An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit

Accepted Manuscript An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit Leopoldo Eduardo Cárdenas-Barrón, Ali Akbar Shaikh, Sunil Tiwari, Gerardo Treviño-Garza PII: DOI: Reference:

S0360-8352(18)30608-9 https://doi.org/10.1016/j.cie.2018.12.004 CAIE 5557

To appear in:

Computers & Industrial Engineering

Please cite this article as: Eduardo Cárdenas-Barrón, L., Akbar Shaikh, A., Tiwari, S., Treviño-Garza, G., An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit, Computers & Industrial Engineering (2018), doi: https://doi.org/10.1016/j.cie.2018.12.004

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An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit Leopoldo Eduardo Cárdenas-Barrón1*, Ali Akbar Shaikh1,2, Sunil Tiwari3,4, Gerardo TreviñoGarza5 1

Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México 2

Department of Mathematics, The University of Burdwan, Burdwan-713104, India

3

Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi 110007, India

4

The Logistics Institute-Asia Pacific, National University of Singapore, 21 Heng Mui Keng Terrace, #04-01, Singapore 119613 5

BNSF Railway Company 2650 Lou Menk Drive

Fort Worth, TX 76131-2830, USA Worth, TX 76131-2830, USA Abstract This paper deals with an economic order quantity (EOQ) inventory model under both nonlinear stock dependent demand and nonlinear holding cost. This inventory model is developed from retailer’s point of view, where the supplier offers a trade credit period to the retailer. In this paper, we relax the traditional assumption of zero ending inventory level to a non-zero ending inventory level. Consequently, the ending inventory level can be positive, zero or negative. When the ending inventory level is negative means that the shortages are permitted and partially backlogged with a constant backlogging rate. Basically, two inventory models are developed: (i) an inventory model with shortage and (ii) an inventory model without shortage. The primary objective of both inventory models is to determine the optimal ordering quantity and the ending inventory level which maximizes the retailer’s total profit per unit time. In order to obtain the optimal solution, lemmas, and theorems are derived along with a solution procedure. The proposed inventory models are a general framework as several previously published inventory models are particular cases of the inventory models derived in this paper. Some numerical examples and a sensitivity analysis are conducted to illustrate the findings of the inventory models and some observations are also discussed. Keywords: Inventory; EOQ inventory model; nonlinear stock dependent holding cost; stock-dependent demand; credit policy.

*

 Corresponding author. Tel. +52 81 83284235, Fax +52 81 83284153. E-mail: [email protected] (L.E. Cárdenas-Barrón), [email protected], [email protected] (A.A. Shaikh), [email protected] (S.Tiwari), [email protected] (G. Treviño-Garza)

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An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit Leopoldo Eduardo Cárdenas-Barrón1*, Ali Akbar Shaikh1,2, Sunil Tiwari3,4, Gerardo TreviñoGarza5 1

Department of Industrial and Systems Engineering, School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México 2

Department of Mathematics, The University of Burdwan, Burdwan-713104, India

3

Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi 110007, India

4

The Logistics Institute-Asia Pacific, National University of Singapore, 21 Heng Mui Keng Terrace, #04-01, Singapore 119613 5

BNSF Railway Company 2650 Lou Menk Drive

Fort Worth, TX 76131-2830, USA Worth, TX 76131-2830, USA Abstract This paper deals with an economic order quantity (EOQ) inventory model under both nonlinear stock dependent demand and nonlinear holding cost. This inventory model is developed from retailer’s point of view, where the supplier offers a trade credit period to the retailer. In this paper, we relax the traditional assumption of zero ending inventory level to a non-zero ending inventory level. Consequently, the ending inventory level can be positive, zero or negative. When the ending inventory level is negative means that the shortages are permitted and partially backlogged with a constant backlogging rate. Basically, two inventory models are developed: (i) an inventory model with shortage and (ii) an inventory model without shortage. The primary objective of both inventory models is to determine the optimal ordering quantity and the ending inventory level which maximizes the retailer’s total profit per unit time. In order to obtain the optimal solution, lemmas, and theorems are derived along with a solution procedure. The proposed inventory models are a general framework as several previously published inventory models are particular cases of the inventory models derived in this paper. Some numerical examples and a sensitivity analysis are conducted to illustrate the findings of the inventory models and some observations are also discussed. Keywords: Inventory; EOQ inventory model; nonlinear stock dependent holding cost; stock-dependent demand; credit policy. 1. Introduction and literature review *

 Corresponding author. Tel. +52 81 83284235, Fax +52 81 83284153. E-mail: [email protected] (L.E. Cárdenas-Barrón), [email protected], [email protected] (A.A. Shaikh), [email protected] (S.Tiwari), [email protected] (G. Treviño-Garza)

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Over the last few years, much attention has been attracted to inventory systems, since the inventory models manage the level of inventories in the companies efficiently and effectively. As it is known, the first inventory model was introduced by Harris (1913). He developed the well-known economic order quantity (EOQ) inventory model which considers that the demand is constant and known. After that, many research works have been done in the field of inventory. Many researchers have extended the Harris (1913)’s inventory model by taking into consideration more convincing assumptions with the aim to model more closely real-life situations. According to Cárdenas-Barrón et al. (2014), Harris is considered as the founding father of the inventory theory. According to Levin et al. (1972), the people show a great interest in purchasing more products when the supermarkets display the products in large quantities at the showroom; this commonly rises the demand of the products. Therefore, the stock of products becomes a significant factor in the inventory analysis. In this direction, Gupta and Vrat (1986) built an inventory model for multi items considering stock-dependent consume rate. Afterward, Baker and Urban (1988) derived an EOQ inventory model for a stock-dependent demand with power-form. After that, Mandal and Phaujdar (1989) developed an economic production (EPQ) inventory model for deteriorating goods considering that the production rate is constant and the demand is stock-dependent. One year later, Datta and Pal (1990) proposed an inventory model assuming that the demand is a function of inventory level. Later, Pal et al. (1993) revisited and extended the Baker and Urban (1988)'s inventory model for deteriorating products. Subsequently, Padmanabhan and Vrat (1995) formulated an inventory model for perishable products with stock-dependent selling rate and shortage. After, Wu et al. (2006) presented an inventory model with stock-dependent demand for non-instantaneous deteriorating products and partial backlogging. It is worth to mentioning that in this line of research are several inventory models; for example: Sarker et al. (1997), Ray and Chaudhuri (1997), Ray et al. (1998), Dye and Ouyang (2005), Lee and Dye (2012), Min et al. (2012), Avinadav et al. (2013), Taleizadeh et al. (2013), Yang (2014), Singh and Sharma (2014), Krommyda et al. (2015), Lu et al. (2016), Mishra et al. (2017), and Jaggi et al. (2018); just to name a few recent research works. In a real condition, the demand rate is not fixed and changes under various factors. Therefore, many researchers in the literature have studied the inventory system with inconstant demand pattern. Maihami and Kamalabadi (2012) studied a lot-sizing model for non-instantaneous deteriorating products by considering partial backlogging where the demand of customer depends on selling price and time. Khanra et al. (2013) presented an inventory model for a single product by considering shortages and permissible delay in payment where customer’s demand is a quadratic function of time. Avinadav et al. (2013) derived an inventory model by considering price and time-dependent demand rate for a perishable product. Their purpose is to determine the optimal pricing, replenishment period, and order quantity. Sarkar et al. (2014) developed a production inventory model in an imperfect production process where the

