Operational Research in Engineering Sciences: Theory and Applications Vol. 4, Issue 3, 2021, pp. 122-141 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta111221122m * Corresponding author. 9285poojameena@gmail.com (P. Meena), anil.sharma.maths@gmail.com (A. K. Sharma), Ganeshmandha1988@gmail.com (G. Kumar) CONTROL OF NON-INSTANTANEOUS DEGRADING INVENTORY UNDER TRADE CREDIT AND PARTIAL BACKLOGGING Pooja Meena 1, Anil Kumar Sharma 2, Ganesh Kumar 1* 1 Department of Mathematics, University of Rajasthan, Jaipur-302004, India 2 Department of Mathematics, Raj Rishi Govt. College, Alwar, India Received: 12 June 2021 Accepted: 08 November 2021 First online: 11 December 2021 Original Scientific paper Abstract: Inventory management is an extremely difficult task. It has become usual practice for a provider during the last few decades to provide a retailer with a credit term. In this article, a non-instantly degradable product inventory system is built with a price-sensitive demand and a Weibull credit term allocation reduction rate. Some backlogged deficiencies are permitted. The aim is to maximize the total profit by taking three cases into account. Numerical examples, graphical representations and sensitivity analysis demonstrate the application of the approach developed in this study. Key words: Inventory control, Weibull deterioration, price-sensitive demand, trade credit, non-instantaneous deterioration. 1. Introduction Everyday life is a prevalent phenomenon in the deterioration of commodities. Some examples of these things are vegetables, fruits, dairy products, drugs and blood bank. Therefore, the tendency of the object to deteriorate is important to take into account. In the real world, most products have a shelf life that allows them to maintain their quality or their original condition for a period of time. During that period of time, there was no deterioration in the situation. Examples of such foods include vegetables and fruits as well as meat, fish, and seafood. This is referred to as "non-instantaneous deterioration" in the scientific literature. First and foremost, Ghare and Schrader (1963) took an important stride in this approach. Giri et al. (2003) have presented a mathematical methodology for Weibull decreasing items. Ghosh and Chaudhury (2004), as well as Roy and Chaudhuri (2009), proposed inventory systems for perishable commodities that are in low supply. Das et al. (2010) created A model for Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 123 an item with variable quality that takes into account random machine failure. An inventory model for small lots was developed by Das et al. (2011), with regular and overtime works being combined to produce the production rates. Kawale and Bansode (2012) developed an inventory system for perishable items under the influence of time dependent holding cost using Weibull rate of deterioration. Barik et al. (2013) created a mathematical approach for deteriorating items under the influence of inflation. Confident weights of experts were used by Das et al. (2014) as part of an algorithmic method for MAGDM problems. In this topic, For m secondary warehouses (SWs) and one primary warehouse, Das et al. (2015) created a multi-item multi-warehouse inventory model for degrading goods. Mahata et al. (2018) and Muriana (2020) also offered several models. Ghosh et al. (2021) studied an EOQ model with full backorder for perishable commodities with varied advance and delayed payment conditions. Non-instantaneous models were investigated by Ouyang et al. (2006) and Wu et al. (2009). Soni (2013) used trade credit to overcome the problem of non-instantaneously decaying inventories (Table 1). Table 1. Literature summary Authors Price dependent demand Deterio ration Trade Credit Constant Holding Cost Non- Instantaneous Giri et al. (2003) No Yes No No No Ghosh and Chaudhury (2004) No Yes No Yes No Roy and Chaudhuri (2009) No Yes No Yes No Jain and Kumar (2010) No Yes No Yes No Geetha and Udayakumar (2016) Yes Yes No Yes Yes Mahata et al. (2018) No Yes Yes Yes No Singh et al. (2020) No Yes Yes Yes No Halim et al. (2021) Yes Yes No Yes No Present Paper Yes Yes Yes Yes Yes Geetha and Udayakumar (2016) employed advertisement dependent demand, Shaikh and Cárdenas-Barrón (2020), and Udayakumar et al. (2020) developed