one of the traders located in the modern market 

of Palembang city. The merchant places orders every day. Unsold red chillies are not stored 

in a special room. The level of damage to red chillies is assumed to follow an exponential 

distribution. The demand for unsold red chillies sold every day is limited, in this study the 

level of demand is assumed to follow the Pareto distribution. 

 



 

 

 

Eka Susanti et al. 

Optimization of Inventory Level Using Fuzzy Probabilistic Exponential Two Parameters Model 

 

126 

2. Research Method 

The data used in this research is primary data. Data were collected by interviewing 

one of the traders. The data collection period is 30 days. Interviews were conducted by 

telephone. Primary data consist of purchase data, the number of red chillies sold every day.  

Following is the procedure for solving the problem of optimization of inventory levels and 

the cycle time for ordering red chillies. 

a. Defining parameters and variables  

There are six cost parameters (𝐶𝑝, 𝐶𝑜, 𝐶𝑏 , 𝐶ℎ , 𝐶𝑑 , 𝐾𝑑 ), namely the cost of purchasing 

unit items for one order cycle, ordering costs, backlogged costs, storage costs, damage 

costs, and goals related to the estimated cost of damage. Based on 30 days records, the 

mean and deviation for each cost were obtained. Cost and deviation data are given in 

table 1. Statistical parameters of the two-parameter exponential distribution and the 

Pareto distribution were estimated using the maximum likelihood method. Estimated 

data are data on the level of damage and the level of demand.  

b. Fuzzification stage 
At this stage, the parameter values are defined in the form of triangular fuzzy numbers. 

The definition of values is based on the data obtained. This parameter is a cost 

parameter. Following are given the values of fuzzy parameters and statistical 

parameters. �̃�𝑝, �̃�𝑜, �̃�𝑏 , �̃�ℎ , �̃�𝑑, �̃�𝑑; 𝜎, 𝜂 

c. Defuzzification stage 
At the defuzzification stage, the fuzzy parameters are transformed into a deterministic 

form using the α-cut concept.  

d. Formulation of a two-parameter fuzzy probabilistic model 
The following gives a fuzzy probabilistic model with a Pareto distribution of demand 

and the level of damage with an exponential distribution introduced by [15]. 

minimize 𝐸(𝑇𝐶(𝑄𝑚 , 𝑛𝑑 , 𝑛1, 𝑁, ))      (1)   
Subject to                

𝐸(𝐷𝐶(𝑄𝑚 , 𝑛𝑑 , 𝑛1, 𝑁, )) ≤ 𝑘�̃�      (2) 
Where  

𝐸(𝑇𝐶)  : expected annual total cost  
𝐸(𝐷𝐶) : expected varying deteriorating cost 
𝐸(𝑉𝐶)  : expected the salvage cost 
𝐸(𝐵𝐶)  : expected backorder cost 
𝑄𝑚   : maximum inventory level (kg) 
𝑛𝑑   : time of deterioration  (days) 
𝑛1   : time of stock out (days) 
𝑁   : time of review cycle (days) 

𝑘�̃�   : the fuzzy goal associated to expected deterioration cost (Rupiah) 
�̃�𝑝  : the fuzzy purchase cost unit item at a cycle ordering (Rupiah) 

�̃�𝑜  : the fuzzy ordering cost unit item at a cycle ordering (Rupiah) 
�̃�𝑏  : the fuzzy backlogged cost unit item at a cycle ordering (Rupiah) 
�̃�ℎ  : the fuzzy storage cost unit item at a cycle ordering (Rupiah) 
�̃�𝑑  : the fuzzy damage cost unit item at a cycle ordering (Rupiah) 

 

e. The nonlinear model solution obtained in Step (4) uses Lingo software 
Model formulations and completion procedures are given in the results and discussion 

sections. 