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demand rate depends on the selling price and the time. Alfares and Ghaithan (2016) proposed an economic order quantity inventory model in which the demand rate is a function of the selling price, the holding cost is a function of storage time, and purchase cost depends on order size by considering allunits quantity discounts. A few researchers have considered the dependency of demand on both price and time. Hsieh and Dye (2017) formulated an inventory model for a deteriorating item by considering the displayed stock level and sale price-dependent demand. Tiwari et al. (2017a) explored the replenishment policy for the retailer that leverages stock-dependent demand for non-instantaneous deteriorating items. Particle swarm optimization is used to derive an optimal solution. Mishra et al. (2017) developed an EOQ inventory model with selling price and stock dependent demand for deteriorating products and determined the optimal price, order quantity, and preservation technology investment from the perspective of the retailer. There exist many research works which assume that the holding cost is constant. It is observed that in order to prevent the deterioration of the products make that the holding cost sometimes increases drastically. It is clear that the extent of the increment in the holding cost depends on the nature of the product that is being conserved. In this context, Weiss (1982) constructed an EOQ inventory model assuming that the inventory holding cost is a convex function of time. After that, Goh (1994) also extended the Baker and Urban (1988)’s inventory model relaxing the supposition that the holding cost is constant. Afterward, Giri and Chaudhuri (1998) generalized Goh (1994)’s inventory model and derived an inventory model considering that the items deteriorate. In recent times, Pando et al. (2012a) developed an inventory model assuming that both demand rate and holding cost depends on the stock level. Their inventory model imposed that the ending inventory model must be zero. In this direction, the reader can see the following research works: Hwang and Hahn (2000), Alfares (2007), Roy (2008), Valliathal and Uthayakumar (2011), Pando et al. (2012b), Pando et al. (2013), Tripathi (2013), Yang (2014) and others. Yang (2014) extends Pando et al. (2012a)’s inventory model by relaxing the assumption that the ending inventory model is zero to non-zero ending inventory level. Pando et al. (2018) studied an inventory model for deteriorating items having stock-dependent demand. In their inventory model, they have taken holding cost as a function of time. Recently Dobson et al. (2018) proposed an EOQ inventory model for perishable items, where the demand for the items/products depends on their age. San-José et al. (2018) developed an EOQ inventory model for a single product where the demand of the product depends on both the time and price. They considered the case of shortages also in their inventory model. Li and Teng (2018) formulated an inventory model for perishable goods whose demand depends on selling and reference price, the freshness of the product and the displayed stock. They discussed two different scenarios of demand behavior named as (i) loss neutrality and (ii) loss aversion.

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Traditionally, on the one hand, the EOQ inventory model assumes that the buyer needs to pay the entire fee to the seller when the lot size is received. On the other hand, if the buyer has the option to pay the products later without interest charges, then the buyer is more interested in purchasing products. This policy is called trade credit policy. In the trade credit policy, the seller offers a delay in payment up to a specific period with the aim to promote the buyer to purchase large quantities of products and therefore to reduce the on-hand stock level. This type of trade credit is also commonly known in the literature as permissible delay in payments. The trade credit policy is an excellent option for a buyer with limited financing opportunities. Possibly, Haley and Higgins (1973) were the first researchers in developing an inventory model with credit policy. After that, Goyal (1985) formulated an inventory model and discussed the impact of credit policy in the EOQ inventory model. Goyal (1985)’s inventory model assumes that the lifetime of the product is infinite within the storage time. Actually, in real life, it is not possible to have a product with an infinite lifetime. Therefore, Aggarwal and Jaggi (1995) modified this condition and extended the Goyal (1985)’s inventory model for the case of deteriorating goods. Examples of some recent papers in this line of research are: Jaggi and Mittal (2012), Chung (2012), Chung (2013), Chung and Cárdenas-Barrón (2013), Taleizadeh et al. (2013), Chen et al. (2014), Chung et al. (2014), Wu et al. (2014), Bhunia and Shaikh (2015), Shah and Cárdenas-Barrón (2015), Bhunia et al. (2015), Bhunia et al. (2016), among others. Tiwari et al. (2017b) developed a lot-size model for defective and deteriorating items with trade credit. They considered the demand to be time-dependent. Tiwari et al. (2018a) determined the optimal pricing and lot-sizing policy for a supply chain system considering deteriorating items under limited storage capacity. Tiwari et al. (2018b) proposed a pricing and inventory model for deteriorating items. They incorporated expiration dates and partial backlogging under two-level partial trade credit in their inventory model. This paper addresses an inventory model with nonlinear holding cost and nonlinear stockdependent demand with credit policy permitting that the ending inventory level can be positive, zero or negative (shortage occurs and is partially backlogged). Therefore, two inventory models are considered as follows: (i) inventory model with shortage and (ii) inventory model without shortage. It is important to remark that both inventory models contain the situation when the ending inventory is zero. Thus, this paper determines the optimal lot size and the ending inventory level in order to maximize the total profit considering that the supplier offers to the buyer a trade credit policy. The rest of this paper is organized as follows. Section 2 provides the assumptions and the notation required to develop the inventory models. Section 3 develops the inventory models with nonlinear holding cost and nonlinear stock-dependent demand with credit policy. Section 4 proposes some lemmas, theorems and provides an algorithm to optimize the total profit of the inventory models. Section 5 identifies some special cases that are contained in the proposed inventory models. Section 6 solves some

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numerical examples. Section 7 presents a sensitivity analysis and discusses some observations. Finally, Section 8 gives some conclusions and future research directions.

2. Assumptions and notation The following assumptions and notation are made in order to develop the mathematical expressions for inventory models. 2.1. Assumptions 1. The demand function is considered as a nonlinear stock dependent demand given by      q(t ) when q(t )  0 D(t )   when q(t )  0  

It is important to remark that the above demand function ( D(t ) ) has been used in Baker and Urban (1988), Alfares (2007), Pando et al. (2013) and Pando et al. (2018). The D(t ) expression indicates that the demand rate is an increasing function of the stock level. This means as the inventory level, q(t), drops, so, the demand rate also drops. In other words, at the start of the cycle, the stock level decays quickly due to the fact that demand is high when there is a higher level of inventory. As more stock is diminished, then the demand rate is also decreased. When the elasticity of the demand,  is zero then the demand is constant and given by  . Notice also that when the inventory level, q(t), is below than zero then the demand rate is only  . 2. The holding cost is a nonlinear function which is dependent on stock level, and it is given by

H (t )  h[q(t )]

where   0

It is worth to mention that holding cost function ( H (t ) ) has been used in Yang (2014), San-José et al. (2015), and Pando et al. (2018). The H (t ) expression implies that the holding cost per unit time is a power function with respect to stock level. This means that when the inventory level declines then the holding cost per unit time also diminishes. It is significant to remark that when the holding cost elasticity,  , is equal to one then the inventory model with constant holding cost is obtained. 3. The planning horizon of the inventory system is infinite. 4. Replenishment rate is instantaneous, and lead-time is negligible. 5. Shortages are permitted, and these are partially backlogged with a backlogging rate  . 6. Single level trade credit policy is considered. Here, the supplier/manufacturer/retailer gives a credit facility to his/her customer up to a specified period under terms and conditions well defined.

