A periodical of the Faculty of Natural and Applied Sciences, UMYU, Katsina
ISSN: 2955 – 1145 (print); 2955 – 1153 (online)
ORIGINAL RESEARCH ARTICLE
Lois Iretide Oluwafemi1* and Zaharaddeen Haruna Aliyu2
1,2Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna – Nigeria
Corresponding Author:
Lois Iretide Oluwafemi. Department of Mathematical Sciences, Nigerian Defence Academy, Kaduna – Nigeria.
Corresponding Author: Lois Iretide Oluwafemi 
loisiretide.oluwafemi2021@nda.edu.ng
In this research, an inventory model is developed for non-instantaneous deteriorating items with time-dependent demand under the retailer’s trade credit and partial backlogging. In the model, the retailer receives no trade credit from the supplier but offers trade credit to customers. Payment strategies for the retailer were adopted in calculating the interest associated with the trade credit. The cost function was obtained and optimized using the gradient method. To illustrate the model, numerical examples are provided. The results show that case 1 scenario 2 with \(t_{1}^{*} = 24.5070\) and \(T^{*} = 31.7335\) produced the highest profit of \({TP}^{*} = 16,000\). Sensitivity analyses carried out to examine the influence of the inventory parameters showed that decreases in h by 10% and N by 20% lead to higher profit, indicating that decreases in holding costs and offering customers trade credit can lead to higher profitability and better decisions. From a managerial point of view, it is advised that retailers should regularly extend trade credit to customers to reduce item overstay and earn higher profits.
KEYWORDS: Non-instantaneous deterioration; Time dependent demand; Downstream Trade Credit; Partial Backlogging; Inventory optimization.
Considered a downstream trade credit, which is prevalent in the present-day markets which is a novel in the literature.
A novel way of calculating interest payable and interest earned by the retailer when downstream trade credit is considered.
The model presented an interplay between realistic business features, such as shortages and trade credit, in a non-instantaneous, deteriorating-asset environment.
Even though retailers receive no trade credit, the model shows that offering trade credit to customers can lead to higher profitability and better decision-making.
Effective inventory management is the tracking of useful goods throughout the entire supply chain, from the purchase of raw materials to production to end sales. Thus, this achieves greater inventory visibility, meets customer demand, understands profitability, improves workflow efficiency, and enables accurate financial reporting. In addition, the nature of the inventory item to be managed plays a crucial role in the supply chain. Therefore, studying the inventory of deteriorating items is crucial owing to their dynamic nature.
Ghare and Schrader (1963) were the first to consider the optimal ordering policies for deteriorating items. They assumed the item to have a constant deterioration rate. Mandal and Phaujdar (1989) assumed a constant production rate and a linearly stock-dependent demand in an inventory model for deteriorating items. Dave and Patel (1981) presented an inventory model for deteriorating items with linearly increasing demand and a constant deterioration rate. Lee and Dye (2012) worked on an inventory model under stock-dependent demand and a controllable deterioration rate for deteriorating items. But instant deterioration does not always represent the nature of some deteriorating items. In practice, many perishable items such as fruits, vegetables, and meat do not deteriorate immediately upon receipt; instead, they exhibit a shelf-life period of no spoilage before deterioration sets in. This is termed in the literature as non-instantaneous deterioration.
Building on this, many stockists and researchers developed EOQ models for non-instantaneous deterioration, incorporating realistic features such as trade credit and shortages. For example, Wu et al. (2006) conducted a study to determine the optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand. Maihami and Kamalabadi (2012) considered joint pricing in an inventory model for non-instantaneous deteriorating items with permissible payment delays. Pal and Chandra (2014) developed an inventory model for non-instantaneous deteriorating items with stock- and time-dependent demand, while adopting a price discount to attract more sales. Pal and Samanta (2018) studied an inventory model with a non-instantaneous deterioration period, assuming the pre-deterioration period is random.
One of the management problems associated with keeping deteriorating items, whether instantaneous or non-instantaneous, is their nature. Keeping them for too long can lead to overstocking and further deterioration. Keeping less led to shortages, a phenomenon of vital importance. Therefore, it is worth noting that some works in the literature on non-instantaneous deteriorating items have studied shortages. When a shortages occur, it is more practical to treat unmet demand as partially backlogged rather than assuming full backlogging. Park (1982) studied an inventory model with partial backorder. Wee (1995) studied A deterministic lot-size inventory model for deteriorating items with shortages and a declining market. Mashud et al. (2017) studied an inventory model that considers price, stock-dependent demand with partially backlogged shortages, and two constant deterioration rates. Sundararajan et al. (2019) developed an inventory system of non-instantaneous deteriorating items with backlogging and time discounting.
In the earliest models, inventory was assumed to be paid for upon receipt of consignments (Goyal, 1985). However, in today’s competitive market, businesses use trade credit to stimulate demand and increase profitability. Trade credit has typically been analyzed at the supplier level or at the two-level trade credit level. Goyal (1985) was the first to propose an EOQ model that incorporates a permissible payment delay. The supplier extended trade credit to the retailer. Pal and Maity (2012) developed an inventory model for deteriorating items with a constant deterioration rate and a trade credit policy. Shortage is allowed and completely backlogged. Chandra et al. (2020) considered optimal replenishment strategies for a constant-demand item under a progressive trade credit policy, accounting for non-instantaneous deterioration. Babangida and Baraya (2022) developed a model for non-instantaneous deteriorating items with two-phase demand rates, time-dependent linear holding costs, and shortages under a a trade credit policy. Vadana and Sharma (2016), proposed an inventory model for non-instantaneous deterioration items under a quadratic demand rate with permissible delay in payment and time dependent deterioration rate. In the model, unmet demand is fully backlogged. Vadana and Sharma (2016) modeled non-instantaneous deterioration with quadratic demand and permissible delay in payment, but assumed full backlogging.
In other studies, such as Aliyu (2022), trade credit is extended from the supplier–retailer level to the retailer– customer level. This is termed as two-level trade credit. Despite the prevalence of this practice of downstream trade credit in the market, it was only recently that Malumfashi et al. (2023) justified that it is worthwhile to consider the trade credit to be between the retailer and the customers only by proposing an inventory model for deteriorating items having linear time-dependent demand under downstream trade credit. In the model, the supplier offers no trade credit to the retailer for goods ordered, and the retailer offers trade credit to the customer for a period. However, this idea of downstream has not been studied when the items are non-instantaneously deteriorating. This needs to be addressed and that’s the theme of this work.
The following are the notations used in the model:
\(D(t)\) – is the time-varying demand of items. It is given as \(D(t) = a + bt\) where \(a\ > \ 0\ \)is the initial demand and \(b\ > \ 0\) is the demand rate.
\(I_{o}\) – is the maximum inventory ordered by the retailer.
\(P,\ \) \(C\) – are the selling price per unit and purchasing cost per unit item, respectively.
\(I_{p},{\ I}_{e}\) – are the interest payable and interest earned return rates paid and received by the retailer, respectively.
\(\theta(t) -\) is the deterioration rate of the item.