 

 

  



 
 

 

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127 

3. Result and Discussions 

In this study, it was determined the optimal supply and ordering time on the 

optimization problem of supply of red chilli from one of the traders. Cost parameters were 

measured for 30 days of recording. The following are given the average and deviation of 

each cost parameter. 
Table 1. Costs Parameter Data 

Cost 

Parameter 

Average 

(Rupiah) 

Deviation 

(Rupiah) 

𝐶𝑝 36800 7500 

𝐶𝑜   6000 2200 
𝐶𝑏  38400    500 
𝐶ℎ 17500 6900 
𝐶𝑑   1900 2000 

𝐾𝑑    7000     1000 

 

In this study, a fuzzy probabilistic model with two-parameter exponential distribution 

and a Pareto distribution was implemented. The cost parameter values are expressed in the 

triangular fuzzy numbers.  Based on the data in table 1, the triangular fuzzy number can be 

determined as introduced by [15].  The following parameter values are given.  

𝐶�̃� =  (𝐶𝑝 − 𝜔1, 𝐶𝑝 , 𝐶𝑝 + 𝜔2) =  (29300; 36800 ;  44400) 

𝐶�̃� =  (𝐶𝑜 − 𝜔3,  𝐶𝑜 , 𝐶𝑜 + 𝜔4) =  (3800;  6000;  8400) 
𝐶ℎ̃ =  (𝐶ℎ − 𝜔5, 𝐶ℎ  , 𝐶ℎ + 𝜔6) =  (17000;  17500;  19000)  
𝐶�̃� =  (𝐶𝑏 − 𝜔7, 𝐶𝑏  , 𝐶𝑏 + 𝜔8) =  (31500;  38400;  45400) 
𝐶�̃� =  (𝐶𝑑 − 𝜔9, 𝐶𝑑  , 𝐶𝑑 + 𝜔10) =  (4000;  6000;  19000) 
𝐾�̃� =  (𝐾𝑑 − 𝜔11, 𝐾𝑑  , 𝐾𝑑 + 𝜔12) =  (6000;  7000 ;  9000) 

      Where  

𝜔𝑖 , 𝑖 = 1,2, … ,12 are arbitrary positive numbers that satisfy the following constraints. 

0 ≤ 𝜔1 ≤ 𝐶𝑝, 𝜔2 ≥ 0  0 ≤ 𝜔3 ≤ 𝐶𝑜, 𝜔4 ≥ 0 

0 ≤ 𝜔5 ≤ 𝐶ℎ , 𝜔6 ≥ 0  0 ≤ 𝜔7 ≤ 𝐶𝑏 , 𝜔8 ≥ 0 

0 ≤ 𝜔9 ≤ 𝐶𝑑 , 𝜔10 ≥ 0  0 ≤ 𝜔11 ≤ 𝐾𝑑 , 𝜔12 ≥ 0 

 

using the concept of α-cut introduced by [15] at the defuzzification stage obtained the 

following parameter values. 

 

• �̃�𝑝 = 𝐶𝑝 +  
1

4
(𝜔2 − 𝜔1)    

�̃�𝑝 = 36800 +  
1

4
(7600 − 7500) 

�̃�𝑝 = 36800 +  
1

4
(100) 

�̃�𝑝 = 36825 

 

• �̃�ℎ = 𝐶ℎ +  
1

4
(𝜔6 − 𝜔5) 

�̃�ℎ = 17500 +  
1

4
(1500 − 500) 

�̃�ℎ = 17500 +  
1

4
(1000) 

�̃�ℎ = 17750 
    



 

 

 

Eka Susanti et al. 

