AN INVENTORY MODEL WITH UNCERTAIN DEMAND UNDER PRESERVATION STRATEGY FOR DETERIORATING ITEMS

. The capacity of a firm to accomplish its goals is financially compromised by degeneration of goods. A suitable preservation strategy to reduce degradation is a vital part of the managerial decisions. This study employs preservation technologies under uncertain demand to frame a continuous review inventory model with full back-ordering and the influence of promotional efforts. Survey of existing research finds few models with synchronised optimization over this entire scenario with all factors.The best values of the preservation cost and the two fractions of the cycle period when inventory is kept against the backorder part are determined to lower the total average cost. A mathematical model is built to incorporate these elements and numerical scenarios are presented to compare three possible approaches. In both crisp and fuzzy contexts, the sensitivity of the solution and decision variables concerning various inventory characteristics is investigated. Backorder duration is inversely proportional to the presence of preservation. The coefficient of preservation has a tipping point below which accepting the impact of undamped deterioration becomes more cost-effective. The total cost at the optimal point is more elastic to a reduction in base deterioration rate and relatively inelastic to its increase. Finally, this study proves that the preservation strategy converges over deterioration for the crisp case rather than the fuzzy case. It is expected the fuzzy case can provide better results, however, the crisp case provides lower total cost than the fuzzy case though it is slightly less efficient in per unit cost.


Introduction
In the present scenario, limited-time procedures and showcasing are closely connected.In this pandemic circumstance due to Coronavirus episode, special procedures forced immense changes in the stock framework.The worldwide pandemic (COVID-19) has driven commercial entities towards revamping present and future promotional activities to maintain a healthy flow of financial returns.Limited time exertion might zero in on various objectives like further developed arrangements or make better thoughts among customers.Extension of in-store traffic and a higher inspiration of the staff are likewise potential advantages.The goal is to clear extra stock and make a higher income.Therefore, inventory management is critical for every modern, sophisticated firm.Properly managed inventories pay out in a variety of ways, such as direct profits and customer loyalty.In addition, the interwoven linkages among these diverse business aims maintain inventory management's dazzling importance.The continuous review has attracted more attention than the periodic review because of its mathematical approach and capacity to tackle a wide range of problems.
Disintegration is a character portrayed by deterioration, harm, decay, hurt or other change in item quality because of natural issues during caching.A couple of examples are items like battery, semiconductor chips, food assortments, unstable fluids, clinical things, i.e, blood face deterioration and gradually lose potential.A huge issue in the stock framework is to control and stay aware of inventories of decaying merchandise.The objective of stock administration is to further develop profit from the venture by diminishing stock waste and the deteriorated items contribute negatively towards this target.The pace of decay of products can be seen as a dependent variable, subject to control utilizing protection innovation.Organisations have understood that they must strictly manage the deterioration losses.One of the common avenues of control is to improve and upgrade storage processes.The retailers can decrease the pace of decay of items through viable capital input along these avenues and thus keep away from superfluous waste, limit financial misfortunes, and upgrade business efficiency.Such models concerning deterioration control have gotten a lot of consideration and have greater concurrence with the actual inventory circumstances.
Perishable products especially require precise preservation and inventory control in today's dynamic markets.For example, the deterioration of food will damage the goodwill and reputation of the retailer.As a significant reason for stock misfortune, weakening builds the association's expense and this manner diminishes the benefit.The weakening interaction brought about by activities of chemicals, oxidation and microorganisms often rely upon ecological conditions, like temperature, stickiness, and environment.Temperature has a significant impact on deterioration and must be controlled to maintain products quality.The protection innovation, for example, temperature controlling gear and imaginative bundling, can influence the weakening rate and thus defer the crumbling cycle.
When management launches a new product then, they have imperfect knowledge about the demand and other factors related to the product.The management relies on experts' analysis.When such counsel is imprecise then demand or other factors related to the expert opinion can be modeled using fuzzy principles and the corresponding environment is known as fuzzy environment.
