Introduction

The trade-credit policy plays a vital role in supply chain management. The existence of trade-credit financing in inventory control problems should not be overlooked in economic order quantity models. In practice, deterioration is also an important factor for cost/profit analysis of the inventory analysis. Ghare and Schrader [6], was the first who introduced a constant deterioration rate on inventory control problem. This model was extended by Covert and Philip [4] considering two-parameter Weibull distribution deterioration rate. In this continuation, we may refer some related articles on deterioration done by Chang and Dye [2], Min et al. [15], Wang and Jiang [34] and Tiwari et al. [33], among others.

Thereafter, Dave and Patel [5] introduced an inventory model with time dependent demand, deterioration and shortages. After a few years’ later, Sachan [18] extended the model of Dave and Patel [5] incorporating shortage. Giri et al. [7] discussed an inventory model for ramp-type demand rate including time-dependent deterioration rate. Skouri et al. [26] improved this model by considering general ramp type demand rate and Weibull distributed deterioration together with partial backlogging. An EOQ inventory model having time-varying deterioration and partial backlogging was developed by Sana [20]. Taleizadeh [27] developed an EOQ model for evaporating items with partial back ordering and advance payments scheme. In this line of research works, an inventory model with time dependent demand and deterioration rate through finite replenishment rate was discussed by Sarkar [21]. Moreover, Sarkar [22] discussed a production-inventory model with three types of continuously disseminated deterioration functions. In this continuation, few related research works done by by Taleizadeh and Pentico [28], Wee [35], Hariga [9], Wee and Law [36], Moon et al. [16], Chung and Wee [3], Sana [19], Shaikh et al. [25], Mashud et al. [13] and Tiwari et al. [32] are referred to the readers.

In trade-credit financing, the pioneering work in inventory literature was done by Goyal [8]. Goyal [8] assumed that the supplier only could offer the trade-credit period but the retailer did not has opportunity to offer any credit-period to his/her customer. This circumstance is identified as single-level trade-credit or one-level-trade-credit policy where each stage contains a member and shares only two stages of the chain. Perceptibly, it would be more interesting to study if the retailer had the power to offer the trade-credit to his/her customer. This singularity is known as two-levels of trade-credit policy where the chain integrates with three stages and one member belongs to each stage. [10] explored the EOQ inventory model under two-levels of trade-credit policy in this extension when the supplier provided to the retailer a permissible delay and then consequently the retailer also offered to its customer a permissible delay. Later, a new factor deterioration is introduced and extended Goyal’s [8] model by Aggarwal and Jaggi [1]. Thereafter, Teng and Goyal [30] proposed an optimal ordering policies for a retailer in a supply chain with consideration of upstream and down-stream trade-credits. Considering deterioration as an important factor, Liao [12] presented an inventory model with two-level trade-credit for non-instantaneous deteriorating items. Teng and Chang [29] investigated two-level trade credit facilities in production inventory model. Min et al. [14] proposed an inventory model for deteriorating items with stock-dependent demand rate considering two-levels of trade-credit policy. Wu et al. [37] added deterioration with expiration dates and developed an inventory model under two-level of trade-credit policy. Mukherjee and Mahata [17] also developed an inventory model considering trade-credit policy and deterioration. Some supplementary related researches on trade-credit policy are Sarkar et al. [23], Sarkar et al. [24], Ting [31] etc. A comparative study is given in Table 1.

Table 1 Related research works with price-dependent demand, shortage and trade-credit
Full size table

This paper investigates an inventory model for a deteriorating item considering two-level of trade-credit policy, different price-dependent demand and fully backlogged shortage. Here, the demand functions of the customers are (i) exponential function of selling price and (ii) negative power of selling price. The deterioration is considered on-hand inventory of the product in the retailer’s warehouse. The shortages are permitted and are fully backlogged. The average profit function is then formulated and maximized analytically as well as numerically. We consider two numerical examples and prove the concavity of each objective function graphically to describe and validate the inventory model. Sensitivity analysis is carried out to study the effect of key parameters on optimal solution of the objective function which are realistic in practice.

Assumption and Notations

The proposed model in this paper is based on the subsequent assumptions.

  1. 1.

    The rate of replenishment is instantaneous and the lead-time is negligible.

  2. 2.

    In this paper, two different price-sensitive demand is considered as follows

    1. (i)

      \( D(p) = ae^{{\left( {\frac{ - p}{\alpha }} \right)}} , \ a > 0,\alpha > 1 \)

    2. (ii)

      \( D(p) = ap^{ - \alpha } ,\,a > 0,\alpha > 1 \)

  3. 3.

    The infinite planning horizon is considered.

  4. 4.

    The shortage is permitted and fully backlogged.

Notations

The following notations are used throughout the paper.

Notations

Description

c o

Replenishment cost per order

c p

Purchasing cost per unit

c h

Holding cost per unit per unit time

c b

Shortage cost per unit per unit time

\( \theta \)

Deterioration rate

S

Maximum stock per cycle

R

Maximum shortage level

I e

Interest which can be earned per $ per year

by retailer

I p

Interest charges per $ in stocks per year by the

supplier

M

The retailer’s trade-credit period offered by

supplier in years

N

The supplier’s trade-credit period offered by

retailer in years

I(t)

Inventory level at any time t where \( 0 \le t \le T \)

\( X_{1} \)

Total inventory cost for the first case

\( X_{2} \)

Total inventory cost for the second case

TPi (p, t1, T)

The total profit per unit time for i = 1,2

*

Optimal values of the parameters

Decision variables

t 1

Time at which the stock is zero

p

Selling price

T

Total cycle length

The Model

The proposed inventory model is developed on the basis of above-mentioned assumptions. Initially, an enterprise purchases goods (S + R) units. This stock is depleted owing to meet up the customers’ demands with products’ deterioration. At time t = t1, stock reaches at zero. Then, shortage appears which is fully backlogged. Therefore, the inventory systems are defined by the following differential equations (Fig. 1).