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2.2. Notation Notation

Units $/per order

Description replenishment cost per order

c s ch cb

$/unit $/unit $/unit/unit time

purchasing cost per unit selling price per unit holding cost per unit per unit time

$/unit/unit time

shortage cost per unit per unit time

cl

$/unit

lost sale cost per unit the elasticity of holding cost;   0

co





demand elasticity; 0    1 partial backlogging parameter; a fraction of the demand within the stockout period that is backlogged,   [0,1]



 q(t) units M unit time Ie %/ unit time Ip %/ unit time TP(Q, B) $/ unit time Decision variables Q Units Units B

Scale parameter of the demand rate inventory level at any time t where 0  t  T the retailer’s trade credit period provided by the supplier Interest earned by the retailer Interest paid by the retailer the total profit per unit time order quantity per cycle the end inventory level at time T

Dependent variables unit time t

time at which inventory level reaches to zero (for shortage case only)

T

the length of the replenishment cycle

1

unit time

3. The mathematical derivation of the inventory model with nonlinear stock dependent holding cost and nonlinear stock dependent demand with trade credit This section derives, in detail, the inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit. First, a brief problem description is provided as follows. The inventory model consists in that there exist Q units of products in stock at the starting of the cycle. A replenishment must be done when the inventory level on hand reaches B units, and the order quantity must be Q-B which raises the inventory level to Q units at the starting time of the next cycle. Additionally, the supplier offers a trade credit period (M) to the retailer. To derive the total profit per unit time two inventory models are considered as follows: (i) an inventory model with shortage ( B  0 ) and (ii) an inventory model without shortage ( B  0 ). These two inventory models are derived in subsection

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3.1 and subsection 3.2, respectively. In order to provide a general perspective of the two inventory models, Table 1 gives the cases that are considered and discussed for each inventory model. Table 1. Cases that occur in the inventory problem An inventory model with shortage  B  0 

An inventory model without shortage  B  0 

Case 3.1.1 Trade credit time is less than or equal to Case 3.2.1 Trade credit time is less than or equal to the cycle time ( T  M )

the cycle time ( T  M )

Case 3.1.2 Trade credit time is greater than or equal Case 3.2.2 Trade credit time is greater than or equal to the cycle time ( T  M )

to the cycle time ( T  M )

3.1. An inventory model with shortage ( B  0 ) Here, shortages are permitted, and within the stock-out time, a fraction (  ) of the demand is backlogged; and the rest fraction ( 1   ) is lost. Additionally, in the inventory model with shortage, the value of the inventory level at T is less than zero (i.e. q(T )  B  0 ). Initially, the retailer purchases an order of Q units. Then, the lot size of Q units decreases by the demand of the customer during the interval [0,T]. At t  t1 , the inventory level is zero. After that, shortage appears with a backlogging rate of

 and the

inventory level starts to drop below zero. The inventory situation can be described easily by some differential equations at any time t over the period [0, T]. Hence, the stock level is governed by the following differential equations which are given by:

dq  t  dt

dq  t  dt

   q(t ) , 

0  t  t1

(1)

t1  t  T

  ,

(2) Solving

the

differential

equation

(1)

and

equation

(2)

with

the

boundary

conditions

q(0)  Q, q(T )  B  0 , the following results are obtained: 1

q  t   Q(1  )  (1   ) t  (1  )

,

0  t  t1

(3)

q  t   B   (T  t ) ,

t1  t

T

(4) Using continuity condition from the equation (3) and equation (4) at the point t  t1 , hence the cycle time T is determined as:

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T

Q(1  ) B   (1   ) 

(5) Also, using the condition q(t1 )  0 in equation (4) it follows that,

t1 

B



T

(6) The total profit for the inventory model with shortage is comprised of the following terms: a) The

ordering

cost

per

order

=

co

(7)

 t1   b) The inventory holding cost ( Chol )  ch    q  t   dt    0   t1

Chol

 ch  Q1   (1   ) t  0

1 

  1  ch dt   Q1   (1   ) t1 Q  (  1   ) 





 1  1 

  

Substituting t1 from equation (6) into the above expression and simplifying, thus, the holding cost becomes to

Chol

chQ 1    (  1   )

(8)

c) The purchasing cost (PC)  c(Q  B) d) The

sale

revenue

during

(9) the

period

(SR)

 s(Q  B)

(10) T

e) The shortage cost ( Csho ) during the inventory cycle  cb

  q(t ) dt t1

   Csho  cb   B   (T  t )dt  cb  B(T  t1 )  (T  t1 ) 2  2   t1 T

Substituting T and t1 from equation (5) and equation (6) respectively into the above expression, hence, the shortage cost reduces to Csho 

cb B 2 2

(11)

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T



f) The opportunity cost ( OCls ) due to the lost sale during the inventory cycle  cl (1   ) dt t1

OCls  cl (1   ) (T  t1 ) Substituting T and t1 from equation (5) and equation (6) respectively into the above expression, therefore, the opportunity cost transforms in

OCls  

cl (1   ) B

(12)



Due to the fact that trade credit policy is a regular business practice in many companies nowadays, thus it is relevant to explore the effect of the trade credit policy on the current inventory problem. According to the trade credit policy, the supplier/manufacturer offers a specific credit time-period (M) to his/her retailer. Comparing M and T the following cases occur: trade credit time is less than or equal to the cycle time ( T

 M ) and trade credit time is greater than or equal to the cycle time ( T  M ). These two

cases are discussed in detail below. Case 3.1.1: Trade credit time is less than or equal to the cycle time ( T

M)

In this case, the supplier offers a credit period ( M ) to his/her retailer, but this credit period is less than or equal to the inventory cycle length. It is obvious that after credit time M, the retailer faces interest charges and consequently must pay the interest during the interval time [ M , T ] . Therefore, the interest paid (IP) is calculated with

T  cI p   q(t )dt  , which reduces to M  2   (1  ) 1  Q  (1   )  M    IP  cI p   (2   )  

    

(13)

Due to credit time period the retailer earns interest up to time t  M . Thus, the interest earned (IE) is M t

determined as follows sI e

    q(u)



dudt , and it results in

0 0

2    1   1 1   Q 2   IE  sI e QM  Q  (1   )  M      (2   )   

(14) Therefore, the total profit per unit time, TP1  Q, B  , is written as

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TP1  Q, B  

X1 T

(15)

where X1 = +- < ordering cost >-- < holding cost > --- Thus, X1  SR  IE  co  c(Q  B)  Chol  Csho  OCls  IP

Problem-1:

(16)

X1   T  T  M 

Maximize TP1  Q , B   subject to

(17)

Case 3.1.2. Trade credit time is greater than or equal to the cycle time ( T

M)

Notice that in this case, the inventory cycle length (T) is less than or equal to the retailer credit period (M). So, in this situation, the retailer must not need to pay interest. On the other hand, the retailer earns the interest due to his credit period offered up to time t  M . Obviously, the interest paid is equal to zero ( IP  0 ). And the interest earned (IE) is computed with t1 t

 sI e     q(u ) dudt  M  t1  which results in 0 0

  Q 2  Q1   IE  sI e  M    (1   )    (1   )(2   )   (18) Consequently, the total profit per unit time is given by

TP2  Q, B  

X2 T

(19)

where X2 = +- < ordering cost >-- < holding cost > --

X 2  SR  IE  co  c(Q  B)  Chol  Csho  OCls

Hence, (20) Problem-2:

X2   T  T M  

Maximize TP2  Q, B   subject to

(21)

3.2 An inventory model without shortage ( B  0 )

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In this case, the inventory level thru time is always greater than or equal to zero. Note that the ending inventory level at T is positive or zero (i.e. q(t )  B  0 ). Therefore, in this case, the shortage cannot occur. The inventory level drops due to demand that occurs during the interval [0, T]. Thus, the inventory behavior at any time t over the period [0, T] is modeled by the following differential equation.

dq  t  dt

   q(t ) , 

0t T

(22)

Solving the differential equation (22) with the boundary condition q(0)  Q , which results in 1

q  t   Q(1  )  (1   ) t  (1  ) ,

0t T

(23)

Using the condition q(T )  B then the cycle time T is given by:

T

Q(1  )  B (1  )  (1   )

(24)

The total profit per unit time is comprised of the following terms: a) The ordering cost per order = co

(25)

T   b) The inventory holding cost ( Chol )  ch    q  t   dt  0   T

Chol  ch  Q1   (1   ) t  0

1 

dt 

  1  ch  Q1   (1   )T Q  (  1   ) 





 1  1 

  

Placing the value of T given in equation (24) into the above expression, consequently, the holding cost is

 Q 1   B 1   Chol  ch     (  1   ) 

(26)

c) The purchasing cost (PC)  c(Q  B)

(27)

d) The sales revenue during the period (SR)  s(Q  B)

(28)

As it was previously mentioned, in trade credit policy the supplier gives a certain credit time-period (M) to his retailer. Consequently, the following two cases occur: trade credit time is less than or equal to the cycle time ( T

 M ) and trade credit time is greater than or equal to the cycle time ( T  M ). These two

cases are described below.