Optimization of Inventory Level Using Fuzzy Probabilistic Exponential Two Parameters Model 

 

128 

• �̃�𝑏 = 𝐶𝑏 +  
1

4
(𝜔8 − 𝜔7) 

�̃�𝑏 = 38400 +  
1

4
(7000 − 6900) 

�̃�𝑏 = 38400 +  
1

4
(1100) 

�̃�𝑏 = 38425 
 

• �̃�𝑑 = 𝐶𝑑 +  
1

4
(𝜔10 − 𝜔9)  

�̃�𝑑 = 6000 +  
1

4
(13000 − 5600) 

�̃�𝑑 = 6000 +  
1

4
(7400) 

�̃�𝑑 = 7850 
 

• �̃�𝑑 = 𝐾𝑑 +  
1

4
(𝜔12 − 𝜔11)  

�̃�𝑑 = 7000 +  
1

4
(2000 − 1000) 

�̃�𝑑 = 7000 +  
1

4
(1000) 

�̃�𝑑 = 7250 
 

The parameter estimation for the level of deterioration has an exponential distribution of 

two parameters and the level of demand has a Pareto distribution. Parameter estimation 

using Maximum Likelihood. 

𝐿(𝜎, 𝜇) = ∏
1

𝜎
𝑒

−
(𝑡−𝜇)

𝜎
𝑛
𝑖=1   

  = (
1

𝜎
)

𝑛
exp (

− ∑(𝑥𝑖−𝜇)

𝜎
) , ∀𝑥𝑖 ≥ 𝜇  

𝑑 ln 𝐿(𝜎,𝜇)

𝑑𝜎
= −

𝑛

𝜎
+

∑(𝑥𝑖−𝜇)

𝜎2
= 0,   

 �̂� =
∑(𝑥𝑖−�̂�)

𝑛
  

 

𝐿(𝜂, 𝛿) = ∏
𝜂𝛿𝜂

(𝑥+𝛿)𝜂+1
𝑛
𝑖=1   

 =  𝜂𝑛 𝛿 𝑛𝜂  ∏
1

(𝑥+𝛿)𝜂+1
𝑛
𝑖=1  

 =  ln 𝜂𝑛 + ln 𝛿 𝑛𝜂 + (−𝜂 − 1) ∑ ln 𝑥𝑖
𝑛
𝑖=1 + 𝛿 

 =
𝜕

𝜕𝜂
[ln 𝜂𝑛 + ln 𝛿 𝑛𝜂 + (−𝜂 − 1) ∑ ln 𝑥𝑖

𝑛
𝑖=1 + 𝛿] 

 =  
𝑛

𝜂
+ 𝑛 ln 𝛿 − ∑ ln 𝑥𝑖

𝑛
𝑖=1 + 𝛿 = 0 

 
𝑛

𝜂
= ∑ ln 𝑥𝑖

𝑛
𝑖=1 + 𝛿 − ln 𝛿 

 �̂�𝐿 =  
𝑛

∑ ln
𝑥𝑖+𝛿

𝛿
𝑛
𝑖=0

,  𝛿 = min 𝑥𝑖  

The estimated value are 𝛿 = 1,47 𝜂 = 0,8618 , 𝜎 = 0,08 . The formulation of a two-

parameter fuzzy probabilistic model on the problem of determining the level of inventory 

red chilis is as follows.  

Min 𝐸(𝑇𝐶) = 𝐸(𝑂𝐶) + 𝐸(𝑃𝐶) + 𝐸(𝐷𝐶) + 𝐸(𝑉𝐶) + 𝐸(𝐵𝐶)   (3) 

Subject to          



 
 

 

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Vol 7, No 2, October 2021, pp. 124-131 
 

129 

𝐸(𝐷𝐶) ≤ �̃�𝑑          (4) 

𝑁 ≥ 0           (5) 

Where 

𝐸(𝑂𝐶) = �̃�0 = 6050  

𝐸(𝑃𝐶) = �̃�𝑝 �̃� ∫ 𝑥 𝑓(𝑥) 𝑑𝑥 

∞

𝑥=0

 

𝐸 (𝑃𝐶)  =  36.825 �̃� ∫ 𝑥 
0,8618 × 1,470,8618

(𝑥 + 1,47)0,8618
 𝑑𝑥 

∞

𝑥=0

 

𝐸 (𝐷𝐶) =  �̃�𝑑  𝑁
𝛽 (�̃�𝑚 − ∫ 𝑥 𝑓(𝑥)𝑑𝑥 

�̃�𝑚

𝑥=0

) 