A unique methodology to improve demand forecasting, which is a major difficulty of a continuous review inventory model, is offered in this work.This study further examines the effect of fuzzy demand within an infinite time horizon.An optimal operating strategy is sought for a continuous review inventory system to reduce the total layout in a fuzzy setup.The mechanism of full-backordering is assumed to balance the component of loss during absence of inventory.A continuous review inventory system is numerically solved for both crisp and fuzzy instances, as well as an analysis of the best policy.The decision maker's strategies under uncertain demand can benefit strongly from the results obtained here and reduce the total inventory cost based on preservation strategy.
The latest investigation is motivated by the fact that while there has been numerous research on each of the features, the overall interaction has not been exhaustively investigated.Previous studies that regulated deterioration by implementing functional forms of investment in maintenance techniques have ignored two important aspects.
-The maximum suitable investment in safety techniques ought to not be based totally on the fee of deterioration.Techniques are mostly base on selling price (and therefore unit price) and degradation fee are more feasible and are the strength of this paper.-Protection time values synchronize with the unit of time are not sustainable in opposition to deterioration for all sorts of goods.Preparations or meals require their personal packaging to lessen expiration.Others can boost up the deterioration of numerous easy gadgets withinside the presence of big quantities of rather oxidized products.
Practical considerations, such as attenuated deterioration, demand affected by both fuzzy base and promotional up-scaling, complicate the approach.It has been proposed to consider analytical convexity without approximating the deterioration factor.Because of the infinite time horizon, the continuous domain for cycle length leads to continuous time duration variables.As a result, these aspects of inventory cost minimization form a helpful aim that motivates to tackle this issue vis-a-vis the following research gap: -For exponential dampening preservation, truncation error is introduced through polynomial approximation.This spills over into the accuracy and analysis of the optimal point.For promotional impact, detached uncertainty in base demand is not modelled.-Convexity is not examined in domain-wide analytical outcome.It is usually coordinate limited in process and scope and has other parametric barriers which weaken the mathematical basis.-Choice of time decision variables and arbitrary horizon is often less than ideal reducing managerial flexibility.
The analysis of the two phases of the cycle as impacted variously by deterioration, preservation, and demand uncertainty is often not clearly reflected.
In light of these considerations, the originality and contribution of the current study is that it compares a continuous review inventory model, which allows for full exponential depreciation attenuation analysis, to an inventory model.The retention strategy is designed for a more realistic scheme in which demand uncertainty is modeled with a trapezoidal fuzzy base and a term that scales with promotion efforts.
The decision process of a priori planning requires the forecasting of demand patterns on the retailer or seller's part.The comprehensive performance of the arrangement can be improved by flexibility in managing resources and operations.The first model hereafter uses the pure deterministic scenario where exact demand information is available to the decision-maker and it serves as a valuable mathematical prototype.The second model incorporates the issue of deterioration control through variable preservation investment and its economic ramifications, whereby this additional budgeting eventually leads to enhanced cost optimization.The third model returns to the original contention of flexibility to tackle uncertainty, by using a fuzzy formulation for the inexactly predicted demand.The three models together provide the researcher a clear and concise look into the mathematical procedure and economic justification of incorporating preservation and fuzziness to cope with deterioration and uncertainty respectively.
The structure of the present article's organization is as follows: the literature review is in Section 2, which goes through earlier research work that helps to frame this work.Section 3 contains the research problem and the assumptions that aid in the planning of the model's outline.The notations used are tabulated.Section 4 discusses the modelling framework and solutions for the crisp and fuzzy environments.The applicability of this paradigm in a real-world situation is clarified using numerical examples in the next Section 5.The inventory system's sensitivity analysis and administrative framework are found in Section 6.In Section 7, avenues for furthering this research as well as its findings are discussed.Finally, in Appendix A, a short introduction to the trapezoidal fuzzy numbers is laid out.