\(h\ -\) is the holding cost per unit item per unit time.
\(\sigma,\ \ S\ \ -\) are the backlogging rate and quantity of maximum backlogged demands.
\(t_{d},{\ t}_{1} -\) are the times when deterioration sets in and the time in which inventory finishes, respectively.
\(T\ -\) is the end of the replenishment cycle.
\(A\ -\) is the ordering cost per order.
\(I_{1}(t),I_{2}(t)\) , \(I_{3}(t)\) – are the inventory levels at a time \(t,\ \ 0 \leq t \leq T,\) before deterioration sets in, after deterioration sets in, and during the shortage period.
\(N\ -\) The retailer’s trade credit period offered to the customer.
\(TP\ -\) The optimal total profit per unit time of the inventory system.
The following assumptions are made in the model:
The retailer offers trade credit to the customers.
The supplier offers no trade credit to the retailer.
Shortages are allowed and partially backlogged.
The item is single and non-instantaneously deteriorating.
The replenishment rate is infinite and instantaneous.
The lead time is zero.
The demand is time-dependent.
We restrict N > \(t_{d}\)
Figure 1: Pictorial representation of the model.
At the beginning of the cycle, \(I_{0}\) Units of items are stocked in the inventory. The items maintained freshness during the period \(\left( 0,t_{d} \right)\) After which, the items start to deteriorate. The deterioration and demand continued during the period \(\left( t_{d},t_{1} \right)\) till when the stocked items are depleted completely. At this time, shortages are occurring because demand is ongoing, indicating the business cycle has not ended. Therefore, from \(t_{1}\) to T, the unmet demands are partially backlogged. Figure 1 represents the scenario. This phenomenon is modelled as follows:
\(\frac{dI_{1}(t)}{dt} = - (a + bt)\) \(0\ < \ t\ \leq \ t_{d}\) (1)
with initial condition \(I_{1}(0) = \ I_{0}\)
\(\frac{dI_{2}(t)}{dt} + \theta I_{2\ }(t) = - (a + bt)\) \(t_{d}\ < \ t\ \leq \ t_{1}\) (2)
with condition \(I_{2\ }\left( t_{1\ } \right)\) = 0
\(\frac{dI_{3}(t)}{dt} = - \sigma(a + bt)\) \(t_{1}\ < \ t\ < \ T\) (3)
with condition \(I_{3\ }\left( t_{1\ } \right) = 0\)
The solutions to equation (1), (2) and (3) using the initial and boundary condition are given as
\(I_{1}(t) = I_{0} - \ at - \ \frac{1}{2}\ bt^{2}\) \(0\ < \ t\ \leq \ t_{d}\) (4)
\(I_{2\ }(t) = \frac{1}{\theta^{2}}\left\lbrack (\ b\ - \ \theta\ (a + bt)) - (\ b\ - \ \theta\ \left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t \right)} \right\rbrack\) \(t_{d}\ < \ t\ \leq \ t_{1}\) (5)
\(I_{3\ }(t) = \sigma a\left( t_{1} - t \right) + \sigma\frac{b\ }{2}\left( {t_{1}}^{2} - t^{2} \right)\) \(t_{1}\ < \ t\ < \ T\) (6)
For the sales to continue after deterioration sets in, a continuity is established at \(t = t_{d}\), by equating equations (4) and (5) and is given as:
\(I_{o} = \ \frac{1}{\theta^{2}}\left\lbrack \left( b - \theta\left( a + bt_{d} \right) \right) - \left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack + at_{d} + \ \frac{1}{2}\ b{t_{d}}^{2}\) (7)
To get the total profit (TP) per annum, we take the difference between the sales revenue per annum with the total incurred costs during the annum. Therefore, we first calculate the following costs
Annual ordering cost (OC)
The ordering cost per order is given as A, therefore, OC per annum is given as
\(OC = \frac{A}{T}\) (8)
Annual holding cost (HC)
The HC is giving as \(h\int_{0}^{T}{I(t)dt = h\int_{0}^{t_{1}}{I_{1}(t)dt + h\int_{t_{1}}^{T}{I_{2}(t)dt}}}\)
After evaluating the definite integral, the annual HC is giving by
\(HC = \frac{h}{T}\left\lbrack \ \left\{ \frac{1}{\theta^{2}}\left( b - \ \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\} t_{d} + at_{d}^{2} + \frac{1}{2}\ bt_{d}^{3} - \frac{{at}_{d}^{2}}{2} - \ \frac{1}{6}bt_{d}^{3} + \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right)\ - \frac{1}{2}\theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack\) (9)
Annual Deteriorating cost (DC)
The number of deteriorated items is \(\theta\int_{t_{1}}^{T}{I_{2}(t)dt}\). The annual DC incurred by the retailer, after some algebraic simplification, is giving as
\(DC = \frac{C\theta}{T}\left\lbrack \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right)\ - \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack\) (10)
Annual Shortage Cost (SC)
The cost due to shortages incurred by the retailer is given as
\(SC = \int_{t_{1}}^{T}{{- \ I}_{3}(t)}dt =\)
\(c\left\lbrack {\sigma a}_{\ }\left( \frac{T^{2}}{2} - Tt_{1\ } + \ \frac{t_{1}^{2}}{2}\ \right) + \frac{\sigma b}{6}\left( T^{3} - 3T{t^{2}}_{1} + {2t}_{1}^{2} \right) \right\rbrack\) (11)
Annual Opportunity Cost (OP)
The annual OP incurred by the retailer due to lost sales is giving us
\(OP = \int_{t_{1}}^{T}{(1 - \sigma)(a + bt)dt}\)
\(= (1 - \sigma\ )\left( aT + \ \frac{bT^{2}}{2} - at_{1} - \ \frac{bt_{1}^{2}}{2} \right)\) (12)
Purchasing cost (PC)
The cost of purchasing the stocked items is given as
\(PC = C\left\lbrack - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) + \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right\rbrack\) (13)
Annual Sales Revenue (SR)
The sales revenue generated by the retailer due to sales during the period \((0,\ \ T)\) is giving as
\(SR = \frac{P}{T}\ \left\lbrack \int_{0}^{t_{1}}{(a + bt)dt - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right)} \right\rbrack = \frac{P}{T}\ \left\lbrack at_{1} + \ \frac{{bt}_{1}^{2}}{2} - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) \right\rbrack\) (14)
Interest Earned (IE) and Interest Payable (IP) by the retailer
Since the retailer did not get the grace of trade credit but gave the grace to his/her customers for a period N, then two cases are possible:
Case 1: when N< T (when the trade credit period given to customers is less than the replenishment cycle) and
Case 2: when N\(\geq T\) (when the trade credit period offered to the customers exceeds the replenishment cycle).
Now,
Case 1: N < T
In this case, the supplier offers no trade credit to the retailer, then the annual total interest payable by the retailer is given as:
\(I_{P1}\) = 0 (15)
For the IE, since there are still unsold items with the customer at the expiration of the trade credit period given, then the retailer may earn interest in two ways:
Scenario 1: when partial payment is acceptable by the retailer from the customer, and it is to be made at 𝑡 = 𝑁 and the rest amount at a time after 𝑡 = 𝑁.