               =  7850 𝑁𝛽 (�̃�𝑚 − ∫ 𝑥 
0,8618 × 1,470,8618

(𝑥 + 1,47)0,8618
𝑑𝑥 

�̃�𝑚

𝑥=0

) 

𝐸 (𝑉𝐶) =  �̃�𝑑  𝛾 𝑁
𝛽 (𝑄𝑚 − ∫ 𝑥 𝑓(𝑥) 𝑑𝑥 

𝑄𝑚

𝑥=0

) 

               =  7850 (0,02) 𝑁𝛽 (𝑄𝑚 − ∫ 𝑥 
0,8618 × 1,470.8618

(𝑥 + 1,47)0,8618
 𝑑𝑥 

𝑄𝑚

𝑥=0

) 

𝐸 (𝐵𝐶) =  �̃�𝑏  [∫
𝑥 −�̃�𝑚

∈
(1 −

1

∈(�̃�−�̃�1)
ln[1+ ∈ (�̃� − �̃�1)]) 𝑓(𝑥)𝑑𝑥

∞

�̃�𝑚
]     

𝐸 (𝐵𝐶) =  38425 [ ∫
𝑥 − �̃�𝑚

0,5
(1

∞

�̃�𝑚

−
1

0,5(�̃� − �̃�1)
ln[1 +  0,5(�̃� − �̃�1)])

0,8618 × 1,470,8618

(𝑥 + 1,47)0,8618
𝑑𝑥] 

The optimal solution is determined for some 𝛽 value. 𝛽 is an arbitrary positive number with  
0 ≤ 𝛽 ≤ 1.  For the 𝛽 = 0,1;  0,3;  0,5;  0,9;  1. Nonlinear models (3), (4), (5) are solved 
using the lingo software, the following solution is obtained. 

 

Table 2. Solution Nonlinear Model  

𝛽 𝑁 𝑄𝑚  𝐸(𝑇𝐶) 𝐸(𝑇𝐶)/𝑄𝑚  
0,1 0 10 397748,6 39774,86 

0,3 0 10 397748,6 39774,86 

0,5 0,128 15 405022 27001,466 

0,9 0,247 15 405240,2 27001,48 

1 0,923 10 405143,6 40514,36 

 
The minimum total cost is influenced by the variable value 𝛽, 𝑁 and 𝑄𝑚. Based on 

table 2, it can be seen that the more the value approaches 1, the ordering cycle is getting 

closer to 1 and the total costs incurred are increasing. The maximum inventory is 15 kg and 

the largest total cost is Rp 405143,6. The largest total cost occurs when the value of 𝑁 
increases. The length of the time of review cycle is affected by the remaining inventory and 

the remaining product incurs storage costs. Base on the results obtained, the optimal 



 

 

 

Eka Susanti et al. 

Optimization of Inventory Level Using Fuzzy Probabilistic Exponential Two Parameters Model 

 

130 

amount of red chillies that can be ordered so that the minimum cost is 10 kg with the length 

of the time of review cycle is 0. A value of 𝑁 equal to zero means that the red chillies 
ordered were sold out immediately or the seller can order as much as 15 kg with a review 

cycle duration of 0,128 days or approximately 3 hours. 

 

4. Conclusion 
 

Based on the results and discussion, it can be concluded that the fuzzy approach can 

be applied to inventory optimization problems with uncertainty. The choice of 𝛽 variable 
value influences the solution obtained. The ordering cycle is getting closer to one day and 

the total costs incurred are increasing for the 𝛽 value close to 1. 𝛽 value affects the value 
of 𝑁. The greater the 𝛽 value, the greater the value of 𝑁. Consequently, the total costs are 
increasing. The maximum inventory level of red chillies is 15 kg. 

 

Acknowledgements 

This Research is supported by Universitas Sriwijaya through Sains Teknologi dan Seni 

(SATEKS) Research Scheme with the number of the research assignment contract number 

0163.175/UN9/SB3.LPPM.PT/2020. 

 

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