Plane Thermoelastic Waves in Infinite Half-Space Caused


Decision Making: Applications in Management and Engineering  
Vol. 2, Issue 1, 2019, pp. 1-12. 
ISSN: 2560-6018 
eISSN: 2620-0104  

 DOI: https://doi.org/10.31181/dmame1901001a 

* Corresponding author. 
 E-mail addresses: krish.math23@gmail.com (K. Adhikary), jaga.math23@gmail.com 
(J. Roy), kar_s_k@yahoo.com (S. Kar) 

Newsboy problem with birandom demand 

Krishnendu Adhikary 1, Jagannath Roy 1* and Samarjit Kar 1 

1 Department of Mathematics, National Institute of Technology Durgapur, Durgapur, 
India 

 
Received: 15 October 2018;  
Accepted: 12 December 2018;  
Available online: 18 December 2018. 

 
Original scientific paper 

Abstract: Estimation of accurate product demand in a single period inventory 
model (SPIM) is an essential prerequisite for successfully managing the supply 
chain in large and medium merchandise. Managers/ decision makers (DMs) 
often find it difficult to forecast the exact inventory level of a product due to 
complex market situations and its volatility caused by several factors like 
customers uncertain behavior, natural disasters, and uncertain demand 
information. In order to make fruitful decisions under such complicated 
environment, managers seek applicable models that can be implemented in 
profit maximization problems. Many authors studied SPIM (also known as 
newsboy problem) considering the demand as a normal random variable with 
fixed mean and variance. But for more practical situations the mean demand 
also varies time to time yielding two-folded randomness in demand 
distribution. Thus, it becomes more difficult for DMs to apprehend the actual 
demand having two-folded random/birandom distribution. A blend of 
birandom theory and newsboy model has been employed to propose birandom 
newsboy model (BNM) in this research to find out the optimal order quantity 
as well as maximize the expected profit. The practicality of the projected BNM 
is illustrated by a numerical example followed by a real case study of SPIM. 
The results will help DMs to know how much they should order in order to 
maximize the expected profit and avoid potential loss from excess ordering. 
Finally, the BNM will enhance the ability of the managers to keep parity of 
product demand and supply satisfying customers’ needs effectively under 
uncertain environment. 

Key words: Newsboy problem, Uncertain variable, Birandom variable, 
Expectation. 

1. Introduction 

The classical newsboy problem (CNP) aims at determining the optimal order 
quantity of products, which minimize the expected total cost and / or maximize the 

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2 

expected profit in a SPIM. Thus, the characteristics of a SPIM become more 
complicated due to complex market situations and its volatility caused by several 
factors like customers uncertain behavior. In response to handle such problems, the 
CNP is extended in many directions and several researchers have proposed modified 
versions of CNP. Some of them made proper and thoughtful use of probability theory 
to solve newsboy problems where product demands follow Poisson and Normal 
distribution (Hadley and Whitin, 1963), Weibull distribution (Tadikamalla, 1978), 
Erlang distribution (Mahoney and Sivazlian, 1980), compound Poisson distribution 
(Dominey and Hill, 2004). Gallego and Moon (1993, 1994) analyzed distribution-free 
newsboy problems where DMs have no idea about the demand distribution. The only 
information they have are mean and variance of demand. Their newsboy models can 
be used as strategic tools in deciding the stock of products that have a limited selling 
period. 

This paragraph is dedicated to articulate the recent developments of SPIM and find 
the literature gap. Agarwal and Seshadri (2000) worked in CNP where they assumed 
the demand distribution as a function of selling price and the objective of the risk-
averse retailers. To maximize DMs expected utility they presented two models for 
comparing the risk-neutral retailers (who charge a higher price for less order) with a 
risk-averse retailer (who charge a low price). The distribution-free newsboy problem 
under the worst-case and best-case scenario was revealed by Kamburowski (2014). 
Further, Kamburowski (2015) studied a newsboy problem where the distribution of 
the random variable is only known when to be non-skewed with given support, mean 
and variance. For the distribution-free newsboy problem, Gler (2014) extended the 
model developed by Lee and Hsu (2011). Here, the authors showed the expected profit 
increases with a proper advertisement policy while an unorganized advertising policy 
can have its backfire effect or make a very small improvement of the optimal profit 
value. Ding (2013) proposed a chance constraint multi-product newsboy problem 
with uncertain demand and uncertain storage capacity. Abdel-Aal et al. (2017) studied 
a multi-product newsboy problem assuming the service level as a constraint to offer 
the DMs to select the market to serve. Watt and Vzquez (2017) considered newsboy 
problem under two new assumptions. First, they assumed that the wholesaler is an 
expert who sets the wholesale price optimally and a newsboy can return the unsold 
item with some salvage value. In the second one, the salvage value acts as a standard 
insurance demand. Sun and Guo (2017) built a newsboy model with fuzzy random 
demand based on fuzzy random expected value model. Vipin and Amit (2017) 
proposed a loss aversion SPIM under alternative option and proved the rationality of 
the decision maker to predict the order quantity by imposing loss aversion in the 
newsboy model with the change of selling price and purchase cost factors. 
Additionally, they showed the models based on utility functions perform better in 
forecasting the rational behavior due to loss aversion. Natarajan et al. (2017) allowed 
asymmetry and ambiguity in newsvendor models. The effects of decision makers 
emergency order in SPIM are discussed and analyzed by Pando et al. (2013) and Zhang 
et al. (2017). Zhang et al. (2017) compared two ways to treat the excess demand and 
came up with the better one. 

