An Inventory Model for Stock and Time-Dependent Demand with Cash Discount Policy Under Learning Effect and Partial Backlogging

Abstract

This study considers an inventory model with stock and time-dependent demand. Stock level always plays a very vital role and affects the demand rate. Vendors usually offer different schemes to attract more customers. In this paper, the scheme of cash discount is working as promotional tool for increasing demand rate. Shortages are allowed with partial backlogging and backlogging rate present in the model is assumed as a waiting time-dependent function. To make the study more realistic learning effect is applied on holding cost. Three cases for the allowed trade credit period are described in the present paper. To illustrate the model numerical example for different cases have been discussed by using Mathematica 11.3. sensitivity analysis with respect to distinct parameters is carried out for the feasibility and the applicability of the model.

2.1 Introduction

The role of demand is very vital while developing an inventory model, Available stock and time are the factors that always influence the demand. Khurana and Chaudhary (2016) proposed an inventory model using stock and price-dependent demand for deteriorating items under shortage backordering. Giri et al. (2017) introduced a vendor–buyer supply chain inventory model using time-dependent demand under preservation technology. Khurana and Chaudhary (2018) developed a deteriorating inventory model for stock and time-dependent with partial backlogging. Bardhan et al. (2019) introduced a non-instantaneous deteriorating inventory model for stock-dependent demand under preservation technology. Handa et al. (2020) worked on an EOQ model with stock-dependent demand for trade credit policy under shortages.

Deterioration is another important factor whose part in the construction of an inventory model is very useful. Deterioration can be defined as the reduction, or spoilage in the original value of the product. Skouri et al. (2009) developed some inventory policies under Weibull deterioration rate for ramp type demand. Chowdhury et al. (2014) formulated an inventory model for price and stock-dependent demand. Mahapatra et al. (2017) introduced a model using deteriorating items based on reliability-dependent demand under partial backlogging. Rastogi et al. (2018) developed an inventory policy for non-instantaneous deteriorating items using price-sensitive demand with partial backordering.

In the construction of an inventory model, the basic assumption is that when the stock out situation occurs then the shortages that take place are either completely lost or completely backlogged which is not realistic. At the arrival of the stock some customers are interested to come back, which is known as partial backlogging. Roy and Chaudhuri (2011) studied an inventory model using price-dependent demand, Weibull deterioration, and partial backlogging. Kumar and Singh (2014) presented a two-warehouse inventory model in which demand depends upon stock level under partial backordering. Geetha and Udayakumar (2016) formulated inventory policies for non-instantaneous deteriorating products under multivariate demand rate and partial backorder. Khanna et al. (2017) proposed an inventory model using selling price-dependent demand for imperfect items under shortage backordering and trade credit. Kumar et al. (2020) studied the effect of preservation and learning on partial backordering inventory model for deteriorating items with the environment of the Covid-19 pandemic.

In today’s competitive market the trade credit period offered by the seller has become a very useful incentive policy for attracting new customers. Singh et al. (2016) proposed an EOQ model allowing stock-dependent demand under trade credit policy. Shaikh (2017) introduced a deteriorating inventory model based on advertisement and price-dependent demand using partial backlogging and mixed type of trade credit. Tripathi et al. (2018) studied an inventory model for time-varying holding cost with stock dependent demand having different. Shaikh et al. (2019) introduce a Weibull distributed deteriorating inventory model allowing multivariate demand rate and trade credit period.

Learning is a realistic phenomenon that occurs naturally. Generally, it is seen that when workers accomplish the same procedure repeatedly then they learn how to performs more efficiently such phenomenon is called learning effect. Singh et al. (2013) presented an inventory model for imperfect products under the effect of inflation and learning. Singh and Rathore (2016) formulated a reverse logistic inventory model with preservation and inflation under learning effect. Goyal et al. (2017) proposed an EOQ model using advertisement-based demand under learning effect and partial backorder. Singh et al. (2020) introduced a reverse logistic inventory model for variable production under learning effect.

This paper represents an inventory model considering stock and time-dependent demand, cash discount, and partial backlogging. To make the study more realistic learning effect is applied on holding cost. Different cases for the allowed trade credit period are also described in the model. To improve the efficiency of the model numerical example for different cases and sensitivity analysis for distinct value of parameters have been discussed.