alternative inventory models for non-instantaneous falling commodities. Models in this direction have also been presented by Ahmad and Benkherouf (2018) and Tripathi and Pandey (2020). Ouyang et al. (2006) introduced a price-dependent inventory system. Goyal and Chang (2009) used stock-based demand to construct inventory policy. Amutha and Chandrasekaran (2013) created an inventory system for perishable items that incorporates the Weibull rate of deterioration and price- based demand. Avinadav et al. (2013), Guchhai et al. (2013), Avinadav et al. (20 14), Feng et al. (2017), and Cheng et al. (2020) followed the work. Halim et al. (2021) devised a strategy for resolving an inventory problem involving decaying products. Sana et al. (2008) developed an inventory model with advertising cost and selling price dependent demand using trade credit. Sarkar (2012) also provided an inventory system for deteriorated commodities purchased on trade credit. By assuming two- level trade credits, Shah et al. (2015) pioneered a novel methodology. Several academics, including Aggarwal and Jaggi (2017), Goyal (2017), and Shah et al. (2017), created several approaches to address inventory problems while taking trade credit into account. Tripathi and Chaudhary (2017) and Singh et al. (2020) employed the Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 124 Weibull deterioration rate to develop distinct inventory models for perishable products with trade credits. Tripathi et al. (2018) suggested mathematical systems with various trade credits. Sundararajan et al. investigated the impact of trade credit under inflation on an EOQ model (2020). Jain and Kumar (2010) created a strategy for perishable items with Weibull deterioration rates and scarcity. Sarkar and Sarkar (2013) improved a partial backlog solution approach for time-dependent perishable commodities. Mishra (2016) and Gupta et al. (2018) employed partial backlogging to generate multiple systems for Weibull degrading goods. Jamal et al. (2017), San-José et al. (2018), Akbar et al. (2019), Rastogi and Singh (2019), and San-José et al. (2020) all made major contributions in this area. Although several researchers have developed inventory models that take the Weibull deterioration rate into account in their work, practitioners have paid less attention to the inclusion of non-instantaneous deterioration. Novelties of present study are as follows: • We focused our efforts on constructing a mathematical system for non- instantaneous Weibull declining products under trade credit. • Demand is thought to be price related. • Shortages are considered partially backlogged are tolerated. • Impact of different input variables is studied. • Concavity of profit functions is shown by graphs. The structure of the paper is in the following format: Segment 2 of this article describes several notations and assumptions. Segment 3 discusses the model formulation. Segment 4 contains the solution technique. Segment 6 demonstrates concavity of profit functions. Segment 7 discusses sensitivity analysis. Segment 8 contains the conclusion. 2. Notations and Assumptions To create the mathematical model, some notations and assumptions are used. 2.1 Notations K The ordering cost /order Q The retailer’s order quantity ( )D p The demand rate m The time in which the item does not decay M Permissible delay period c Purchasing price /unit p Selling price /unit h Unit holding price s Unit shortage cost/order l c Lost sale cost/ unit Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 125 p I Rate of interest payable/dollar/unit time e I Rate of interest earned/dollar/unit time  The time at which the inventory level becomes zero T Replenishment period ( )t Deterioration rate 1 ( )I t Inventory level in period 0 t m  2 ( )I t Inventory level in period m t   3 ( )I t Inventory level in period t T   ( )Z  Total profit  Optimal value 2.2 Assumptions • The replenishment rate is assumed to be limitless. • The time stamp begins at zero. • Shortages that are partially backlogged are allowed. The pace of backlogging is determined by the time required for subsequent replenishing. As a result, during the stock-out period, it is denoted as ( ) ( ) T t B t e − − = , where 0 1  . • Articles within the cycle period cannot be replaced or repaired in any way. • The demand D depends on selling price p and ( ) µD p p −= , 0, 0   . • After the interval [0, ]m the goods begin to deteriorate with the Weibull deterioration rate, ( ) 1; 0, 0t t    −=   . • For a specified term the supplier gives commercial credit to the retailer. 