Effect of Carbon Emission and Shelf-Life on Random Emission and Random Price Dependent Demand of a Perishable Product in Interval Environment

An inventory of suppliers dealing with supply of raw materials from the natural resources like sugarcane, beetroot, vegetable scraps etc. need to choose the product for supply to the production industries with a great consciousness due to their finite shelf-life and carbon emission property. A product with small shelf-life and carrying the burden of carbon tax need to be sold quickly so that overall profit earn can be maximize in such a situation. Further the situation deteriorates when the selling price of a product is affected by the carbon emission and its absolute value reduces with time. There we need a formal mathematical design that can include all these issues of price and carbon tax so that the supplier can select the best product for sale. In this paper we develop and inventory model to investigate the earnings made by the supplier, selling the product with price- and stock- dependent demand and study the effect of carbon emission and shelf-life. Further we obtain the optimal preservation cost and green investment to reduce the deterioration and carbon emission. We study the model under different cases; deterministic demand, probabilistic demand with random emission and probabilistic demand with random price. We perform the sensitivity analysis. Abstract An inventory of suppliers dealing with supply of raw materials from the natural resources like sugarcane, beetroot, vegetable scraps etc. need to choose the product for supply to the production industries with a great consciousness due to their finite shelf-life and carbon emission property. A product with small shelf-life and carrying the burden of carbon tax need to be sold quickly so that overall profit earn can be maximize in such a situation. Further the situation deteriorates when the selling price of a product is affected by the carbon emission and its absolute value reduces with time. There we need a formal mathematical design that can include all these issues of price and carbon tax so that the supplier can select the best product for sale. In this paper we develop and inventory model to investigate the earnings made by the supplier, selling the product with price- and stock- dependent demand and study the effect of carbon emission and shelf-life. Further we obtain the optimal preservation cost and green investment to reduce the deterioration and carbon emission. We study the model under different cases; deterministic demand, probabilistic demand with random emission and probabilistic demand with random price. We perform the sensitivity analysis. Words


Introduction
In the production industries of sugar and ethanol, the use of natural resources as a raw material in a common practice. However, exploiting natural resources like sugarcane, Woodstock, vegetable scrap etc. can increase the risk of carbon emission in the environment due to fast rate of deterioration. Further, over utilization of natural resources can cause natural disasters and therefore a controlled supply of these products to the production industries is important in order to earn a good profit and simultaneously minimizing the carbon tax.Apart from carbon emission the products stated above have a small shelf-life so the supplier must sale the product as early as possible. Incorporation and exploitation of green technologies and modern preservation techniques can reduce both carbon tax and deterioration cost. Inventory modelling is an active are of research since decades. Recently the research work on the application and analysis of carbon footprints and its effect on the demand of a product is an interesting topic for the researchers. Saha et. al (2021), proposed a supply chain model to study the competition on green products retailing.A study on green policies is proposed by Dutta et. al (2019). Ritha and Poongidisathiya (2018) proposed a green inventory model to minimize the total cost of the system and optimize the ordering quantity of the carbon sensitive products under green policies. Hovelaque and Bironneau (2015) proposedan EOQ model with emission dependent demand and carbon constraint. De-la-Cruz-Marquez et. al (2021), propose an inventory model with price dependent demand and considering carbon emission with imperfect quality. For more research on EOQ inventory model under carbon emission we referChristata and Daryanto (2020). Dutta (2017), made an extensive research on effect of green investment on production inventory with carbon emission under bounded production limit the research illustrate that, carbon emission cost has positive effect on environment and negative effect on GDP.A three-stage dynamic game model is developed by Tang et. al (2020) and proposed an inventory model to optimize sustainable transportation cost, wholesale price and a carbon tax policy. An inventory model for perishable items is proposed by Macias-Lopez et. al (2021) with price, stock and time dependent demand with quadratic holding cost, the demand functions considered in the model are linear, iso-elastic, exponential, logit, logarithmic and polynomial function.Carbon reduction and price dependent demand is investigated by Lu et. al (2020) for the deteriorating items under game theory methodology. A sustainable inventory modelling is developed by Sarkar et. al (2018) with multi-trade-credit-period and backordering. Stochastic inventory and supply chain models were developed to study a parameter under different probabilistic distributions such as uniform, triangular, beta, gamma etc. Most of the stochastic model is developed by considering probabilistic deterioration we refer the following research work in this direction Palanivel et al. (2015) and Sarkar (2013), Sarkar & Sarkar (2013) and Singh et al. (2013). For inventory models assuming the carbon emission in the demand, supply chain and as a constraint we refer Allabadi et al. (2018), Arkin &Jammernegg (2014) and Benjaafar and Daskin (2013).