Promotional impact
Discounts, credit period, free goods, and after-sale services are some of the common promotional strategies.Practitioners and researchers have investigated the importance of promotional policies.Taleizadeh et al. [49] formulated an inventory system by considering partial backlogging and delayed payment with shortages.Rajan and Uthayakumar [48] considered a deterministic inventory model by considering full backorder and holding cost that grow exponentially under permitted payment delays and promotional effort.Other researchers (Mahapatra et al. [15], Soni and Suthar [47], Liao et al. [14]) had considered inventory systems with demand sensitive to promotional strategies.Nouri et al. [20] analysed the benefit of investing for promotion in a supplier-retailer chain model where demand was stochastic.Recently, Pal [23] established optimal pricing while offering incentive decisions for a supply chain model.Zhou et al. [55] formulated an inventory system by calculating joint pricing by considering a deep reinforcement learning algorithm.Mahapatra et al. [16] formulated three different types of economic order quantity (EOQ) model by considering time-dependent deterioration rate, promotional effect, preservation technology, and uncertainty learning.An inventory system's promotional activities effects the whole system's sales volume, and hence on the earnings of other members.As a result, in the inventory model, it's important to organize promotional decisions.

Deterioration
Deterioration has been deeply explored in inventory management.Food spoilage through microbes or oxidation is ubiquitous.Electronics goods storage must account for damage through electrostatic discharge, moisture, and contaminants.Pervin et al. [24] determined EOQ model for perishable goods with demand varying with stock level and holding costs that was time-dependent.Pervin et al. [25] designed a multi-item inventory model that took into account trade credits, on-demand, and constant rate of deterioration.Barman et al. [1] analysed an economic production quantity (EPQ) model with shortages and inflation in a fuzzy environment using fixed rate of deterioration and incorporating time-dependent demand.Roy et al. [30] developed a two warehouse, two credit level, and probabilistic setup for deteriorating items.Roy et al. [31] proposed an imperfect production system with partial backlog under a credit policy for a deteriorating product.In the practical scenario, it is seen that variable deterioration rates better represent the rate of failure and the life span of various goods.Many practitioners like (Ouyang et al. [22], Soni and Patel [46], Zhang et al. [54], Mishra et al. [18]) worked with this phenomenon but in different directions.By taking into account trade credit policy, partial backlog, and preservation strategy, Shaikh et al. [44] developed an inventory control system for deteriorating product by considering time-varying demand.Khan et al. [11] analysed a setup with constant deterioration and variable demand pattern along with delayed payment.Recently, Shah et al. [43] investigated the case where the goods deteriorate and their demand fluctuates with the selling price by considering the greening effect.

Preservation investment
Financially, the weakening of items contrarily affects the accounts and brand image of a firm.High deterioration rates are entitled to higher yearly expenses, deficiencies, and lost deals.On this record, business associations are keen on appreciating the reasons for crumbling and creating ways to deal with save their delivered merchandise and increment benefit.Antimicrobial (e.g.sulfites in fruits) and antioxidant (e.g.butylated hydroxytoluene in edible oils) agents are commonly employed in a wide range of biologically derived products.Recent advances have employed hydrophilic and lipophilic nanosystems for edible preservative coating in food.Clean-room paradigms are used to control deterioration in pharmaceuticals and electronics sectors.
Dye and Hsieh [5] applied preservation procedures to control the deterioration rate for stored goods.Jani et al. [10] explored a production inventory system for lifetime-dependent products by using preservation technology.Pervin et al. [26] utilised an EPQ system with decaying products, stock preservation technology, and demand that was sensitive to the selling price, with partial backlog.Both preservation and inspection policy are employed in Pervin et al. [27] to optimise a quadratic demand vendor-buyer model.Das et al. [3] proposed a multi-period credit policy and stock and price varying demand for an item with noninstantaneous spoilage with preservation.Shah et al. [42] formulated a model for optimum inventory under stock-dependent rate of depletion with a constant degree of deterioration controlled by preservation investment.Priyamvada et al. [28] took into account the deterioration of products under the effect of cost and stock ward.They limited deterioration using preservation processes.Priyamvada et al. [29] proposed a realistic preservation formulation that varies the investment with the unit price and inventory level.The impact of strategic inventories on a supply chain was studied by Saha et al. [32] who took into account how two producers or a common retailer and a manufacturer could collaborate through wholesale pricing in the supply chain.Shah et al. [41] developed an EPQ model for deterioration products as price sensitive demand to reduce emissions by using green technology and preservation investment.Sarkar et al. [34] considered a variable demand-based supply chain model for sustainable manufacturing and investment for autonomation.Recently, Sarkar et al. [38] presented a substitutable product manufacturing process through a dual-channel and ensure environmental sustainability by using carbon tax along with carbon cap.Reducing waste and meeting emissions targets are important goals for inventory systems.Sarkar et al. [37] introduced a multi-phase production system to regulate defective products and reduce waste through a reworking process in order to optimize profit.Kumar et al. [13] designed a supply chain model for manufacturers and consumers that was based on demand and the carbon emissions process.Various sustainability measures were useful investments in keeping with the biodegradable products without preservation philosophy as considered recently by Sarkar et al. [35].

Fuzzy modeling of uncertainty
Apart from promotional efforts, decaying commodities, and preservation technology, demand uncertainty stems from the inventory model's many unknown aspects.However, in real-world settings, the uncertain parameters like demand, various relevant expenses, lead time, preservation cost, may have a higher possibility of deviating from the exact value, leading to a situation in which these uncertain parameters do not follow any probability distribution.Initially, Zadeh [53] first developed the concept of fuzzy set.After that many pioneer's researchers like (Yao et al. [52], Glock et al. [7], Shah and Soni [39]) captured the impreciseness by developing various fuzzy inventory models.The model analyzed by Garai et al. [6] had holding cost that scaled with time and price dependent demand by considering trapezoidal fuzzy numbers.Shah and Patel [40] used preservation technology and developed a model for inventory to reduce rate of spoilage under a cloud fuzzy prescription.Yadav et al. [51] considered an flexible production system with a variable pollution control for a fuzzy environment.A learning environment for dense fuzzy demand was employed by De and Mahata [4] under the overlap of order with rework batches.Time varying demand for decaying items was modeled by Kumar and Paikray [12] in crisp and fuzzy formulations with three different scenarios under total backlogging.A multithreaded neural network was efficiently employed by Sarkar et al. [36] to tackle a fuzzy inflationary model.