Here, we assume the customer pays some amount (partial payment) at 𝑡 = 𝑁 to the retailer and will pay the remaining amount C – CR after 𝑁 (𝑡 > 𝑁). The total amount paid by customer CR during the period [0, 𝑁] is
CR\(= \ C\int_{0}^{N}{D(t)dt}\ = C\left\lbrack aN + \ \frac{bN^{2}}{2}\ \right\rbrack\ \) (16)
Let the rest amount to be paid to the retailer at time \(t\ = \ M\) be (\(CI_{0}\ - \ C_{R}),\) where \(M\ > \ N\). Then total annual interest payable by the customer to the retailer at time 𝑡 = M is given as
\(I_{E11} = \ \frac{1}{T}\left\lbrack \left( CI_{0} - \ C_{R} \right) + I_{e}\left( CI_{0} - C_{R} \right)(M - N) \right\rbrack = \ \frac{1}{T}\left\lbrack \left( CI_{0} - \ C_{R} \right)\ {(1 + I}_{e}(M - N)) \right\rbrack\)
After some algebraic simplifications,
\(I_{E11} = \frac{C}{T}\ \left\lbrack \left( \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left\lbrack b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack + at_{d} + \frac{1}{2}b{t_{d}}^{2} - \left\lbrack aN + \ \frac{bN^{2}}{2}\ \right\rbrack \right){(1 + I}_{e}(M - N))\ \right\rbrack\) (17)
Therefore, the TP for case 1 scenario 1 is
\({TP}_{11} = \frac{1}{T}\left( SR + I_{E11} - OC - HC - DC - I_{P1} - SC - OP - PC \right)\)
Using equations (8), (9), (10), (11), 12), (13), 14), (15), and (17) and simplifying
\({TP}_{11} = \ \frac{1}{T}\ \left\{ P\left\lbrack at_{1} + \frac{{bt}_{1}^{2}}{2} - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\left( t_{1}^{2} - \ T^{2} \right) \right) \right\rbrack + C\left\lbrack \left( \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left\lbrack b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack + at_{d} + \frac{1}{2}b{t_{d}}^{2} - \left\lbrack aN + \ \frac{bN^{2}}{2}\ \right\rbrack \right)\ {(1 + I}_{e}(M - N)) \right\rbrack - A - h\left\lbrack \ \left\{ \frac{1}{\theta^{2}}\left( b - \ \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\} t_{d} + at_{d}^{2} + \frac{1}{2}\ bt_{d}^{3} - \frac{{at}_{d}^{2}}{2} - \ \frac{1}{6}bt_{d}^{3} + \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right)\ - \frac{1}{2}\theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - \ c\theta\ \left\lbrack \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right)\ - \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - c\left\lbrack {\sigma a}_{\ }\left( \frac{T^{2}}{2} - Tt_{1\ } + \ \frac{t_{1}^{2}}{2}\ \right) + \frac{\sigma b}{6}\left( T^{3} - 3T{t^{2}}_{1} + {2t}_{1}^{2} \right) \right\rbrack\ - c\left\lbrack (1 - \sigma\ )\left( aT + \ \frac{bT^{2}}{2} - at_{1} - \ \frac{bt_{1}^{2}}{2} \right) \right\rbrack - C\left\lbrack - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) + \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right\rbrack \right\}\) (18)
Scenario 2: (when partial payment is not accepted at t >N)
In this scenario, due to the retailer not willing to accept partial payment from the consumer, full payment is to be made after \(t\ = \ N\). Let full payment be made at time \(t\ = \ M_{2}\), where \(M_{2}\ > \ N\). Therefore, the interest payable \(I_{E12}\) to the retailer by the consumer is given by
\(I_{E12} = CI_{0} + CI_{0}I_{e}(M - N) = CI_{0}\left( 1 + I_{e}(M - N) \right)\)
Substituting \(I_{0}\) in equation (7), we see that
\(I_{E12} = C\ \left\lbrack \left( \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left\lbrack b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right)\ \ {(1 + I}_{e}(M - N)) \right\rbrack\) (19)
Therefore, the TP for case 1 scenario 2 is given by
\[{TP}_{12} = \frac{1}{T}\left( SR + I_{E12} - OC - HC - DC - I_{P1} - SC - OP - PC \right)\]
Using equation (8), (9), (10), (11), (12), (13), (14), (15) and (19) we get
\({TP}_{12} = \ \frac{1}{T}\ \left\{ P\left\lbrack at_{1} + \ \frac{{bt}_{1}^{2}}{2} - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) \right\rbrack + C\left\lbrack \ \left( \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left\lbrack b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right)\ \ {(1 + I}_{e}(M - N)) \right\rbrack - A - h\left\lbrack \left( \ \frac{1}{\theta^{2}}\ \left( b - \ \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}(\ b\ - \ \theta\ \left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} \right)t_{d} + at_{d}^{2} + \frac{1}{2}\ bt_{d}^{3} - \frac{{at}_{d}^{2}}{2} - \ \frac{1}{6}bt_{d}^{3} + \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right) - \ \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - \ c\theta\ \left\lbrack \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right) - \ \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - c\left\lbrack {\sigma a}_{\ }\left( \frac{T^{2}}{2} - Tt_{1\ } + \ \frac{t_{1}^{2}}{2}\ \right) + \frac{\sigma b}{6}\left( T^{3} - 3T{t^{2}}_{1} + {2t}_{1}^{2} \right) \right\rbrack\ - c\left\lbrack (1 - \sigma\ )\left( aT + \ \frac{bT^{2}}{2} - at_{1} - \ \frac{bt_{1}^{2}}{2} \right) \right\rbrack - C\left\lbrack - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) + \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right\rbrack \right\}\ \) (20)
Case 2: when 𝑁 ≥ 𝑇
Since the supplier offers no trade credit to the retailer, then the annual total interest payable by the retailer for this case says \(I_{P2}\) is given by
\(I_{P2}\)= 0 (21)
On the other hand, since 𝑁 ≥ 𝑇, the customer pays no interest to the retailer. Therefore, the interest earned by the retailer says \(I_{e2}\) is given by
\(I_{E2}\) = 0 (22)
Therefore, to get the total profit relevant per cycle for case 2
\({TP}_{2}\)= \(\frac{1}{T\ }\) \(\left( SR + I_{E2} - OC - HC - DC - I_{P2} - SC - OP - PC \right)\)
Using equation (8), (9), (10), (11), (12), (13), (14), (21) and (22) we get