In the aforementioned works, the product demand is assumed to be either 
normally distributed with 𝑁(𝜇, 𝜎 2) or exponentially distributed with constant mean 
(𝜆) or somewhere distribution free. But the DMs face difficulties to forecast the exact 
demands of products in many practical problems. The demand distribution changes 
dynamically from time to time, which yields randomness in the mean demand. For 
example, the demand (𝐷) is normally distributed with 𝐷 ~ 𝑁(𝜇 ;  400) where 
𝜇 ~ 𝑈(3000;  4000) or 𝐷 ~ 𝑁(𝜇 ;  400) with 𝜇 ~ 𝑒𝑥𝑝(0.0003). However, in newsboy 



Newsboy problem with birandom demand 

3 

 

problems, it is largely appreciated to consider demand variable having the standard 
normal distribution. From the probabilistic viewpoint and the above arguments, it 
would be more realistic to assume the values of 𝜇 and 𝜎 2 should also be treated as 
random variables. For such cases, it is more convincing and practical to represent the 
product demand as birandom variable effectively captures two-folded randomness. 
Traditional probabilistic approaches cannot handle such complicated real world 
problems. In response, Peng and Liu (2007) developed a birandom theory to tackle 
such problems. Zhaojun and Liu (2013) showed the common formula on birandom 
variable and further, several researchers used this theory to solve inventory problems 
in the birandom environment. In the recent years, birandom theory is widely accepted 
as the mathematical language of uncertainty. Some more notable extensions and 
applications of newsboy problem can be found in the recent literature (Abdel-Malek 
and Otegbeye 2013; Chen and Ho 2013). 

From its inception until today, birandom theory has progressed and been applied 
to different areas. Xu and Zhou (2009) introduced a class of multiple objective decision 
making problems using birandom variables and by transforming the birandom 
uncertain problem into its crisp equivalent form through expected value operator and 
used it in the flow shop scheduling problem. A Portfolio selection problem is analysed 
by Yan (2009) assumed the security returns as birandom variables. Xu and Ding 
(2011) developed the general chance constrained multi objective linear programming 
model with birandom parameters for solving a vendor selection problem. They 
presented a crisp equivalent model for a special case and gave a traditional method to 
solve the crisp model. Wang et al. (2012) established a class of job search problem 
with birandom variables, where the job searcher examined job offers from a finite set 
jobs having equivalent probability. A multi-mode resource constrained project 
scheduling problem (Zhang and Xu, 2013) of drilling grounding construction projects 
considering the uncertain parameters as birandom variables. In the earlier year, Xu et 
al. (2012) used birandom theory to develop the nonlinear multi objective bi-level 
models for finding the minimum cost in a network flow problem dealing a large scale 
construction project. Tavana et al. (2013) measured the efficiencies of decision making 
units after developing a data envelopment analysis (DEA) model with birandom input 
and output data. Nevertheless, many more real life applications can be found in the 
following literature: a multi-objective birandom inventory problem (Tao and Xu, 
2013), a birandom multi-objective scheduling problem (Xu et al., 2013) in ship 
transportation, optimal portfolio selection with birandom returns(Cao and Shan, 
2013), a modified genetic algorithm (Maity et al., 2015), chance-constrained 
programming model for municipal waste management with birandom variables (Zhou 
et al., 2015), and the CCUS (carbon capture, utilization, and storage) management 
system in birandom environment (Wang et al., 2017). 