Research problem
The proposed inventory model herein aims to determine the lowest total cost per unit time for the cases of crisp and fuzzy demand by computing the most effective replenishment policy and optimal investment in preservation technology for a deteriorating item.A model without preservation is used as a baseline formulation to compare and contrast with the goal of the model.Figure 1 is an illustrative inventory profile of the model.In the three models, most of the parameters are common while some are specific to the model as required.The assumptions made are itemised below.The decision variables and other specifications are laid out in Table 2.

Assumptions
(1) The demand rate is () =  0 +  1 , where  0 is fixed base demand while  1  measures the effect of promotional activity  with  1 as a positive, constant scaling term.Here, in model-3 it is assumed that   0 is a trapezoidal fuzzy number (TpFN) as follows: (2) A single type of item is analyzed in this model, as well as shortages and full backlogs.When replenishment occurs, backlogged demand is initially met.(3) The constant deterioration rate is  0 (0 <  0 < 1).Deterioration is proportional to the inventory.Per unit time, a percentage of goods in the inventory deteriorate.Repair or replenishment of such items does not occur within a cycle.(4) The promotional effort cost (PEC) is   which collect from the leading results of the cost-of-promotion function, where  and  are constant.(Mahapatra et al. [15], Soni and Suthar [47]) (5) The impact of preservation investment  per unit time is that it reduces the effective rate of deterioration and is modeled by a term using a strictly monotonic increasing function,  0 (1 −  − ) for  ≥ 0, where  = positive constant.This ensures that there is no impact when  = 0 and as  increases, the effect increase but is bounded by  0 , i.e., the maximum impact as  → ∞ is  0 (Shah et al. [42]).Realistically, the decision maker has budgetary limits for preservation investment and hence an upper bound  ≤  , is a necessary numerical constraint.(6) The effective deterioration rate after subtracting the impact of the preservation cost is ,  →  0 if  → 0 (when without preservation) and  → 0 if  → ∞ (the limiting case where deterioration fully reduced by preservation).( 7) The lead time is negligible together with infinite replenishment rate and the order quantity is finite.

Model development
The models and their solution methodology are laid out in this section.The governing differential equations are setup and solved to obtain the total cost function.Necessary and sufficient criterion for convexity and global optimality are imposed on this objective function.Defuzzification is used when fuzzy parameters are employed.

A Continuous review inventory model with crisp demand and constant deterioration rate (Model-1)
A continuous review EOQ setup is constructed with the above assumptions.An instantaneous replenishment starts the cycle at  = 0 and the inventory level jumps to its maximum value,  = (0).() reduces in time interval [0,   ] as units are consumed to fulfill the demand while some are lost to deterioration.All units are used up at  =   so (  ) = 0. Backorders are kept during the period [  ,   +   ] which are fully backlogged to be satisfied from the next replenishment.In Model-1 there is no preservation investment ( = 0).Based on these assumptions, the differential equations governing by the following cases: Case 1 ( 0 ≤  ≤   ): Consumption due to demand and loss due to deterioration depletes the inventory, hence the equation governing the inventory level () is The deterioration term () is proportional to the existing on-hand inventory and  =  0 is a constant in Model-1.In the differential equation (4.1), the boundary condition (  ) = 0 is applied to obtain the level of inventory Using equation (4.2) the highest inventory level  is at  = 0 as follows: Now, equation (4.3) is different compared to the case of no deterioration where  =   .It is exactly this difference which gives the amount of inventory lost due to deterioration, as in the buyer's inventory model considered in Wee et al. [50], given by Case 2 (  ≤  ≤   +   ): Consider shortages during the period, as well as backorders caused by a negative inventory level ().The only term is due to demand because orders are entirely backlogged.As a result, the differential equations that follows Using the boundary condition (  ) = 0; equation (4.5) leads to the following expression for the inventory level This negative inventory level actually implies that the accumulated backorder at time  (in Here  (  +   ) = −  is the lowest inventory level and the maximum backorder is therefore   .The ordered quantity  fulfills this backorder first and the remaining inventory provides the maximum inventory level.
In equation (4.7), the order quantity is larger than that in the classical backorder model without deterioration [(  +   )], as it has to satisfy the demand together with the items lost to deterioration.
Therefore, the total inventory cost has the following constituents.
, where  and  are positive parameters and  is the promotional effort.
(e) Cost due to the deteriorated items ( is the per unit deterioration cost) (DC) As a result, by considering deterioration but not preservation costs, the total inventory cost per cycle is as follows:

Optimization methodology (Model-1)
The classical optimization process yields the decision variable values for the least total cost ($/cycle).The computational procedure has two parts.
Step 1: Obtain critical point Step 2: Verify the convexity of  (  ,   ) by proving that (in the feasible region) The factors not containing the decision variables in equation (4.8) are collected and grouped for algebraic convenience.

