Plane Thermoelastic Waves in Infinite Half-Space Caused Operational Research in Engineering Sciences: Theory and Applications Vol. 5, Issue 2, 2022, pp. 85-98 ISSN: 2620-1607 eISSN: 2620-1747 DOI: https://doi.org/10.31181/oresta210322106s * Corresponding author. nitahshah@gmail.com (N. H. Shah), kavitagalchar1994@gmail.com (K. Rabari), ektapatel1109@gmail.com (E. Patel) A DETERIORATING INVENTORY MODEL UNDER OVERTIME PRODUCTION AND CREDIT POLICY FOR STOCK- AND PRICE SENSITIVE DEMAND FUNCTION Nita H. Shah, Kavita Rabari, Ekta Patel Department of Mathematics, Gujarat University, Ahmedabad-380009, Gujarat, India Received: 28 October 2021 Accepted: 15 March 2022 First online: 21 March 2022 Research paper Abstract: This paper develops an overtime production model for demand rate to be a function of price and stock-level. Companies manufacturing rate fluctuates with the change in stock-level and demand rate. To control the deterioration up to some extent, the system introduces preservation technology investment. The article permits a two- level credit policy for flexible financing. The model calculates feasible profit value under preservation technology investment, production period and selling price. Conclusively, a sensitivity analysis related to different inventory parameters is performed to study the dependency of optimal values on parameters. Key words: Overtime production, Production period, Conservation investment time, Deterioration, Stock- and vending worth dependent time, Sensitivity. 1. Introduction Inventory management is challenging when the model pacts with perishable products. Products such as chemicals, pharmaceuticals, and dairy products are at high risk and deteriorate with time. Henceforth, the products are discarded over time as they cannot satisfy customer's expectations and demands. Consequently, the harm cannot be ignored. One can control the decay rate by preservation technology investment, that is the company may capitalize further to store the goods with better cure and freshness keeping efforts. Instead of a deterioration rate, the system also changes with change in consumption rate. These days’ people are more interested in buying the products from the showroom having large quantities; this supports to lift the consumption rate. Along with stock-level, one can notice that sales of the goods are inversely proportional to the selling price. More will be priceless will be the demand. Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 86 Many inventory models were developed under constant manufacturing rate. But, in reality the order may change according to the situation and demand. Moreover, to grab the attention, the company develops a certain strategy to conceive consumer’s attention. The model deals with a two-level credit system to enhance the sale by permitting them to pay the bill under a certain allowable period. This paper introduces a perishable inventory model, assuming demand rate to be a function of stock and selling price with overtime production. The paper provides managerial insights corresponding to sensitivity analysis. The purpose is to evaluate optimum profit value. Countless scholars functioned for the overtime manufacture rate under constant consumption rate. Thus, for the first time, the study talks the overtime manufacture model for unpreserved goods, taking into consideration the simultaneous outcome of trade credit policy and preservation investments on a company’s revenue function. The paper is designed as: Section 2 contains literature review. Required notations and assumptions are introduced in Section 3. Section 4 discussed inventory model. Computational algorithm is defined in section 5. A numerical example with sensitivity analysis is included in section 6. Finally, section 7 provides a conclusion along with future scope. 