A Two-Warehouse Ordering Policy for Non-Instantaneous Deteriorating Items with two Phase Demand Rates, Two-Tiered Pricing and Shortages Under Trade Credit Policy

A Two-Warehouse Ordering Policy for Non-Instantaneous Deteriorating Items with two Phase Demand Rates, Two-Tiered Pricing and Shortages Under Trade Credit Policy

B. Babangida, Z. Muazu, M. L. Malumfashi and T. A. Yusuf

Department of Mathematics and Statistics, Umaru Musa Yar’adua University, PMB 2025, Katsina Nigeria

DOI: https://doi.org/10.51584/IJRIAS.2024.911040

Received: 17 November 2024; Accepted: 23 November 2024; Published: 17 December 2024

ABSTRACT

The classical inventory models for non-instantaneous deteriorating items tacitly assumed that the selling price before and after deterioration sets in is the same. However, when items start decaying, the vendor might resolved to decrease the selling price in order to boost additional sales, reduces the cost of holding stock, attracts new clients and reduces lost due to deterioration. In this research, the vendor’s best refill approach for non-instantaneous decaying goods with two-phase demand rates, two-storage facilities, and shortages under an allowable payment delay has been determined. The unit selling price before deterioration sets in is greater than that after deterioration sets. Though a constant consumption rate is considered as soon as deterioration begun, the consumption rate before items starts decaying supposed to be time-dependent quadratic. Shortages are permitted depending on how long it will take before next replenishment. The model determines the best cycle time, optimal order quantity, and optimal time at which the inventory level in the owned ware-house reaches zero in order to increase the overall profit per unit of time. The solution’s existence and uniqueness are both checked by establishing the necessary and sufficient conditions. The model is validated by conducting some numerical experiments, after which sensitivity analysis is carried out which offer some managerial insights.

Keywords: Two-warehouse, Two Phase Demand Rates, Two-tiered Pricing, Shortages, Trade Credit Policy.

INTRODUCTION

Inventory models for non-instantaneous deteriorating items, in some cases, assumed that the unit selling price before and after deterioration sets in is the same. However, in real practice, the unit selling price before and after deterioration sets in differs and this assumption need to be considered in developing inventory policies for non-instantaneous deteriorating items, where the objective function is to maximize the total profit of the inventory system. Tsao and Sheen (2008) presented dynamic pricing, promotion and replenishment policies for deteriorating items when a delay in payment is permissible. Lee and Hsu (2009) developed a two warehouse production model for instantaneous deteriorating inventory items with time-dependent demands rates, a finite replenishment rate within a finite planning horizon. Tsao (2010) developed two-phase pricing and inventory decisions for deteriorating and fashion goods under permissible delay in payment. The demand rate vary with price or time, shortages are allowed and partially backlogged, and the objective function is to maximize profit. Chen and Kang (2010) developed integrated inventory models considering the two-level trade credit policy and a price negotiation scheme, in which customers’ demand is sensitive to the buyers’ price. Wang et al. (2015) proposed a dynamic pricing inventory model for non-instantaneous deteriorating items, where the objective function is to maximize the total profit per unit time, and both uniform pricing and two-stage pricing models are developed and a comparative study between dynamic and uniform pricing shows the advantage of dynamic pricing over uniform pricing. Tsao et al. (2017) developed two-level pricing and ordering policies for non-instantaneous deteriorating items with price-sensitive demand rates under permissible delay in payment. Babangida and Baraya (2021) developed an EOQ model for non-instantaneous deteriorating items with two-phase demand rates and two level pricing strategies under trade credit policy. The cost of holding items in the stock is assumed to be constant and shortages are not allowed. Pang et al. (2022) develop an inventory model for perishable items with two-stage pricing. Moreover, some related studies on inventory models with two-phase or two-period pricing strategies can be found in Dye (2012), Dye and Hsieh (2013), Sainathan (2013), Chang et al. (2015). Herbon (2015) and so on.