\({TP}_{2} = \frac{1}{T}\ \left\{ P\left\lbrack at_{1} + \ \frac{{bt}_{1}^{2}}{2} - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) \right\rbrack - A - h\left\lbrack \ \left( \frac{1}{\theta^{2}}\left( b - \ \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right)t_{d} + at_{d}^{2} + \frac{1}{2}\ bt_{d}^{3} - \frac{{at}_{d}^{2}}{2} - \ \frac{1}{6}bt_{d}^{3} + \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right) - \ \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - \ c\theta\ \left\lbrack \frac{1}{\theta^{2}}\left( b\left( \ t_{1} - \ t_{d} \right) - \theta a\left( t_{1} - t_{d} \right) - \ \frac{1}{2}\ \theta b\left( t_{1}^{2} - t_{d}^{2} \right) \right) - \left( b - \theta\left( a + bt_{1} \right)\frac{1}{\theta}\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right) \right\rbrack - c\left\lbrack {\sigma a}_{\ }\left( \frac{T^{2}}{2} - Tt_{1\ } + \ \frac{t_{1}^{2}}{2}\ \right) + \frac{\sigma b}{6}\left( T^{3} - 3T{t^{2}}_{1} + {2t}_{1}^{2} \right) \right\rbrack\ - c\left\lbrack (1 - \sigma\ )\left( aT + \ \frac{bT^{2}}{2} - at_{1} - \ \frac{bt_{1}^{2}}{2} \right) \right\rbrack - C\left\lbrack - \left( \sigma a\left( t_{1} - T \right) + \frac{\sigma b}{2}\ \left( t_{1}^{2} - \ T^{2} \right) \right) + \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{d} \right) \right) - \frac{1}{\theta^{2}}\left( b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) + at_{d} + \frac{1}{2}b{t_{d}}^{2} \right\rbrack \right\}\) (23)
Considering the two decision variables \(t_{1}\) and T,
The necessary conditions for \({TP}_{11\ }\)to be maximized are \(\frac{\partial{TP}_{11}}{\partial t_{1}} = 0\ \ and\ \frac{\partial{TP}_{11}}{\partial T} = 0.\ \)
Using equation (18)
\(\frac{{\partial TP}_{11}}{\partial T} = \frac{1}{T}\ \left\{ P\lbrack\sigma a + \sigma bT\rbrack - C\left\lbrack \sigma a\left( T - t_{1} \right) + \frac{\sigma b}{6}\left( 3T^{2} - 3t_{1}^{2} \right) \right\rbrack - c\left\lbrack (1 - \sigma)(a + bT) \right\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - {TP}_{11} \right\}\) = 0 (24)
and
\(\frac{\partial{TP}_{11}}{\partial t_{1}} = \frac{1}{T}\ \left\{ P\left\lbrack \left( a + {bt}_{1} \right) - \left( \sigma a + \sigma{bt}_{1} \right) \right\rbrack + c\left\lbrack - \frac{1}{\theta^{2}}\left( ( - \theta b)e^{\theta\left( t_{1} - t_{d} \right)} + \theta\left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right)\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta be^{\theta\left( t_{1} - t_{d} \right)} \right)t_{d} + \frac{1}{\theta^{2}}\left( b - \theta a - {\theta bt}_{1} \right) - \left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} + b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}(b - \theta a - {\theta bt}_{1}) - (b - \theta\left( a + {bt}_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \ b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\left\lbrack (1 - \sigma)\left( - a - {bt}_{1} \right) \right\rbrack - c\left\lbrack \sigma a\left( - T + t_{1} \right) + \frac{\sigma b}{6}\left( - 6Tt_{1} + 4t_{1} \right) \right\rbrack - c\left\lbrack - \left( \sigma a + \sigma bt_{1} \right) - \frac{1}{\theta^{2}}\left( ( - \theta b)e^{\theta\left( t_{1} - t_{d} \right)} + \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) \right\rbrack \right\} = 0\) (25)
The solution to the non-linear equation (24) and (25) gives the value of \(t_{1}\) and T. To confirm that the optimal solution \(\left( t_{1}^{*}T^{*} \right)\) exist and unique, we show that
\(\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} \right) < 0,\) \(\left( \frac{{\partial^{2}TP}_{11}}{\partial T^{2}} \right)\ \)< 0 and \(\left( \frac{{\partial^{2}TP}_{11}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{11}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}\partial T} \right)|_{\left( t_{1}^{*}T^{*} \right)} > 0\)
Differentiating equation (24) further
\(\frac{{\partial^{2}TP}_{11}}{\partial T^{2}} = \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack - 2\frac{{\partial TP}_{11}}{\partial T} \right\}\) (26)
Evaluating equation (26) at \(\left( t_{1}^{*},T^{*} \right)\), and since \(\frac{{\partial TP}_{11}}{\partial T}\)}\(|_{\left( t_{1}^{*}T^{*} \right)} = 0\), we see that
\(\frac{{\partial^{2}TP}_{11}}{\partial T^{2}}|_{\left( t_{1}^{*}T^{*} \right)\ } < 0\ \) (27)
Now differentiating equation (25) with respect to \(t_{1}\), we see that,
\(\frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} = \frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack + c\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta^{2}(b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\}\) (28)
When equation (28) is evaluated at \(\left( t_{1}^{*}T^{*} \right)\)
\(\frac{{\partial^{2}TP}_{11}\left( t_{1}^{*}T^{*} \right)}{\partial t_{1}^{2}}\) < 0 (29)
Likewise,
\(\frac{{\partial^{2}TP}_{11}}{\partial{T\partial t}_{1}} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) (30)
Also, from equation (25)
\(\frac{{\partial^{2}TP}_{11}}{\partial t_{1}\partial T} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) (31)
Also,
\(\left( \frac{{\partial^{2}TP}_{11}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{11}}{{\partial t}_{1}\partial T} \right) = \ \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} > 0\) (32)
Evaluating,
\(\left( \frac{{\partial^{2}TP}_{11}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{11}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}\partial T} \right) = \left( - \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack \right\} \right)\left( \frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack + c\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta^{2}(b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\}\ \right) - \left( \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} \right)\) \(> 0\) (33)
Since \(\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} \right) < 0\), \(\left( \frac{{\partial^{2}TP}_{11}}{\partial T^{2}} \right) < 0\ and\left( \frac{{\partial^{2}TP}_{11}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{11}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{11}}{\partial t_{1}\partial T} \right)\) > 0 We conclude that \({TP}_{11}^{*}\) at the point \(t_{1}^{*}\) and \(T^{*}\) is a maximum. Therefore equation (27) (29) (30) and (31) satisfied the Hessian matrix. Therefore \({TP}_{11}\) is negative definite.