2. Literature Review Demand function plays a vital role in making tactics for inventory models. It fluctuates with respect to different parameters like time, quality, promotional offers etc. Large displays in shops tend to grab consumer’s attention. Moreover, the consumption level is directly proportional to the product’s selling price. Setting higher power reduces demand rate. Dey et al. (2018) investigated rebate, stock-level and price dependent demand. They analysis a comparison between static and dynamic rebate. Chang (2013) revisit burwell contribution and put a note for quantity and freight discounts where demand is price-sensitive. Ouyng et al. (2008) developed a non-instantaneous deteriorating problem under stock-dependent demand, considering all unit quantity discount. Jaggi et al. (2017) established perishable products ordering policy under selling price dependent consumption rate. Li et al. (2019) proposed a replenishment policy for perishable goods. Mishra et al. (2018) presented an inventory model under credit periods considering preservation investment and pricing policy. Liu et al. (2015) considered a time-dependent assessing policy where the goods decay with period. Seifert et al. (2013) had reviewed trade credit literature. Yang et al. (2015) examined a deteriorating system seeing conservation strategy and credit plan. Halim et al. (2021) established an overtime strategy for the inventory problems dealing with decay products. Khan et al. (2020) deals with time-sensitive stock cost and advanced payment policy under advertisement scenario. Lee and Dye (2021) developed a simple algorithm to solve replenishment schedules and preservation technology costs for decay products assuming stock-sensitive demand rate. Tsao et al. (2021) proposed a network design problem assuming credit policy and freshness-keeping strategy. The model evaluated the total cost function by continuous approximation. Dye and Hseih (2012) formulated a worsening stock problem with salvation strategies and partial backordering. Mishra et al. (2017) suggested a model for partly and totally tolerable A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 87 backordering under demand as a function of store level. Geetha and Udayamkumar (2015) formulated a credit model for dynamic demand. Nowadays, many business enterprises focus on various promotional strategies to increase their sales. In inventory models, promotional policy is the basic requirement in business scenarios to compete with others. In the trade credit scheme, the seller offers some delay period to the retailer at the same time the retailer also permits the consumer to pay the bill within some permissible time interval to stimulate consumer’s need and demand. During the permissible credit period, retailers can earn interest from the sold products and gather revenue. A high interest is charged, if the amount is not settled at the end of the cycle period. Credit period strategy makes financial sense and allows retailers to settle the account at the last moment of the allowable time. The approach is used to promote the products. Chung et al. (2014) introduced an article, to minimize total price under permissible credit periods for decay items. Soni et al. (2013) developed a stock-sensitive demand function under replenishment and credit policies when stock is limited. Shaikh et al. (2020) studied an EPQ model for credit policy and permissible shortages. The model calculated optimization problems using the gradient method. Pervin et al. (2020) formulated a production model considering safeguarding assets to overcome the decay rate for storage-and cost-related claims. The model evaluated the total profit function for optimal preservation investment and cycle length. Rapolu and Kandpal (2020) designed an inventory model for decay products having three-parameter Weibull distribution. A simple algorithm is generated to evaluate optimal decision variables. The model is developed incorporating advertisement policy and joint assessing policy to increase profit level. Mahata and De (2016) proposed an ordering strategy model for cost sensitive consumption rate. 3. Notations and Assumptions The article uses the following assumptions and notations given in Table 1: Table 1. Notations np C Per unit normal production cost (in $) op C Overtime making charge per piece (in $) d C Worsening rate per piece (in $) P Vending worth per piece (in $) A Set up fee per demand (in $) h Stock charge per element per year (in $)  Preservation investment rate per cycle  Constant deterioration rate, 0 1   Scale demand, 0   Stock-dependent parameter, 0,    R(P, I( t )) Stock and Price sensitive demand Q Order quantity p t production period (in years) T Rotation period (in years) Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 88 ( )I t Inventory at a time t during the interval [0, ]T e I Interest made per year (in $/ year) c I Interest paid per year (in $/year) M Time given by trader to vendor(in years) N Time given by vendor to customer (in years) TP Total profit per unit time (in $/year) • The system is generated for a single product. • The demand rate is a function of products price and stock available and is specified through 0 0 0R(P,I( t )) ( I( t )) P , , , ,         − = +     . • The model is formulated for the overtime production rate. The expression is  o rP ( t ) P ( I(t) R(P,I(t))) = + − + , where 0rP , and 0 1    . • The article does not permit refunding, replacement, or reworking of imperfect products. • The deterioration rate is a constant function. • To overawed destruction rate, the system includes salvation investment. 