Most inventory models are developed based on the assumption for a single warehouse with unlimited capacity. However, this assumption is debatable in most business set up. Retailer may purchase large quantity of items at a time as result of price discount (for bulk purchase), quantity discount, fear of inflation, uncertainty in demand, stock outs and so on. These items may not be stoked in the existing storage called owned warehouse (OW) with limited capacity due to its bulkiness. The retailer may rent another store called the rented warehouse (RW). Items are moved from the rented warehouse to owned warehouse and sold. This is because the holding cost in rented warehouse is assumed to be higher than that in owned warehouse due to better preserving facility which lower deterioration rate. Hence it is more economical to consume the items of rented warehouse earlier. Chandra et al. (2017) developed ordering policies for non-instantaneous deteriorating items with price dependent demand and two storage facilities under trade credit policy, where shortages are allowed and completely backlogged and the objective function is to maximize profit. Palanivel et al. (2016) developed a two-warehouse economic order quantity model for non-instantaneously deteriorating items with stock-dependent demand under the effects of inflation and the time value of money is presented. Also in this model, shortages are allowed and partially backlogged. The backlogging rate is dependent on the waiting time for the next replenishment. The objective of this model is to minimize the total inventory cost of the retailer by finding the optimal intervals and the optimal order quantity. Udayakumar and Geetha (2018) developed an EOQ model for non-instantaneous deteriorating items with constant demand rate and two levels of storage under trade credit policy, where shortages are not allowed and the objective function is to minimize cost. Babangida and Baraya developed an inventory model for non-instantaneous deteriorating items with time dependent quadratic demand, two storage facilities and shortages under trade credit policy. The demand rate before deterioration sets in is assumed to be time dependent quadratic and that is considered as a constant after deterioration sets in. Shortages are allowed and completely backlogged, and the model determine length of time at which the inventory level reaches zero in OW, cycle length and order quantity simultaneously such that total variable cost has a minimum value.

In the classical inventory model, shortages are not allowed. However, sometimes customers’ demands cannot be fulfilled by the supplier from the current stocks, this situation is known as stock out or shortage condition. In real-life situations, stock out is unavoidable due to various uncertainties. According to Sharma (2003), allowing shortages to occur increases cycle length, spread the ordering cost over a long time and hence reducing the total variable cost. Choudhury et al. (2013) developed an inventory model for non-instantaneous deteriorating items with stock-dependent demand rate, time-varying holding cost and shortages that are completely backlogged.

However, when shortages occur, one cannot be certain that all customers are willing to wait for a backorder due to customers’ impatient and dynamic nature of human beings. When shortages occur, some customers whose needs are not critical at that time may wait for the back-orders to be fulfilled, while others may opt to buy from other sellers. Consequently, the opportunity cost due to lost sales should be considered.

For most items, such as fashionable goods, electronics, automobiles and its spare parts, photographic films, seasonal products and so on, the length of the waiting time for the next replenishment would determine whether the backlogging will be accepted or not. Therefore, the backlogging rate should be variable and depend on the waiting time for the next replenishment. That is, the longer the waiting time, the lower the backlogging rate will be and vice versa. Sarkar and Sarkar (2013) developed an inventory model for deteriorating items with stock-dependent demand rate and time-varying deterioration. Shortages are allowed and partially backlogged; the backlogging rate depends on the waiting time for the next replenishment. Dutta and Kumar (2015) discussed a time-dependent partially backlogged inventory model for deteriorating items with a time-varying demand rate and holding cost. Nath and Sen (2022) developed a partially backlogged inventory model for time-deteriorating items using penalty cost and time-dependent holding cost. Babangida and Baraya (2022) developed an EOQ model for non-instantaneous deteriorating items with two-phase demand rates, linear holding cost and time-dependent partial backlogging rate under trade credit policy. Babangida et al (2023) developed retailer’s ideal replenishment strategy for non-instantaneous decaying goods with two-phase demand rates, two storage facilities, and shortages under a permissible payment delay. Whether or not the backlog will be accepted depends on how long it will be until the next replenishment. As a result, the backlogging rate fluctuates and depends on how long it takes for the next refill.

It could be observed from the above reviews that non-instantaneous deterioration, two-phase demand rates, two-ware house, two-level pricing strategies, shortages and trade credit policy are the most realistic features to consider in developing inventory policies for items, such as electronics, automobiles, seasonal products, fashionable goods, and so on.

This research work investigates EOQ Models for non-instantaneous deteriorating items with two-phase demand rates, two-tiered pricing strategy and time dependent partial backlogging rates under trade credit policy. The demand rate before deterioration sets in is assumed to be time-dependent quadratic after which it is considered as constant up to when the inventory is completely depleted. It is also assumed that the unit selling price before and after deterioration sets in is not the same. The necessary and sufficient conditions for optimality of the solution have been established. The optimal length of time at which the inventory level reaches zero in OW, cycle length and order quantity that maximizes the total profit per unit time will be determined. Some numerical examples have been given to illustrate the theoretical results of the models. Sensitivity analysis of some parameters of the proposed models had been carried out on the decision variables and suggestions towards maximizing the total profit have also been given.

Model Description and Formulation

This section provides the model notation, assumptions, and formulation.