The necessary conditions for T\(P_{12}\) to be maximized are \(\frac{\partial{TP}_{12}}{\partial t_{1}} = 0\ ,\ \frac{\partial{TP}_{12}}{\partial T} = 0.\ \)
Using equation (28)
\(\frac{{\partial TP}_{12}}{\partial T} = \frac{1}{T}\ \left\{ P\lbrack\sigma a + \sigma bT\rbrack - C\left\lbrack \sigma a\left( T - t_{1} \right) + \frac{\sigma b}{6}\left( 3T^{2} - 3t_{1}^{2} \right) \right\rbrack - c\left\lbrack (1 - \sigma)(a + bT) \right\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - {TP}_{12} \right\}\) = 0 (34)
And
\(\frac{{\partial TP}_{12}}{\partial t_{1}} = \frac{1}{T}\ \left\{ P\left\lbrack \left( a + {bt}_{1} \right) - \left( \sigma a + \sigma{bt}_{1} \right) \right\rbrack + c\left\lbrack - \frac{1}{\theta^{2}}\left( ( - \theta b)e^{\theta\left( t_{1} - t_{d} \right)} + \theta\left( b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right)\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta be^{\theta\left( t_{1} - t_{d} \right)} \right)t_{d} + \frac{1}{\theta^{2}}\left( b - \theta a - {\theta bt}_{1} \right) - \left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} + b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}(b - \theta a - {\theta bt}_{1}) - (b - \theta\left( a + {bt}_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \ b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\left\lbrack (1 - \sigma)\left( - a - {bt}_{1} \right) \right\rbrack - c\left\lbrack \sigma a\left( - T + t_{1} \right) + \frac{\sigma b}{6}\left( - 6Tt_{1} + 4t_{1} \right) \right\rbrack - c\left\lbrack - \left( \sigma a + \sigma bt_{1} \right) - \frac{1}{\theta^{2}}\left( ( - \theta b)e^{\theta\left( t_{1} - t_{d} \right)} + \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) \right\rbrack \right\} = 0\) (35)
The solution to the non-linear equations (34) and (35) gives the value of \(t_{1}\) and T. To confirm that the optimal solution \(\left( t_{1}^{*},\ T^{*} \right)\) exist and unique, we show that the determinant of the corresponding hessian matrix of the profit function is negative definite \(\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} \right) < 0,\ \left( \frac{{\partial^{2}TP}_{12}}{\partial T^{2}} \right) < 0\) then
\(\left( \frac{{\partial^{2}TP}_{12}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{12}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}\partial T} \right)|_{\left( t_{1}^{*}T^{*} \right)} > 0\)
Using equation (34)
\(\frac{{\partial^{2}TP}_{12}}{\partial T^{2}} = \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack - 2\frac{{\partial TP}_{12}}{\partial T} \right\}\) (36)
Evaluating equation (36) at \(\left( t_{1}^{*}T^{*} \right)\)
\(\frac{{\partial^{2}TP}_{11}\left( t_{1}^{*}T^{*} \right)}{\partial T^{2}} < 0\) (37)
Now using equation (35)
\(\frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} = \frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack + c\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta^{2}(b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\}\) (38)
Evaluating (38) at \(\left( t_{1}^{*}T^{*} \right)\)
\(\frac{{\partial^{2}TP}_{12}\left( t_{1}^{*}T^{*} \right)}{\partial t_{1}^{2}} < 0\) (39)
using equation (34)
\(\frac{{\partial^{2}TP}_{12}}{\partial{T\partial t}_{1}} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) (40)
Also, from equation (35)
\(\frac{{\partial^{2}TP}_{12}}{\partial t_{1}\partial T} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) (41)
Using equations (40) and (41)
\(\left( \frac{{\partial^{2}TP}_{12}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{12}}{{\partial t}_{1}\partial T} \right) = \ \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} > 0\) (42)
Using equations (37), (39) and (42)
\(\left( \frac{{\partial^{2}TP}_{12}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{12}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}\partial T} \right) = \left( - \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack \right\} \right)(\frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack + c\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta^{2}(b - \theta\left( a + bt_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})\left( 1 + I_{e}(M - N) \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\} - \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} > 0\) (43)
Since \(\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} \right) < 0,\left( \frac{{\partial^{2}TP}_{12}}{\partial T^{2}} \right) < 0\ and\left( \frac{{\partial^{2}TP}_{12}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{12}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{12}}{\partial t_{1}\partial T} \right)\) > 0 We conclude that \({TP}_{12}\) at the point \(t_{1}^{*}\) and \(T^{*}\) is a maximum. Therefore, equations (36), (38), (40), and (41) satisfied the hessian matrix. then \({TP}_{12}\) is negative definite.
The necessary conditions for T\(P_{2}\) to be maximized are \(\frac{\partial{TP}_{2}}{\partial t_{1}} = 0\ ,\ \frac{\partial{TP}_{2}}{\partial T} = 0.\ \)
Using equation (31)
\(\frac{{\partial TP}_{2}}{\partial T} = \frac{1}{T}\ \left\{ p\lbrack\sigma a + \sigma bT\rbrack - C\left\lbrack \sigma a\left( T - t_{1} \right) + \frac{\sigma b}{6}\left( 3T^{2} - 3t_{1}^{2} \right) \right\rbrack - c\left\lbrack (1 - \sigma)(a + bT) \right\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - {TP}_{2} \right\}\) = 0 (44)
Also,
\(\frac{\partial{TP}_{2}}{\partial t_{1}} = \frac{1}{T}\ \left\{ P\left\lbrack \left( a + {bt}_{1} \right) - \left( \sigma a + \sigma{bt}_{1} \right) \right\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}\left( \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta be^{\theta\left( t_{1} - t_{d} \right)} \right)t_{d} + \frac{1}{\theta^{2}}\left( b - \theta a - {\theta bt}_{1} \right) - \left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} + b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}(b - \theta a - {\theta bt}_{1}) - (b - \theta\left( a + {bt}_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \ b\left( e^{\theta\left( t_{1} - t_{d} \right)} - 1 \right) \right\rbrack - c\left\lbrack (1 - \sigma)\left( - a - {bt}_{1} \right) \right\rbrack - c\left\lbrack \sigma a\left( - T + t_{1} \right) + \frac{\sigma b}{6}\left( - 6Tt_{1} + 4t_{1} \right) \right\rbrack - c\left\lbrack - \left( \sigma a + \sigma bt_{1} \right) - \frac{1}{\theta^{2}}\left( ( - \theta b)e^{\theta\left( t_{1} - t_{d} \right)} + \theta\left( b - \theta\left( a + {bt}_{1} \right) \right)e^{\theta\left( t_{1} - t_{d} \right)} \right) \right\rbrack \right\} = 0\) (45)