4. Mathematical Model Here, the model is developed for dynamic production rate, taking into account the rate to be dependent on demand and stock level i.e.,  o rP ( t ) P ( I(t) R(P,I(t))) = + − + for rP to be continuous and ( I(t) R(P, I(t)) − + is overtime manufacturing rate. The demand rate is a non-linear function of product vending worth and storage level. i.e., R(P,I( t )) ( I( t )) P    − = + for 0 0,   . In the beginning, the storage amount is supposed to be zero. The manufacture takes place during period 0t = and continuous up to time p t t= , where the stock reaches its saturation point. Hence, the manufacturing rate terminates at the time p t t= , that becomes zero at the end of cycle period i.e. T due to the simultaneous effect of demand and deterioration. The inventory during the period interval [ ,T] p t and [ ,T] p t is taken as ( )1I t and ( )2I t respectively. The mathematical expression for the different inventory are as follows: ( ) ( )( ) ( ) ( ) ( )( )1 1 o dI t m I t P t R P,R t dt   + − = − (1) ( ) ( )( ) ( ) ( )( )2 2 dI t m I t R P,R t dt   + − = − (2) A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 89 With boundary conditions ( )1 0I t = at 0t = and ( )2 0I t = at t T= . At time pt t= the inventory level preserves the continuity i.e. ( ) ( )1 2p pI t I t= . Using boundary conditions, the solution of equation (1) and (2) are mentioned as follows: ( ) ( ) ( )1 1 1 r Bt P P I t e B    − − + − = − (3) ( ) ( )( )2 1 A T tP I t e F   − −− = − (4) Using the continuity at p t t= , the cycle time is defined as: ( )( )( )1 11 p p At r t A P P eF P T log F B FP e        −− − −−   + − −   = +        (5) where ( )( ) ( )( ) ( )1 1 1F m P ,B m P .        − −= − + = − + − − The order quantity is ( )( ) ( )( )1 2 0 p p t T t Q I t P dt I t P dt       − − = + + +  (6) Moreover, the holding cost for the entire cycle period is given by ( ) ( )1 2 0 p p t T t HC h I t dt I t dt   = +     (7) The production cost, preservation investment cost and ordering cost are as shown: ( ) ( )( )( )1 1 0 pt np r p op PC C P t C I t I t P dt      −  = + − + +    (8) ( ) ( )1 2 0 p p t T t PIC I t dt I t dt   = +     (9) OC A= (10) The cost related to deteriorated items over the entire cycle period is evaluated as ( ) ( )1 2 0 p p t T d t DC C I t dt I t dt    = +     (11) The sales revenue for the proposed inventory model is Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 90 ( )( ) ( )( )1 2 0 p p t T t SR P I t P dt I t P dt       − −   = + + +     (12) The two cases take place under a two-level credit period system i.e., case 1 and case 2. Case 1: M N Here, we have three different possibilities defined as (1.1) 0 p M N t T    , (1.2) 0 p t M N T    , and (1.3) 0 p t T M N    . First, let’s discuss each subcase. Sub-case 1.1. 0 p M N t T    With  0,M , retailers earned some interest over the revenue function defined as ( )( )11 1 0 M e IE PI I t P tdt    −   = +     Here, the retailer will finance all the costs between  M ,T . Hence, the interest charged is calculated as ( ) ( ) ( )11 1 2 p p t T np op c op c M t IC C C I I t dt C I I t dt     = + +            Hence, the entire revenue function is ( )11 11 11 1 TP SR OC PC HC DC PIC IE IC T = − − − − − + − (13) Sub-case 1.2. 0 p t M N T    In this case, the interest is earned for the time interval  0,M , and the interest is charged for the interval  M ,T . The expression for interest made and interest paid are ( )( ) ( )( )12 1 2 0 p p t M e e t IE PI I t P tdt PI I t P tdt       − −     = + + +            ( )12 2 T np c M IC C I I t dt   =      Here, the total profit function in this case is ( )12 12 12 1 TP SR OC PC HC DC PIC IE IC T = − − − − − + − (14) A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 91 Subcase 1.3. 0 p t T M N    One can observe that in this case M T implies that the products are vended formerly the permitted credit period. So, the total interest paid is nil. i.e., 13 0IC .