Notation and Assumptions

Notation

• \( \text{OC} \): Ordering cost per order.
• \( \text{PC} \): Purchasing cost per unit per year (\$/unit/year).
• \( SP_1 \): Unit selling price during the interval \([0, p_d]\).
• \( SP_2 \): Unit selling price during the interval \([p_d, P]\), where \( SP_1 > SP_2 > C \).
• \( C_b \): Shortage cost per unit per unit of time.
• \( hc_o \): Holding cost per unit per unit time in own warehouse (\$/unit/year).
• \( hc_r \): Holding cost per unit per unit time in rented warehouse (\$/unit/year).
• \( I_c \): Interest charged by the supplier per dollar per year (\$/unit/year).
• \( I_e \): Interest earned per dollar per year (\$/unit/year), where \( I_c \geq I_e \).
• \( T \): Trade credit period (year).
• \( \omega_o \): Constant deterioration rate in own warehouse, \( 0 < \omega_o < 1 \).
• \( \omega_r \): Constant deterioration rate in rented warehouse, \( 0 < \omega_r < 1, \omega_r < \omega_o \).
• \( p_d \): Time where the product exhibits no deterioration.
• \( p_r \): Time at which the inventory level reaches zero in rented warehouse.
• \( p_o \): Time at which the inventory level reaches zero in the owned warehouse.
• \( P \): Length of the replenishment cycle (time unit).
• \( R_m \): Maximum positive inventory level per cycle.
• \( R_d \): Capacity of the owned warehouse.
• \( R_m – R_d \): Capacity of the rented warehouse.
• \( N_m \): Backorder level during the shortage period.
• \( R \): Order quantity during the cycle length, \( R = R_m + N_m \).
• \( q_o(p) \): Inventory level in the owned warehouse at time \( p \), \( 0 \leq p \leq P \).
• \( q_r(p) \): Inventory level in the rented warehouse at time \( p \), \( 0 \leq p \leq P \).
• \( q_s(p) \): Shortage level at time \( p \), \( p_o \leq p \leq P \).

Model Assumptions

1. The replenishment rate is instantaneous.
2. The lead time is zero.
3. A single non-instantaneous decaying item is considered.
4. Warehouses have fixed capacities: \( R_d \) for owned and \( R_m – R_d \) for rented.
5. Holding cost in the rented warehouse is higher, and deterioration rate is lower.
6. There is no replacement or repair for deteriorated goods.
7. Demand before deterioration begins is quadratic:
   \[
   \eta + \mu p + \sigma p^2, \quad \text{where } \eta \geq 0, \mu \neq 0, \sigma \neq 0.
   \]
8. After deterioration begins, demand is constant and given by \( \text{\textit{ն}} \).
9. During the trade credit period \( T \), generated sales revenue earns interest until payment.
10. Shortages are partially backlogged with a dynamic backlogging rate:
    \[
    N(p) = \frac{1}{1 + \zeta (P – p)}, \quad 0 < \zeta < 1, \, p_o \leq p \leq P.
    \]

Formulation of the Model

The retailer’s ideal replenishment strategy for non-instantaneous decaying commodities with two-phase demand rates, two-storage facilities, and shortages within a permissible payment delay has been taken into consideration in this article. Allowable payment delays encourage retailers to stock up on more goods since they boost sales, increase cash flow, lower the cost of stock holding, draw in new customers, or just retain their current ones. When the quantity exceeds the merchant’s ware-house capacity, the retailer may choose to rent a ware-house to store the excess inventory. In this inventory system, R_m units of a single product arrive at the inventory at the beginning of the cycle in which R_d units are stored in their own ware-house and the remaining (R_m-R_d ) units in a rented ware-house. Thus, in order to find the optimal replenishment policy of the inventory system, two cases of when p_d<p_r and when p_d>p_r are discussed and are as follows.

Case I: p_d<p_r (when items start deteriorating before the inventory level in rented ware-house completely depleted to zero)

Figure 3.1 designates the behaviour of the inventory system. During the time interval [0,p_d], the inventory level q_r (p) in rented ware-house is depleting gradually due to market demand only and it is assumed to be a quadratic function of time p whereas in the owned ware-house inventory level remains unchanged. At time interval [p_d,p_r] the inventory level q_r (p) in the rented ware-house is depleting due to combined effects of constant market demand rate ն and deterioration while the inventory level in the owned ware-houses gets used up due to deterioration only. At time interval [p_r,p_o], the inventory level q_o (p) in the owned ware-house depletes to zero due to the combined effects of consumer demand and deterioration. Shortages occur at the time p=p_o and are partially backlogged in the interval [p_o,P]. The whole process of the inventory system is repeated.

Figure 3.1: Two-ware-house inventory system when p_d<p_r

The differential equations that describe the inventory level in both rented ware-house and owned ware-house at any time p over the period [0,P] are given by

\[
\frac{dq_r(p)}{dp} = -(\eta + \mu p + \sigma p^2), \quad 0 \leq p \leq p_d. \tag{1}
\]
\[
\frac{dq_r(p)}{dp} + \omega_r q_r(p) = -\text{\textit{ն}}, \quad p_d \leq p \leq p_r. \tag{2}
\]
\[
\frac{dq_o(p)}{dp} + \omega_o q_o(p) = 0, \quad p_d \leq p \leq p_r. \tag{3}
\]
\[
\frac{dq_o(p)}{dp} + \omega_o q_o(p) = -\text{\textit{ն}}, \quad p_r \leq p \leq p_o. \tag{4}
\]
\[
\frac{dq_s(p)}{dp} = -\frac{\text{\textit{ն}}}{1 + \zeta (P – p)}, \quad p_o \leq p \leq P. \tag{5}
\]