Case 3.2.1. Trade credit time is less than or equal to the cycle time ( T

M)

12

In this case, the credit period (M) is less than or equal to the inventory cycle length (T). According to this case, the retailer must pay interest to the supplier during the interval time [ M , T ] .

T  As a result, the interest paid (IP) is determined by cI p   q(t )dt  . Hence, M  2   (1  ) 1   B 2   Q  (1   )  M   IP  cI p   (2   )  

    

(29)

As it is known, during the credit time-period the retailer earns the interest up to time t  M . M t

Hence, the interest earned (IE) is obtained with sI e

    q(u)



dudt . Therefore,

0 0

2    1   1 1   Q 2   IE  sI e QM  Q  (1   )  M     (2   )   

(30)

Thus, the total profit per unit time is expressed as

TP3  Q, B  

X3 T

(31)

where X3 = +- < ordering cost >-- < holding cost > - Then,

it

is

equivalent

to

X 3  SR  IE  co  c(Q  B)  Chol  IP

(32) Problem-3:

X3   T  T M  

Maximize TP3  Q, B   subject to

(33)

Case 3.2.2. Trade credit time is greater than or equal to the cycle time ( T

M)

In this case, the inventory cycle length (T) is less than or equal to the retailer’s credit period (M). Thus, the retailer does not need to pay interest. On the contrary, the retailer earns interest due to credit period up to time t  M . It is easy to see that the interest paid (IP) is zero. Thus,

IP  0

(34) T t

The interest earned is calculated with sI e

    q(u)



dudt  M  T  and it results in

0 0

 Q 2  QB (1  ) B 2   IE  sI e     M  T    (1   )(2   ) (1   ) (2   ) 

(35)

13

Therefore, the total profit per unit time of the inventory cycle is given by

TP4  Q, B  

X4 T

(36)

where X4 = +- < ordering cost >-- < holding cost > -

X 4  SR  IE  co  c(Q  B)  Chol  IP

Hence, (37) Problem-4:

X4   T  T M  

Maximize TP4  Q, B   subject to

(38)

4. Theoretical results and optimization procedure It is important to say that the functions (15), (19), (31) and (36) are highly nonlinear optimization problems which are complex and difficult to optimize. Due to this fact, it is not easy to determine the closed-form solution for the decisions variables. Additionally, the concavity property of the total profit per unit time cannot be demonstrated mathematically. Therefore, the optimization of the total profit per unit time is done thru a search algorithm. 4.1. An inventory model with shortage ( B  0 ) This subsection derives the optimal solution for Q and B that maximizes the total profit per unit time in equations (15) and (19). The optimization procedure is as follows. The equations (15) and (19) are written as    sIe  cI p  Q1  (1   ) M  (2(1 ))  s  c  Q  B      (2   )     TP1  Q, B   (1  )   (   1   ) 2 2   Q B   chQ cb B cl (1   ) B  Q  c     (1   )      sI e QM    (2   )  o  (  1   ) 2     

(39)

  sI eQ 2  Q1    M   s  c  Q  B     (1   )(2   )   (1   )     TP2  Q, B     Q (1  ) B   chQ ( 1  ) c B 2 c (1   ) B   b  l  (1   )     co   (  1   ) 2     

(40)

Differentiating partially equations (39) and (40) with respect B, hence

14

  sI e  cI p  Q1  (1   ) M  (2(1 ))   s  c  Q    (2   )      (  1  ) 2  =0 2  TP1  Q, B   cQ cB    sI e QM  Q   co  h  b    2 1   B  (2   )   (  1   ) 2   Q  B    (1   )     1  cb B      Q  ( s  c)   cl (1   )   (1   )       

(41)

and   sI eQ 2  Q1    M  s  c  Q     (1   )(2   )   (1   )       =0 TP2  Q, B  c Q ( 1  ) c B2   co  h    b 2 B  (  1   ) 2   Q1  B    (1   )     Q1  cb B        ( s  c )    c (1   )   l  (1   )    

(42)

Now, differentiating partially of equations (39) and (40) with respect Q, thus    sI e  cI p  Q1  (1   ) M  (11 )    s  c  Q    1     B   Q          Q  c Q    (1   )      sI e  MQ     h          (2   )  sI e  cI p  1   TP1  Q, B  Q   (1   )     s  c  Q  B   2   Q  (1   ) M   1  Q  (2   )  Q B    (1   )     (  1  ) 2 2   chQ cb B cl (1   ) B     sI QM  Q c     e   (2   )  o  (  1   ) 2          

=0

(43) and   sI e  Q1  Q 2     M    s  c  Q      1   (1   )   (1   )  (2   )    Q B         c Q   (1   )      h       2   1  TP2  Q, B    sI eQ Q Q    s  c  Q  B     M   c o   2  Q  (1   )(2   )   (1   )   Q1  B    (1   )     (  1  ) 2 cb B cl (1   ) B      chQ   (  1   )  2           

=0

(44)

15

It is well known that the necessary conditions to find the optimal value for B and Q are

TP1  Q, B   0, B

TP1  Q, B  TP2  Q, B  TP2  Q, B   0,  0,  0. Q B Q Unfortunately, with these conditions, it is not possible to find the closed form for Q and B. Therefore, a search procedure is required to obtain the value for Q and B jointly. For convenience, let   sI e  cI p  Q1  (1   ) M  (2(1 ))   s  c  Q    (2   )     (  1  ) 2 2    c Q c B Q h b  H1 ( B)    sI e QM    co   (  1   )  2    (2   )     1  cb B    Q  ( s  c)   cl (1   )   (1   )       

(45) Using the equation (45) and putting B=0, the following result is obtained   sI e  cI p  Q1  (1   ) M  (2(1 ))   s  c  Q    (2   )      (  1  ) 2    c Q Q h  1  H1 (0)    sI e QM   c     (2   )  o  (  1   )    1   Q   (1   ) ( s  c)  cl (1   )   

(46) Now, similarly, let   sI eQ 2  Q1    M   s  c  Q    (1   )(2   )   (1   )      c Q ( 1  ) c B2  H 2 ( B)   co  h  b  (  1   ) 2     1  cb B    Q   ( s  c )    c (1   )   l  (1   )    

(47) Using the equation (47) and substituting B=0, thus

16

   sI eQ 2  Q1   s  c Q  M         (1   )(2   )   (1   )    2  H 2 (0)   ( 1  ) 1    cQ Q  co  h  (s  c)  cl (1   )  (  1   ) (1   )   (48) Considering the above theoretical results, equations (45)-(48) then the following lemma is proposed.