The solution to the non-linear equations (44) and (45) gives the value of \(t_{1}\) and T. To confirm that the optimal solution \(\left( t_{1}^{*}T^{*} \right)\) exist and unique, we show that corresponding hessian matrix \(\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} \right) < 0,\) \(\left( \frac{{\partial^{2}TP}_{2}}{\partial T^{2}} \right) < 0,\ and\) \(\left( \frac{{\partial^{2}TP}_{\ \ \ \ \ \ \ 2}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{2}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}\partial T} \right)|_{\left( t_{1}^{*}T^{*} \right)} > 0\)
Using (44),
\(\frac{{\partial^{2}TP}_{2}}{\partial T^{2}} = \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack - 2\left\{ \frac{{\partial TP}_{2}}{\partial T} \right\} \right\}\) (46)
Evaluating equation (46) at \(\left( t_{1}^{*}T^{*} \right)\)
\(\frac{{\partial^{2}TP}_{2}}{\partial T^{2}}|_{\left( t_{1}^{*}T^{*} \right)\ } < 0\) (47)
Using equation (45),
\(\frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} = \frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\}\) (48)
Evaluating equation (48) at \(\left( t_{1}^{*}T^{*} \right)\)
\(\frac{{\partial^{2}TP}_{2}\left( t_{1}^{*}T^{*} \right)}{\partial t_{1}^{2}} < 0\) (49)
Using equation (44)
\(\frac{{\partial^{2}TP}_{2}}{\partial{T\partial t}_{1}} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) since \(\frac{{\partial TP}_{2}}{\partial t_{1}} = 0\) (50)
Also, from equation (45)
\(\frac{{\partial^{2}TP}_{2}}{\partial t_{1}\partial T} = \frac{1}{T}\left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}\) (51)
\(\left( \frac{{\partial^{2}TP}_{2}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{2}}{{\partial t}_{1}\partial T} \right) = \ \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} > 0\) (52)
Using equations (47), (49), (50) and (51)
\(\left( \frac{{\partial^{2}TP}_{2}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{2}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}\partial T} \right) = \left( - \frac{1}{T}\ \left\{ P\lbrack\sigma b\rbrack - c\lbrack\sigma a + \sigma bT \right\rbrack - c\left\lbrack b(1 - \sigma) \right\rbrack - c\lbrack\sigma b\rbrack \right)\left( \frac{1}{T}\left\{ P\lbrack b - \sigma b\rbrack - h\left\lbrack \frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right)e^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)})t_{d} + \frac{1}{\theta}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\theta\left\lbrack \frac{1}{\theta^{2}}( - \theta b) - \theta(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} + \theta be^{\theta\left( t_{1} - t_{d} \right)} \right\rbrack - c\left\lbrack ( - b)(1 - \sigma) \right\rbrack - c\left\lbrack \sigma a + \frac{\sigma b}{6}(4 - 6T) \right\rbrack - c\left\lbrack - (\sigma b)\frac{- 1}{\theta^{2}}(\theta^{2}(b - \theta\left( a + bt_{1} \right){)e}^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)} - \theta^{2}be^{\theta\left( t_{1} - t_{d} \right)}) \right\rbrack \right\} \right) - \ \left( \frac{1}{T}\ \left\{ c\sigma\left\lbrack a + bt_{1} \right\rbrack \right\}^{2} \right) < 0\) (53)
Since \(\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} \right) < 0,\ \left( \frac{{\partial^{2}TP}_{2}}{\partial T^{2}} \right) < 0\ and\left( \frac{{\partial^{2}TP}_{2}}{\partial T^{2}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}^{2}} \right) - \left( \frac{{\partial^{2}TP}_{2}}{\partial{T\partial t}_{1}} \right)\left( \frac{{\partial^{2}TP}_{2}}{\partial t_{1}\partial T} \right) > 0\) We conclude that \({TP}_{2}\) at the point \(t_{1}^{*}\) and \(T^{*}\) is a maximum. Therefore, equation (47), (49), (50) and (51) satisfied the hessian matrix. Therefore, \({TP}_{2}\) is negative definite.
Consider an inventory system with the following input parameters. A=150; a=200; b=3; P=150; h=10; M=40; N=35; c=10; Ie=0.4; td=25; \(\sigma\)=0.8; r = \(\theta\)=0.2 and td=30 for Case 2.
Using MAPLE software based on the parameter values provided for case 1 (scenario 1), it was found that \(\mathbf{t}_{\mathbf{1}}^{\mathbf{11*}}\) =24.5070, \(\mathbf{T}_{\mathbf{11}}^{\mathbf{*}}\)= 31.7335 and \(\mathbf{TP}_{\mathbf{11}}^{\mathbf{*}}\)= $16000, which showed that offering customers with trade credit and not accepting partial payments can lead to higher profitability. The result for all the cases is provided below:
Table 1: Results for the numerical Analysis for all cases.
| CASES | \[\mathbf{t}_{\mathbf{1}}\] | T | TP (in dollars) |
|---|---|---|---|
| Case 1 (scenario 1) | 25.2374 | 36.2394 | 8200 |
| Case 1 (scenario 2) | 24.5070 | 31.7335 | 16000 |
| Case 2 | 29.1121 | 40.6600 | 7200 |
The result shows that when a partial payment is not acceptable at the expiration of initial agreed permissible delay in payment period, N, but complete payment later at M>N, presents the optimal situation with \(\mathbf{t}_{\mathbf{1}}\mathbf{= 24.5070}\), \(\mathbf{T} = 31.7335\) and a profit of $16000. This implies that the retailer should always ensure items are fresh to achieve higher profits. This is because the optimal case is when a large chunk of goods has been sold before deterioration sets in, and the cycle ends immediately once items begin to deteriorate. Meaning less inventory in the store during the period. The results in Table 1 are presented in Figures 2, 3, and 4.
Figure 2: Graphical Representation of the Numerical Analysis for Case 1 (scenario 1).
Figure 3: Graphical Representation of the Numerical Analysis for Case 1 (scenario 2).
Figure 4: Graphical Representation of the Numerical Analysis for Case 2.
The sensitivity is performed on some parameters A, h, N, \(\sigma\), and \(\theta\). The result is represented in Table 2, 3 and 4.