𝑇 𝐴𝐶 (𝜏
where The total cost per time unit varies continuously with the positive inventory time   and negative inventory time   .This objective function is minimized using the decision variables   and   .From equation (4.9), the first order partial derivatives of   (  ,   ) are The second order partial derivatives of   (  ,   ) are ) ) Using this inequality in equation (4.14), which becomes The necessary condition for the objective function to attain minimum cost are that the first order partial derivative must be zero (Step-1 above).when sufficient conditions (Step-2 above) are met; setting these partial derivative equal to zero gives the best solution, that is    = 0 and     = 0.
Further, to make certain optimality, the enough situation should be satisfied.Hence forth the corresponding fundamental minors should be positive definite.The Hessian determinant is From equations (4.12), (4.13) and (4.14), we get (4.17) Hence, equations (4.9) and (4.20) give Simplifying equation (4.21), one can get Equation (4.20) implies From equations (4.20) and (4.7), the minimum total cost per unit time and the economic order quantity are From equation (4.4), effective rate of loss (deterioration per unit time averaged over a cycle) is

A continuous review inventory model by considering deterioration items and preservation effects (Model-2)
Proper maintenance strategies reduce the impact of inventory system degradation.As fewer items are lost, this translates to savings.The less items you lose, the more you save.However, investments in such technologies should be included in the overall cost.In this subsection, an optimal point is found between these two conflicting economic effects.
In this model, preservation strategies cost is used by the retailer as an additional investment of  per unit time and this reduces the rate at which items deteriorate by  0 (1 −  − ), where the coefficient of preservation is  > 0 (as in Shah et al. [42]).This is a strictly monotonic increasing function of the preservation investment, bounded above by  0 , and below by 0.
The effective deterioration coefficient is (for  > 0) Now, Model-1 is extended to incorporate the preservation technology.This variable has two effects as the preservation cost adds to the total cost, but dampens the deterioration through .Solving the differential equations (4.1) and (4.5) for () using the boundary condition (  ) = 0; the inventory level () with the consideration of preservation investment are as follows: Equations (4.26) and (4.27) give the inventory at any point in the cycle (0 ≤  ≤   +  ).Maximum inventory is at (0) and maximum backorder is at (  +   ).
Over a complete cycle, number of items lost to deterioration is (as in equation (4.4)) The total cost per cycle in this inventory model now contains the preservation cost  per unit time.Hence, the preservation cost per cycle (PC) is (  +   ).
The other cost components are as defined in Model-1.Thus, the total inventory cost per cycle with additional preservation cost and deterioration dampened by preservation is as follows:  (  ,   , ) = The total cost per cycle time varies continuously with the positive inventory time   , negative inventory time   , and the preservation investment .This is the objective function to be minimized using decision variables   ,   , and .

Optimization methodology (Model-2)
In this study, the positive inventory time   , negative inventory time   , and preservation investment  that minimizes   are found.This investment reduces the deterioration to a certain degree, but beyond the point at which the benefits of the preservation outweigh the costs, the cost of preserving the investment might be more than the savings from reducing the number of deteriorated items.
The model aims to strike a balance between the marginal cost of preservation investment and the deteriorated quantity.This subsection chalks out the classical optimization pathway step by step for the projected scenario to derive the optimal solution as follows: Step 1: To achieve least total cost under this model, necessary conditions are to be satisfied, i.e, the first order partial derivatives must vanish at the optimum point ( *  ,  *  ,  * ).
Step 2: Convexity of  (  ,   , ) in the feasible region is investigated at this step through the application of the sufficient conditions involving the second order partial derivatives.
In  (  ,   , ) there are two time duration variables and one preservation cost variable.First the convexity of  (  ,   |) is considered for any particular feasible value of the preservation cost and then the convexity of  ( |  ,   ) is considered for any particular feasible values of the time duration variables.
The process for checking convexity of  (  ,   |) is similar to the derivation in Model-1.Note that, here  =  0  − contains the third decision variable .For any particular given feasible value of  =  * , this cost function is checked for convexity.
Hence, in the following analysis, equation (4.30) is rewritten and the compact algebraic form of  (  ,   | * ) is as follows: where The first order partial derivatives of  ( Using the second order partial derivatives from equations (4.32), (4.33), and (4.34) In the feasible region, where   > 0, the terms   (1 −   ) 2 +   − 2 and  2 − 2    − 1 can be simplified to obtain the following inequality from equation (4.36)where Computing second order partial derivative with respect to  of equation (4.38) (using chain rule) gives is convex in the feasible range of  for any particular feasible values of   =  *  ,   =  *  .The convex nature of the objective function is further illustrated below which is shown in Figure 3.
These results prove the validity of the optimization process.Solving the equations obtained by setting the first partial derivatives of equation (4.30) with respect to the three decision variables (  ,   and ) to zero gives the optimal values for a continuous review inventory model considering deterioration and preservation effects and crisp demand (Model-2).