*

*

Lemma 1. For any given value of Q1 & Q2

(i). If 1  0, then the solution for Case 3.1.1 (say B1 ) that satisfies equation (41) not only exists but it is *

also unique. (ii). If  2  0, then the solution for Case 3.1.2 (say B2 ) that satisfies equation (42) not only exists but it *

is also unique. (iii). If both 1 ,  2  0 then the solution that satisfies equation (41) and equation (42) does not exist. As a result, B=0. Proof: The detailed proof of Lemma 1 is given in Appendix A1. Using Lemma 1 the following theorem is proposed. Theorem 1. For any given value of Qi* , the optimal value for Bi* that optimizes the total profit per unit time TPi  Qi , Bi   i  1, 2 is:

  * B   

B1* , for Case3.1.1,if 1  0 B2* , for Case3.1.2,if  2  0 0, for both cases when 1 ,  2  0

Proof: The detailed proof of Theorem 2 is shown in Appendix A2. Now, it is necessary to determine the optimal value for Q in order to maximize the total profit per unit time TP  Q, B  . Let

17

   sI e  cI p  Q1  (1   ) M  (11 )    s  c  Q    1    B   Q       Q  c Q    (1   )      sI e  MQ     h          (2   ) sI e  cI p  1     (1   ) Q  (1   ) M   3     s  c  Q  B     (2   )      chQ ( 1  ) cb B 2 cl (1   )B  Q 2   c      sI e QM    (2   )  o  (  1   ) 2          

and   sI e  Q1  Q 2     M     s  c  Q      1   (1   )   (1   )  (2   )    Q B        c Q    (1   )      h       2  1    sI Q Q e   4     s  c  Q  B   M   c    (1   )(2   )   (1   )  o   (  1  ) 2 cb B cl (1   ) B   chQ    (  1   )  2           

Taking into account  3 and  4 the Lemma 2 is established as follows. Lemma 2. For a given value of B1* & B2* (i). If 3  0, then the solution for Case 3.1.1 (say Q1* ) that satisfies equation (43) not only exists but it is also unique. (ii). If  4  0, then the solution for Case 3.1.2 (say Q2* ) that satisfies equation (44) not only exists but it is also unique (iii). If both  3 ,  4  0 then the solution that satisfies equation (43) and equation (44) does not exist. Proof: The proof of Lemma 2 is presented in Appendix A3. By using Lemma 2 the following theorem is stated. Theorem 2. For a given value for Bi* , then an optimal value for Qi* that optimizes the total profit

TPi  Qi , Bi   i  1, 2 is:

18

 Q1* , for Case3.1.1,if 3  0  Q*   Q2* , for Case3.1.2,if  4  0  does not exist for both cases,if  ,   0 3 4  Proof: The proof of Theorem 2 is in Appendix A4. 4.2. An inventory model without shortage ( B  0 ) This subsection determines the optimal solution for Q and B which maximizes the total profit per unit time is given in equations (31) and (36). The optimization process is explained as follows. The equations (31) and (36) are expressed as    sIe  cI p  Q1  (1   ) M  (2(1 ))  s  c  Q  B       (2   ) (1   )   TP3  Q, B   1  1   2   (   1   ) (   1   ) 2   Q  B   c [Q B ] cI p B   sI QM  Q   co  h  e      (2   )  (   1   )  (2   )  

(49)

and    s  c  Q  B     2  1  2  1  1      Q  B   (1   ) Q QB B  TP4  Q, B   1   sI   M      e   Q  B1      (1   )(2   )  (1   )  (2   )     (1   )     (  1  )  B ( 1  ) ]  c  ch [Q   o   (  1   )

(50)

Taking the partial derivatives of equations (49) and (50) with respect B, the following results are obtained

TP3  Q, B  (1   ) B    2 B  Q1  B1 

    sIe  cI p  Q1  (1   ) M  (2(1 ))  ( s  c)(Q  B)      (2   )     (1   )  2  2  (  1  ) (  1  ) cI B  Q  =0 Q B   p   sI e  QM   (2   )    (2   )  co  ch   (  1   )           cI B     c B p 1  1    h     Q  B  ( s  c ) B        

(51)

and

19

TP4  Q, B  (1   ) B    2 B  Q1  B1 

   Q 2  QB1  B 2        ( s  c)(Q  B)  sI e       (1   )(2   )  (1   )  (2   )    (1   )    1  1  (  1  )  B ( 1  )   M  (Q  B )   c  c  Q      o h     (1   )  (   1   )        1  1     (Q  B )   Q B   ( s  c) B  sI e     M      (1   )          Q1   B1       Q 2  QB1  B 2   ch B   sI e          (1   )(2   )  (1   )  (2   )        

=0

(52) Taking the partial derivatives of equations (49) and (50) with respect Q, then the results are

TP3  Q, B  (1   )Q    2 Q  Q1  B1 

1   ( sI e  cI p ) 1     Q  (1   ) M  1     ( s  c)Q     Q1   B1       1      c Q Q  h  sI Q M      =0   e           (2   )   sI e  cI p  1    (1   ) Q  (1   ) M   s  c  Q  B      (2   )    (1   )   2  (  1  ) (  1  ) 2   cI B   ch [Q B ] Q   p     sI e QM   (2   )   co   (  1   )  (2   )     

(53)

and

TP4  Q, B  (1   )Q    2 Q  Q1  B1 

1      Q1   B1     B1      Q   ( s  c)Q  sI eQ   M      1     (1   )     (1   )   1     Q  B     2  1  2   chQ  Q QB B  sI e       (1   )(2   )   (1   )   (2   )       =0            s  c  Q  B          Q1   B1    Q 2  QB1  B 2     (1   )  sI e      M    (1   )      (1   )(2   )  (1   )  (2   )          (  1  ) (  1  ) B ]  co  ch [Q    (   1   )    

The necessary conditions to find the optimal value of B and Q are

(54)

TP3  Q, B   0, B

TP3  Q, B  TP4  Q, B  TP4  Q, B   0,  0,  0. Q B Q Notice that with these conditions it is not possible to derive the closed form for Q and B. Thus, it is required to apply a search algorithm in order to obtain the optimal values for Q and B simultaneously. Let

20

    sI e  cI p  Q1  (1   ) M  (2(1 ))  ( s  c)(Q  B)     (2   )      (1   )  2   2  (  1  ) (  1  ) cI B     Q Q  B   p H 5 ( B)    sI e  QM   (2   )    (2   )  co  ch   (  1   )           cI p B ch B    1  1         Q  B  ( s  c ) B        

(55)

By substituting B=0 into the equation (55), it reduces    sIe  cI p  Q1  (1   ) M  (2(1 ))   ( s  c)Q    (2   )     5  H 5 (0)  (1   )   (  1  ) 2   chQ Q     sI e  QM   co       (2   )   (  1   )    

(56)

Now, let    Q 2  QB1  B 2        ( s  c)(Q  B)  sI e       (1   )(2   )  (1   )  (2   )     (1   )  1  1  (  1  ) ( 1  )    ( Q  B ) Q  B     M   (1   )   co  ch   (  1   )         H 6 ( B)    1  1     (Q  B )   Q B   ( s  c) B  sI e     M      (1   )          Q1   B1       sI e  Q 2  QB1  B 2   ch B        (1   )(2   )   (1   )   (2   )         

   chQ ( 1  )  Q 2  Q1     (1   ) ( s  c)Q  sI e  M    co   (1   )(2   )   (1   )   (  1   )      6  H 6 (0)     sI eQ 2  Q Q1      Q1     sI e  M          (1   )  (1   )(2   )        

(57)

(58)

Taking into consideration the above theoretical results, equations (55) to (58), thus the following lemma is established. * * Lemma 3. For any given value of Q3 & Q4 * (i). If 5  0, then the solution for Case 3.2.1 (say B3 ) that satisfies equation (51) not only exists but it is

also unique. * (ii). If  6  0, then the solution for Case 3.2.2 (say B4 ) that satisfies equation (52) not only exists but it is

also unique. (iii). If both 5 , 6  0 then the solution that satisfies equation (51) and (52) does not exist. As a result, B=0.