Fig. 1
figure 1

Inventory versus time

Full size image

Inventory Model with Respect to the Demand Function \( D = ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \, \)

$$ \frac{{dI_{1} (t)}}{dt} + \theta I(t) = - ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \quad 0 < t \le t_{1} $$
(1)
$$ \frac{{dI_{2} (t)}}{dt} = - \,ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \quad t_{1} < t \le T\,\, $$
(2)

Solving the above (1) and (2) differential equations with the boundary conditions \( I_{1} (t) = S\,at\,t = 0 \) \( \,and\,\,I_{2} (t) = - R\,at\,t = T\,and\,I(t)\,is\,continuous\,\,at\,t = t_{1} \), we have

$$ I_{1} (t) = \frac{a}{\theta }e^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \left[ {e^{{\theta (t_{1} - t)}} - 1\,} \right]\quad t_{1} < t \le T $$
(3)

Using \( I_{1} (t) = S \) at \( t = 0 \) in the above, we have

$$ S = \frac{a}{\theta }e^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \left[ {e^{{\theta t_{1} }} - 1\,} \right] $$
(4)

From Eq. (2) we have

$$ I_{2} (t) = ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \left[ {t_{1} - t} \right]\quad $$
(5)

Using \( I_{2} (t) = - R \) at \( t = T \), we have

$$ R = ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \left[ {T - t_{1} \,} \right] $$
(6)

Now, \( t_{1} \, = T - \frac{R}{a}e^{{\left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \)

Total sales revenue is

$$ \begin{aligned} SR &= p\left[ {ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} t_{1} + ae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \left[ {T - t_{1} \,} \right]} \right] \hfill \\ & = pae^{{ - \left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} T \hfill \\ \end{aligned} $$
(7)

The cost components per cycle are as follows:

  1. (a)
    $$ {\text{Ordering}}\,{\text{cost}} = {\text{C}} $$
  2. (b)
    $$ \begin{aligned} {\text{Holding}}\,{\text{cost}} & = C_{h} \left[ {\int\limits_{0}^{{t_{1} }} {I_{1} (t)dt} } \right] \\ i.e., & = \frac{{C_{h} a}}{{\theta^{2} }}e^{ - (p/\alpha )} \left[ {e^{{\theta t_{1} }} - \theta t_{1} - 1} \right] \\ \end{aligned} $$
    (8)
  3. (c)
    $$ \text{The}\,\text{purchase}\,\text{cost} = \, c_{p} (S + R) $$
  4. (d)
    $$ \begin{aligned} {\text{The}}\,{\text{shortage}}\,{\text{cost}} & = C_{b} \left[ {\int\limits_{{t_{1} }}^{T} { - I_{2} (t)dt} } \right] \\ {\text{i}}.{\text{e}}., & = \frac{1}{2}C_{b} ae^{{\left( {\frac{ - p}{\alpha }} \right)}} (T - t_{1} )^{2} \\ \end{aligned} $$
    (9)

Annual Interest Payable to Supplier

Here, there are three cases may arise where the retailer charges interest for keeping the items in stock per year. These three cases are studied separately as follows

  • Case I \( T \ge M \)

    In this case, two sub-cases may arise, (I-1). \( M \le t_{1} \le T \) and (I-2). \( t_{1} \le M \le T \).

    These are discussed separately below.

    • Subcase (I-1) According to Fig. 2, if the trade-credit period M is at the left of t1, then Eq. (6) implies that \( T \ge t_{1} = T - \frac{R}{a}e^{{\left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \ge M \).

      Fig. 2
      figure 2

      Inventory versus time

      Full size image

      Thus, the payable annual interest is

      $$ \begin{aligned} IC_{11} &= \frac{{c_{p} I_{p} }}{T}\left[ {\int\limits_{M}^{{t_{1} }} {I_{1} (t)dt} } \right] \hfill \\ & = \frac{{c_{p} I_{p} }}{T}\left[ {\frac{D}{\theta }\int\limits_{M}^{{t_{1} }} {\left[ {e^{{\theta (t_{1} - t)}} - 1\,} \right]dt} } \right] \hfill \\ & = \frac{{c_{p} I_{p} D}}{{T\theta^{2} }}\left[ {e^{{\theta (t_{1} - M)}} - \theta \left( {t_{1} - M} \right) - 1} \right] \hfill \\ \,\,\,\,\,\,\, \hfill \\ \end{aligned} $$
      (10)

    • Subcase (I-2): According to Fig. 3, if trade-credit period M is at the right of t1 then Eq. (6) implies that \( T \ge M \ge t_{1} = T - \frac{R}{a}e^{{\left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \)

      Fig. 3
      figure 3

      Inventory veruss time

      Full size image

      Thus, the annual payable interest is zero

  • Case II \( N \le T \le M \)

    The payable annual interest for this case is zero.

  • Case III \( 0 \le T \le N \)

    The payable annual interest is zero also in this case.

Interest Earn by Retailer Annually

Here, there are three cases arise where supplier charges interest per year. These three cases are as follows:

  • Case I: \( T \ge M \)

    When \( T \ge M \) occurs then there are two sub-cases which are sub-case (I-1) \( M \le t_{1} \le T \) and

    • Subcase (I-1): According to the Fig. 2, if M lies at the left of t1 then Eq. (6) indicates that \( M \le t_{1} = T - \frac{R}{a}e^{{\left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \le T \).