2.2 Assumptions

• 1. Demand used in the model is a function of stock and time i.e. $(\delta + \beta t + \gamma E_{1} (t))$.

• 2. Items used in the model are of decaying nature.

• 3. No replacement policy is allowed for deteriorating products in whole cycle period.

• 4. Shortages are considered with partial backlogging.

• 5. Deteriorating rate is constant.

• 6. Backlogging rate present in the model is assumed as a waiting time-dependent function.

• 7. This model incorporates the effect of learning on holding cost.

• 8. Trade credit period is allowed in the model.

2.3 Notations

Notations used in the model.

$E(t)$:

level of inventory at any time t

$\delta ,\beta ,\gamma$:

coefficients of demand

$Q_{1}$:

initial stock level

$Q_{2}$:

backorder quantity during stock out

$k$:

rate of deterioration

$\phi (\eta )$:

rate of backlogging

$\eta$:

waiting time up to next arrival lot

$T$:

cycle time

$u_{1}$:

time at which level of inventory becomes zero

$h_{f} + \frac{{h_{g} }}{{n^{\lambda } }}$:

per unit holding cost under learning effect where $\lambda > 0$

$s_{r}$:

shortage cost per unit

$d$:

per unit deterioration cost

$l_{r}$:

per unit lost sale cost

$c$:

purchasing cost per unit

$A$:

per order ordering cost

$p$:

selling price per unit

$U.T.P_{x} .$:

unit time profit

$M$:

allowed trade credit period

$I_{c}$:

rate of interest charged

$I_{e}$:

rate of interest earned

y:

rate of cash discount.

2.4 Mathematical Modelling

Figure 2.1 represents the behavior of inventory system with respect to time. $Q_{1}$ denotes the initial inventory level at $t = 0$. Level of inventory depletes in the interval $[0,u_{1} ]$ for the reason of deterioration and demand. At $t = u_{1}$, inventory level turns into zero, and after that shortages occur with partial backlogging. The depletion of the inventory is shown in Fig. 2.1

Fig. 2.1

Inventory time graph of system

Differential equations of the inventory system can be represented as follows

$$\frac{{{\text{d}}E_{1} }}{{{\text{d}}t}} + kE_{1} = - \left( {\delta + \beta t + \gamma E_{1} (t)} \right)\quad 0 \le t \le u_{1}$$

(2.1)

$$\frac{{{\text{d}}E_{2} }}{{{\text{d}}t}} = - \left( {\delta + \beta t} \right)\quad u_{1} \le t \le T$$

(2.2)

Boundary equations are given as follows:

$$E_{1} (u_{1} ) = E_{2} (u_{1} ) = 0$$

Solution of Eqs. (2.1) and (2.2) are given by

$$\begin{aligned} E_{1} (t) & = \left[ {\delta \left( {u_{1} - t} \right) + \frac{\beta }{2}\left( {u_{1}^{2} - t^{2} } \right) + \left( {k + \gamma } \right)\left\{ {\frac{\delta }{2}} \right.\left( {u_{1}^{2} - t^{2} } \right)} \right. \\ & \quad \left. {\left. { + \frac{\beta }{2}\left( {u_{1}^{3} - t^{3} } \right)} \right\}} \right]{\text{e}}^{{ - (k + \gamma )t}} \quad 0 \le t \le u_{1} \\ \end{aligned}$$

(2.3)

$$E_{2} (t) = \left[ {\delta \left( {u_{1} - t} \right) + \frac{\beta }{2}\left( {u_{1}^{2} - t^{2} } \right)} \right]\quad u_{1} \le t \le T$$

(2.4)

2.5 Associated Costs

Ordering Cost:

Ordering cost per order for the system is taken as follows:

$$O.C_{x} . = A$$

(2.5)

Purchasing Cost:

$Q_{1}$ denotes the initial inventory level at $t = 0$ and $Q_{2}$ for the duration $[u_{1} ,T]$.

$$E_{1} (0) = Q_{1} = \left\{ {\delta u_{1} + \beta \frac{{u_{1}^{2} }}{2} + \left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{2} }}{3}} \right)} \right\}$$