21

Proof: The proof of Lemma 3 is shown in the Appendix A5. From Lemma 3, the following theorem is given. Theorem 3. For any given value of Qi* , the optimal value of Bi* which optimizes the total profit

TPi  Qi , Bi   i  3, 4 is:  B3* , for Case3.2.1,if 5  0  B*   B4* , for Case3.2.2,if  6  0  0, for both cases when  ,   0 5 6  Proof: The proof of Theorem 3 is in Appendix A6. To find the value of Q that optimizes the total profit

TP  Q, B  is necessary to let

1   ( sI e  cI p ) 1     Q  (1   ) M  1     ( s  c)Q     Q1   B1       1      c Q Q  sI Q  M    h     e         7   (2   )   sI e  cI p  1    (1   ) Q  (1   ) M   s  c  Q  B      (2   )     (1   )   2  (  1  ) ( 1  ) 2   cI B   c [ Q  B ] Q   p h     sI e QM   (2   )   co   (  1   )  (2   )     

and 1      Q1   B1     B1        Q ( s  c ) Q  sI Q M          e    1     (1   )     (1   )   1     Q  B     2  1  2   chQ  Q QB B  sI e           (1   )(2   )  (1   )  (2   )               8    s  c  Q  B      2  1  2  1  1           Q QB B Q  B   (1   )   sI   M   e        (1   )(2   )  (1   )  (2   )     (1   )        (  1  ) (  1  ) c [ Q  B ]  h co     (   1   )     

Considering the above-mentioned two expressions, the following lemma is proposed. Lemma 4. For a given value of B3* & B4*

22

(i). If  7  0, then the solution for Case 3.2.1 (say Q3* ) that satisfies equation (53) not only exists, but it is also unique. (ii). If 8  0, then the solution for Case 3.2.2 (say Q4* ) that satisfies equation (54) not only exists, but it is also unique. (iii). If both 7 , 8  0 then the solution that satisfies equation (53) and equation (54) does not exist. Proof: The proof is exposed in Appendix A7 The Lemma 4 motivates to the following theorem. Theorem 4. For a given value of Bi* , the optimal solution to Qi* that optimizes the total profit per unit time TPi  Qi , Bi   i  3, 4 is:

 Q3* , for Case3.2.1,if  7  0  Q*   Q4* , for Case3.2.2,if 8  0 does not exist for both cases,if  ,   0 7 8  Proof: The proof is presented in Appendix A8.

4.3. Solution procedure In order to determine the optimal solution for the lot size (Q) and ending inventory level (B) an iterative algorithm is built using the theoretical results (Lemmas and Theorems). Algorithm Step 1. Input all parameters value and set  =0.000001. Step 2. Star with j=0 and set the initial values for Q1 j & Q2 j with any positive value greater than 1 for Case 3.1.1 and Case 3.1.2 in equation (45) and (47) respectively. * * Step 3. For a given value of Q1 j & Q2 j calculate the value for 1 j &  2 j respectively.

Step 4. * * (i). If 1 j  0, set B1 j  B1 , by solving equation (41) * * (ii). If  2 j  0, set B2 j  B2 , by solving equation (42)

(iii). If 1 j ,  2 j  0, go to Step 15. * * Step 5. For given values of B1 j & B2 j compute 3 j &  4 j respectively.

Step 6.

23

(i). If 3 j  0, set Q1*j  Q1* , by solving equation (43) (ii). If  4 j  0, set Q2* j  Q2* , by solving equation (44) (iii). If 3 j ,  4 j  0, go to Step 15. Step

7.

Calculate

the

difference

Q1*j 1  Q1 j & Q2* j 1  Q2 j .

between

If

Q1*j 1  Q1 j   & Q2* j 1  Q2 j   then set Q1*  Q1*j ; B1*  B1*j & Q2*  Q2* j ; B2*  B2* j as the optimal solution for Case 3.1.1 and Case 3.1.2, respectively. Otherwise, set j=j+1 and go to Step 3. Step 8. Begin with j=0 and initialize the starting values for Q3 j & Q4 j with any positive value greater than 1 for Case 3.2.1 and Case 3.2.2 in equations (55) and (57) respectively. Step 9. For a given value of Q3*j & Q*4 j determine the value of 5 j & 6 j respectively. Step 10. (i). If 5 j  0, set B3* j  B3* , by solving equation (51) (ii). If  6 j  0, set B4* j  B4* , by solving equation (52) (iii). If 5 j , 6 j  0, go to Step 15. * * Step 11. Using the value of B3 j & B4 j for Case 3.2.1 and Case 3.2.2, calculate 7 j & 8 j

Step 12. (i). If  7 j  0, set Q3*j  Q3* , by solving equation (53) * * (ii). If 8 j  0 set Q4 j  Q4 , by solving equation (54)

(iii). If both 7 j , 8 j  0 go to Step 15. Step

13.

Compute

the

difference

Q3* j 1  Q3 j and Q4* j 1  Q4 j .

between

If

Q3* j 1  Q3 j   and Q4* j 1  Q4 j   then set Q3*  Q3* j ; B3*  B3* j & Q4*  Q4* j ; B4*  B4* j as the optimal solution for Case 3.1.1 and Case 3.1.2, respectively. Otherwise, set j=j+1 and go to Step 9.





Step 14. Compute the maximum total profit TPj (Q j , B j )  Max TPi (Q j , B j ) for Case 3.1.1 and Case *

*

*

*

*

i 1,2

3.1.2

of

the

inventory

model

with

shortage;

calculate

the

maximum

total

profit

TPj* (Q*j , B*j )  Max TPi (Q*j , B*j ) for Case 3.2.1 and Case 3.2.2 of inventory model without shortage and i 3,4

select the optimal solution that corresponds to the maximum total profit.

24

Step 15. Stop.

5. Special cases The proposed inventory models contains the following inventory models: (i). When the trade credit is not considered (i.e. M = 0) as a result it reduces to the Yang (2014)’s inventory model. (ii). If the trade credit is not included (i.e. M = 0) and the ending inventory level is imposed to be zero thus it becomes to Pando et al. (2012a)’s inventory model. (iii). When the trade credit is not taken into account (i.e. M = 0), the ending inventory level is constrained to be zero,  =0 and  =1 therefore it converts to the classical Harris (1913)’s EOQ inventory model. (iv). When the trade credit is not involved (i.e. M=0), the ending inventory level is restricted to be zero,

 =0 and  >1 hence it turns into Naddor (1982)’s inventory model in p.107 and Chikán (1990)’s inventory model in p. 123. (v). If the trade credit is not incorporated (i.e. M = 0), the ending inventory level is equal to zero and s=c then it fits to Goh (1994)’s inventory model. (vi). When the trade credit is not considered (i.e. M = 0), the ending inventory level is forced to be zero, s>c and  =1 consequently it reduces to Baker and Urban (1988)’s inventory model without shortage.

6. Numerical examples The purpose of this section is to present the solutions to some numerical examples. These examples are solved by applying the algorithm developed in Section 4.3. Example 1. This example illustrates the inventory model when B<0. Consider the following values for the input parameters.

co  $50 / order , s  $70 / unit , c  $50 / unit , ch  $0.5 / unit / year ,   1.2,   0.1,   1,   0.8, I p  12% / year , I e  9% / year , M  180 / 365 year, cb  $20 / unit / year, cl  $10 / unit. The maximum optimal total profit corresponds to Case 3.1.1 of the inventory model with shortage. The optimal solution is given below.