3. Mathematical Formulation During the period  0, m , there is no deterioration. The inventory level in the period  , m  is consumed by both demand and deterioration. In the period  , T , shortages occur which are partially backlogged. The change of inventory level ( )I t in different time durations given by Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 126 ( )1 0, µ dI t p t m dt  − = −   (1) ( ) ( ) 2 1 2 , µ dI t p t I t m dt     − − = − −  (2) ( ) ( )3 , T tµ dI t p e t T dt    − −− = −   (3) The solution of equations (1), (2), and (3) with boundary conditions ( ) ( ) ( )1 2 30 , 0I Q I I = = = , are ( ) 1 ( )I t p t Q   − = − + . (4) ( ) ( ) ( ) ( )1 12 1 µ I t p t t t t          − + +  = − + − − −  +  . (5) ( ) ( ) ( )3 1 2 µ I t p t T t      −   = − − + +    . (6) Using boundary conditions, ( ) ( ) ( )1 2 3,I m I m I T E= = − we get ( ) ( )1 1 1 µ Q p m m m          − + +  = + − − −  +  . (7) ( ) ( )1 2 µ E p T T T      −   = − − + +    (8) The total annual profit/cycle is obtained by including the following: 1. Ordering cost (O) = K . 2. Inventory holding cost (H) ( ) ( ) 1 2 0 m m h I t dt I t dt  = +  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 6 4 6 41 2 1 1 5 2 2 2 2 2 3 1 1 1 1 2 1 µ x x mx x x m m x x x m h m m p x x mm m m p                       + + + + + +   − −  − + −      + = − − − − −      −   + − − +  +     where Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 127 2 1 (3 2 ) y x p b b= + + , 2 (1 ) b x zx b az= + + , 3 2 ( 1) y x p b= + , 4 ( 2)x zax b= + , 5 ( 1) y x p b= + , 6 ( 1)x ax b= + . 3. Purchase cost (P) ( )c Q E= − ( ) ( )1 1 21 1 2 2 µ c T m m m T T p             + +   = − − − + − + −   +    . 4. Sales revenue ® ( ) ( ) ( )  0 T T t p D p dt D p e dt    − − = +  ( ) 1 T µ p e p      − − − = −    . 5. Shortage cost (S) ( )3 T s I t dt  = − ( ) ( ) 2 2 2 3 6 µ s T T p    = − − + . 6. Lost sale cost (L) ( ) ( )( )1 T T t l c D p e dt   − − = − ( )( )1Tl c e T p        − − = − + − . 7. Interest payable (i) When 0 M m  ( ) ( ) 1 1 2 m p M m IP cI I t dt I t dt  = +  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 6 4 6 41 2 1 1 5 2 2 2 2 3 1 1 1 1 2 2 2 2 1 µ x x mx x x m m x x x b cIp m m x x M m M m m p m                       + + + + + +     − −  − + −     +  = − − − −     + − −      + − +   −  +    . (ii) When m M   Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 128 ( ) 2 1p M IP cI I t dt  =  ( ) ( ) ( ) ( ) ( ) ( ) 1 6 4 6 41 2 1 1 5 2 2 2 2 3 1 1 x x mx x x m M x x x cIp b M M x x            + + + +  − − − − −    =   +  + − + −     . (iii) When M T   , 3 0IP = . 8. Interest earned (i) When 0 M m  ( ) 21 0 2 M e eµ p IE pI tD p dt I M p  = = . (ii) When m M   ( ) 22 0 2 M e eµ p IE pI tD p dt I M p  = = . (iii) When M T   ( ) ( ) ( ) 3 0 0 e IE pI tD p dt M D p dt   = + −  2 µ p Ie m p    = −    . The total profit/unit time ( )Z  is written as ( ) ( ) ( ) ( ) 1 2 3 ; 0 ; ; Z M m Z Z m M Z M T           =      ( ) 1 11 R IE O P H S L IP Z T  + − − − − − − = 1 X T = . (9) Where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 1 1 2 1 2 6 4 6 41 2 1 1 5 2 2 2 2 3 1 1 2 1 1 2 2 2 1 T µ µ µ µ p e p X IeM K p p c T m m m T T p x x mx x x m m m x x x p h m m x x m m m p                                         − − + + + + + +     −  = − + −               − − − − + − + −    +     − − − + − − + − − − − − + − − + ( )1 1 1 m   + +                            −       +       Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 129 ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1 6 4 6 41 2 1 1 5 2 2 2 2 3 1 1 1 2 2 3 1 6 1 2 2 2 2 1 Tl µ µ cs T T e T p p x x mx x x m m x x x b cIp m m x x M m m M m m p                                 − − + + + + + +     − − − + − − + −                  − −  − + −     +  − − − − −      + − −    + −  +   −  +                               ( ) 2 22 R IE O P H S L IP Z T  + − − − − − − = 2 X T = . (10) Where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 1 1 2 1 6 4 6 41 2 1 1 5 2 2 2 2 2 1 1 1 2 1 1 2 2 1 2 T µ µ µ µ p e p X IeM p p m m m c K p T T T x x mx x x m m x x x m h