To the best of our knowledge, no researcher has investigated the SPIM with 
birandom demand till date. With these considerations, we discuss an inventory model 
with single period considering the demand for the birandom variable. We solve a real 
problem using the presented model in searching the optimum order quantity for 
maximum profit by using the expected value model. The principal aim of this paper is 
to deliver basic knowledge and suggest precise results of a complex inventory 
practical problem for the management. 
 

The remaining part of our paper is presented in the following way. Section 2 
presents some basic knowledge of birandom variable and related theorems. Section 3 
introduces the birandom simulation for finding the expected value of the birandom 

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variable. In section 4, we provide a mathematical model for newsboy problem with 
birandom demand. A numerical example is discussed in section 5. To validate the 
applicability of the proposed model we discuss a real case study in section 6. Finally, 
the conclusion and future research directions are presented in section 7. 

2. Preliminaries 

In this section we discuss the basic notations of birandom variables. 

2.1. Birandom Variable 

Roughly speaking a birandom variable is a random variable of a random variable, 
i.e., a function defined from a probability space to a collection of random variables is 
said to be a birandom variable. The formal definition of birandom variable and related 
theorems are defined in the following way. 

 
Definition 1 (Peng and Liu, 2007). A birandom variable 𝜉 is a mapping from a 

probability space (𝛺, 𝐴, 𝑃𝑟) to a collection 𝑆 of a random variable such that for any 
Borel subset 𝐵 of the real line ℜ the induced function 𝑃𝑟 {𝜉(𝜔) ∈ 𝐵} is a measurable 
function with respect to 𝜔. 

For each given Borel subset 𝐵 of the real line ℜ, the function 𝑃𝑟 {𝜉(𝜔) ∈ 𝐵} is a 
random variable defined on the probability space (𝜔, 𝐴, 𝑃𝑟). 

 

Lemma 1 (Peng and Liu 2007). Let an n dimensional birandom vector 𝜉 =
(𝜉1, 𝜉2, … , 𝜉𝑛 ) and 𝑓: ℜ

𝑛 → ℜ be a measurable function. Then 𝑓(𝜉) is a birandom 
variable. 

Let two probability spaces (Ω1, 𝐴1, Pr1) and (Ω2, 𝐴2, Pr2), 𝜉1 and 𝜉2 be two 
birandom variables respectively taken from that probability spaces. Then 𝜉 = 𝜉1 + 𝜉2 
is a birandom variable on (Ω1 × Ω2, 𝐴1 × 𝐴2, Pr1 × Pr2) defined by 

       1 2 1 1 2 2 1 2 1 2, ,  , Ω Ω              

Widely, for the n-tuple operation on birandom variables defined as follows. Let a 
Borel measurable function defined as 𝑓: ℜ𝑛 → ℜ and 𝜉𝑖  be birandom variable defined 
on (Ω𝑖 , 𝐴𝑖 , Pri), 𝑖 = 1,2, … , 𝑛 respectively. Then 𝜉 = 𝑓(𝜉1, 𝜉2, … , 𝜉𝑛 ) is birandom 
variable on (Ω1 × Ω2 × … × Ω𝑛 , 𝐴1 × 𝐴2 × … × 𝐴𝑛, Pr1 × Pr2 × … × Prn), defined by 

        

 

1 2 1 1 2 2

1 2 1 2

, , , , , .,  

 , , ., Ω Ω Ω

n n n

n n

f         

  

  

    
 

2.2. Expected Value of birandom variables 

We can transform the complex uncertain problems into their equivalent crisp 
models, which will be easier to solve. Generally, expected value operator is applied to 
transform the birandom problem into its deterministic value model for calculating the 
objective functional value. First, we present the definition of the expected value 
operator of a birandom variable and then the expected value model of SPIM. The 
effective tool of an uncertain variable is expectation, which is applied in a different 
field of applications. Therefore, the idea of expected value of birandom variable is 
useful. 

The expected value operator of birandom variable is defined as follows. 



Newsboy problem with birandom demand 

5 

 

 

Definition 2 (Peng and Liu, 2007). Let 𝜉 be a birandom variable defined on the 
probability (Ω, 𝐴, 𝑃𝑟). Then the expected value of birandom variable 𝜉 is defined as  

       
0

0

Pr Ω| Pr Ω|E E t dt E t dt      




                (1) 

Provided that at least one of the above two integrals is finite. 
 