TP(Q* , B* )  5.171458, Q*  3.099273, B*  4.414271, T *  8.593157, t1*  3.075318, 1  51.50789  0 & 3  0. Example 2. This example shows the inventory model when B>0. The values for the input parameters are as follows:

25

co  $20 / order , s  $70 / unit , c  $50 / unit , ch  $0.5 / unit / year ,   0.9,   0.4,   1, I p  12% / year , I e  9% / year , M  60 / 365 year. The maximum optimal total profit occurs in Case 3.2.1 of the inventory model without shortage. The optimal solution is as follows: TP(Q* , B* )  7.597666, Q*  5.070562, B*  0.1123477, T *  3.965570, 5  19.35  0 and 7  0.

Example 3. This example shows the inventory model when B=0. The values for the input parameters are given below.

co  $20 / order , s  $70 / unit , c  $55 / unit , ch  $0.5 / unit / year ,   1.2,   0.3,   1, I p  12% / year , I e  9% / year , M  180 / 365 year. The optimal solution is expressed below. TP(Q* , B* )  2.457996, Q*  3.255263, B*  0,T *  3.263715.

7. Sensitivity analysis In this section the numerical example 1 is used to study the impact of under or overestimation of input parameters on the optimal values of the lot size (Q), ending inventory level (B), time when the stock reaches to zero (t1), cycle length (T) and the total profit per unit time (TP(Q, B)) of the inventory system. The percentage changes in the above mentioned optimal values are taken as measures of sensitivity. The sensitivity analysis is carried out by changing (increasing and decreasing) the parameters by -20% to +20%. The results are determined by changing one parameter at a time and keeping the other parameters at their original values. The results of these analyses are given in Table 2. Table 2. Sensitivity analysis for example 1 Parameters



co

ch

% change in

%change parameters

TP* (Q*, B*)

Q*

B*

t1*

T*

-20

-42.63

-15.73

-6.74

7.15

13.20

-10

-21.76

-7.81

-3.11

3.27

6.08

10

22.54

7.72

2.65

-2.80

-5.29

20

45.78

15.35

4.90

-5.24

-9.96

-20

28.23

-6.75

-13.96

-6.09

-11.14

-10

11.56

-3.27

-6.77

-2.94

-5.40

10

-10.97

3.08

6.43

2.77

5.12

20

-21.43

6.01

12.55

5.39

9.99

-20

1.17

1.83

-0.69

1.65

0.15

26

s

c







cb

cl

M

-10

0.58

0.91

-0.34

0.81

0.07

10

-0.57

-0.89

0.33

-0.80

-0.07

20

-1.13

-1.76

0.66

-1.58

-0.14

-20

-234.49

-17.98

10.49

-16.34

0.89

-10

-117.87

-9.17

5.61

-8.29

0.64

10

119.26

9.44

-6.43

8.46

-1.10

20

240.07

19.05

-13.76

16.99

-2.75

-20

184.09

33.58

-17.22

29.77

0.40

-10

90.23

15.21

-7.55

13.59

0.02

10

-78.88

-11.48

5.43

-10.40

-0.24

20

-173.01

-23.22

10.73

-21.17

-0.68

-20

-47.33

13.21

-26.65

11.81

-1.11

-10

-24.00

6.73

-13.13

6.04

-0.07

10

24.63

-6.98

12.76

-6.30

0.65

20

49.9

-14.22

25.14

-12.9

1.87

-20

-0.80

-2.83

0.47

-2.54

-0.61

-10

-0.41

-1.44

0.24

-1.29

-0.31

10

0.42

1.50

-0.25

1.32

0.31

20

0.86

3.05

-0.51

2.67

0.63

-20

0.84

2.40

-0.49

2.15

0.46

-10

0.44

1.25

-0.26

1.13

0.24

10

-0.47

-1.36

0.28

-1.23

-0.26

20

-0.97

-2.83

0.57

-2.55

-0.55

-20

12.16

-3.44

16.09

-3.10

9.22

-10

5.76

-1.63

7.36

-1.46

4.20

10

-5.23

1.47

-6.31

1.32

-3.58

20

-10.10

2.81

-11.78

2.53

-6.66

-20

4.99

-1.41

1.61

-1.27

0.58

-10

2.49

-0.70

0.81

-0.63

0.29

10

-2.48

0.70

-0.81

0.63

-0.30

20

-4.94

1.39

-1.64

1.25

-0.60

-20

-4.09

-2.46

2.40

-2.21

0.75

-10

-2.06

-1.22

1.21

-1.10

0.38

10

2.10

1.22

-1.23

1.09

-0.40

27

20

4.23

2.42

-2.48

2.18

-0.81

Based on the results shown in Table 2, the following observations are stated: (i). It is visible that with the increase in the value of the scale parameter of the demand rate   , the





optimal order quantity  Q  , the ending inventory level  B  and the total profit TP* (Q*, B*) increase. Because when the demand at retailer’s end rises, then he or she has to order a large quantity in order to fulfill the demand of his/her customers. Which ultimately boost the sale revenue of retailer, so as his/her total profit. It is clear that with the increment in the value of   , the time at which inventory level attains zero  t1  and cycle length T  decrease. (ii). It can be observed that with the increase in the value of replenishment cost  co  , the order quantity

 Q  , the ending inventory level  B  , the time at which inventory level reaches zero  t1  and cycle





time T  increase but the total profit TP* (Q*, B*) decrease. It can be seen that when the holding cost

 ch  increases, the order quantity  Q  , the time at which inventory level achieves zero  t1  , the cycle





length (T) and the total profit TP* (Q*, B*) decrease. This way it also becomes cost-effective for the retailer as the inventory holding cost gets diminished by fast replenishment of the items. It is easy to see that with an increment in the value of  ch  , the ending inventory level  B  rises. (iii). As the selling price  s  increases, the order quantity  Q  , the time at which inventory level is zero

 t1  ,and





the total profit TP* (Q*, B*) increase but the ending inventory level  B  and the cycle

length (T) decrease. When the purchasing cost  c  increases then the order quantity  Q  , the time at



which inventory level is equal to zero  t1  , the cycle length (T) and the total profit TP* (Q*, B*)



decrease, but the ending inventory level  B  increases. This is quite an apparent result. (iv). With the increment in the value of the partial backlogging parameter   , the total profit

TP* (Q*, B*)  , the ending inventory level  B  and the cycle length (T) increase but the order quantity  Q  and the time at which inventory level is equal to zero  t1  decrease.

28

(v). As the value of demand elasticity    increases, on the one hand, the order quantity  Q  , the time at





which inventory level reaches zero  t1  , cycle time T  and the total profit TP* (Q*, B*) increment, and on the other hand, the ending inventory level  B  decreases. For the case of elasticity of holding cost    it gives the opposite results; this means with the increase in the value of elasticity of holding cost    , the order quantity  Q  , the time at which inventory level attains zero  t1  , cycle time T 





and the total profit TP* (Q*, B*) decrease but the ending inventory level  B  increases. (vi). It can be concluded that with an increment in the value of shortage cost  cb  and lost sale cost

 cl  , the ending inventory level  B  , the cycle length (T) and the total profit TP* (Q*, B*)  decrease but the order quantity  Q  and the time at which inventory level attains zero  t1  increase. (vii). It is visible that as the credit period of the retailer (M) increases, the value of order quantity  Q  ,





the time at which inventory level is zero  t1  and total profit TP* (Q*, B*) increase but the ending inventory level  B  and the cycle time (T) decrease. With the increase in retailer’s credit limit, he or she can put the revenue in an interest-bearing account for a longer duration and hence boost up his/her profit values considerably. To make use of the trade credit policy efficiently, the retailer orders more products.