m m p x x m m p                                     − − + + + + + +     − = − +             − − −  +   − −     + − + −        − − − + − + − − − − − − + − ( ) ( ) ( ) ( ) ( )( ) 1 1 2 1 2 2 3 6 1 µ Tl m m s T T p c e T p                 + + − −                         −   − +   +         − − − +      − − + −    Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 130 ( ) ( ) ( ) ( ) ( ) ( ) 1 6 4 6 41 2 1 1 5 2 2 2 2 3 1 1 x x mx x x m M x x x cIp b M M x x            + + + +   − − − − −      −    +  + − + −       ( ) 3 33 R IE O P H S L IP Z T  + − − − − − − = 3 X T = . (11) Where ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 1 1 2 1 6 4 6 41 2 1 1 5 2 2 2 2 2 3 1 1 2 1 1 2 2 1 2 T µ µ µ µ p e p X Ie m p p m m m c K p T T T x x mx x x m m x x x m h m m p x x m                                 − − + + + + + +     −   = − + −               − − −  +   − −     + − + −        − − − + − + − − − − − − + ( ) ( ) ( ) ( ) ( )( ) 1 1 2 1 2 2 3 6 1 µ Tl m m m p s T T p c e T p                      + + − −                         −   − − +   +         − − − +      − − + −    4. Solution Procedure The aim of this article is to maximize total profit. The following condition must be fulfilled by ( )iZ  for maximization: ( ) 0 i dZ d   = , 1, 2, 3i = (12) Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 131 Solving the equation (12) for , we get optimal value *  of , for which ( ) * 2 | 0 ; 1, 2, 3 i d Z i d     =  = 5. Numerical Examples Example 5.1 When 0 M m  10000, 10, 20, 100, 0.08, 0.05, 0.125, 2, 6000, 0.9, 0.20, 0.25, 1, .7, 60, 70 e p l K h c p m M µ I I T s c     = = = = = = = = = = = = = = = = Putting these values in equation (9), we obtain the optimal solutions * 1 0.6924 = * 1 2371Z = * 1 67.1094Q = Example 5.2 When m M   10000, 10, 20, 100, 0.05, 0.07, 0.125, 2, 6000, 0.9, 0.20, 0.25, 1, .7, 60, 70 e p l K h c p m M µ I I T s c     = = = = = = = = = = = = = = = = Putting these values in equation (10), we obtain the optimal solutions * 2 0.6930 = * 2 2363.2Z = * 2 67.1989Q = Example 5.3 When M T   10000, 10, 20, 100, 0.05, 0.9, 0.125, 2, 6000, 0.9, 0.20, 0.25, 1, .7, 60, 70 e p l K h c p m M µ I I T s c     = = = = = = = = = = = = = = = = Putting these values in equation (11), we obtain the optimal solutions * 3 0.7335 = * 3 1530Z = * 3 71.2940Q = Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 132 To solve the above examples, MATLAB software is used. 6. Concavity of Profit Functions Figures 1, 2, and 3 depict the concavity of the profit functions, and this finding corresponds to the theoretical idea of a profit functions with concavity. Figure 1. Concavity of profit function 1 ( )Z  Figure 2. Concavity of profit function 2 ( )Z  Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 133 Figure 3. Concavity of profit function 3 ( )Z  7. Sensitivity Analysis and Observations For sensitivity analysis, we have used the preceding cases. The impacts of parameter modifications on optimal values of *, *Z  and *Q in this segment have been studied. The results are summarized in Tables 2, 3 and 4. Observations and Managerial Implications From Table 2, Table 3 and Table 4, we observe that • As we increase the parameter h by 10% and 20% we observe that the total profit * Z remains almost constant and the optimal values of *  and *Q decrease. We can therefore suggest to the firm that they are free to accept any type of lot as long as the profit remains constant in accordance with the above parameters. • Increasing the value of K by 10% and 20% increases the value of * Z very rapidly but the optimal values of *  and *Q decreases. In order to increase profits, a company will increase the value of K parameter. • Enlarge of c by 10% and 20% results increase in * Z and decrease in *Q while it drops the value of *  very sharply. In order to increase profits, a company will increase the value of c parameter. • When p increases by 10% and 20%, it makes a decrease in * Z and an increase in *  but it causes the value of total inventory *Q to drop very rapidly. In order to increase profits, a company will decrease the value of p parameter. Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 134 • When we increase the parameters s and l c by 10% and 20% we see that the optimal values of *  and *Q show the same behavior (they increase) while the total profit remains unchanged. We can therefore suggest to the firm that they are free to accept any type of lot as long as the profit remains constant in accordance with the above parameters. Figures 4, 5, and 6 demonstrate the effect of various factors on profit functions. Table 2. Change in * , * 1 Q and * 1 Z with respect to parameters Parameters %change in parameters * * 1 Q * 1 Z h -20 0.7013 68.0065 2323.9 -10 0.6968 67.5528 2347.6 +10 0.6881 66.6765 2394.1 +20 0.6837 66.2338 2416.9 K -20 0.6924 67.1094 371.041 -10 0.6924 67.1094 1371.0 +10 0.6924 67.1094 3371.0 +20 0.6924 67.1094 4371.0 c -20 0.7415 72.0749 2168.0 -10 0.7169 69.5820 2274.3 +10 0.6680 64.6568 2458.4 +20 0.6437 62.2236 2536.6 p -20 0.6668 78.8904 2922.2 -10 0.6801 72.4239 2624.5 +10 0.7037 62.6383 2151.2 +20 0.7142 58.8202 1957.5 s -20 0.6739 65.2490 2322.3 -10 0.6834 66.2037 2347.3 +10 0.7010 67.9762 2393.6 +20 0.7092 68.8038 2415.0 l c -20 0.6749 65.3494 2327.7 -10 0.6839 66.2539 2350.0 +10 0.7004 67.9157 2391.1 +20 0.7080 68.6826 2410.1 Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 135 Figure 4. Changes in * 1 Z with respect to parameters Table 3. Change in * , * 2 Q and * 2 Z with respect to parameters Parameters %change In parameters * * 2 Q * 2 Z h -20 0.7018 68.0864 2316.0 -10 0.6974 67.6425 2339.8 +10 0.6886 66.7557 2386.4 +20 0.6843 66.3228 2409.2 K -20 0.6930 67.1989 363.2157 -10 0.6930 67.1989 1363.2 +10 0.6930 67.1989 3363.2 +20 0.6930 67.1989 4363.2 c -20 0.7420 72.1570 2161.0 -10 0.7175 69.6729 2266.8 +10 0.6686 64.7449 2450.3 +20 0.6444 62.3204 2528.2 p -20 0.6674 78.9980 2913.5 -10 0.6807 72.5215 2616.3 +10 0.7043 62.7211 2143.7 +20 0.7148 58.8972 1950.3 s -20 0.6745 65.3374 2314.6 -10 0.6840 66.2926 2339.6 +10 0.7016 68.0662 2385.7 +20 0.7097 68.8842 2407.0 l c -20 0.6755 65.4379 2320.0 -10 0.6845 66.3430 2342.2 +10 0.7010 68.0056 2383.2 +20 0.7086 68.7730 2402.1 Meena et al./Oper. Res. Eng. Sci. Theor. Appl. 4 (3) (2021) 122-141 136 Figure 5. Changes in * 2 Z with respect to parameters Table 4. Change in * , * 3 Q and * 3 Z with respect to parameters Parameters %change In parameters * * 3 Q * 3 Z h -20 0.7418 72.1367 1477.1 -10 0.7376 71.7101 1503.7 +10 0.7294 70.8782 1556.0 +20 0.7253 70.4626 1581.7 K -20 0.7335 71.2940 -470.0094 -10 0.7335 71.2940 529.9906 +10 0.7335 71.2940 2530.0 +20 0.7335 71.2940 3530.0 c -20 0.7741 75.4279 1322.1 -10 0.7538 73.3572 1429.7 +10 0.7132 69.2379 1623.0 +20 0.6930 67.1989 1708.8 p -20 0.7092 84.1433 2092.8 -10 0.7220 77.1037 1789.2 +10 0.7440 66.4120 1304.7 +20 0.7536 62.2385 1105.9 s -20 0.7188 69.8044 1492.7 -10 0.7263 70.5639 1511.8 +10 0.7403 71.9843 1547.3 +20 0.7469 72.6551 1563.8 l c -20 0.7200 69.9259 1497.3 -10 0.7269 70.6247 1514.1 +10 0.7398 71.9335 1545.2 +20 0.7457 72.5331 1559.7 Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging 137 Figure 6. Changes in * 3 Z with respect to parameters 8. Conclusion and Future Scope In present article Weibull rate of deterioration is influenced by trade credit and demand depends on selling price. The model is assessed with the variable time and optimized. Stagnant shortages are acceptable with backlogged allowances. Numerical examples and sensitivity analysis illustrate the constructed model. Our findings demonstrate: • Increasing holding cost increases the overall profit. • The optimal overall profit grows as the shortage costs increase. The supplied model can be used to keep inventory of things that do not perish quickly, such as electronic products, and fashion items. In retail trading, the approach is useful for optimizing unit time profit when partial backlogging occurs. Future work in this area will examine freight charges and other factors. This is also applicable when all the parameters are clear and precise, but if there is any uncertainty in the future, we can use fuzzy mathematics to deal with the situation. 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Control of Non-instantaneous Degrading Inventory under Trade Credit and Partial Backlogging Pooja Meena 1, Anil Kumar Sharma 2, Ganesh Kumar 1* 1. Introduction 2. Notations and Assumptions 2.1 Notations 2.2 Assumptions 3. Mathematical Formulation 4. Solution Procedure 5. Numerical Examples 6. Concavity of Profit Functions 7. Sensitivity Analysis and Observations 8. Conclusion and Future Scope References