In a competitive market, the supplier or even retailer must sell his products as soon as possible; to do so, he must employ strategies that allow him to sell the product sooner or find a way to increase his profit
From the known details, in the literature, the model for imperfect quality deteriorating items with two levels of inspection and learning with partial credit payment
The limitations of the developed model is where there would be a possibility of finding defective items after the learning in inspection from the retailer’s side. So, that could be the possible drawback of the model. Similarly, the credit period offers may create some goodwill loss to the retailer among the customer’s side. The promotional dependent demand may cause some loss to the retailer, if the promotion is not utilised properly.
With all this previous literature works, we have analyzed and developed a model for imperfect quality deteriorating items, with promotional dependent demand, learning in in section and offering the partial credit policy offers for old and new customers is not developed anywhere and here we present our model through the following section, in section 1, we have discussed the possibilities of the inventory management strategies over years and current idea and motivation explained, in section 2, in section 3, problem description is given, in section 4, the notations and assumptions of the developed mathematical model is given, in section 5 the inventory level of the model in various time period is given and the profit function is derived in this section. In section 7 and 6, the numerical examples to prove the reliability of the developed model and sensitivity analysis in section 9 to verify the result obtained in numerical example is given. Finally in section 8 , 10 result analysis, conclusion and future research directions are given.
In this article, the three echelon model for imperfect quality deteriorating items, with shortages are developed, as in previous literature they have worked for imperfect production inventory model
1. Single item inventory model for supplierretailercustomer is developed
2. The items deteriorates at a constant rate. The retailer performs an inspection to separate the imperfect quality products from the lot to avoid any inconvenience. There are possible chances for any misclassifcation of products while inspection.
3. To avoid the misclassification, learning from the previous inspection process, in this model inspection rate is to be presented as,
4. Supplier offers the retailer a full credit period of
5. The retailer classifies the clients as good and bad ones who pays the amount while buying as a single payment and the customer who doesn’t paid well at the correct time he offered.
6. Shortages are allowed and completely backordered
7. Demand is dependent on the promotional frequency,
In the inspection, there might be a possibility of errors, that can happen in two ways, there are
Classifying the perfect product as imperfect
Classifying the imperfect product as perfect
The classification can be represented in mathematical form as,
The demand is the sum of perfect items and mistakenly classified as perfect items, by learning process, the lot size can be derived by,
Let
with boundary condition
Before learning, the number of defective items found out in the screening process at time
The inventory level at
By applying b.c’s,
Then, for
where
Profit can be calculated by finding the difference between sales revenue and other costs. The total cost components are given below,
1. Sales Revenue: In this model, sales revenue, is not just a single part, we have to calculate the revenue by calculating the income through perfect goods, presumed income through misclassified item, retailer’s lost income due to bad debt, and finally the revenue obtained through selling of scrap items are given below respectively.
8. Holding cost and Deterioration cost: The holding and deterioration cost for the retailer’s inventory cycle during,
According to credit period offers
Case 1:
Here are some other possibilities regarding the length of the inventory cycle
Here, WKT,
From the diagram, the retailer can get some revenue after
The retailer has to pay the interest between
Retailer’s interest earned during
Retailer has to pay the interest amount between
Retailer also earned some interest between
Interest to be paid by the retailer between
TP1
Retailer’s interest, between
The interest retailer could earn by sales, and since
TP_{2}=
Retailer’s interest for the damaged items, miscalculated as perfect items is,
where,
TP_{3}=
The retailer’s interest have to be paid during this time is given here,
Here,
TP_{4}=
Since the time period is less than
TP_{5}=
To find the solution for the derived mathematical model, and to calculate the global optimal values, which gives the maximum profit to the retailer, with respect to the time
The values of the above referred derivative functions of the respective equations, is concave for all the subcases and the optimal values are obtained and unique.