Lemma 2 (Peng and Liu, 2007). Let 𝜉 be a birandom variable defined on the 
probability (𝜔, 𝐴, 𝑃𝑟). If the expected value 𝐸[𝜉(𝜔)] of the random variable 𝜉(𝜔) is 
finite for each 𝜔, then 𝐸[𝜉(𝜔)] is a random variable on (𝜔, 𝐴, 𝑃𝑟). 

Lemma 3 (Peng and Liu, 2007). Let us consider two birandom variable 𝜉 and 𝜂 with 
finite expected value, then for any two real numbers a and b, we have 

     E a b aE bE       

We know that a function from a probability space (Ω, 𝐴, 𝑃𝑟)  to a collection of 
random variables is called a birandom variable, from the definition of birandom 
variable. Birandom variables are two type. They are either discrete birandom variable 
or continuous birandom variable. Expectation theory of birandom variables will be 
discussed in the current subsection. 

Definition 3 (Xu et al., 2009). For a birandom variable 𝜉, in the probability space 
(𝛺, 𝐴, 𝑃𝑟), if 𝜉(𝜔) is a random variable with a continuous distribution function when 
𝜔 ∈ 𝛺 and its expected value 𝐸[𝜉(𝜔)] is a birandom variable. Then we call 𝜉 
continuous birandom variable. 

Definition 4 (Xu et al., 2009). Suppose 𝜉 is a birandom variable, then 𝜉(𝜔) is a 
random variable. If 𝑓(𝑥, 𝜉) be the density function of 𝜉(𝜔), and  

   
Ωx

E xf x dx 


        (2) 

Then the density function of birandom variable 𝜉 is 𝑓(𝑥, 𝜉). 
Definition 5 (Xu et al., 2009). For the continuous birandom variable 𝜉, if its density 

function is 𝑓(𝑥), we can define the expected value of 𝜉 as follows 

     
0

0 Ω Ωx x

E Pr xf x r dr Pr xf x r dr


  

      
      

      
      (3) 

Definition 6 (Xu et al., 2009). If the density function of a birandom variable 𝜉 is 
𝑓(𝑥, 𝜉) and 𝑔(𝑥) is a continuous function. Then expectation for the birandom variable 
𝑔(𝜉) is defined as 

         
0

0 Ω Ωx x

E g Pr g x f x r dr Pr g x f x r dr


  

      
         

      
      (4) 

Theorem 1 (Xu et al., 2009). Let 𝑓(𝑥, 𝜉) is the density function of the birandom 
variable 𝜉. Then the expected value of 𝜉 exists if only if the expected value of random 
variable 𝜉(𝜔) exists. 

Theorem 2 (Xu et al., 2009). The expectation of a birandom variable. 𝜉~𝑁(𝜇, 𝜎 2), 

where, 𝜇~U(a, b) is 
𝑎+𝑏

2
. 

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6 

Theorem 3. The expectation of a birandom variable 𝜉~𝑁(𝜇, 𝜎 2) where 𝜇~exp (𝜆) is 
1

𝜆
. 

 
Proof: By definition (2), we know  

       
0

0

Pr | Pr |  E E t dt E t dt      




               

Since 𝜇~ exp(𝜆), and obviously 𝐸[𝜉(𝜔)] = 𝜇, by definition 4 and 5, the above 
function can be transformed as follows, 

     
0

0

Pr PrE t dt t dt  




      

Since, 𝜇~ exp(𝜆), and we know that the density function and the distribution 
function of exponential distribution are as follows,  

   , 0,   xf x e x and     

   1 ,  0, .xF x e x     

According to the definition of the distribution function we can obtain the following 
two functions from the distribution function 

Pr(𝜇 ≤ 𝑥) = 1 − 𝑒 −𝜆𝑥 ,   0 ≤ 𝑥 < ∞ 𝑎𝑛𝑑 
 

Pr(𝜇 ≥ 𝑥) = 𝑒 −𝜆𝑥 , 0 ≤ 𝑥 < ∞ 

Obviously, ∫ Pr{𝜇 ≤ 𝑡} 𝑑𝑡 = 0
0

−∞
 

Therefore  

𝐸[𝜉] = ∫ Pr{𝜇 ≥ 𝑡} 𝑑𝑡 = ∫ 𝑒 −𝜆𝑡 𝑑𝑡 = [
𝑒 −𝜆𝑡

𝜆
]

0

∞

=
1

𝜆

∞

0

∞

0

. 

 

However, it is very hard to accomplish the mathematical expression of expected 
value for all types of birandom variables. But using birandom simulation, we could 
calculate the expected value of birandom variables, with the help of Strong Number 
Law. 