The solutions to these equations are as follows:

\[
q_r(p) = R_m – R_d – \left( \eta p + \frac{\mu p^2}{2} + \frac{\sigma p^3}{3} \right), \quad 0 \leq p \leq p_d. \tag{6}
\]
\[
q_r(p) = \frac{\text{\textit{ն}}}{\omega_r} \left( e^{\omega_r (p_r – p)} – 1 \right), \quad p_d \leq p \leq p_r. \tag{7}
\]
\[
q_o(p) = R_d e^{\omega_o (p_d – p)}, \quad p_d \leq p \leq p_r. \tag{8}
\]
\[
q_o(p) = \frac{\text{\textit{ն}}}{\omega_o} \left( e^{\omega_o (p_o – p)} – 1 \right), \quad p_r \leq p \leq p_o. \tag{9}
\]
\[
q_s(p) = -\frac{\text{\textit{ն}}}{\zeta} \left[ \ln(1 + \zeta (P – p_o)) – \ln(1 + \zeta (P – p)) \right], \quad p_o \leq p \leq P. \tag{10}
\]

From continuity of \( q_o(p) \) at \( p = p_r \), it follows:

\[
R_d = \frac{\text{\textit{ն}}}{\omega_o} \left( e^{\omega_o (p_o – p_d)} – e^{\omega_o (p_r – p_d)} \right), \quad p_o \leq p \leq P. \tag{11}
\]

From continuity of \( q_r(p) \) at \( p = p_d \), it follows:

\[
R_m = \frac{\text{\textit{ն}}}{\omega_o} \left( e^{\omega_o (p_o – p_d)} – e^{\omega_o (p_r – p_d)} \right) + \left( \eta p_d + \frac{\mu p_d^2}{2} + \frac{\sigma p_d^3}{3} \right) + \frac{\text{\textit{ն}}}{\omega_r} \left( e^{\omega_r (p_r – p_d)} – 1 \right), \quad p_o \leq p \leq P. \tag{12}
\]

The maximum backordered units \( N_m \) at \( p = P \) is:

\[
N_m = -q_s(P) = \frac{\text{\textit{ն}}}{\zeta} \left[ \ln(1 + \zeta (P – p_o)) \right]. \tag{13}
\]

The total order size \( R \) across the time period \([0, P]\) is given by:

\[
R = R_m + N_m = \frac{\text{\textit{ն}}}{\omega_o} \left( e^{\omega_o (p_o – p_d)} – e^{\omega_o (p_r – p_d)} \right) + \left( \eta p_d + \frac{\mu p_d^2}{2} + \frac{\sigma p_d^3}{3} \right) + \frac{\text{\textit{ն}}}{\omega_r} \left( e^{\omega_r (p_r – p_d)} – 1 \right) + \frac{\text{\textit{ն}}}{\zeta} \left[ \ln(1 + \zeta (P – p_o)) \right]. \tag{14}
\]

The sales revenue is calculated as follows:

\[
SR = SP_1 \left[ \int_0^{p_d} (\eta + \mu p + \sigma p^2) \, dp \right] + SP_2 \left[ \int_{p_d}^{p_r} \text{\textit{ն}} \, dp + \int_{p_r}^{p_o} \text{\textit{ն}} \, dp + \int_{p_o}^P \frac{\text{\textit{ն}}}{1 + \zeta (P – p)} \, dp \right].
\]

After integration:

\[
SR = SP_1 \left( \eta p_d + \frac{\mu p_d^2}{2} + \frac{\sigma p_d^3}{3} \right) + SP_2 \text{\textit{ն}} (p_o – p_d) + SP_2 \frac{\text{\textit{ն}}}{\zeta} \left[ \ln(1 + \zeta (P – p_o)) \right]. \tag{15}
\]

The purchasing cost is:

\[
PCQ = PC \left[ \frac{\text{\textit{ն}}}{\omega_o} \left( e^{\omega_o (p_o – p_d)} – e^{\omega_o (p_r – p_d)} \right) + \left( \eta p_d + \frac{\mu p_d^2}{2} + \frac{\sigma p_d^3}{3} \right) + \frac{\text{\textit{ն}}}{\omega_r} \left( e^{\omega_r (p_r – p_d)} – 1 \right) + \frac{\text{\textit{ն}}}{\zeta} \left[ \ln(1 + \zeta (P – p_o)) \right] \right]. \tag{16}
\]