      Thus, the annual interest earned is

      $$ \begin{aligned} IE_{11} &= \frac{{pI_{e} }}{T}\int\limits_{N}^{{t_{1} }} {Dtdt} \hfill \\ & = \frac{{pI_{e} D}}{T}\int\limits_{N}^{{t_{1} }} {tdt} \hfill \\ & = \frac{{pI_{e} D}}{2T}\left[ {t_{1}^{2} - N^{2} } \right] \hfill \\ \end{aligned} $$
      (11)

    • Subcase (I-2): Accordingly to Fig. 3, if the trade-credit period M lies at the right side of t1 then Eq. (6) implies that \( t_{1} = T - \frac{R}{a}e^{{\left({\raise0.7ex\hbox{$p$} \!\mathord{\left/ {\vphantom {p \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}\right)}} \le M \le T \) and we have

      $$ \begin{aligned} IE_{12} &= \frac{{pI_{e} }}{T}\left[ {\int\limits_{N}^{{t_{1} }} {Dtdt + \int\limits_{N}^{{t_{1} }} {Dt_{1} dt} } } \right] \hfill \\ &= \frac{{pI_{e} D}}{T}\left[ {\frac{{\left( {t_{1}^{2} - N^{2} } \right)}}{2} + t_{1} \left( {t_{1} - N} \right)} \right] \hfill \\ \end{aligned} $$
      (12)

  • Case II: \( N \le T \le M \)

    The annual interest earn for this case is

    $$ \begin{aligned} IE_{13} &= \frac{{pI_{e} }}{T}\left[ {\int\limits_{N}^{T} {DTdt + \int\limits_{T}^{M} {DTdt} } } \right] \hfill \\ & = \frac{{pI_{e} D}}{2T}\left[ {T(T - N) + T(M - T)} \right] \hfill \\ & = \frac{{pI_{e} D}}{2}\left[ {M - N} \right]\,\, \hfill \\ \end{aligned} $$
    (13)

  • Case III \( 0 < T \le N \)

    The annual interest earn for this case is

    $$ \begin{aligned} IE_{14} &= \frac{{pI_{e} }}{T}\int\limits_{N}^{M} {DTdt} \hfill \\ & = \frac{{pI_{e} }}{T}\int\limits_{N}^{M} {ae^{{{\raise0.7ex\hbox{${ - p}$} \!\mathord{\left/ {\vphantom {{ - p} \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}}} Tdt} \hfill \\ & = pI_{e} ae^{{{\raise0.7ex\hbox{${ - p}$} \!\mathord{\left/ {\vphantom {{ - p} \alpha }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\alpha $}}}} \left[ {M - N} \right] \hfill \\ \end{aligned} $$
    (14)

Therefore, \( Total\,inventory\,\cos t,\,X_{1} \) = <ordering cost >+<purchase cost >+<holding cost >+<shortage cost >+<interest payable per cycle >-< interest earn per cycle>

Total profit = \( Total \, sales \, revenue,SR - Total\,inventory\,cost,\,X_{1} \)

Therefore, the constrained optimization problem of this proposed model is as follows:

Problem 1

Maximize

$$ TP_{1} (p,t_{1} ,\,T) = \frac{1}{T}\left\{ \begin{aligned} &TP_{11} (p,t_{1} ,\,T),\quad when\,T \ge t_{1} \hfill \\ &TP_{12} (p,t_{1} ,\,T)\,\quad when\,\,t_{1} \le M \le T \hfill \\ &TP_{13} (p,t_{1} ,\,T),\quad when\,N \le T \le M \hfill \\ &TP_{14} (p,t_{1},\,T),\quad when\,0 < T \le N \hfill \\ \end{aligned} \right. $$
(15)

Subject to \( o \le t_{1} \le T \)

Where,

$$ \begin{aligned} TP_{11} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{1} } \right] - \left[ {IC_{11} - IE_{11} } \right]\,\,\, \hfill \\ TP_{12} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{1} } \right] + \left[ {IE_{12} } \right] \hfill \\ TP_{13} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{1} } \right] + \left[ {IE_{13} } \right] \hfill \\ TP_{14} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{1} } \right] + \left[ {IE_{14} } \right] \hfill \\ \end{aligned} $$

Inventory Model with Respect to the Demand Function \( D = ap^{ - \alpha } \, \)

The governing differential equation are as follows:

$$ \frac{{dI_{1} (t)}}{dt} + \theta I(t) = - ap^{ - \alpha }\quad 0 < t \le t_{1} \,\,\, $$
(16)
$$ \frac{{dI_{2} (t)}}{dt} = - \,ap^{ - \alpha }\quad t_{1} < t \le T\, $$
(17)

From Eq. (16), we have

$$ I_{1} (t) = \frac{a}{\theta }p^{ - \alpha } \left[ {e^{{\theta (t_{1} - t)}} - 1} \right]\quad t_{1} < t \le T $$
(18)

Using \( I_{1} (t) = 0\,at\,t = t_{1} \,\,and \)\( I_{1} (t) = S \) at \( t = 0 \) conditions, one can obtain

$$ S = \frac{a}{\theta }p^{ - \alpha } (e^{{\theta t_{1} }} - 1) $$
(19)

From Eq. (17), we have

$$ I_{2} (t) = ap^{ - \alpha } (t_{1} - t)\quad t_{1} < t \le T $$
(20)

Using \( I_{2} (t) = 0\,at\,t = t_{1} \,\,and \) \( I_{2} (t) = - R \) at \( t = T \), we have

$$ R = - ap^{ - \alpha } (t_{1} - T) $$
(21)

This implies

$$ t_{1} = T - \frac{R}{a}p^{\alpha } $$
(22)

Total sales revenue is

$$ \begin{aligned} SR &= p\left[ {ap^{ - \alpha } t_{1} + ap^{ - \alpha } \left[ {T - t_{1} \,} \right]} \right] \hfill \\ &= ap^{ - \alpha + 1} T \hfill \\ \end{aligned} $$
(23)

The cost factors per cycle are as follows:

  1. (a)
    $$ {\text{Ordering}}\,{\text{cost}} = {\text{C}} $$
  2. (b)
    $$ \begin{aligned} {\text{Holding}}\,{\text{cost}} & = C_{h} \left[ {\int\limits_{0}^{{t_{1} }} {I_{1} (t)dt} } \right] \\ {\text{i}}.{\text{e}}., & = \frac{{C_{h} a}}{{\theta^{2} }}p^{ - \alpha } \left[ {e^{{\theta t_{1} }} - \theta t_{1} - 1} \right] \\ \end{aligned} $$
    (24)
  3. (c)
    $$ \text{The}\,\text{purchase}\,\text{cost} = \, c_{p} (S + R) $$
  4. (d)
    $$ \begin{aligned} {\text{The}}\,{\text{shortage}}\,{\text{cost}} & = C_{b} \left[ {\int\limits_{{t_{1} }}^{T} { - I_{2} (t)dt} } \right] \\ {\text{i}}.{\text{e}}. & = \frac{1}{2}C_{b} ap^{ - \alpha } (T - t_{1} )^{2} \\ \end{aligned} $$
    (25)

Annual Interest Payable to Supplier

Here, there are three cases to occur in interest charged for the items kept in stock per year. These three cases are examined separately as follows

  • Case I \( T \ge M \)

    There are two sub-cases such as (I-1) \( M \le t_{1} \le T \) and (I-2) \( t_{1} \le M \le T \). We discuss them accordingly as below.

    • Subcase (I-1) As per Fig. 2, if M is situated at the left side of t1 then Eq. (22) provides \( T \ge t_{1} = T - \frac{R}{a}p^{\alpha } \ge M \).

      Therefore, the annual interest payable is

      $$ \begin{aligned} IC_{21} &= \frac{{c_{p} I_{p} }}{T}\left[ {\int\limits_{M}^{{t_{1} }} {I_{1} (t)dt} } \right] \hfill \\ & = \frac{{c_{p} I_{p} }}{T}\left[ {\frac{{ap^{ - \alpha } }}{\theta }\int\limits_{M}^{{t_{1} }} {(e^{{\theta (t_{1} - t)}} - 1)dt} } \right] \hfill \\ & = \frac{{c_{p} I_{p} D}}{{T\theta^{2} }}\left[ {e^{{\theta (t_{1} - M)}} - \theta \left( {t_{1} - M} \right) - 1} \right] \hfill \\ \,\,\,\,\,\,\, \hfill \\ \end{aligned} $$
      (26)

    • Subcase (I-2) As per Fig. 3 if M is located at the right side of t1 as shown in the figure, then Eq. (22) implies that \( T \ge M \ge t_{1} = T - \frac{R}{a}p^{\alpha } \)

      Subsequently, the annual interest payable is zero

  • Case II \( N \le T \le M \)

    The annual interest payable is zero here.

  • Case III \( 0 < T \le N \)

    The annual interest payable is zero here also.

Annual Interest Earn by the Retailer

Here, annual interest is earned by the retailer which is offered by the supplier. In this situation, there are three cases may arise for interest earned per year. These three cases are calculated independently below.

  • Case I \( T \ge M \)

    When this condition holds then there are another two sub-cases which are sub-case (I-1) \( M \le t_{1} \le T \) and sub-case (I-2) \( t_{1} \le M \le T \). All are discussed separately as below

    • Subcase (I-1) According to the Fig. 2, if M is lies at the left side of t1 then Eq. (22) gives \( T \ge t_{1} = T - \frac{R}{a}p^{\alpha } \ge M \).

      Consequently, the annual interest earn is

      $$ \begin{aligned} IE_{21} &= \frac{{pI_{e} }}{T}\int\limits_{N}^{{t_{1} }} {Dtdt} \hfill \\ &= \frac{{pI_{e} }}{T}\int\limits_{N}^{{t_{1} }} {ap^{ - \alpha } tdt} \hfill \\ & = \frac{{pI_{e} ap^{ - \alpha } }}{2T}\left[ {t_{1}^{2} - N^{2} } \right] \hfill \\ \end{aligned} $$
      (27)

    • Subcase (I-2)

      Accordingly Fig. 3, if M is located at the right hand side of t1 then Eq. (22) infers that \( T \ge M \ge t_{1} = T - \frac{R}{a}p^{\alpha } \).

      $$ \begin{aligned} IE_{22}& = \frac{{pI_{e} }}{T}\left[ {\int\limits_{N}^{{t_{1} }} {Dtdt + \int\limits_{N}^{{t_{1} }} {Dt_{1} dt} } } \right] \hfill \\ & = \frac{{pI_{e} ap^{ - \alpha } }}{T}\left[ {\frac{{\left( {t_{1}^{2} - N^{2} } \right)}}{2} + t_{1} \left( {t_{1} - N} \right)} \right]\, \hfill \\ \end{aligned} $$
      (28)

  • Case II: \( N \le T \le M \)

    The annual interest earned for this case is

    $$ \begin{aligned} IE_{23}& = \frac{{pI_{e} }}{T}\left[ {\int\limits_{N}^{T} {DTdt + \int\limits_{T}^{M} {DTdt} } } \right] \hfill \\ & = \frac{{pI_{e} ap^{ - \alpha } }}{2T}\left[ {T(T - N) + T(M - T)} \right] \hfill \\ &= \frac{{pI_{e} ap^{ - \alpha } }}{2}\left[ {M - N} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\, \hfill \\ \end{aligned} $$
    (29)

  • Case III \( 0 < T \le N \)

    The annual interest earned for this case is

    $$ \begin{aligned} IE_{24} &= \frac{{pI_{e} }}{T}\int\limits_{N}^{M} {DTdt} \hfill \\ &= \frac{{pI_{e} }}{T}\int\limits_{N}^{M} {ap^{ - \alpha } Tdt} \hfill \\ & = pI_{e} ap^{ - \alpha } \left[ {M - N} \right] \hfill \\ \end{aligned} $$
    (30)

Therefore, \( Total\,inventory\,\cos t,\,X_{2} \) = <ordering cost >+<purchase cost >+<holding cost >+<shortage cost >+<interest payable per cycle >-< interest earn per cycle>