(2.6)

$$Q_{2} = \int\limits_{{u_{1} }}^{T} {\left( {\delta + \beta t} \right)\phi (\eta ){\text{d}}t}$$

(2.7)

$$= \left\{ {\frac{\delta }{2}\left( {T^{2} - u_{1}^{2} } \right) + \frac{\beta }{3}\left( {T^{3} - u_{1}^{3} } \right)} \right\}$$

(2.8)

$$P.C_{x} . = \left\{ {Q_{1} + Q_{2} } \right\}c$$

(2.9)

Hence, the purchasing cost of the system is given by

$$P.C_{x} = \left\{ {\delta u_{1} + \beta \frac{{u_{1}^{2} }}{2} + \left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{2} }}{3}} \right) + \frac{\delta }{2}\left( {T^{2} - u_{1}^{2} } \right) + \frac{\beta }{3}\left( {T^{3} - u_{1}^{3} } \right)} \right\}c$$

(2.10)

Sales Revenue:

Sales revenue can be taken as follows:

$$S.R_{x} . = \left( {Q_{1} + Q_{2} } \right)p$$

(2.11)

$$S.R_{x} . = \left\{ {\delta u_{1} + \beta \frac{{u_{1}^{2} }}{2} + \left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{2} }}{3}} \right) + \frac{\delta }{2}\left( {T^{2} - u_{1}^{2} } \right) + \frac{\beta }{3}\left( {T^{3} - u_{1}^{3} } \right)} \right\}p$$

(2.12)

Holding Cost:

Holding cost is considered in the duration when the system holds the inventory. Holding cost is taken as follows:

$$H.C_{x} . = \left( {h_{f} + \frac{{h_{g} }}{{n^{\lambda } }}} \right)\int\limits_{0}^{{u_{1} }} {E_{1} (t){\text{d}}t}$$

(2.13)

$$H.C_{x} . = \left( {h_{f} + \frac{{h_{g} }}{{n^{\lambda } }}} \right)\left\{ {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{3} }}{3} + \left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{3} }}{6} + \beta \frac{{u_{1}^{4} }}{8}} \right)} \right\}$$

(2.14)

Shortage Cost:

In the inventory system shortages occur during the stock out condition when goods are not available to fulfil the customers demand. Shortage cost of the system is taken as follows:

$$S.C_{x} . = s_{r} \int\limits_{{u_{1} }}^{T} {(\delta + \beta t){\text{d}}t}$$

(2.15)

$$S.C_{x} . = \left\{ {\delta \left( {T - u_{1} } \right) + \frac{\beta }{2}\left( {T^{2} - u_{1}^{2} } \right)} \right\}s_{r}$$

(2.16)

Lost Sale Cost:

In the inventory system lost sale cost is considered during the stock out condition when some customers fulfil their demand from other places. Lost sale cost is taken as follows:

$$L.S.C_{x} . = l_{r} \int\limits_{{u_{1} }}^{T} {(\delta + \beta t)(1 - \phi (\eta )){\text{d}}t}$$

(2.17)

$$L.S.C_{x} . = l_{r} \left\{ {\delta \frac{{T^{2} }}{2} + \beta \frac{{T^{3} }}{6} - \delta u_{1} T - \beta T\frac{{u_{1} ^{2} }}{2} + \delta \frac{{u_{1} ^{2} }}{2} + \beta \frac{{u_{1} ^{3} }}{3}} \right\}$$

(2.18)

Deterioration Cost:

Deterioration cost is considered for those products that are deteriorated or decayed in the system. The deterioration cost is taken as follows:

$$D.C_{x} . = d\left\{ {E_{1} (0) - \int\limits_{0}^{{u_{1} }} {(\delta + \beta t){\text{d}}t} } \right\}$$

(2.19)

$$D.C_{x} . = d\left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{3} }}{3}} \right)$$

(2.20)

2.6 Permissible Delay

Trade credit period is the useful incentive policy for attracting more customers. In this time period vendor allows a certain time limit to retailer to pay all his dues. If the retailer pays all his dues before the credit limit then there will be no interest charged otherwise interest will be charged on unpaid amount. Retailer can also earn interest on sales revenue.