The total profit per unit time \( ATPPU(p_o, P) \) is given by:

\[
ATPPU(p_o, P) =
\begin{cases}
ATPPU_{11}(p_o, P), & \text{Sub-case 1.1: } 0 < T \leq p_d \\
ATPPU_{12}(p_o, P), & \text{Sub-case 1.2: } p_d < T \leq p_r \\
ATPPU_{13}(p_o, P), & \text{Sub-case 1.3: } p_r < T \leq p_o \\ ATPPU_{14}(p_o, P), & \text{Sub-case 1.4: } T > p_o.
\end{cases}
\tag{17}
\]

Where:

\[
ATPPU_{11}(p_o, P) = \frac{1}{P} \left\{ SR – \text{Purchasing Cost} – OC – \text{Holding Costs} – \text{Backordered Costs} – \text{Interest Charges} + \text{Interest Earned} \right\}.
\]

Optimal Decision

The necessary condition for the total profit per unit time \( \text{ATPPU}_{ij}(p_o, P) \) to be minimum are:
\[
\frac{\partial \text{ATPPU}_{ij}(p_o, P)}{\partial p_o} = 0 \quad \text{and} \quad
\frac{\partial \text{ATPPU}_{ij}(p_o, P)}{\partial P} = 0,
\]
for \( i = 1 \) when \( p_r > p_d \) and \( j = 1, 2, 3, 4 \).

Sufficient Conditions:

The value of \( (p_o, P) \) obtained from the equations
\[
\frac{\partial \text{ATPPU}_{ij}(p_o, P)}{\partial p_o} = 0 \quad \text{and} \quad
\frac{\partial \text{ATPPU}_{ij}(p_o, P)}{\partial P} = 0,
\]
for which the sufficient condition is satisfied:
\[
\left( \frac{\partial^2 \text{ATPPU}_{ij}(p_o, P)}{\partial p_o^2} \right)
\left( \frac{\partial^2 \text{ATPPU}_{ij}(p_o, P)}{\partial P^2} \right)
– \left( \frac{\partial^2 \text{ATPPU}_{ij}(p_o, P)}{\partial p_o \, \partial P} \right)^2 > 0,
\]
gives a minimum for the total profit per unit time \( \text{ATPPU}_{ij}(p_o, P) \).

Optimality condition for sub-case 1.1: 0<M≤t_d

Similarly

Lemma 1.1

(i) If〖 ∆〗_11≤0, then the solution of p_o∈[p_d ┤,├ ∞) (say p_o11^*) which satisfies equation (25) not only exists but also is unique.

(ii) If〖 ∆〗_11>0, then the solution of p_o∈[p_d ┤,├ ∞) which satisfies equation (25) does not exist.

Proof of (i): From equation (25), a new function F_11 (p_o ) is defined as follows

Hence F_11 (p_o ) is a strictly increasing of p_o in the interval [p_d ┤,├ ∞). Moreover, lim┬(p_o→∞)⁡〖F_11 (p_o )=∞〗 and F_11 (p_d ) 〖 =∆〗_11≤0.

Therefore, by applying intermediate value theorem, there exists a unique p_o say p_11^*∈[p_d ┤,├ ∞) such that F_11 (p_o11^* )=0. Hence p_o11^* is the unique solution of equation (23). Thus, the value of p_o (denoted by p_o11^*) can be found from equation (23) and is given by

Once p_o11^* is obtained, then the value of P (denoted by P_11^*) can be found from equation (21) and is given by

Equations (27) and (28) give the optimal values of p_o11^* and P_11^*for the cost function in equation (18) only if X_11 satisfies the inequality given in equation (29)

Proof of (ii): If〖 ∆〗_11>0, then from equation (26), F_11 (p_o )>0. Since F_11 (p_o ) is a strictly increasing function of p_o∈[p_d ┤,├ ∞), F_11 (p_o )>0 for all p_o∈[p_d ┤,├ ∞). Thus, a value of p_o∈[p_d ┤,├ ∞) cannot be found such that F_11 (p_o )=0. This completes the proof.

Theorem 1.1

(i) If〖 ∆〗_11≤0, then the total profit 〖ATPPU〗_11 (p_o,P) is convex and reaches its global minimum at the point (〖p_o11^*,P〗_11^* ), where (〖p_o11^*,P〗_11^* ) is the point which satisfies equations (25) and (22).

(ii) If ∆_1>0, then the total profit〖 ATPPU〗_11 (p_o,P) has a minimum value at the point (〖p_o11^*,P〗_11^* ) where p_o11^*=p_d and P_11^*=1/(((C_b+C_π Ϛ)+(〖SP〗_2-PC)Ϛ) ) (W_11 p_d-X_11 )

Proof of (i): When ∆_11≤0, it is observed that p_o11^* and P_11^* are the unique solutions of equations (25) and (22) from Lemma l.1(i). Taking the second derivative of 〖ATPPU〗_11 (p_o,P) with respect to p_o and P, and then finding the values of these functions at the point (〖p_o11^*,P〗_11^* ) yields

It is therefore concluded from equation (28) and Lemma 1.1 that 〖ATPPU〗_11 (〖p_o11^*, P〗_11^* ) is convex and (〖p_o11^*, P〗_11^* ) is the global minimum point of 〖ATPPU〗_11 (p_o, P). Hence the values of p_o and P in equations (27) and (28) are optimal.