Therefore, the constrained optimization problem is as follows

Problem 2

Maximize

$$ TP_{2} (p,t_{1},\,T) = \frac{1}{T}\left\{ \begin{aligned}& TP_{21} (p,t_{1},\,T),\quad when\,\,\,\,T \ge t_{1} \hfill \\& TP_{22} (p,t_{1},\,T),\quad when\,\,\,\,t_{1} \le M \le T \hfill \\ &TP_{23} (p,t_{1},\,T),\quad when\,\,\,N \le T \le M \hfill \\& TP_{24} (p,t_{1},\,T),\quad when\,\,\,0 < T \le N \hfill \\ \end{aligned} \right. $$
(31)

Subject to \( o \le t_{1} \le T \)

Where,

$$ \begin{aligned} TP_{21} (p,t_{1},T) &= \frac{1}{T}\left[ {SR - X_{2} } \right] - \left[ {IC_{21} - IE_{21} } \right] \hfill \\ TP_{22} (p,t_{1},T) &= \frac{1}{T}\left[ {SR - X_{2} } \right] + \left[ {IE_{22} } \right] \hfill \\ TP_{23} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{2} } \right] + \left[ {IE_{23} } \right] \hfill \\ TP_{24} (p,t_{1} ,T) &= \frac{1}{T}\left[ {SR - X_{2} } \right] + \left[ {IE_{24} } \right] \hfill \\ \end{aligned} $$

Solution Procedure

In order to obtain the profit function, we need to calculate sales revenue, the ordering cost, the holding cost, the purchasing cost, the interest earned and the interest payable for the whole chain. The retailer’s average profit per unit time can be written as

$$ TP(p,t_{1} ,T) = \frac{1}{T}\left[ {Sales\,revenue - \left\{ \begin{aligned} & ordering\,cost + Holding\,cost + Purchase\,\,cost \hfill \\ &\quad +\, Shortage\,cost + interest\,payable \hfill \\&\quad -\, interest\,earn \hfill \\ \end{aligned} \right\}} \right] $$

The necessary and sufficient conditions to find out the optimal values are

$$ \begin{aligned} \frac{{\partial TP_{1} \left( {p,t_{1} ,T} \right)}}{\partial p} = 0,\frac{{\partial TP_{2} \left( {p,t_{1} ,T} \right)}}{\partial p} = 0,\,\,\,\,\,\frac{{\partial TP_{1} \left( {p,t_{1} ,T} \right)}}{{\partial t_{1} }} = 0,\frac{{\partial TP_{2} \left( {p,t_{1} ,T} \right)}}{{\partial t_{1} }} = 0,\,\,{\text{and}} \hfill \\ \frac{{\partial^{2} TP_{1} \left( {p,t_{1} ,T} \right)}}{{\partial p^{2} }} < 0,\frac{{\partial^{2} TP_{2} \left( {p,t_{1} ,T} \right)}}{{\partial p^{2} }} < 0,\,\frac{{\partial^{2} TP_{1} \left( {p,t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} < 0,\frac{{\partial^{2} TP_{2} \left( {p,t_{1} ,T} \right)}}{{\partial t_{1}^{2} }} < 0\, \hfill \\ \end{aligned} $$

The hessian matrix Hi of TPi at p, t1, and T should have all negative eigenvalues for concavity of TPi.

For Case 1: The average profit function \( TP_{1} (p,t_{1} ,\,T) \) of the whole system is

$$ TP_{1} (p,t_{1} ,\,T) = \frac{1}{T}\left\{ \begin{aligned} &TP_{11} (p,t_{1} ,\,T),\quad when\,T \ge t_{1} \hfill \\& TP_{12} (p,t_{1} ,\,T),\quad when\,\,t_{1} \le M \le T \hfill \\& TP_{13} (p,t_{1} ,\,T),\quad when\,N \le T \le M \hfill \\& TP_{14} (p,t_{1} ,\,T),\quad when\,0 < T \le N \hfill \\ \end{aligned} \right. $$

Total profit function \( TP_{1} (p,t_{1} ,\,T) \) has four branch functions. More accurately the maximum values of those four branch functions will be the required solution.

From overhead expression some continuity relation will be rise such as:

$$ TP_{11} (p,t_{1} ,T) = TP_{12} (p,t_{1} ,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,t_{1} \,point. $$
$$ TP_{12} (p,t_{1} ,\,T)\, = TP_{13} (p,t_{1} ,\,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,M\,{\text{point}}. $$
$$ TP_{13} (p,t_{1} ,T) = TP_{14} (p,t_{1} ,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,N\,{\text{point}}. $$

The extreme value of \( TP_{1} (p,t_{1} ,\,T) \) is determined by the local extreme points or the boundary points of \( (p,t_{1} ,\,T)\, \), exactly when one need to search for \( (p,t_{1} ,\,T)\, \) directly. One could easily checked whether \( (p,t_{1} ,\,T)\, \) is originate within the allowed range, i.e.\( T \ge t_{1} \), \( t_{1} \le M \le T \), \( N \le T \le M \), \( 0 \le T \le N \).One can select the optimal point \( (p^{*} ,t_{1}^{ * } ,\,T^{ * } )\, \) such that \( TP_{1}^{ * } (p,t_{1} ,\,T) = \hbox{max} \left\{ {TP^{ * }_{11} (p,t_{1} ,\,T),\,TP^{ * }_{12} (p,t_{1} ,\,T),TP^{ * }_{13} (p,t_{1} ,\,T),\,TP^{ * }_{14} (p,t_{1} ,\,T)} \right\} \)

To show the concavity of the profit function \( TP_{1} (p,t_{1} ,T) \) some theoretical results are develop as follows.

Theorem 1

The total profit function \( TP_{11} (p,t_{1} ,T) \) attains its maximum value with respect to \( p \), when other decision variables \( t_{1} \), T are fixed.