Two cases for allowed trade credit period are given as follows:

Case 1: When M $\ge u_{1}$ (Fig. 2.2).

Fig. 2.2

Inventory time graph when (M $\ge u_{1}$)

For this case vendor has enough amount to settle all his payments since the credit limit period is more than the period of sold out all the stock. In this case, interest charged would be zero and interest earned in the duration [0, M] is given as follows.

$$I.V_{1} . = pI_{e} \int\limits_{0}^{{u_{1} }} {\left( {\delta + \beta t + \gamma E(t)} \right)} {\text{d}}t + \left( {M - u_{1} } \right)\int\limits_{0}^{{u_{1} }} {\left( {\delta + \beta t + \gamma E(t)} \right)} {\text{d}}t$$

(2.21)

$$\begin{aligned} pI_{e} & \left\{ {\frac{{\delta u_{1} ^{2} }}{2} + \frac{{\beta u_{1} ^{3} }}{3} - \gamma \left( {\frac{{\delta (k + \gamma ) + \beta }}{8}u_{1} ^{4} + \frac{{\delta u_{1} ^{3} }}{6} + \frac{{\beta (k + \gamma )}}{{10}}u_{1} ^{5} } \right.} \right. \\ & \quad - \frac{{\delta (k + \gamma )}}{{12}}u_{1} ^{4} - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{{15}}u_{1} ^{5} \\ & \quad \left. {\left. { - \frac{{\beta (k + \gamma )^{2} }}{9}u_{1} ^{6} } \right)} \right\} + \left\{ {\left( {M - u_{1} } \right)} \right. \\ & \quad \left( {\delta u_{1} + \frac{{\beta u_{1} ^{2} }}{2} - \gamma \left( {\frac{{\delta u_{1} ^{2} }}{2} + \frac{{\delta (k + \gamma ) + \beta }}{3}u_{1} ^{3} - \delta \left( {k + \gamma } \right)\frac{{u_{1} ^{3} }}{6}} \right.} \right. \\ & \quad \left. {\left. { - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{8}u_{1} ^{4} - \frac{{\beta (k + \gamma )^{2} }}{{10}}u_{1}^{5} } \right)} \right\} \\ \end{aligned}$$

(2.22)

And interest charged is given as follows:

$$I.C_{1} = 0$$

Case 2: When M $< u_{1}$ (Fig. 2.3)

Fig. 2.3

Inventory time graph when M $< u_{1}$

For this case, vendor has to settle all his payments before to sold out all the stock. For interest earned and interest charged two following cases take the place:

Case 2.1: When M $< u_{1}$ and

$$pD\,[0,M] + I.V_{{2.1}} \,[0,M] \ge cE\,(0){:}$$

(2.23)

For this case, vendor has enough amount to settle all his payments. Interest charged would be zero for this case, but interest would be earned in the duration [0, M].

$$I.C_{{2.1}} = 0$$

(2.24)

$$I.V_{{2.1}} = pI_{e} \int\limits_{0}^{M} {(\delta + \beta t + \gamma E(t)} )t{\text{d}}t$$

(2.25)

$$\begin{aligned} & = pI_{e} \left\{ {\frac{{\delta M^{2} }}{2} + \frac{{\beta M^{3} }}{3} - \gamma \left( {\frac{{\delta (k + \gamma ) + \beta }}{8}M^{4} } \right.} \right. \\ & \quad + \frac{{\delta M^{3} }}{6} + \frac{{\beta (k + \gamma )}}{{10}}M^{5} - \frac{{\delta (k + \gamma )}}{{12}}M^{4} \\ & \quad \left. {\left. { - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{{15}}M^{5} - \frac{{\beta (k + \gamma )^{2} }}{9}M^{6} } \right)} \right\} \\ \end{aligned}$$

(2.26)

Case 2.2: When M $< u_{1}$ and

$$pD\,[0,M] + I.V_{{2.2}} \,[0,M] < cE\,(0){:}$$

(2.27)