Proof of (ii): When〖 ∆〗_11>0, then F_11 (p_o )>0 for all p_o∈[p_d ┤,├ ∞). Thus, (∂〖ATPPU〗_11 (p_o, P))/∂P=(F_11 (p_o))/P^2 >0 for all p_o∈[p_d ┤,├ ∞) which implies 〖ATPPU〗_11 (p_o, P) is an increasing function of T. Thus 〖ATPPU〗_11 (p_o, P) has a minimum value when T is minimum. Therefore, 〖ATPPU〗_11 (p_o, P) has a minimum value at the point (〖p_o11^*, P〗_11^* ) where p_o11^*=p_d and P_11^*=1/(((C_b+C_π Ϛ)+(〖SP〗_2-PC)Ϛ) ) (W_11 p_d-X_11 ). This completes the proof.

Optimality condition for sub-case 1.2: t_d<M≤t_r

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o12^*) and P (denoted by P_12^*) are given by

Equations (31) and (32) give the optimal values of p_o12^* and P_12^* for the cost function in equation (17) only if X_12 satisfies the inequality given in equation (33)

Optimality condition for sub-case 1.3: t_r<T≤t_o

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o13^*) and P (denoted by P_13^*) are given by

Equations (34) and (35) give the optimal values of p_o13^* and P_13^* for the cost function in equation (18) only if X_13 satisfies the inequality given in equation (36)

Optimality condition for sub-case 1.4: M>t_o

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o14^*) and P (denoted by P_14^*) are given by

Equations (37) and (38) give the optimal values of p_o14^* and P_14^* for the cost function in equation (19) only if X_14 satisfies the inequality given in equation (39)

Thus, the economic order quantity (EOR) corresponding to the optimal cycle length P^* will be computed as follows:
〖EOR〗^*=The maximum inventory +the backordered units during the shortage period.

Numerical Examples

This section provides some numerical examples to illustrate the model established.

Example 3.1.1 (Sub-case 1.1)

Given the following input parameters: OC=$550/order, PC=$55/unit/year, S=$75/unit/year, 〖hc〗_o=
$10/unit/year, 〖hc〗_r=$17/unit/year, ω_o=0.08 units/year, ω_r=0.04 units/year, η=1080 units, µ=280 units, σ=25 units, ն=550 units, p_d=0.2971 year (109 days), p_r=0.3126 year (114 days), C_b=$25/unit/year, C_π=$10/unit/year, Ϛ=0.75, T=0.0884 year (32 days), I_c=0.11, I_e=0.09. It is observed that T≤p_d, ∆_11=-47.1059<0, X_11^2=0.6696, 2W_11 Y_11=116.1989 and hence X_11^2<2W_11 Y_11. Substituting the above values in equations (27),(28),(18) and (40), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows:p_o11^*=0.5247 year (191 days), P_11^*=0.7002 year (256 days), 〖ATPPU〗_11 (p_o11^*,〖 P〗_11^* )=$2775.5607 per year and 〖EOR〗_11^*=767.4678 units per year.

Example 3.1.2 (Sub-case 1.2)

The data are adopted as in Example 3.1.1 apart from T=0.2998 year (110 days). It is observed that T>p_d, 〖 ∆〗_12=-55.6558<0, X_12^2=3.5659, 2W_12 Y_12=109.0987 and hence X_12^2<2W_12 Y_12. Substituting the above values in equations (31),(32),(19) and (40), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o12^*=0.4654 year (170 days), P_12^*=0.5857 year (214 days), 〖ATPPU〗_12 (p_o12^*,〖 P〗_12^* )=$2372.4383 per year and 〖EOR〗_12^*=993.5549 units per year.

Example 3.1.3 (Sub-case 1.3)

The data are adopted as in Example 3.1.1 apart from T=0.32764 year (120 days). It is observed that T>p_r,〖 ∆〗_13=-44.0696<0, X_13^2=2.4398, 2W_13 Y_13=94.0811 and hence X_13^2<2W_13 Y_13. Substituting the above values in equations (34),(35),(20) and (40), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o13^*=0.4250 year (155 days), P_13^*=0.5689 year (208 days), 〖ATPPU〗_13 (p_o13^*,〖 P〗_13^* )=$1853.7399 per year and 〖EOR〗_13^*=1011.6565 units per year.