See “Appendix B”

Theorem 2

The total profit function \( TP_{11} (p,t_{1} ,T) \) attains its maximum value with respect to \( t_{1} \) , when other decision variables \( p,T \) are fixed.

See “Appendix B”.

Theorem 3

The total profit function \( TP_{11} (p,t_{1} ,T) \) attains its maximum value with respect to T, when other decision variables \( p,t_{1} \) are fixed.

See “Appendix B”.

For Case 2:

$$ TP_{2} (p,t_{1} ,\,T) = \frac{1}{T}\left\{ \begin{aligned} &TP_{21} (p,t_{1} ,\,T),\quad when\,\,\,\,T \ge t_{1} \hfill \\ &TP_{22} (p,t_{1} ,\,T),\quad when\,\,\,\,t_{1} \le M \le T \hfill \\ &TP_{23} (p,t_{1} ,\,T),\quad when\,\,\,N \le T \le M \hfill \\ &TP_{24} (p,t_{1} ,\,T),\quad when\,\,\,0 < T \le N \hfill \\ \end{aligned} \right. $$

Total profit function \( TP_{2} (p,t_{1} ,\,T) \) has four branch functions. More precisely the maximum values of those four branch functions will be the required solution.

$$ TP_{21} (p,t_{1} ,\,T)\, = TP_{22} (p,t_{1} ,\,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,t_{1} \,{\text{point}}. $$
$$ TP_{22} (p,t_{1} ,\,T)\, = TP_{23} (p,t_{1} ,\,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,M\,{\text{point}}. $$
$$ TP_{23} (p,t_{1} ,\,T)\, = TP_{24} (p,t_{1} ,\,T)\, {\text{is}}\,{\text{continuous}}\,{\text{at}}\,N\,{\text{point}}. $$

The extreme value of \( TP_{2} (p,t_{1} ,\,T) \) is determined by the local extreme points or the boundary points of \( (p,t_{1} ,\,T)\, \), exactly when one need to search for \( (p,t_{1} ,\,T)\, \) directly. \( (p,t_{1} ,\,T)\, \). One could easily check whether \( (p,t_{1} ,\,T)\, \) is initiate inside the legal range, i.e.\( T \ge t_{1} \), \( t_{1} \le M \le T \), \( N \le T \le M \), \( 0 \le T \le N \).One can select the optimal \( (p^{*} ,t_{1}^{ * } ,\,T^{ * } )\, \) points such that \( TP_{2}^{ * } (p,t_{1} ,\,T) = \hbox{max} \left\{ {TP^{ * }_{21} (p,t_{1} ,\,T),\,TP^{ * }_{22} (p,t_{1} ,\,T),TP^{ * }_{23} (p,t_{1} ,\,T),\,TP^{ * }_{24} (p,t_{1} ,\,T)} \right\} \)

To show the concavity of the profit function \( TP_{2} (p,t_{1} ,T) \), some theorem are develop which are as follows.

Theorem 4

The total profit function \( TP_{21} (p,t_{1} ,T) \) attains its maximum value with respect to \( p \) , when other decision variables \( t_{1} \) , T are fixed.

See “Appendix C”.

Theorem 5

The total profit function \( TP_{21} (p,t_{1} ,T) \) attains its maximum value with respect to \( t_{1} \) , when other decision variables \( p,T \) are fixed.

See “Appendix C”.

Theorem 6

The total profit function \( TP_{21} (p,t_{1} ,T) \) attains its maximum value with respect to T, when other decision variables \( p,t_{1} \) are fixed.

See “Appendix C”.

Numerical Illustrations

Here, the solution of the projected inventory model is discussed. This model is more relevant for the deteriorating product like vegetables, fruits, sweets etc. During any season the demand of this item increases linearly or exponentially with the dependencies on selling price. One of the popular example of proposed model is ice-cream. This is the practical example of the anticipated model.

To demonstrate the inventory model, some numerical examples solved. The values of the input parameters for examples considers for the model are as follows:

For Case 1:

Example 1

(\( T \ge t_{1} \))We consider the numerical values of the parameters in appropriate units as follows: Ordering costs C0 = $ 150 per order, a = 600, \( \alpha = 7 \),holding cost rate \( C_{h} = \$ \,2 \), backorder cost rate for inventory \( C_{b} = \$ \,3 \), purchase cost of different item \( C_{p} = \$ \,6 \), deterioration rate \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.09 \), interest earn rate \( I_{p} = \$ \,0.06 \), delay period of retailer \( M = 3.5\,\,years \), the supplier’s trade-credit period offered by retailer in years \( N = 0.5 \).

Example 2

(\( t_{1} \le M \le T \))We consider the numerical values of the parameters in appropriate units as follows: Ordering costs is C0 = $ 150 per order, a = 600, \( \alpha = 7 \), holding cost rate is \( C_{h} = \$ \,2 \), backorder cost rate for inventory is \( C_{b} = \$ \,3 \),purchase cost of different item is \( C_{p} = \$ \,6 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.09 \), delay period of retailer is \( M = 1.5\,{\text{years}} \), the supplier’s trade-credit period offered by retailer in years \( N = 0.85 \).

Example 3

(\( N \le T \le M \)) We consider the numerical values of the parameters in appropriate units as follows: Ordering costs is C0 = $ 150 per order, a = 600, \( \alpha = 7 \), holding cost rate is \( C_{h} = \$ \,2 \), backorder cost rate for inventory is \( C_{b} = \$ \,3 \),purchase cost of different item is \( C_{p} = \$ \,6 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.09 \), delay period of retailer is \( M = 3.5\,{\text{years}} \), the supplier’s trade-credit period offered by retailer in years \( N = 0.5 \).