= The interest rate is ( )( ) ( )( )13 1 2 0 p p t T M e t T IE PI I t P tdt I t P tdt T Qdt       − −    = + + + +        The entire revenue function is ( )13 13 13 1 TP SR OC PC HC DC PIC IE IC T = − − − − − + − (15) Case 2: N M Here, we have only one case to discuss .i.e., 0 p N t M T    The retailer must have to pay the charge to the products that are not sold after credit period. Therefore, the interest charge is ( )2 2 T op c M IC C I I t dt=  Here, The interest earned during interval  0,M is ( )( ) ( )( )2 1 2 0 p p t M e t IE PI I t P tdt I t P tdt       − −    = + + +       The entire revenue function is ( )2 2 2 1 TP SR OC PC HC DC PIC IE IC T = − − − − − + − (16) Here, the problem is 11 12 13 2 0 0 0 0 p p p p TP , if M N t T TP , if t M N T Maximize TP TP , if t T M N TP , if N t M T          =           (17) The ultimate aim is to evaluate the entire revenue function related to is to calculate the production period, vending worth, conservation investment cost. Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 92 5. Computational Algorithm: The ambition is to exploit profit level. To solve problem, model uses a classical optimization algorithm. The algorithm is defined as follows: Step 1. Initially, consign some values to different variables. Step 2. Calculate the derivative of equation (17), of all profit functions that vary according to the credit policies with respect to vending worth, preservation technology investment, and production period. That stands, 0, 0, 0 p TP TP TP t P    = = =    (18) equation (18) yields the value of decision variables that are being used in equation (17) to calculate the extreme revenue price. 6. Numerical example and Sensitivity analysis Example 1: Consider  =5000,  =0.2,  =0.03,  =0.2, h =$1.5/unit/year, A =$200/order, 100 r P = , dC =$20/unit, npC =$20/unit, opC =$6.5/unit, 15. = , e I =0.10, c I =0.15, 07k .= , M = 0.0822, N= 0.164. Using these numerical values, the decision variables are P = $82.77/unit,  = 0.307, p t =0.226 years, T = 3.221 years, and TP = $299.82/ year. Figure 1 represents the convex nature of profit function with respect to decision variables selling price, preservation investment cost and production time. Next, we compare two level trade credit cases in Table 2, for given inventory system. Chart amid  and P Chart amid p t and P Chart amid p t and  Figure 1. Concavity of objective function A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 93 Table 2. Shows the possible cases, when the relation between credit period varies. The sensitivity analysis is performed for case 1. A sensitivity of inventory parameters is performed by varying a particular inventory parameter by - 10%,10%,20%, -20%, keeping other variables constant. Figure 2. Change in selling price related to inventory parameters. The purpose in Figure 2 is to exploit the entire revenue function. By an upsurge in deterioration charge, the vending worth increases. Moreover, higher deterioration cost reduces the damage rate which in turn decreases preservation cost. The increase in d C will have a negative impact on profit function. P (Selling price)  (Preservation investment cost) p t (Production time) T (Cycle time) TP (Total profit) 1 0    p case . M N t T 82.77 0.307 0.226 3.221 299.82 2 0    p case . t M N T 81.01 0.264 0.195 2.725 293.03 3 0    p case . t T M N 57.57 0.030 0.186 1.495 378.75 4 0    p case . N t M T 78.15 0.237 0.237 3.104 307.27 Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 94 Figure 3. Change in preservation cost related to inventory parameters. It is depicted from the Figure 3 that a large ordering cost, especially when the manufacturing rate is high, will have a negative effect. The company may introduce some inventory policy to overcome the loss that occurs due to high setup cost. The model uses a credit policy where the delay in payment is permissible up to a certain time. Commonly, the demand for perishable products declines thru period. To avoid a loss that occurs due to damage and spoilage, preservation cost is one of the crucial strategies. An increase in preservation investment cost results in a decrease in profit level. The increase is not advisable. Figure 4. Change in production time related to inventory parameters. A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 95 From Figure 4 it is advised that the company may boost the sale by introducing overtime production techniques. The firm will not face shortages and the products are available whenever needed. Figure 5. Change in cycle time related to inventory parameters. Figure 6. Change in total profit related to inventory parameters. Figure 