A continuous review inventory model with fuzzy demand by considering deterioration items and preservation effects (Model-3)
Incorporating the effect of the promotional effort, in the previous models the annual demand has been taken in the form  () =  0 +  1  .In this subsection, fuzzy demand is considered where  0 is a trapezoidal fuzzy number, i.e, ̃︁  0 =    ( 0 − ∆ 2 ,  0 − ∆ 1 ,  0 + ∆ 1 ,  0 + ∆ 2 ).This enhances the flexibility in modelling real scenarios.
In the present article, the function principle and signed distance defuzzification procedure (see Jaggi et al. [9], Sharma and Govindaluri [45]) are considered.Now, the membership function of ̃︁  0 is as follows: With this fuzzy demand, equation (4.30) is transformed to Then by the function principle (Mahata and Goswami [17]), as the demand has a    base value, the above expression reveals that the total cost per unit time becomes a    as shown in Figure 4.
The parameters of this    are themselves real valued functions.For any feasible values of   ,   ,  this holds: where Applying the median formula, the total cost per time unit in Model-3 is estimated as Thus, the median estimate of the total cost per time unit for Model-3 (fuzzy demand with preservation) exists as the sum of two parts.The first term is identical to the total cost per time unit for Model-2 (crisp demand with preservation).
and the Hessian of  (  ,   |) are constructed.As in Model-2; the terms are grouped.
For any particular given feasible value of  =  * (hence  =  0  − * below) in equation (4.52) where From equation (4.53), the first order partial derivatives of  (  ,   | * ) with respect to   and   are The second order partial derivatives of  (  ,   | * ) with respect to   and   then become In equation (4.53), the inequality Substituting this in equation (4.56) The Hessian determinant of  (  ,   | * ) is Substituting the expressions for the second order partial derivatives from equations (4.54), (4.55), and (4.56) For any particular feasible value of  =  * this can be simplified as in equation (4.36). where • The first and second order partial derivatives of  (| *  ,  *  ) with respect to  are computed from equation (4.60) (where  =  0  − ).The iteration scheme for optimization of  (  (  ,   , )) is obtained from these three equations as follows )︃ (4.63) As in Model-2, using the optimal time duration values  *   ,  *   and the optimal preservation cost per unit time  *  obtained from the above scheme (Eqs.(4.63), (4.64), and (4.65)) with  *  =  0  − *  , gives the optimal results for the Model-3.The optimal   and  for Model-3 are )︃ .(4.67)

Input parameters
A numerical example is employed to illustrate the models developed here.A decision maker may extract useful insights by analysing the results.Numerical values of the parameters used (Mahapatra et al. [15]) here are given below:  = 80, ℎ = 0.4,  0 = 1000,  1 = 20,  = 10,  = 2,  = 2.5,  = 6,  = 3, ∆ 2 = 75, ∆ 1 = 50, ∆ 1 = 100, ∆ 2 = 150 and  0 = 0.15.The preservation technology investment () dampens the deterioration rate as () =  0  − , where the positive parameter  is a coefficient measuring the dampening of deterioration rate for every unit of such investment .The value of  is considered as  = 0.02 in this study.The upper bound of the preservation investment is  = 200 where  ≤  .The values of the decision variables: time duration for inventory level to fall to zero after replenishment (  ), time between zero inventory and replenishment that is fully backlogged, (  ) and preservation cost () for Model-1, Model-2 and Model-3 and corresponding order quantity and total cost at the optimal point are tabulated (Tab.3).
The column  *  *  + *  −  gives the number of units lost per unit time over a cycle as a measure of the effective rate of deterioration.In Table 3, the per unit cost per unit time at the optimal point is   *  * = 0.907, 0.587, 0.585 for Models-1,2,3, respectively, which proves that the fuzzy formulation (Model-3) is the most efficient on a per unit basis even though it has a slightly higher   * than Model-2.
Figure 5 shows the variation in   * for various cycle lengths ( *  +  *  ).It exhibits two distinct phases.In the first phase, for very small cycle lengths, the impact of deterioration is not prominent and all three models yield similar results.A substantial fraction of the total inventory cost goes into preservation investment in Model-2 and Model-3.The percentage investment in preservation is shown in Figure 6.For longer cycle lengths, Model-3 has a slightly lower fraction of the cost under this head.