1000 

600 

0.05 

0.02 

0.1 

0.05 

0.05 

0.01 



0.2 

0.2 

0.4 

0.95 

0.001 

0.001 

283.97 

83340 

0.7 









0.7 
0.7 
0.7 
0.7 
0.7 

0.9225 
0.8965 
0.9215 
0.9202 
0.9345 

2915082453 
2921892347 
2920969058 
2918390864 
2915082453 
With the numerical values given in the table 1, the optimal values of the developed model is obtained in the table 2.
From the obtained results , in the numerical example 1, we have obtained the profit values of the subcases of case 1 and 2, out of this subcase 1.2 has the profit value higher than the other cases with the minimum possible time period of the inventory cycle. The deterioration rate of the inventory model is fixed and constant, for all the possibilities discussed. Here the shortages are allowed in the considerable rate and those are completely backlogged. Considerably, the subcase 1.3 has the second highest profit value and the minimum time period
The inspector’s inspection rate
After learning process, the impact of production of imperfect products may decrease, whether during this time if the inspection time period is higher it would affect the product’s deterioration level
By changing the frequency of advertisement after learning, helps us to sell the products as soon as possible, but if the error
Demand rate






DC 
27992.87 
33387.13 
38715.94 
43980.83 
49183.49 

2917837060 
2918175605 
2918510381 
2918840890 
2919166676 

2921219026 
2921557571 
2921892347 
2922222856 
2922548642 

2920295737 
2920634282 
2920969058 
2921299567 
2921625354 

2917717543 
2918056088 
2918390864 
2918721373 
2919047159 

2914409132 
2914747677 
2915082453 
2915412962 
2915738748 







2,91,97,64,877 
2,91,85,10,381 
2,91,92,28,7674 
2,91,81,56,549 
2,91,76,20,440 

2,91,27,76,394 
2,91,22,40,285 
2,91,15,21,899 
2,91,11,68,067 
2,91,06,31,957 

2,92,22,23,554 
2,92,15,20,731 
2,92,09,69,058 
2,91,85,06,758 
2,91,78,03,936 

2,91,96,45,360 
2,91,91,09,250 
2,91,83,90,864 
2,91,80,37,032 
2,91,75,00,923 

2,91,63,36,949 
2,91,58,00,839 
2,91,50,82,453 
2,91,50,82,453 
2,91,41,92,511 







2860699428 
2869846408 
2878735162 
2887383148 
2895806644 

2864911431 
2874589726 
2884009797 
2893189101 
2902143913 

2863922035 
2873557299 
2882939535 
2892086199 
2901013570 

2860694985 
2869841965 
2878730719 
2887378705 
2895802201 

2858101536 
2867779832 
2877199903 
2886379206 
2895334019 
In general, firms tend to minimize the time period of production, and to reduce the ratio of production of imperfect products, to maximize the profit and to improve their standard in the market. The products must be inspected and have to separate the imperfect products from the perfect ones, in our model, we have considered the deteriorating items, with imperfect quality products present in the lot, the inspector have to inspect carefully the whole lot. But there are always the possibilities to have the error that he can classify the imperfect product as perfect one and perfect product as imperfect one, during this inspection process , if the learning of the inspector can help the firm to avoid the misclassification, it also avoid the uncomfortable situation to the retailer and it induces the customer to buy more, the lot has become null in short period. Shortages are considered in this model, and backlogged. The promotional effort of the demand pattern induces the sales and we also considered two levels of delay in payment it helps the retailer and customer to face the different uncertainties. The solution procedure given in this study is to find the global optimal solution for the firm.
For the future research, one can change the demand pattern as stochastic in nature, price dependent, green product consideration, or some other discount policies and pre payment offers to be discussed. Various shortage situations can be added to make the model more realistic.
Declaration Presented in 9th INTERNATIONAL CONFERENCE ON DISCRETE MATHEMATICS AND MATHEMATICAL MOD ELLING IN DIGITAL ERA (ICDMMMDE2023) during March 2325, 2023, Organized by the Department of Mathe matics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram  624302, Dindigul, Tamil Nadu, India. ICDMMMDE23 was supported by GRIDTBU, CSIR.