Table 2: Sensitivity analysis result of case 1(Scenario 1)
| Parameter | % change in parameter | New \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | % in \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | New \(\mathbf{T}^{\mathbf{*}}\) | % in \(\mathbf{T}^{\mathbf{*}}\) | New \(\mathbf{TP}_{\mathbf{11}}^{\mathbf{*}}\) | % in \(\mathbf{TP}_{\mathbf{11}}^{\mathbf{*}}\) |
|---|---|---|---|---|---|---|---|
| +20 | 25.2374 | 0 | 36.2399 | 0.0014 | 8200 | 0 | |
| +10 | 25.2374 | 0 | 36.2396 | 0.0006 | 8200 | 0 | |
| A | +0 | 25.2374 | 0 | 36.2394 | 0 | 8200 | 0 |
| -10 | 25.2373 | 0.0004 | 36.2392 | -0.006 | 8200 | 0 | |
| -20 | 25.2373 | 0.0004 | 36.2389 | -0.0014 | 8200 | 0 | |
| +20 | 24.7162 | -2.0652 | 37.6661 | 3.9369 | 4000 | -51.2195 | |
| +10 | 24.9598 | -1.1100 | 36.9564 | 1.9785 | 6000 | -26.8293 | |
| h | +0 | 25.2374 | 0 | 36.2394 | 0 | 8200 | 0 |
| -10 | 25.5583 | 1.2715 | 35.5192 | -1.9873 | 10.400 | 26.8293 | |
| -20 | 25.9362 | 2.7689 | 34.8023 | -3.9655 | 12,500 | 52.4900 | |
| +20 | 24.3860 | -3.3736 | 34.3078 | -5.3301 | 10400 | 26.8293 | |
| +10 | 24.8447 | -1.5560 | 35.5324 | -1.9509 | 8800 | 7.3170 | |
| N | +0 | 25.2374 | 0 | 36.2394 | 0 | 8200 | 0 |
| -10 | 25.5805 | 1.3595 | 36.4997 | 0.7183 | 8400 | 2.4390 | |
| -20 | 25.8830 | 2.5581 | 36.3514 | 0.3091 | 9400 | 14.6341 | |
| +20 | 25.3554 | 0.4676 | 36.4463 | 0.5709 | 10200 | 24.3902 | |
| +10 | 25.2963 | 0.2337 | 36.3468 | 0.2964 | 9200 | 12.1951 | |
| \[\sigma\] | +0 | 25.2374 | 0 | 36.2394 | 0 | 8200 | 0 |
| -10 | 25.1787 | -0.2326 | 36.1214 | -0.3256 | 7200 | -12.1951 | |
| -20 | 25.1203 | -0.4640 | 35.9884 | -0.6929 | 6000 | -26.8293 | |
| +20 | 25.0214 | -0.8559 | 36.0178 | -0.6115 | 8200 | 0 | |
| +10 | 25.1118 | -0.4977 | 36.1114 | -0.3532 | 8200 | 0 | |
| \[\theta\] | +0 | 25.2374 | 0 | 36.2394 | 0 | 8200 | 0 |
| -10 | 25.4158 | 0.7069 | 36.4179 | 0.4923 | 8200 | 0 | |
| -20 | 25.6763 | 1.7395 | 36.6732 | 1.1970 | 8200 | 0 |
Table 3: Sensitivity analysis result of case 1 (scenario 2)
| Parameter | % change in parameter | New \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | % in \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | New \(\mathbf{T}^{\mathbf{*}}\) | % in \(\mathbf{T}^{\mathbf{*}}\) | New \(\mathbf{TP}_{\mathbf{12}}^{\mathbf{*}}\) | % in \(\mathbf{TP}_{\mathbf{12}}^{\mathbf{*}}\) |
|---|---|---|---|---|---|---|---|
| +20 | 24.5071 | 0.0004 | 31.7341 | 0.0019 | 16000 | 0 | |
| +10 | 24.5071 | 0.0004 | 31.7337 | 0.0006 | 16000 | 0 | |
| A | +0 | 24.5071 | 0 | 31.7335 | 0 | 16000 | 0 |
| -10 | 24.5070 | 0 | 31.7332 | -0.0009 | 16000 | 0 | |
| -20 | 24.5070 | -0.0004 | 31.7329 | -0.0019 | 16000 | 0 | |
| +20 | 24.1093 | -1.6228 | 33.5389 | 5.6893 | 10000 | -37.5 | |
| +10 | 24.2962 | -0.8602 | 32.6502 | 2.8887 | 12000 | -25 | |
| h | +0 | 24.5070 | 0 | 31.7335 | 0 | 16000 | 0 |
| -10 | 24.7498 | 0.9862 | 30.7888 | -2.9770 | 18000 | 12.5 | |
| -20 | 25.0314 | 2.1398 | 29.8178 | -6.0368 | 20000 | 25 | |
| +20 | 24.3266 | -0.7361 | 33.9358 | 6.9400 | 10,000 | -37.5 | |
| +10 | 24.2662 | -0.3648 | 32.8650 | 3.5656 | 12000 | -25 | |
| N | +0 | 24.5070 | 0 | 31.7335 | 0 | 16000 | 0 |
| -10 | 24.7487 | 0.3501 | 30.5307 | -3.7903 | 18000 | 12.5 | |
| -20 | 25.6720 | 0.6733 | 29.2428 | -7.8488 | 20000 | 25 | |
| +20 | 24.5458 | 0.5663 | 36.6494 | 2.8862 | 16000 | 0 | |
| +10 | 24.5769 | 0.2852 | 32.2301 | 1.5649 | 16000 | 0 | |
| \[\sigma\] | 0 | 24.5070 | 0 | 31.7335 | 0 | 16000 | 0 |
| -10 | 24.4356 | -0.2913 | 31.1309 | -1.8989 | 14000 | -12.5 | |
| -20 | 24.3618 | -0.5925 | 30.3774 | -4.2734 | 14000 | -12.5 | |
| +20 | 24.3909 | -0.4737 | 31.5884 | -0.4572 | 16000 | 0 | |
| +10 | 24.4347 | -0.2950 | 31.6447 | -0.2798 | 16000 | 0 | |
| \[\theta\] | 0 | 24.5070 | 0 | 31.7335 | 0 | 16000 | 0 |
| -10 | 24.6244 | 0.4790 | 31.8722 | 0.4371 | 16000 | 0 | |
| -20 | 24.8150 | 1.2568 | 32.0890 | 1.1203 | 15000 | -6.25 |
Table 4: Sensitivity analysis result of case 2
| Parameter | % change in parameter | New \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | % in \(\mathbf{t}_{\mathbf{1}}^{\mathbf{*}}\) | New \(\mathbf{T}^{\mathbf{*}}\) | % in \(\mathbf{T}^{\mathbf{*}}\) | New \(\mathbf{TP}_{\mathbf{2}}^{\mathbf{*}}\) | % in \(\mathbf{TP}_{\mathbf{2}}^{\mathbf{*}}\) |
|---|---|---|---|---|---|---|---|
| +20 | 29.1122 | 0.0003 | 40.6604 | 0.0010 | 7200 | 0 | |
| +10 | 29.1121 | 0 | 40.6602 | 0.8132 | 7200 | 0 | |
| A | 0 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 |
| -10 | 29.1121 | 0 | 40.6598 | -0.0005 | 7200 | 0 | |
| -20 | 29.1121 | 0 | 40.6596 | -0.0010 | 7200 | 0 | |
| +20 | 28.7201 | -1.3465 | 42.5954 | 4.7600 | 1500 | -79.1667 | |
| +10 | 28.9053 | -0.7104 | 41.6405 | 2.4122 | 4500 | -37.5 | |
| h | +0 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 |
| -10 | 29.3459 | 0.8031 | 39.6529 | -2.4768 | 10000 | 38.8889 | |
| -20 | 29.6154 | 1.7247 | 38.6204 | -5.0162 | 12000 | 77.7778 | |
| +20 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 | |
| +10 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 | |
| N | +0 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 |
| -10 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 | |
| -20 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 | |
| +20 | 29.2750 | 0.5596 | 40.7765 | 0.2865 | 9400 | 30.556 | |
| +10 | 29.1936 | 0.2800 | 40.7164 | 0.387 | 8200 | 13.8889 | |
| \[\sigma\] | 0 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 |
| -10 | 29.0305 | -0.2803 | 40.6084 | -0.1269 | 6000 | -16.6667 | |
| -20 | 28.9487 | -0.5613 | 40.5638 | -0.2366 | 5000 | -30.5556 | |
| +20 | 29.1160 | 0.0134 | 40.6500 | -0.0246 | 7200 | 0 | |
| +10 | 29.1076 | -0.0154 | 40.6471 | -0.0317 | 7200 | 0 | |
| \[\theta\] | 0 | 29.1121 | 0 | 40.6600 | 0 | 7200 | 0 |
| -10 | 29.1394 | 0.0938 | 40.6993 | 0.0966 | 7200 | 0 | |
| -20 | 29.2068 | 0.3253 | 40.7838 | 0.3045 | 7200 | 0 |
It is observed from Tables 2, 3, and 4 that:
As the ordering cost A increases \(t_{1}^{*}\) , T∗ increase while TP∗ remains unchanged for all cases. In real life, when ordering is high, the retailer tends to order more goods, which leads to an an increase in the replenishment cycle time and total profit. But the rise in holding costs offsets fewer orders, keeping the total profit unchanged.