3. Birandom Simulation 

Let (𝛺, 𝐴, 𝑃𝑟), be a probability space and a 𝑓: ℜ𝑛 → ℜ be a measurable function. 
Consider that 𝜉 is an 𝑛 − dimension birandom vector on the given probability space. 
Now we have to find the expectation 𝐸[𝑓(𝜉)] of birandom variable. Using stochastic 
simulation, we can find the expected value for every ω ∈ Ω. Here, we have used an 
algorithm for birandom simulation to find the expectation of 𝐸[𝑓(𝜉(𝜔))], which is 

defined as follows. 
 

Algorithm: 
 

Step 1. Start 

Step 2. Set l=0 and N= Number of iteration 

Step 3. Sample 𝜔 from 𝛺 according to the probability measure 𝑃𝑟 



Newsboy problem with birandom demand 

7 

 

Step 4. 𝐸[𝑓(𝜉(𝜔))] is find by the stochastic simulation. 

Step 5. Then 𝑙 ← 𝑙 + 𝐸[𝑓(𝜉(𝜔))]  

Step 6. Repeat the steps from second to fifth steps 𝑁 ttimes. 

Step 7. 𝐸[𝑓(𝜉(𝜔))] =
𝑙

𝑛
. 

Step 8. Stop 

4. Mathematical Formulation 

We are assuming a single period inventory problem with single product. Here all 
the costs (buying cost and selling cost) are deterministic. Salvage value is taken which 
is also deterministic. But the demand is birandom variable. The mathematical notation 
of a birandom newsboy problem is defined as follows: 

𝜉 ̅   : The demand of market, a birandom variable 
𝑥   : The quantity which to be order, a decision variable 
𝑝 = 𝑐(1 + 𝑚)  : Selling price per unit 
𝑠 = 𝑐(1 − 𝑑)   : Salvage value per unit 
𝑐   : Purchasing cost per unit 

𝑔 (𝑥, 𝜉 ̅)  : The profit for the order quantity 𝑥 and demand 𝜉 ̅  

𝜇   : Expected value of birandom demand 𝜉 ̅  

𝜎 2   : Variance of the birandom demand 𝜉̅   
𝑚  : Mark-up, i.e., return per dollar on unit sold 
𝑑   : Discount rate, i.e., loss per dollar on unit unsold 
𝑥 + = max{𝑥, 0} :  The positive part of x 
 

Then the profit can be expressed as 

     ,  min ,g x p x s x cx  


    
  

  (5) 

Now min (𝑥, 𝜉 ̅ ) = 𝜉̅  − (𝜉̅  − x)
+

 

Where 

     x x x  
 

        (6) 

      ,  g x p s x cx p s x   


         (7) 

Since the demand for the product is birandom, the profit function 𝑔 (𝑥, 𝜉̅ ) is also 

consists of birandom variable. Hence, the expectation criteria is used for handling the 
birandom variable. Therefore, to find the optimal quantity, the decision maker will 
maximize the total expected value. 
 

We can write the expected profit as 

         Π x p s s c x p s E x 


         (8) 

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Or using the definition of m and d, as 

       Π [x c m d xd m d E x 


        (9) 

The information of 𝜉 ̅is known. To maximize the profit function, we need the 
following lemma. 

Lemma 4. For given a birandom variable 𝜉,̅ we have the following inequality, 

 
   

1
22 2

2

x x
E x

  



    
 

    (10) 

Proof: Notice that (𝜉̅ − 𝑥)
+

=
 |�̅̃�−𝑥|+(�̅̃�−𝑥)

2
 

The result follows by taking expectations and by using the Cauchy-Schwarz 
inequality 

𝐸 [𝜉̅ − 𝑥] ≤ [𝐸 |𝜉̅ − 𝑥|
2

]

1
2

= [𝜎 2 + (𝑥 − 𝜇)2]
1
2 

By using the lemma (4) the equation (9) will be rewritten as 

     
   

1
22 2

Π
2

x x
x c m d xd m d

  


 
     
 

     
 
  

  (11) 

It is easy to validate that equation no (11) is strictly convex in 𝑥. Upon setting the 
derivative to zero and solving for 𝑥 we obtain the ordering rule 

1 1

2 2
*

2

m d
x

d m




 
             
 

   (12) 

5. Numerical Example  

Assume that the unit purchase price of a perishable product is 𝑐 =  $40, the unit 

selling price is 𝑝 =  $60, and there is no salvage value (𝑠 =  0). Thus, 𝑚 =
𝑝

𝑐
 −  1 =

60

40
 −  1 =

1

2
. Discount rate 𝑑 =  1 −

𝑠

𝑐
 =  1. Further assume that the product demand 

is a birandom variable with normal distribution 𝑁(µ1, 400), and µ1  ∼  𝑒𝑥𝑝(0.0003). 
From theorem (3), we have 𝜉 ∼  𝑁(µ1, 400), where, 𝜆 =  0.0003.  