For this case, vendor has not enough amount to settle all his payments so interest would be charged on unpaid amount. In the duration [0, M] earned interest is given by as follows:

$$\begin{aligned} I.V_{{2.2}} & = pI_{e} \int\limits_{0}^{M} {(\delta + \beta t + \gamma E(t)} )t{\text{d}}t \\ & = pI_{e} \left\{ {\frac{{\delta M^{2} }}{2} + \frac{{\beta M^{3} }}{3} - \gamma \left( {\frac{{\delta (k + \gamma ) + \beta }}{8}M^{4} } \right.} \right. \\ & \quad + \frac{{\delta M^{3} }}{6} + \frac{{\beta (k + \gamma )}}{{10}}M^{5} - \frac{{\delta (k + \gamma )}}{{12}}M^{4} \\ & \quad \left. {\left. { - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{{15}}M^{5} - \frac{{\beta (k + \gamma )^{2} }}{9}M^{6} } \right)} \right\} \\ \end{aligned}$$

(2.28)

Interest charged on unpaid amount is given by as follows:

$$I.C_{{2.2}} = B.I_{c}$$

(2.29)

$$B = cE_{1} (0) - \left\{ {pD\,[0,M] + I.V_{{2.2}} \,[0,M]} \right\}$$

(2.30)

$$\begin{aligned} & = \left\{ {\left[ {c\left( {\delta u_{1} + \frac{{\beta u_{1} ^{2} }}{2} + \left( {k + \gamma } \right)\left( {\frac{{\delta u_{1} ^{2} }}{2} + \frac{{\beta u_{1} ^{3} }}{3}} \right)} \right.} \right] - p\left[ {\delta M + \frac{{\beta M^{2} }}{2}} \right.} \right. \\ & \quad - \gamma \left( {\frac{{\delta M^{2} }}{2} + \frac{{\delta (k + \gamma ) + \beta }}{3}M^{3} + \frac{{\beta (k + c)}}{4}M^{4} - \delta \left( {k + \gamma } \right)\frac{{M^{3} }}{6}} \right. \\ & \quad \left. {\left. {\left. { - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{8}M^{4} } \right) - \frac{{\beta (k + \gamma )^{2} }}{{10}}M^{5} } \right)} \right] \\ & \quad - pI_{e} \left[ {\frac{{\delta M^{2} }}{2} + \frac{{\beta M^{3} }}{3} - \gamma \left( {\frac{{\delta (k + \gamma ) + \beta }}{8}M^{4} } \right.} \right. \\ & \quad + \frac{{\delta M^{3} }}{6} + \frac{{\beta (k + \gamma )}}{{10}}M^{5} - \frac{{\delta (k + \gamma )}}{{12}}M^{4} \\ & \quad \left. {\left. {\left. { - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{{15}}M^{5} - \frac{{\beta (k + \gamma )^{2} }}{9}M^{6} } \right)} \right]} \right\} \\ \end{aligned}$$

Case 3: When cash discount facility is given:

For this case, retailer provides cash discount at a rate of y% to settle all his dues at the arrival of the stock. Interest earn would be

$$I.V_{3} = pI_{e} \int\limits_{0}^{T} {(\delta + \beta t + \gamma E(t)} ){\text{d}}t$$

(2.31)

$$\begin{aligned} pI_{e} & \left\{ {\delta T + \frac{{\beta T^{2} }}{2} - c\left( {\frac{{\delta u_{1} ^{2} }}{2} + \frac{{\delta (k + \gamma ) + \beta }}{3}u_{1} ^{3} + \frac{{\beta (k + \gamma )}}{4}u_{1} ^{4} } \right.} \right. \\ & \quad \left. {\left. { - \delta \left( {k + \gamma } \right)\frac{{u_{1} ^{3} }}{6} - \frac{{(\delta (k + \gamma )^{2} + \beta (k + \gamma ))}}{8}u_{1} ^{4} } \right)} \right\} \\ \end{aligned}$$

Purchasing cost for this case would be

$$\begin{aligned} P.C_{x} & = \left\{ {\delta u_{1} + \beta \frac{{u_{1}^{2} }}{2} + \left( {k + \gamma } \right)\left( {\delta \frac{{u_{1}^{2} }}{2} + \beta \frac{{u_{1}^{2} }}{3}} \right)} \right. \\ & \quad \left. { + \frac{\delta }{2}\left( {T^{2} - u_{1}^{2} } \right) + \frac{\beta }{3}\left( {T^{3} - u_{1}^{3} } \right)} \right\}c\left( {1 - \frac{y}{{100}}} \right) \\ \end{aligned}$$