As the holding cost h increases \(t_{1}^{*}\) decreases, T∗ increases, and TP∗ decreases for all cases. In real life, when holding costs rise, retailers tend to order smaller batches, leading to shorter cycle lengths and shorter cycle times, but also to higher inventory costs, which may cause a fall in profit.
As the downstream credit period N increases, \(t_{1}^{*}\ ,T*\ \)𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒s, T∗ while TP∗ increases for case 1 (scenario 1). Then, \(t_{1}^{*}\) decrease, T∗ increases, and TP∗ decreases for case 1 (scenario 2). In real life, an extended trade credit period can decrease cycle time and increase sales volume, leading to higher total profit. But sometimes, the cost associated with higher credit periods can reduce the business total profit.
As the backlogging rate \(\sigma\) increases, \(t_{1}^{*}\), T∗ TP∗ increases for all cases. This aligns with the expectation of real life. When the backlogging rate is high, it can lead to an increase in the replenishment cycle time and the total profit.
As the deterioration rate 𝜃 increases, \(t_{1}^{*}\), T∗ decreases while TP∗ remain unchanged for all cases. In real-life scenarios, when deterioration is high, the cycle time and the replenishment length actually decrease. But since sales remain constant and demand fulfilment is unaffected, profit stays unchanged.
The model effectively captures both operational and financial dynamics by incorporating trade credit, delayed deterioration, and customer behavior during shortages. This study presents an inventory model for non-instantaneous deterioration items under a retailer’s trade credit with partial backlogging. This model builds on the work of Vadana and Sharma (2016) and Malumfashi et al. (2023) by incorporating the retailer’s trade credit and partial backlogging for non-instantaneous, deteriorating items. Based on the numerical examples provided, case 1 (scenario 1) it was found that \(\mathbf{t}_{\mathbf{1}}^{\mathbf{11*}}\) =24.5070, \(\mathbf{T}_{\mathbf{11}}^{\mathbf{*}}\)= 31.7335 and \(\mathbf{TP}_{\mathbf{11}}^{\mathbf{*}}\)=$16000 is the optimal policy with optimal values of the total profit and the decision variables.
For managerial implications, the model shows that it is more cost-effective for the retailer to order items with a longer period of maintaining freshness to earn maximum profit. This is because the optimal case is when a large portion of goods has been sold before deterioration sets in, and the cycle ends immediately once items begin to deteriorate, which means less inventory in the store during the deterioration period. This model offers a more realistic framework that reflects actual market conditions, where trade credit and shortages do not always lead to a loss of sales.
For future research, this can be extended to include multiple items, stochastic demand or multi-echelon supply chains. Also, for a more comprehensive analysis, inflation and discount can be incorporated. Furthermore, the model can be extended to include dynamic pricing strategies under a trade credit policy.
Aliyu, Z. H. (2022). Two-warehouse inventory model for deteriorating items under two-level trade credit financing. Applied Mathematics and Computational Intelligence (AMCI), 11(2), 437–452. Retrieved from [Link]
Babangida, B., & Baraya, Y. M. (2022). A model for non-instantaneous deteriorating items with two phase demand rate, time dependent linear holding cost and shortages under trade credit policy. Abacus (Mathematics Science Series), 49(2), 91–125.
Chandra, K. J., Monalisha, T., Anuj, K. S., & Geetanjali, S. (2020). An inventory model for non-instantaneous deteriorating item under progressive trade credit policy. Revista Investigacion Operacional, 4(6), 804–825.
Dave, U., & Patel, L. K. (1981). (T, Si) - Policy inventory model for deteriorating items with time proportional demand. Journal of the Operational Research Society, 32, 137–142. [Crossref]
Ghare, P. M., & Schrader, G. F. (1963). A model for an exponentially decaying inventory. Journal of Industrial Engineering, 15, 238–243.
Goyal, S. K. (1985). Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society, 36(4), 335–338. [Crossref]
Lee, Y. P., & Dye, C. Y. (2012). An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate. Computers & Industrial Engineering, 63(2), 474–482. [Crossref]
Malumfashi, A. A., Aliyu, Z. H., & Dari, S. (2023). Inventory model for deteriorating items having linear time dependent demand under downstream trade credit. Academy Journal of Science and Engineering (AJSE), 17(1), 60–76. Retrieved from www.academyjsekad.edu.ng
Mandal, B. N., & Phaujdar, S. (1989). An inventory model for deteriorating items and stock-dependent consumption rate. Journal of the Operational Research Society, 40, 483–488. [Crossref]
Maihami, R., & Kamalabadi, I. N. (2012). Joint control of inventory and its pricing for non-instantaneous deteriorating items under permissible delay in payment and partial backlogging. International Journal of Production Economics, 136(1), 116–122. [Crossref]
Mashud, A. H. Md., Khan, Md. A., Sharif Uddin, M., & Nazrul Islam, M. (2017). A non-instantaneous inventory model having different deterioration rates with stock and price dependent demand under partially backlogged shortages. Uncertain Supply Chain Management. [Crossref]
Pal, M., & Maity, H. K. (2012). An inventory model for deteriorating items with permissible delay in payment and inflation under price dependent demand. Pakistan Journal of Statistics and Operational Research, 8(3), 583–592. [Crossref]
Pal, M., & Chandra, S. (2014). Inventory model for non-instantaneous deteriorating items with stock and time dependent demand, price discount and partial backlogging. IAPQR Transactions, 39(2), 153–168.
Pal, M., & Samanta, A. (2018). Inventory model for non-instantaneous deteriorating items with random pre-deterioration period. International Journal of Inventory Research, 5(1), 3–12. [Crossref]
Park, K. S. (1982). Inventory model with partial backorders. International Journal of Systems Science, 13(2), 1313–1317. [Crossref]
Sundararajan, R., Palanivel, M., & Uthayakumar, R. (2019). An inventory system of non-instantaneous deteriorating items with backlogging and time discounting. International Journal of Systems Science: Operations & Logistics, 7(3), 233–247. [Crossref]
Vandana, & Sharma, B. K. (2016). An inventory model for non-instantaneous deteriorating items with quadratic demand rate and shortages under trade credit policy. Journal of Applied Analysis and Computation, 6(3), 720–737. [Crossref]
Wee, H. M. (1995). A deterministic lot-size inventory model for deteriorating items with shortages and a declining market. Computers & Operational Research, 22, 345–356. [Crossref]
Wu, K. S., Ouyang, L. Y., & Yang, C. T. (2006). An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partial backlogging. International Journal of Production Economics, 101, 369–384. [Crossref]