Hence, by theorem (3) we can say that the mean (µ) of the birandom variable (𝜉) 

is 
1

𝜆
 =

1

0.0003
 =  3333.33, and 𝜎 2  =  400. Now it remains to calculate the optimal order 

quantity and expected profit. For this purpose, we apply equation (12) and obtained 
the optimal order quantity, 𝑥 ∗  =  3326. Hence, the expected profit is, 𝛱 = $66100. 



Newsboy problem with birandom demand 

9 

 

6. A Case Study 

To endorse the model developed in this study, we sent our projected framework to 
five leading fish merchants in West Bengal, India. They sell only the freshest and best 
quality fishes, and maintains quality control at every stage of packaging and delivery 
in many different parts or areas of West Bengal. Among them two firms positively 
responded to explore this research proposal and we conducted necessary 
preliminary tasks on these companies. We selected a reputed fish merchant (“Lakshmi 
fish enterprise”, name changed), situated in the ”southern” West Bengal, which has 
several operational units nationwide. Our objective is to incorporate the perceptions 
of all participants (customer/retailer/company mangers) in the fish industry and to 
achieve its comprehensive outcomes since this research is purely grounded on 
birandom product demand information obtained from experts in the business. In this 
paper, we consider the perspectives of a wholesale fish merchant who buys fishes from 
the company, having a large market share in Kolkata zone. 

In West Bengal, fish merchants generally sell a special fish in monsoon. The name 
of this fish is Hilsa. The business of this fish is a good example of SPIM. The business of 
this fish totally depends on its demand and supply in the monsoon season. Merchants 
have to decide how many fishes should be purchased from his or her supplier 
depending on the customer’s demand. Buying more amount of Hilsa may not bring him 
more profit. Rather it can cause him a great loss since it cannot be preserved for long 
periods and the expired fish has no market value. If he/she buys too few amount of 
Hilsa he/she will lose the opportunity of making a higher profit. Thus, the actual 
inventory level cannot be determined precisely in such complex situation. It may be 
assumed for simplicity that the fish demand follows normal distribution. But under 
such circumferences, the manager looks after of some previous data, and finds the 
mean demand of ”Hilsa” is also a random variable. This leads us to consider the ”Hilsa” 
demand as a birandom variable. Each such fish sells for $60 and costs for the shop 
owner $40. Investigating the previous year data, the decision maker decides the 
demand follows the two types two- folded random variable (birandom). 

Scenario 1: Normal distribution 𝑁~(𝜇1, 400), with 𝜇1~𝑈(3000, 4000). 
Scenario 2: Normal distribution 𝑁~(𝜇1, 400), with 𝜇1~𝑁(3500, 500). 
Therefore, according to our proposed model we have 𝑐 =  40, 𝑝 =  60, and there 

is no salvage value i.e., 𝑠 =  0. In scenario 1, by theorem (2) the mean of the birandom 
variable is µ =  3500, and 𝜎 2  =  400. Therefore the optimal order quantity 𝑥 ∗  =
 3492. Expected profit 𝛱 =  $69434. And in scenario 2, using birandom simulation we 
get the mean of the birandom simulation µ =  3508, and 𝜎 2  =  400. Therefore, the 
optimal order quantity 𝑥 =  3500, expected profit = $69594. Finally, we shared the 
outcomes of this research work with the managers of our case enterprise, they are 
satisfied with the outcomes and willing to accept this result for their monsoon 
business of ”Hilsa”. 

7. Conclusion 

In this paper we have proposed a newsboy problem where the demand is 
considered as birandom variable. The market volatility and uncertainty in customers’ 
behavior make the demand of the product a birandom variable. We use the expected 
value model for handling this birandom variable and to convert the BNM into its 
equivalent deterministic model. We discuss a case study of fish merchant to validate 
the usefulness and applicability of the proposed model. In this proposed model 

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10 

demand is the only parameter considered as birandom variable. However, in reality 
cost may be assume as a birandom variable. This is the one aspects to be considered 
for further investigation in the current model. Also, for future research 
work, one can develop chance constraint technique to convert the birandom model 
into deterministic one. 

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