5 and Figure 6 suggest that the system is temperately delicate to k and r P . The increase in the cost function is not advisable as it tends to decrease the profit level. The entire revenue function is exceedingly delicate toward the scale demand i.e.,  as large volumes provide choices and quality to the customers. It is observed Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 96 that with an increase in credit periods, cycle time increases. The stock sensitive demand helps manufacturers to fluctuate profit amounts and reflects product requirements in the market as well. An increase in deterioration rate tends to reduce cycle time, which ensure that one should order less in case to avoid damage rate. To cure the damage, the company requires more money for preservation technology. The total profit decreases. 7. Conclusion In this paper, the model evaluates overall profit revenue considering the overtime production rate for perishable products. The demand rate is a function of vending worth and stock and selling price taking conservation technology assets to decrease the degree of decline. The model also provides some managerial insights related to key parameters and the problem is solved using classical optimization methods. The goal is to exploit the entire revenue function associated with vending worth, preservation technology investment and production period. To compete with the modern world, the model takes into account credit periods. The revenue parameter is exceedingly sensitive to scale demand. The preservation technology investment will have a positive impact on the deterioration rate and is negatively related to profit function. The overtime production helps to fulfill the demand on time. Most of the researchers who study overtime production model consider demand rate to be a constant function but in reality it depends on certain factors such as price, time and stock. This work addresses the different research questions. What is the effect of deterioration on retailer’s profit? Because of the overtime production process, the company needs to invest more during production time to neglect the situation of stock out. Trade credit policy is being used along with the preservation investment to maximize the profit value. The model can apply to the problems, dealing with perishable products under permissible overtime production, credit periods and preservation investment to control decay rate. The model has some limitations as the problem is solved for constant rate of deterioration due to the complexity of the problem whereas deterioration rate changes with time. The article can further be extended assuming ramp-type demand or introducing advertisement policy. Additionally, one may incorporate multiple products and multiple buyers under allowable shortages. References Chang, H. C. (2013). A note on an economic lot size model for price-dependent demand under quantity and freight discounts. International Journal of Production Economics, 144(1), 175-179. https://doi.org/10.1016/j.ijpe.2013.02.001 Chung, K. J., Cárdenas-Barrón, L. E., and Ting, P. S. (2014). An inventory model with non-instantaneous receipt and exponentially deteriorating items for an integrated three-layer supply chain system under two levels of trade credit. International Journal of Production Economics, 155, 310-317. https://doi.org/10.1016/j.ijpe.2013.12.033 https://doi.org/10.1016/j.ijpe.2013.02.001 https://doi.org/10.1016/j.ijpe.2013.12.033 A Deteriorating Inventory Model under Overtime Production and Credit Policy for Stock- and Price Sensitive Demand Function 97 Dey, K., Chatterjee, D., Saha, S., & Moon, I. (2019). Dynamic versus static rebates: an investigation on price, displayed stock level, and rebate-induced demand using a hybrid bat algorithm. Annals of Operations Research, 279(1), 187-219. Dye, C. Y., and Hsieh, T. P. (2012). An optimal replenishment policy for deteriorating items with effective investment in preservation technology. European Journal of Operational Research, 218(1), 106-112. https://doi.org/10.1016/j.ejor.2011.10.016 Geetha, K. V., and Udayakumar, R. (2015). Optimal replenishment policy for deteriorating items with time-sensitive demand under trade credit financing. American Journal of Mathematical and Management Sciences, 34(3), 197-212. https://doi.org/10.1080/01966324.2014.1000551 Halim, M. A., Paul, A., Mahmoud, M., Alshahrani, B., Alazzawi, A. Y., & Ismail, G. M. (2021). An overtime production inventory model for deteriorating