introduction of single stage trade credit policy in the inventory system. Huang (2003) extended the concept between supplier and retailer. He assumed that retailer’s offered credit period to the customer is less than the credit period offered by the supplier to the retailer. Jaggi et al. (2008) introduced credit linked demand rate in their model. Chung (2013) implemented the concept of two-level trade credit periods in economic production quantity model with storage capacity constraints. Chung et al. (2014) considered the facility of permissible delay in payment in economic production quantity model for deteriorating items. Giri and Sharma (2016) incorporated the concept of permissible delay in payment in a model where demand rate increases linearly with time. Pal (2018) examined the effect of trade credit policy with partial backlogging of shortages. Giri and Sharma (2019) developed inventory model with partial trade credit policy. Tiwary et al. (2018) introduced two level partial trade credit policy in a three-echelon supply chain associated with perishable items. Mandal et al. (2020) introduces reliability in a production inventory model with two tier credit policy. An inventory model with two stage deterioration under the atmosphere of permissible delay option was formulated by Pal et al. (2021). 2. NOTATIONS
3. BASIC ASSUMPTIONS ·
A single item is considered ·
Time horizon is infinite ·
Retailer has some capital to arrange
set-up cost. But he has to go for bank loan for
ordering quantity .
Banks provide a credit period. After that period bank will charge interest for
remaining unpaid amount. Retailer can enjoy interest from sells revenue up to
the time . ·
Obtained interest rate is less than paid
interest rate. ·
Retailer faces two types of customers. One
category wants to pay immediately after purchase of a thing while another
category wants to avail some time delay for making payment. ·
Retailer offers a credit period to the customer and sells all items by the
time.
From the point of category of customer enjoying permissible delay in payment scheme, if buys item at time arranges payment at time . ·
Retailer’s offered credit period is less
than his cycle length ·
Demand rate is a monotonic function which decreases
linearly with selling price . Exact form of demand rate is with . 4. MODEL FORMULATION WITH JUSTIFICATION
Retailer
ordering quantity is .
This amount depletes at the rate . The
differential equation which governs inventory level of the retailer is given
below. subject to initial condition
(1) Solution is given by
(2) Terminal
condition gives,
(3) Total
holding cost is given as = [
using (2) and (3)] (4) Retailer
goes for bank loan for purchasing amount . So, bank loan amount is . Two cases come up. Either
Case1. or
Case2. . Now, Case 2 can be subdivided into
three Sub Cases on the basis of the fact that retailer
collects his final payment from the customer at time . Possible three Sub Cases are given
below. Sub Case2.1 Sub
Case 2.3 Now we will discuss all situations with
figures. Case 1.
For
immediate payment, retailer enjoys the interest from sales revenue for the
period .
He has to pay interest for the period . Interest
obtained = ,
Interest paid = For
permissible delay in payment scheme, retailer does not enjoy any interest from
sales revenue. Interest paid = Case
2.1:
For
immediate payment, Interest obtained = , Interest
paid = . For
permissible delay in payment scheme, Interest
obtained = , Interest
paid = Case
2.2:
For
immediate payment, Interest obtained = Interest
paid = 0. For
permissible delay in payment scheme, Interest
obtained = Interest
paid = Case
2.3:
Interest
obtained = Interest
paid = 0 For
permissible delay in payment scheme, Interest
obtained = Interest
paid = 0 Retailer’s
average profit = (Selling price – Bank loan- Set up cost- Holding cost +
Interest obtained from sales revenue –Interest paid to the bank) / Cycle
length. Average
profit function of the retailer for different situations is given in the
following table.
5. SOLUTION METHODOLOGY We
first solve the following partial differential equations Then
we get solutions for cycle length and selling price as .
These solutions are optimal if the eigen values of the following Hessian matrix
at are negative. 6. NUMERICAL EXAMPLE Values
of parameters for different situations are given below. Common
parameters to all situations.
Parameter
varies along with situations.
Optimal
results obtained
Now we show three-dimensional plotting of retailer’s
average profit function with respect to two decision variables selling price
and cycle length for Case1, Sub Case 2.1, Sub Case 2.2, Sub Case 2.3 respectively.