Example 4

(\( 0 < T \le N \)) Now, Ordering costs is C0 = $ 150 per order, a = 600, \( \alpha = 7 \), holding cost rate is \( C_{h} = \$ \,2 \), backorder cost rate for inventory is \( C_{b} = \$ \,3 \),purchase cost of different item is \( C_{p} = \$ \,6 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.09 \), delay period of retailer is \( M = 4.5\,{\text{years}} \), the supplier’s trade-credit period offered by retailer in years \( N = 2.5 \).

For Case 2:

Example 5

(\( T \ge t_{1} \)) Here, Ordering costs is C0 = $ 250 per order, a = 400,000, \( \alpha = 2.5 \), holding cost rate is \( C_{h} = \$ \,0.5 \), backorder cost rate for inventory is \( C_{b} = \$ \,1 \),purchase cost of different item is \( C_{p} = \$ \,5 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.07 \), interest earn rate \( I_{p} = \$ \,0.06 \), delay period of retailer is \( M = 2.5\,{\text{years}} \), the supplier’s trade-credit period offered by retailer in years \( N = 0.5 \).

Example 6

(\( t_{1} \le M \le T \)) In this cases, Ordering costs is C0 = $ 250 per order, a = 400,000, \( \alpha = 2.5 \), holding cost rate is \( C_{h} = \$ \,0.5 \), backorder cost rate for inventory is \( C_{b} = \$ \,1 \),purchase cost of different item is \( C_{p} = \$ \,5 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.07 \), delay period of retailer is \( M = 2.5\,{\text{years}} \), retailer offered delay period to suppliers is in years \( N = 0.5 \) (Table 2).

Table 2 Results for each inventory model
Full size table

Example 7

(\( N \le T \le M \)) In this case, Ordering costs is C0 = $ 250 per order, a = 400,000, \( \alpha = 2.5 \), holding cost rate is \( C_{h} = \$ \,0.5 \), backorder cost rate for inventory is \( C_{b} = \$ \,1 \),purchase cost of different item is \( C_{p} = \$ \,5 \),deterioration rate is \( \theta = 0.2 \), interest charged rate \( I_{e} = \$ \,0.07 \), delay period of retailer is \( M = 4.5\,{\text{years}} \), retailer offered delay period to suppliers is in years \( N = 2.5 \).

Example 8

(\( 0 < T \le N \)) Here, Ordering costs is C0 = $ 250 per order, a = 400,000, \( \alpha = 2.5 \), holding cost rate is \( C_{h} = \$ \,0.5 \), backorder cost rate for inventory is \( C_{b} = \$ \,1 \),purchase cost of different item is \( C_{p} = \$ \,5 \),deterioration rate is \( \theta = 0.2 \), interest charged rate is \( I_{e} = \$ \,0.07 \), delay period of retailer is \( M = 2.5\,{\text{years}} \), retailer offered delay period to suppliers is in years \( N = 0.5 \).

The concavity of profit function TP1

figure a
figure b
figure c

The concavity of profit function TP2

figure d
figure e
figure f

Example 9

(For different types of Demand function)

  • Sub Case I: Considering another price-dependent demand \( D = a - bp \).

    Taking all the values same as of Example 4 except the value b = 15.

    The optimal solution is:

    $$ T^{*} = 0. 8 5 6 0 9 3 0 { },\,p^{*} = 2 2. 8 2 6 9 2 { },\,t_{1}^{*} = 0. 4 0 5 5 1 6 9 { },\,\,\,TP^{*} (T^{*} ,p^{*} ,\,t_{1}^{*} ) = 5 0 4 4. 7 7 4. $$

Similarly we can prove it for Example 8.

  • Sub Case II: Considering selling price and the advertisement reliant demand \( D = A^{\gamma } (a - bp) \).

    Taking all the values same as of Example 4 except the value \( A_{c} = 2;\gamma = 0.4; \).

    The optimal solution is:

    $$ T^{*} = 0. 7 4 5 6 8 1 2 { },\,p^{*} = 2 2. 7 8 9 8 1 { },\,t_{1}^{*} = 0. 3 5 4 1 8 3 3 { },\,\,\,TP^{*} (T^{*} ,p^{*} ,\,t_{1}^{*} ) = 6 7 1 6. 4 5 9. $$

Similarly we can prove it for Example 8.

Example 10

For different types of deterioration.

  • Sub Case I: The Weibull distribution deterioration rate is

    \( \theta (t) = jk\left( {t - \gamma } \right)^{k - 1} \). Here in this paper \( \gamma = t_{1} \, = 0\,(deterioration\,\,\,free\,\,\,time) \).

    If \( j = 0.4;\,k = 2;\,\gamma = 0 \). The optimal solution is:

    $$ T^{*} = 1. 1 0 0 0 4 4,\,p^{*} = 1 3. 1 0 0 2 3 { },\,t_{1}^{*} = 0. 2 9 2 0 1 5 6,\,\,\,TP^{*} (T^{*} ,p^{*} ,\,t_{1}^{*} ) = 6 2 6. 3 6 2 7. $$

Similarly we can prove it for Example 8.

  • Sub Case II: Considering variable deterioration rate \( \theta (t) = 0.2 + 0.1t \) and taking all the values similar as stated in Example 4, the optimal solution is

    $$ T^{*} = 1. 3 4 7 4 4 0,\,p^{*} = 1 3. 0 3 5 1 1,\,t_{1}^{*} = 0. 5 7 5 4 9 2 9,\,\,\,TP^{*} (T^{*} ,p^{*} ,t_{1}^{*} ) = 6 5 8. 5 2 8 2. $$

Similarly, we can prove it for Example 8.

Example 11

For different purposes.

  1. (i)

    When \( t_{1} = T \)(i.e. no shortages) then the projected model (both Case-1 and Case-2) will converted an inventory model with no shortages.

  2. (ii)

    If \( M = 0\,\,and\,\,N = 0 \) then this model reduces to traditional instantaneous inventory model (both Case-1 and Case-2) with fully backlogged shortages.