Example 3.1.4 (Sub-case 1.4)

The data are adopted as in Example 3.1.1 apart from T=0.3859 year (141 days). It is observed that 〖 ∆〗_14a=-46.6787<0,〖 ∆〗_14b=8.9876>0, X_14^2=1.0959, 2W_14 Y_14=68.0493. Here 〖 ∆〗_14a≤0≤〖 ∆〗_14b and X_14^2<2W_14 Y_14. Substituting the above values in equations (37),(38),(21) and (40), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o14^*=0.3588 year (131 days), P_14^*=0.4716 year (172 days), 〖ATPPU〗_14 (p_o14^*,〖 P〗_14^* )=$2598.7098 per year and 〖EOR〗_14^*=1198.6876 units per year. It is also seen that T>p_o.

Case II: when t_d>t_r (Deterioration starts after the inventory level in the rented ware-house becomes zero)

Figure 3.2 designates behaviours of the inventory system. During the time interval [0,p_r], the inventory level q_r (p) in the rented ware-house is depleting gradually due to market demand only and it is assumed to be a quadratic function of time p whereas in the owned ware-house inventory level remains unchanged. At time interval [p_r,p_d] the inventory level q_o (p) in the owned ware-house is depleting due to demand from the consumers and is also assumed to be a quadratic function of time p. At time interval [p_d,p_o], the inventory level q_o (p) in the owned ware-house depletes to zero due to the combined effects of demand from the consumers and deterioration. Shortages occur at the time p=p_o and are partially backlogged in the interval [p_o,P]. The whole process of the inventory is repeated.

Figure 3.2: Two-ware-house inventory system when p_d>p_r

The differential equations that describe the inventory level in both rented ware-house and owned ware-house at any time p over the period [0,P] are given by

Optimal Decision

In order to find the optimal ordering policies that minimize the total profit per unit time, the necessary and sufficient conditions are established. The necessary condition for the total profit per unit time 〖ATPPU〗_ij (p_o,P) to be minimum are (∂〖ATPPU〗_ij (p_o,P))/(∂p_(o ) )=0 and (∂〖ATPPU〗_ij (p_o,P))/∂P=0 for i=2 when p_d>p_r and j=1,2,3,4. The value of (p_o,P) obtained from (∂〖ATPPU〗_ij (p_o,P))/(∂p_(o ) )=0 and(∂〖ATPPU〗_i j(p_o,P))/∂P=0 and for which the sufficient condition {((∂^2 〖ATPPU〗_ij (p_o,P))/(∂p_o^2 ))((∂^2 〖ATPPU〗_ij (p_o,P))/(∂P^2 ))-((∂^2 〖ATPPU〗_ij (p_o,P))/(∂p_(o ) ∂P))^2 }>0 is satisfied gives a minimum for the total profit per unit time 〖ATPPU〗_ij (p_o,P).

Optimality condition for sub-case 2.1: 0<T≤t_r

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o12^*) and P (denoted by P_12^*) are given by

Optimality condition for sub-case 2.2: t_r<T≤t_d

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o12^*) and P (denoted by P_12^*) are given by

Optimality condition for sub-case 2.3: t_d<T≤t_o

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o12^*) and P (denoted by P_12^*) are given by

Optimality condition for sub-case 2.4: M>t_o.

Applying the same procedure as in sub-case 1.1, the value of p_o (denoted by p_o12^*) and P (denoted by P_12^*) are given by

Thus, the economic order quantity (EOR) corresponding to the optimal cycle length P^* will be computed as follows:
〖EOR〗^*=The maximum inventory +the backordered units during the shortage period.

Numerical Examples

This section provides some numerical examples to illustrate the model established.

Example 3.2.1 (Sub-case 2.1)

The data are adopted as in Example 3.1.1 apart from p_d=0.3201 year (117 days). It is observed that T≤p_r, ∆_21=-85.2345<0, X_21^2=0.8771, 2W_21 Y_21=143.3421 and hence X_21^2<2W_21 Y_21. Substituting the above values in equations (58),(59),(54) and (70), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o21^*=0.4527 year (165 days), P_21^*=0.6153 year (225 days), 〖ATPPU〗_21 (p_o21^*,〖 P〗_21^* )=$1996.9865 per year and 〖EOR〗_21^*=701.8182 units per year.

Example 3.2.2 (Sub-case 2.2)

The data are adopted as in Example 3.2.1 apart from T=0.3152 year (115 days). It is observed that T>p_r, 〖 ∆〗_22=-82.4679<0, X_22^2=7.6981, 2W_22 Y_22=115.8946 and hence X_22^2<2W_22 Y_22. Substituting the above values in equations (61),(62),(55) and ((70), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o22^*=0.4321 year (158 days), P_22^*=0.5538 year (202 days), 〖ATPPU〗_22 (p_o22^*,〖 P〗_22^* )=$1798.7077 per year and 〖EOR〗_22^*=876.6575 units per year.