8. Conclusion

Nowadays, with the advent of technology, the world is shrinking. Not only the flow of information but the products are highly accessible across organizations around the world. Thus, the enterprises need to expand beyond their peripheries with strategic and technological innovations in the supply chain in which these are immerse. The contemporary global markets with variable production and volatile markets force to the different stakeholders of the supply chain to harness the clout of technology for customized supply chain decisions. With the world economy growing of diverse markets, the management of supply chain issues related to stocks, trade credit and customer satisfaction is challenging because of business practices and government regulations. Within of this context, in this paper, efforts are taken to include the nonlinear stock holding cost, non-linear demand and trade credit. 29

This paper examines the retailer’s optimal strategy based on non-linear holding and stock-dependent demand of his/her product when he/she is receiving a credit period from his/her supplier. Basically, this research work builds two inventory models with nonlinear stock dependent holding cost, nonlinear stock dependent demand and single level credit policy with relaxed ending inventory level; where shortages are permitted and partially backlogged. The primary objective of both inventory models is to determine the optimal ordering quantity and the ending inventory level which maximizes the retailer’s total profit per unit time. In order to obtain the optimal solutions, lemmas, and theorems are derived along with a solution procedure.

It is normal to think that with the rapid change in market scenario there is fundamental shift from conventional supply chain models to customized ones. For this reason, the proposed inventory models consider various features of real market scenario, which helps the retailers to collaborate with their business partners. Additionally, the proposed inventory models permit to the retailers harness the business and manage the stocks effectively and efficiently. It is important to remark that the two proposed inventory models are a general framework as several previously published models are particular cases of these. Moreover, these inventory models are quite useful for retail industries, shopping, 3PL organizations, among others companies. For future research directions, the following guidelines are suggested: two-level trade credit policy, partial trade credit for credit risk customer, credit dependent demand function, inflation, fuzzy-valued inventory cost, interval valued inventory costs, among others. Additionally, it may be worth to consider an integrated inventory model with several suppliers/distributors/retailers in the supply chain. Also, an exciting extension is to generalize the proposed inventory models which are the single player solution to two players noncooperative Nash or Stackelberg solution. These are some ongoing research lines that the researchers and academicians can explore in the near future.

Appendix A1 The proof of Lemma 1 is as follows. Proof of part (i). The derivative of equation (45) with respect to B is

dH1 ( B) cb B Q1  cb   dB   (1   )  Q1  B   cb     cbT   (1   )   (A1.1)

30

From equation (A1.1), it is shown that H1(B) is a strictly decreasing function during the interval (, 0] . Also, it is observed that lim H1 ( B)   . If 1  0, according to the intermediate value theorem, there B 

exists a unique value of B (say B1) such that H1(B1)=0. Proof of part (ii). Similarly, the derivative of equation (47) with respect to B is

dH 2 ( B) cb B Q1  cb   dB   (1   )  Q1  B   cb     cbT   (1   )   (A1.2) From equation (A1.2), it is proven that H2(B) is a strictly decreasing function during the interval (, 0] . Also, it is observed that lim H 2 ( B)   . If 1  0, according to the intermediate value theorem, there B 

exists a unique value of B (say B2) such that H1(B2)=0. Proof of part (iii). If both 1 ,  2  0, and it is understood that, lim H1 ( B)  , lim H 2 ( B)   then B 

B 

there is no solution for B.

Appendix A2 In this appendix, the proof of Theorem 1 is given. Taking the derivative of equation (41) and equation (42) with respect to B, thus 

 cb B Q1  cb    2TP1 (Q, B)         2 2 B   B  B1  Q1  B     (1   )    (1   )    



 Q1  c B    cb     b  0.  2 1    T   (1   )    Q B  (1   )     

(A2.3) and,

31



 cb B Q1  cb    2TP2 (Q, B)        2  B 2   B  B2  Q1  B     (1   )   (1   )     



 Q1  c B    cb     b  0.  2 1   (1   )   T   Q B  (1   )     

(A2.4) According to Lemma 1, If 1 ,  2  0, then the total profit per unit time is maximized at the point B  B1* and B  B2* . Otherwise, the equations (41) and (42) are greater than zero. 

TP1  Q, B    H1 ( B)  0 if 1  0. 2 1  B  Q B  (1   )     

and TP2  Q, B  B



   Q1  B  (1   )     

2

H 2 ( B)  0 if  2  0.

This implies that TP1  Q, B  & TP2  Q, B  are strictly increasing functions within the interval (, 0] . So, the maximum total profit per unit time is obtained at the point B=0.

Appendix A3 Here, the proof of Lemma 2 is given. Proof of part (i). If the derivative of equation (41) is taken, hence ( sI  cI p ) d 3  B*  1 h Q 1   Q1     ( s  c)Q  1  e  sI e  ( MQ  1  )   1  dQ       (1   )  

Let

(A3.5)

d 3  0. Therefore  3 is a strictly decreasing function. Likewise, it is observed that lim 3   . If B dQ

3  0, according to intermediate value theorem there exists a unique value of Q (say Q1) such that 3 (Q1* )  0 . Proof of part (ii). The proof is similar to the proof of part (i) stated above. Proof of part (iii). If

d 3  0 and 3  0 then it is not possible to apply the intermediate value theorem. dQ

So, it does not exist a solution for Q.

32

Appendix A4 This appendix contains the proof of Theorem 2. Proof of part(i). Considering Lemma 2, it follows that   TP1  Q, B     Q 2   B  B1 2

 ( sI  cI p )    ( s  c)Q  1  e  *  1     Q B  Q      1   0  1  * 2  1  (1   )   1 h Q    Q B1    1   (1   )       sI e  ( MQ   )       





If 3  0, then the total profit per unit time TP1 Q, B1* has a maximum value at the point Q1* . Otherwise, it is not possible to find the maximum value of the total profit per unit time. Proof of part(ii). The proof is similar to the proof of part (i) stated above.

Appendix A5 This appendix presents the proof of Lemma 3. Proof of part (i). Taking the derivative of equation (55) with respect to B, then cI  dH 5 ( B) h B 1    Q1   B1     ( s  c) B  1  p   0. dB    

Therefore, the function H 5 ( B) is a strictly decreasing function. If 5  0 then according to intermediate





theorem there exists a unique B say B3* and this value maximizes the total profit per unit time. Proof of part (ii). The proof is similar to the proof of part (i) stated above Proof of part (iii). If the function H 5 ( B) is a strictly decreasing function and 5  0 then it is not possible to apply the intermediate value theorem. Therefore, H 5 ( B) has its maximum value at B=0.

Appendix A6 The proof is similar to Appendix A2. Therefore it is omitted here.

Appendix A7 The proof is similar to Appendix A3. Therefore it is omitted here.

Appendix A8 The proof is similar to Appendix A4. Therefore it is omitted here.

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33

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Acknowledgements The first and second authors were supported by the Tecnológico de Monterrey Research Group in Optimization and Data Science 0822B01006.

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This paper derives an EOQ model with nonlinear holding cost, nonlinear stock dependent demand

The inventory model considers trade credit policy and assumes the non-zero ending inventory level

Shortages are permitted and partially backlogged with a constant backlogging rate

Lemmas, theorems and an algorithm are derived to determine the optimal inventory policy

Numerical examples are presented to illustrate the applicability of the inventory model

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