Sensitivity Analysis

The above numerical example is used to study the effect of inventory system parameters on the optimal values of the initial stock level, maximum shortage level, cycle length, selling price and the optimum total profit of the system. The percentage changes in the above stated optimal values are taken as processes of sensitivity. By changing the parameters (increasing and decreasing) from − 20 to + 20% the analysis is carried out. The outcomes are found by changing one parameter at a time and keeping the other parameters at their original values. The outcomes of these investigation are given in Tables 3 and 4.

Table 3 Sensitivity analysis for the demand function \( D(p) = ae^{{\left( {\frac{ - p}{\alpha }} \right)}} \) for \( 0 < T \le N \)
Full size table
Table 4 Sensitivity analysis for the demand function \( D(p) = ap^{ - \alpha } \) for \( 0 < T \le N \)
Full size table

Table 3, gives following observations.

  • The total profit is extremely delicate with respect to the purchase cost \( c_{p} \), a, trade-credit period M, \( \alpha \) and is ascetically delicate with respect to all other parameters. When the purchase cost decrease then total profit (TP) and maximum shortage (R) increases, together with selling price is also increase which is analogous to Tiwari et al. [32].

  • The utmost on-hand stock-level S is exceedingly subtle with respect to the purchase cost \( c_{p} \), a, \( \alpha ,C_{0} \) and is temperately sensitive with respect to all other parameters.

  • The peak shortage level R is vastly complex with respect to the shortage cost \( c_{b} \), a, \( \alpha ,C_{0} \) and is discreetly delicate with respect to all other parameters. When the value of holding cost increases then the amount of shortages increases which is analogous to Li et al. [11].

  • The selling price is extremely delicate with respect to the purchase cost \( c_{p} \,\,and\,\,\alpha \). It is abstemiously subtle with respect to other parameters. The control of \( c_{p} \) on p*, AP* is quite spontaneous and actually quite weighty. When the purchase cost cp is increased then it leads to a significant increase in optimal selling price p* and therefore an important decrease in demand which is parallel to Li et al. [11].

  • When the holding cost is increased then total profit of the chain is decreased and also observed that with the increase in deterioration rate lessen the profit which is analogous to Tiwari et al. [33].

  • When the constant demand rate (a) increases then the total profit increase, while as the price-dependent demand rate \( \alpha \) increases, the selling price and optimal total profit per unit time decrease which is analogous to Tiwari et al. [32].

  • The period of deterioration time \( t_{1} \) is discreetly subtle with respect to all other parameters although decidedly sensitive with respect to the holding cost \( C_{h} \), ordering cost \( C_{0} ,a,\alpha \)

  • Total cycle time \( T \) is exceedingly complex with respect to ordering cost \( C_{0} ,a\,\,and\,\,\,\alpha \) and soberly delicate with respect to all other parameters.

Some observations made from Table 4.

  • The total profit is exceedingly sensitive with respect to the purchase cost \( c_{p} \), a, trade-credit period M, \( \alpha \) and is moderately sensitive with respect to all other parameters. When purchase cost decrease then total profit (TP) and shortage (R) increases which is analogous to Tiwari et al. [32].

  • When the constant demand rate (a) increases then the total profit increase, however as the price-dependent demand rate \( \alpha \) increases as a consequence the price and the optimal total profit per unit time will decrease which is alike to Tiwari et al. [32].

  • The uppermost on-hand stock-level S is extremely sensitive with respect to the purchase cost \( c_{p} \), a, \( \alpha ,C_{0} \) and is discreetly sensitive with respect to all other parameters.

  • The maximum shortages level R is exceedingly delicate with respect to the shortage cost \( c_{b} \), a, \( \alpha ,C_{0} \) and is ascetically sensitive with respect to all other parameters. It is noted that when the holding cost is increased then the amount of shortages also increased which is analogous to Li et al. [11].

  • The selling price is extremely delicate with respect to the purchase cost \( c_{p} \,\,and\,\,\alpha \) and discreetly sensitive with respect to the other parameters. The influence of \( c_{p} \) on p*, AP* is quite instinctive and actually quite substantial. An increase in the purchase cost cp pointers to a imperative increase in p* and hence an imperative decreases in the demand which is analogous to Li et al. [11].

  • When holding cost is increased then total profit of the chain is decreased and also observed that with the increase in deterioration rate lessen the profit which is analogous to Tiwari et al. [33].

  • The time \( t_{1} \) is reasonably exquisite with respect to all other parameters and is highly sensitive with respect to holding cost \( C_{h} \), ordering cost \( C_{0} ,a,\alpha \)

  • When the interest earned rate (Ie) is higher, it results more revenue with less quantity ordered and thus the total profit drops. When a higher interest paid rate is added to the total costs, it leads to a lesser amount of profit.

  • Total cycle time \( T \) is discreetly sensitive with respect to all other parameters while highly sensitive to ordering cost \( C_{0} ,a\,\,and\,\,\,\alpha \).

Concluding Remarks

The demand function is considered exponentially with the price for the first problem in this paper. In second problem, demand function is used as a negative power of price. All these demand are considered under the trade-credit policy which has a great impact on total profit. The key decision factors/variables of the proposed model are total cycle length, optimal selling price, optimal profit and different scenarios for a trade-credit period where a retailer can invest safely for optimal profit in different types of demand. We have used Taylor series expansion to obtain the closed form of the cycle length, selling price and total profit. Finally, it is shown graphically the concavity of profit function with the help of MATLAB software and also analyzed the sensitivity of different parameters by changing one and keeping the others same, which helps to the business organization to make better managerial decisions.

For further research, one can easily extend the proposed model in numerous techniques. It can be expanded for different types of variable demand which are dependent on displayed stock-level, time and others. On the other way, this model also could be generalized by considering three level of trade-credit policy. In addition, it can be also extended in fuzzy environments.