All these diagrams ensure maximization of retailer’s
profit function. This section has been performed with the help of MATHEMATICA
SOFTWARE. 7. CONCLUSION In this work we have presented a mathematical model which maximizes retailer average profit depending on decision variables namely cycle time and selling price. Different situations associated with two stage trade credit have been considered. It has been observed from numerical results that retailer profit increases as length of the credit period offered by the bank increases. Here we take demand pattern based on selling price linearly. This work may be extended by taking non-linear demand pattern. Also credit linked as well as selling price dependent demand pattern may be a possible extension of this modelling frame work. REFERENCES Abad, P.L. (1996), Optimal pricing and lot-sizing under conditions of perishability and partial backordering, Management Science, 42(8) :1093-1104. Retrieved from https://doi.org/10.1287/mnsc.42.8.1093 Abad, P.L. (2001), Optimal price and order size for a reseller under partial backordering, Computers and Operations Research, 28, 53-65. Retrieved from https://doi.org/10.1016/S0305-0548(99)00086-6 Chung, K-J, (2013), The EPQ model under conditions of two levels trade credit and limited storage capacity in supply chain management, International Journal of System Science,44(9),1675-1691. Retrieved from https://doi.org/10.1080/00207721.2012.669864 Chung, K-J, Cardenas-Barron, L.E., Ting, P-S., (2014), An inventory model withnon-instantaneous receipt and exponentially deteriorating item for an integrated three layer supply chain under two levels of trade credit, International Journal of Production Economics,155,310-317. Retrieved from https://doi.org/10.1016/j.ijpe.2013.12.033 Giri, B.C., Sharma, S., (2016), Optimal ordering policy for an inventory system with linearly increasing demand and allowable shortages under two levels of trade credit financing, Operational Research, 16(1),25-50. Retrieved from https://doi.org/10.1007/s12351-015-0184-y Giri, B.C., Sharma, S., (2019), Optimising an integrated production-inventory system under cash discount and retailer partial trade credit policy. Journal of System Science : Operations and Logistics,6(2), 99-118. Retrieved from https://doi.org/10.1080/23302674.2017.1371358 Goyal, S.K. (1985), Economic order quantity under conditions of permissible delay in payments, Journal of Operational Research Society,36(4),335-338. Retrieved from https://doi.org/10.1057/jors.1985.56 Harris, F.W. (1913), How many parts to make at once, Factory, The Magazine of Management,10(2) ,135-136. Huang,Y-F.(2003),Optimal retailer's ordering policies in the EOQ model under trade credit financing, Journal of Operational Research Society,54,1011-1015. Retrieved from https://doi.org/10.1057/palgrave.jors.2601588 Jaggi,C.K., Goyal, S.K, Goel, S.K.,(2008), Retailer's optimal replenishment decisions with credit-linked demand under permissible delay in payments, European Journal of Operational Research, 190,130-135. Retrieved from https://doi.org/10.1016/j.ejor.2007.05.042 Mandal, A., Pal, B., Chaudhuri, K.S. (2020), Unrelaible EPQ model with variable demand under two-tier credit financing, Journal of Industrial and Production Engineering,37(7),370-386. Retrieved from https://doi.org/10.1080/21681015.2020.1815877 Pal, B. (2018), Optimal production model with quality sensitive market demand, partial backlogging and permissible delay in payment, RAIRO-Operations Research, 52, 499-512. Retrieved from https://doi.org/10.1051/ro/2017068 Pal, B., Mandal, A., Sana, S.S. (2021), Two-phase deteriorated supply chain model with variable demand and imperfect production process under two-stage credit financing. RAIRO-Operations Research, 55, 457-580. Retrieved from https://doi.org/10.1051/ro/2021008 Tiwary, S., Cardenas-Barron, L.E., Goh, M., Shaikh, A. A., (2018), Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits supply chain, International Journal of Production Economics,200,16-36. Retrieved from https://doi.org/10.1016/j.ijpe.2018.03.006
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