(2.32)

2.7 Unit Time Profit

Unit time profit for the system is given by as follows:

$$\begin{aligned} U.T.P_{x} & = \frac{1}{T}\left\{ {S.R_{x} . - P.C_{x} . - H.C_{x} . - D.C_{x} .} \right. \\ & \quad \left. { - L.S.C_{x} . - S.C_{x} . - O.C_{x} . - I.C. + I.V} \right\} \\ \end{aligned}$$

(2.33)

2.8 Numerical Example

Case 1: When M $\ge u_{1}$

A = 300 per/order, c = 22 Rs/unit, d = 21, $k$ = 0.001, T = 28 days, M = 22 days, $\delta$ = 300 units, $\beta$ = 0.1, $\gamma$ = 0.01, $l_{r}$ = 7 Rs/unit, $s_{r}$ = 5 Rs/unit, $h_{f}$ = 0.22 Rs/unit, $h_{g}$ = 0.15 Rs/unit, p = 30 Rs/unit, $I_{e}$ = 0.02, n = 2, $\lambda$ = 0.1.

After solving Eq. (2.33) with the help of corresponding parameters optimal value of $u_{1}$ = 19.6461 days and $U.T.P_{x}$ = 2077.92 Rs. and optimal ordered quantity $Q =$ 8702.17 units.

The behavior of the system for $U.T.P_{x} .$ is given by Figs. 2.4 and 2.5 with the help of Mathematica 11.3.

Fig. 2.4

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$ and $Q$

Fig. 2.5

Behavior of $U.T.P_{x} .$ with rest to $u_{1}$

Case 2.1: When M $< u_{1}$ and

$$pD\,[0,M] + IV_{{2.1}} \,[0,M] \ge cE\,(0){:}$$

A = 300 per/order, c = 22 Rs/unit, d = 21, $k$ = 0.001, T = 28 days, M = 17 days, $\delta$ = 300 units, $\beta$ = 0.1, $\gamma$ = 0.01, $l_{r}$ = 7 Rs/unit, $s_{r}$ = 5 Rs/unit, $h_{f}$ = 0.22 Rs/unit, $h_{g}$ = 0.15 Rs/unit, p = 30 Rs/unit, $I_{e}$ = 0.02, n = 2, $\lambda$ = 0.1.

After solving Eq. (2.33) with the help of corresponding parameters optimal value of $u_{1}$ = 18.5963 days and $U.T.P_{x}$ = 1533.71 Rs. and optimal ordered quantity $Q =$ 8535.0 units.

The behavior of the system for $U.T.P_{x} .$ is given by Figs. 2.6 and 2.7 with the help of Mathematica 11.3.

Fig. 2.6

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$ and $Q$

Fig. 2.7

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$

Case 2.2: When M $< u_{1}$ and

$$pD\,[0,M] + IV_{{2.2}} \,[0,M] < cE\,(0){:}$$

A = 300 per/order, c = 22 Rs/unit, d = 21, $k$ = 0.001, T = 28 days, M = 17 days, $\delta$ = 300 units, $\beta$ = 0.1, $\gamma$ = 0.01, $l_{r}$ = 7 Rs/unit, $s_{r}$ = 5 Rs/unit, $h_{f}$ = 0.22 Rs/unit, $h_{g}$ = 0.15 Rs/unit, p = 30 Rs/unit, $I_{e}$ = 0.02, n = 2, $\lambda$ = 0.1, $I_{c}$ = 0.016.

After solving Eq. (2.33) with the help of corresponding parameters optimal value of $u_{1}$ = 18.5961 days and $U.T.P_{x}$ = 1627.54 Rs. and optimal ordered quantity $Q =$ 8534.97 units.

The behavior of the system for $U.T.P_{x} .$ is given by Figs. 2.8 and 2.9 with the help of Mathematica 11.3.