Effect of unit shortage cost (𝑠)
The length of the backorder phase   increases with the decrease in shortage cost  and there is a small saving in  .It becomes profitable to keep larger optimal backorder in a cycle.In Model-2, preservation technology ensures that   is much less elastic.The fuzzy values mimic the movement of the crisp values.The fuzziness of demand is balanced out by a small increase in EOQ without much impact on the  .

Impact of unit holding cost (ℎ)
The reduction of holding cost has a proportionately greater impact than an increase of similar magnitude.In sensitivity computation, ℎ complements  as this time   is more affected.The preservation investment allows greater variation in   ,   , and a much larger decrease in   and increases in EOQ when ℎ is reduced as bigger orders become optimal.The median fuzzy output shows a displacement in line with the crisp boundaries set by the decision-maker.

Coefficient of preservation (𝑎)
When the coefficient of preservation () increases, it leads to a higher EOQ and larger positive inventory time (  ) while still reducing  .Model-3 uses a larger investment  to compensate for the fuzziness in demand.There is a change of optimal pattern below  = 0.017.Here,  decreases as the system reach a different optimal strategy with more items lost as preservation investment has a sharp reduction in efficiency towards dampening deterioration.

Effect of unit deterioration cost (𝑤)
The positive inventory time and EOQ both increase to take advantage of reduced  and even with more items lost, the   is reduced in Model-1.When preservation is included, the effect on   and EOQ is smaller.The preservation investment fluctuates in parallel with  and almost completely shields the time duration variables from its effects.Model-3 is marginally less affected in this case than Model-2.

Effect of ordering cost (𝐴)
The ordering cost  has a consistent impact across all three models.There is a small but significant increase in the magnitude of change of positive inventory time and EOQ, but a decrease in the change of backorder time   , and   when preservation is introduced.Both these movements are marginally less in Model-3.
6.0.6.Effect of constant deterioration rate ( 0 ) The   increases as  0 increases but the decrease is more pronounced as this parameter is reduced.The same nature of change occurs in the preservation investment  and for the chosen example this nullifies the effect on the time duration variables.In Model-1 (without this preservation variable), when deterioration increases it is profitable to keep larger backorders, and for low deterioration, larger EOQ and   are optimal.

Effect of promotional effort (𝜌)
A larger value of  (promotional effort) increases the  , EOQ, and all three decision variables as it directly increases demand.Without preservation, backorder time duration   fluctuates marginally more but with preservation, positive inventory duration changes by a larger margin.For increased demand and reasonably effective preservation, it is optimal to order more and satisfy direct demand rather than keep bigger back orders.The Model-3 shows marginally more stability on a point-to-point basis than Model-2.
The above analysis is graphically reinforced in Figure 7 which separately explores the sensitivity of each of the decision variables   ,   (for Model-1 and Model-2),  (for Model-2), and the objective function ( (  ,   ) and  (  ,   , )).Each curve represents the effect of changing one of the parameters among unit shortage cost (), unit holding cost (ℎ), coefficient of preservation (), unit deterioration cost (), ordering cost (), constant deterioration rate ( 0 ), and promotional effort ().The movement of the decision variables and the objective function, as they attain new optimal values are shown when individual parameters change by −50%, −25%, +25%, +50% about their values as specified in the numerical example above while other parameters remain unchanged.

Managerial insights
Distilling the impact of parameter variation on decision variables and objective function (Tab.3, Figs. 5, 6, and 7) leads to comprehension of trends beyond the numerical results.The following points provide managerial guidelines for the decision makers: -In the absence of preservation (Model-1), promotional activities can help to mitigate adverse customer impact during stockouts as it is profitable to keep larger backorders to balance deterioration.-In the presence of preservation, the decision maker need not be conservative with the predicted cycle length as the optimal cost only rises gradually beyond the global minima (Model-2, Model-3).-Shorter cycle lengths are sometimes forced due to real-life limitations (storage constraints, erratic demand patterns, supply limitations).This renders the preservation investment uneconomical.-The fraction of total cost optimally invested in preservation does not remain constant.It appears to increase in step with the cycle length, but then attains a maximum level around 22% beyond which it diminishes very gradually.This observation may provide a budgetary clue for the decision makers.-The preservation investment almost completely shields the time duration variables from the rate and cost of deterioration.When the preservation investment is reasonably effective, decision makers can reduce the total cost and it is optimal to order more and satisfy direct demand during positive inventory rather than keep bigger backorders.-There is a change of optimal pattern when preservation effectiveness falls below a threshold and here the decision makers should reduce the preservation investment as the system reaches a different optimal strategy despite more items lost to deterioration.