items with nonlinear price and stock-dependent demand. Alexandria Engineering Journal, 60(3), 2779-2786. Jaggi, C. K., Tiwari, S., and Goel, S. K. (2017). Credit financing in economic ordering policies for non-instantaneous deteriorating items with price-dependent demand and two storage facilities. Annals of Operations Research, 248(1-2), 253-280. https://doi.org/10.1007/s10479-016-2179-3 Khan, M. A. A., Shaikh, A. A., Konstantaras, I., Bhunia, A. K., and Cárdenas-Barrón, L. E. (2020). Inventory models for perishable items with advanced payment, linearly time-dependent holding cost, and demand dependent on advertisement and selling price. International Journal of Production Economics, 230, 107804. Lee, Y. P., and 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. https://doi.org/10.1016/j.cie.2012.04.006 Li, G., He, X., Zhou, J., and Wu, H. (2019). Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items. Omega, 84, 114-126. https://doi.org/10.1016/j.omega.2018.05.001 Liu, G., Zhang, J., and Tang, W. (2015). Joint dynamic pricing and investment strategy for perishable foods with price-quality dependent demand. Annals of Operations Research, 226(1), 397-416. https://doi.org/10.1007/s10479-014-1671-x Mahata, G. C., and De, S. K. (2016). An EOQ inventory system of ameliorating items for price-dependent demand rate under retailer partial trade credit policy. Opsearch, 53(4), 889-916. https://doi.org/10.1007/s12597-016-0252-y Mishra, U., Cárdenas-Barrón, L. E., Tiwari, S., Shaikh, A. A., and Treviño-Garza, G. (2017). An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment. Annals of operations research, 254(1), 165-190. https://doi.org/10.1007/s10479-017-2419-1 Mishra, U., Tijerina-Aguilera, J., Tiwari, S., and Cárdenas-Barrón, L. E. (2018). Retailer’s joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments. Mathematical Problems in Engineering, 2018. https://doi.org/10.1155/2018/6962417 Ouyang, L. Y., Wu, K. S., & Yang, C. T. (2008). Retailer's ordering policy for non- instantaneous deteriorating items with quantity discount, stock-dependent demand https://doi.org/10.1016/j.ejor.2011.10.016 https://doi.org/10.1007/s10479-016-2179-3 https://doi.org/10.1016/j.cie.2012.04.006 https://doi.org/10.1016/j.omega.2018.05.001 https://doi.org/10.1007/s10479-014-1671-x https://doi.org/10.1007/s12597-016-0252-y https://doi.org/10.1007/s10479-017-2419-1 https://doi.org/10.1155/2018/6962417 Shah et al./Oper. Res. Eng. Sci. Theor. Appl. 5(2) 2022 85-98 98 and stochastic backorder rate. Journal of the Chinese Institute of Industrial Engineers, 25(1), 62-72. https://doi.org/10.1080/10170660809509073 Pervin, M., Roy, S. K., and Weber, G. W. (2020). Deteriorating inventory with preservation technology under-price-and stock-sensitive demand. Journal of Industrial & Management Optimization, 16(4), 1585. Rapolu, C. N., and Kandpal, D. H. (2020). Joint pricing, advertisement, preservation technology investment and inventory policies for non-instantaneous deteriorating items under trade credit. Opsearch, 57(2), 274-300. Seifert, D., Seifert, R. W., and Protopappa-Sieke, M. (2013). A review of trade credit literature: Opportunities for research in operations. European Journal of Operational Research, 231(2), 245-256. https://doi.org/10.1016/j.ejor.2013.03.016 Shaikh, A. A., Cárdenas-Barrón, L. E., and Tiwari, S. (2020). Economic production quantity (EPQ) inventory model for a deteriorating item with a two-level trade credit policy and allowable shortages. In Optimization and inventory management (pp. 1- 19). Springer, Singapore. Soni, H. N. (2013). Optimal replenishment policies for non-instantaneous deteriorating items with price and stock-sensitive demand under permissible delay in payment. International Journal of Production Economics, 146(1), 259-268. https://doi.org/10.1016/j.ijpe.2013.07.006 Tsao, Y. C., Zhang, Q., Zhang, X., and Vu, T. L. (2021). Supply chain network design for perishable products under trade credit. Journal of Industrial and Production Engineering, 1-9. Yang, C. T., Dye, C. Y., and Ding, J. F. (2015). Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model. Computers & Industrial Engineering, 87, 356-369.ticle. Journal of Scientific Communications, 163, 51–59. https://doi.org/10.1016/j.cie.2015.05.027 © 2022 by the authors. Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1080/10170660809509073 https://doi.org/10.1016/j.ejor.2013.03.016 https://doi.org/10.1016/j.ijpe.2013.07.006 https://doi.org/10.1016/j.cie.2015.05.027