Fig. 2.8

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$ and $Q$

Fig. 2.9

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$

Case 3: When cash discount facility is given:

A = 300 per/order, c = 22 Rs/unit, d = 21, $k$ = 0.001, T = 28 days, $\delta$ = 300 units, $\beta$ = 0.1, $\gamma$ = 0.01, $l_{r}$ = 7 Rs/unit, $s_{r}$ = 5 Rs/unit, $h_{f}$ = 0.22 Rs/unit, $h_{g}$ = 0.15 Rs/unit, p = 30 Rs/unit, $I_{e}$ = 0.02, n = 2, $\lambda$ = 0.1, y = 0.02.

After solving this model with the help of corresponding parameters optimal value of $u_{1}$ = 18.4906 days and $U.T.P_{x}$ = 826.307 Rs. and optimal ordered quantity $Q =$ 8517.72 units.

The behavior of the system for $U.T.P_{x} .$ is given by Figs. 2.10 and 2.11 with the help of Mathematica 11.3.

Fig. 2.10

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$ and $Q$

Fig. 2.11

Behavior of $U.T.P_{x} .$ with respect to $u_{1}$

2.9 Sensitivity Analysis

Sensitivity analysis for distinct parameters are specified as follows:

Case 1: When M $\ge u_{1}$ (Tables 2.1, 2.2, 2.3, 2.4, 2.5 and 2.6).

Table 2.1 Variation in optimal solution for demand parameter ($\delta$)

Table 2.2 Variation in optimal solution for shortage parameter ($s_{r}$)

Table 2.3 Variation in optimal solution for lost sale cost parameter ($l_{r}$)

Table 2.4 Variation in optimal solution for deterioration cost parameter (d)

Table 2.5 Variation in optimal solution for deterioration parameter ($k$)

Table 2.6 Variation in optimal solution for interest earned parameter ($I_{e}$)

Case 2: When M $< u_{1}$ (Tables 2.7, 2.8, 2.9, 2.10, 2.11 and 2.12).

Table 2.7 Variation in optimal solution for demand parameter ($\delta$)

Table 2.8 Variation in optimal solution for lost sale cost parameter ($l_{r}$)

Table 2.9 Variation in optimal solution for deterioration cost parameter (d)

Table 2.10 Variation in optimal solution for shortage cost parameter ($s_{r}$)

Table 2.11 Variation in optimal solution for deterioration parameter ($k$)

Table 2.12 Variation in optimal solution for interest earned parameter ($I_{e}$)

2.10 Observations

• Tables 2.1 and 2.7 represent the effect of a on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in a, value of $u_{1}$ in both the tables remain unaffected while some decrement in $U.T.P_{r}$ in Table 2.1 and some increment in $U.T.P_{r}$ in Table 2.7 are detected.

• Tables 2.2 and 2.10 represent the effect of $s_{r}$ on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in $s_{r}$, some increment in $u_{1}$ and some decrement in $U.T.P_{r}$ in both the tables are detected.

• Tables 2.3 and 2.8 represent the effect of $l_{r}$ on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in $l_{r}$, some increment in $u_{1}$ and some decrement in $U.T.P_{r}$ in both the tables are detected.

• Tables 2.4 and 2.9 represent the effect of d on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in d, some decrement in $u_{1}$ and $U.T.P_{r}$ in both the tables are detected.

• Tables 2.5 and 2.11 represent the effect of k on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in k, some increment in $u_{1}$ and $U.T.P_{r}$ in Table 2.5 while some decrement in $u_{1}$ and $U.T.P_{r}$ in Table 2.11 are detected.

• Tables 2.6 and 2.12 represent the effect of $I_{e}$ on $u_{1}$ and on $U.T.P_{r}$, it is observed that after an increment in $I_{e}$, value of $u_{1}$ remains unaffected in Table 2.12 while some increment in $u_{1}$ in Table 2.6 and $U.T.P_{r}$ in both the tables are detected.

2.11 Conclusions

Present paper is concerned with inventory policies for variable demand under some real-life situations like cash discount and learning effect. Shortages are also allowed with partial backlogging and backlogging rate present in the model is assumed as a waiting time-dependent function. All these facts together make this study very unique and straight forward. To improve the efficiency of the model numerical examples for different cases and sensitivity analysis for distinct value of parameters have been discussed with the help of Mathematica 11.3. This Model further can be modified for different demands, deterioration, and more cases of backlogging rate. Also, can be extended for different realistic approaches such as inflationary environment and preservation technology.