Conclusions
This study expanded the existing models and solutions on EOQ with preservation cost to control deteriorating inventory levels.The per unit time cost was considered with both the period of on-hand inventory and the time with stockout condition in a reorder sequence to determine the best reorder size and cycle length, while the expected total average cost was minimized against the preservation investment and the time duration variables in the presence of promotional effort.The effect of deterioration increased the total cost as items were lost.Model-2 was expanded by modelling uncertainty in demand to generate Model-3.A useful formulation was developed to obtain the ideal values for the reorder process and the cost of preservation technology where the decision makers had imprecise information about the demand.This model illustrated the impact of preservation efforts to control the spoilage and the decision makers could compute the optimal investment for this purpose.The signed distance method was applied in the model with uncertain demand to defuzzify the fuzzy cost function.In both Model-2 and Model-3, an algorithm emerged that identified the optimal solutions from the analytical results presented therein.
A numerical setup was investigated to validate the developed models and examine their sensitivity against key parameters to determine the specific effect on the model.In the classical inventory model, shortage and holding costs had correlated effect on shortage time and positive inventory duration respectively.Analysis revealed that the preservation investment skewed this variation in favour of positive inventory duration.Effective promotional effort induced larger optimal order quantity without significant impact on costs thus economic benefits could be obtained by boosting the promotional effort.The total cost showed higher sensitivity to decrease in deterioration rate than to its increase.The following aspects can be expanded for future work.Several avenues present themselves to augment this study, e.g., replenishment that takes finite time, deteriorated items being reworked or substituted, uncertainty and randomness in other facets, multi-item inventory and learning effects, and a deterioration rate depending on the expiration date.The paradigm of renewable energy from the animal fat waste as deteriorated items (Habib et al. [8]) may provide an alternative model to recoup some of the deterioration impact.
This model can be extended to consider measures to reduce emissions, such as carbon offsets and CO 2 quotas.Some of the cost parameters may be sensitive to carbon emissions levels.It is possible to study the environmental impact of degradation and conservation technologies for sustainable supply chains through such inclusions.Analysis of the effects of non-linearity and time dependent operating costs, the presence of incomplete items, prepayments, trade credit, and economic policy inflation [31] are some of the possible generalizations of the presented models.

Figure 1 .
Figure 1.Inventory model with backorder and deterioration.

Figure 4 .
Figure 4. Fuzziness of demand parameter and objective function.

Figure 5 .
Figure 5. Variation in optimum total cost with cycle length.

Figure 6 .
Figure 6.Preservation investment percentage with respect to cycle length.

Table 1 .
Analysis of this model in comparison to prior models.
Sarkar et al. [38] Price and crosselasticity dependent CN CN CN Crisp Priyamvada et al. [29] Price-sensitive demand Constant CN Fully Crisp This study Uncertainty with promotional effort Constant Fully Fully Crisp & fuzzy Notes.CN: Contribution Not-Available relative to present analysis.
effective deterioration rate, dampened by use of preservation strategy ( = 0 − )  maximum positive "inventory level" at time  = 0  maximum preservation outlay ($/ per unit time) () level of inventory at time   (,   ) total cost in absence of preservation (Model-1)($/per unit time)  (,   , ) total cost with preservation (Model-2) ($/per unit time)  (,   , ) total cost for fuzzy demand with preservation (Model-3) ($/per unit time) Decision variables  time duration for inventory level to fall to zero after replenishment   time between zero inventory and replenishment, during which orders are fully backlogged (denoted by negative inventory level)  preservation investment as cost per unit time ($/per time) .23) The iterative numerical solution of equation (4.22) and 4.23 yields the appropriate values of  *  and  *  for optimization.
,   | * ) with respect to   and   from equation (4.31) areThe second order partial derivatives of  (  ,   | * ) with respect to   and   are computed to check the nature of the Hessian matrix for positive definiteness as follows.At the optimal point where the ℎ terms are zero, the mixed second derivative with respect to   ,   is therefore zero.
.37) Hence, from equation (4.35) and (4.37), it can be concluded that  (  ,   | * ) is convex in feasible range of   ,   for any particular feasible value of  =  * .Similarly, the convexity of  (|  ,   ) for any particular feasible values of   ,   is proved analytically.Rearrange equation (4.30) for calculating the partial derivative of the function  (|  ,   ) with respect to , while noting that  =  0  − and   =  *  ,   =  *  .