Example 3.2.3 (Sub-case 2.3)

The data are adopted as in Example 3.2.1 apart from T=0.3444 year (126 days). It is observed that T>p_d, 〖 ∆〗_23=-87.0998<0, X_23^2=8.9098, 2W_23 Y_23=106.7717 and hence X_23^2<2W_23 Y_23. Substituting the above values in equations (64),(65),(56) and ((70), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o23^*=0.4357 year (159 days), P_23^*=0.5821 year (213 days), 〖ATPPU〗_23 (p_o23^*,〖 P〗_23^* )=$1721.3664 per year and 〖EOR〗_23^*=887.6543 units per year.

Example 3.2.4 (Sub-case 2.4)

The data are adopted as in Example 3.2.1 apart from T=0.4109 year (150 days). It is observed that 〖 ∆〗_24a=-168.4024<0,〖 ∆〗_24b=19.0177>0, X_24^2=01.0998, 2W_24 Y_24=85.6297. Here hence〖 ∆〗_24a≤0≤〖 ∆〗_24b and X_24^2<2W_24 Y_24. Substituting the above values in equations (67),(68),(57) and ((70), the value of optimal time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and economic order quantity are respectively obtained as follows: p_o24^*=0.3908 year (143 days), P_24^*=0.4878 year (178 days), 〖ATPPU〗_24 (p_o24^*,〖 P〗_24^* )=$1676.0974 per year and 〖EOR〗_24^*=912.9875 units per year. It is also seen that T>p_o.

Therefore,

Thus, the optimal solution is: p_o^*=0.3588 year (131 days), P^*=0.4716 year (172 days), ATPPU(p_o^*,P^* )=$2598.7098 per year and 〖EOQ〗^*=1198.6876

Sensitivity Analysis

The sensitivity analysis related in conjunction with different parameters is performed by changing each of the parameters from -20% to 20% taking one parameter at a time and keeping the remaining parameters unchanged. The effects of these parameters on length of time at which the inventory level reaches zero in the owned ware-house, cycle length, total profit and the economic order quantity per cycle for the optimal solution has been presented in the figures below.

Figure 3.3: effect of change of selling price before deterioration sets in (SP1) on decision variables

Figure 3.4: effect of change of selling price after deterioration sets in (SP2) on decision variables

Figure 3.5: effect of change of shortage cost (Cb) on decision variables

Figure 3.6: effect of change of lost sales (Cπ) on decision variables

RESULTS AND DISCUSSION

Based on the computational results shown in Tables and figures above, the following managerial insights are obtained.

From figure 3.3, it is apparently seen that as the unit selling price before deterioration sets in (〖SP〗_1) increases, the optimal time at which the inventory level reaches zero in the owned ware-house (p_o^* ), cycle length 〖(P〗^*) and order quantity 〖(EOQ〗^*) decrease while the total profit(ATPPU(p_o^*,P^* )) increases and vice versa. This implies that as the selling price increases the retailer will order less quantity to enjoy the benefits of trade credit more frequently.

From figure 3.4, it is evidently seen that as the unit selling price after deterioration sets in (〖SP〗_2) increases, the optimal time at which the inventory level reaches zero in the owned ware-house (p_o^* ), cycle length 〖(P〗^*), order quantity 〖(EOQ〗^*) and the total profit (ATPPU(p_o^*,P^* )) increase and vice versa. This implies that as the selling price is increasing the retailer maximizes higher profit.

From figure 3.5, it is clearly seen that as the shortage cost (C_b) increases the total profit (ATPPU(p_o^*,P^* )), the economic order quantity 〖(EOQ〗^*), the optimal cycle length 〖(P〗^*) decreases while the the optimal time at which the inventory level reaches zero in the owned ware-house (p_o^* ), increases.
From figure 3.6, it evidently seen that as the unit cost of lost sales per unit

〖(C〗_π) increases, the optimal time at which the inventory level reaches zero in the owned ware-house (p_o^* ), also increases while cycle length 〖(P〗^*), order quantity 〖(EOQ〗^*) and the total profit (ATPPU(p_o^*,P^* )) decrease.This implies that the retailer should order less quantity when the unit cost of lost sales is high.

CONCLUSION

In this research, a two-warehouse ordering policy for non-instantaneous deteriorating items with two phase demand rates, two-tiered pricing and shortages under trade credit policy has been established. The demand rate before deterioration sets in is assumed to be a time-dependent quadratic function after which it is considered as a constant function up to when the inventory is completely used up. Shortages considered which are partially backlogged. The length of the waiting time would determine whether backlogging will be accepted or not, hence, the backlogging rate is variable and depends on the waiting time for the next replenishment. The optimal time at which the inventory level reaches zero in the owned ware-house, cycle length and order quantity that minimizes total profit has been determined. Some numerical examples have been given to demonstrate the assumed set of results of the model. Then Sensitivity analyses of some model parameters on optimal solutions have been performed and finally, suggestions toward minimizing the total profit of the inventory system have been provided. The model can be extended by taking more realistic assumptions such as variable deterioration, inflation rates, reliability of goods and so on.

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