Description of Problem
In demand and supply chain of perishable raw materials to produce biofuel, ethanol, sugar, paper etc. the vendor faces to major problem related to deterioration and carbon emission. A perishable product has a short shelf-life and must be distributed before the sell-by period. Further the production industries prefer the raw materials whose rate of carbon emission is low, therefore the demand of the raw materials depends on both shelf-life and rate of carbon emission. In our present paper, we made an attempt to develop an inventory model to study the effect of shelf-life and carbon emission on the demand of the product. The different cases that we study under given inventory model are as follows; a) To analyse the profit earns under random emission. b) To analyse the profit earns under probabilistic selling price. c) To analyse the profit under variable shelf-life and rate of carbon emission d) To analyse the profit under different pattern of carbon emission

Methodology
In order to solve the model for optimization, first we convert the profit function from crisp environment to interval environment. Then solve the model in interval environment using three forms of particle swarm optimization algorithm, that is, Weighted PSO, Constriction PSO and Quantum PSO and the solution is accepted after statistical verification. We employ linear differential equation to develop the inventory model. We perform sensitivity analysis of the parameters.
Where ∈ (0,1) is the scaling factor that measures the effect of carbon emission on the demand of a product  Demand function where 0 < < 1 is the price sensitivity and ( ) is function of rate of carbon emission representing the pattern of carbon emission.  The selling price is a function of time and it is defined as;  The rate of carbon emission is random.  The shelf-life is random  Replenishment quantity is constant and instantaneous.  Deterioration is a function of preservation cost and rate of carbon emission ( , ) = (1 − −( ( )− ) ) where, 0 < < 1 is the sensitivity of preservation cost  Carbon emission cost in inventory is a function of green investment ( , ) = −( − ( )) , where 0 < < 1 is sensitivity of green investment  Shortages are not allowed.  Planning horizon is infinite and < .

Model Formulation
The differential equation representing the flow of material from the stock to the market is given by; Subject to the condition (0) = 0 The inventory level at any time is given by;  Subject to constraint; < Where, 6 ( ( , ) + ( )) 6 ( ( , ) + ( )) Due to extreme nonlinearity and higher dimensional profit function we apply particle swarm optimization (CPSO) algorithm with constriction factor, particle swarm optimization (WPSO) algorithm and genetic algorithm (GA)to solve the problem. In order to apply algorithms in the interval environment we propose the following solution procedure;

Solution Procedure
1) The objective function in the interval environment consists of two parts and . We consider these two functions as two different objective functions. 2) Let be the decision variable and be the variable representing the weight for objective function. 3) We define the objective functions as; ( , ) = ( ) + (1 − ) ( )

4)
The decision variable must satisfy the constraint < 5) We run the algorithm to maximize ( , 0.5) and obtain the optimal value * such that ( * ) < ( * )

Algorithm
The particle swarm algorithm with constriction factor and genetic algorithm are two direct search methods from two different categories, that is, swarm algorithm and evolutionary algorithm respectively. Using the above solution procedure in section 8, we obtain the optimal solution of the multi-objective function using both the algorithms and implementing student's t-test we justify that there is no significant difference in the mean of the solution obtained by applying CPSO, WPSO and GA.

Genetic Algorithm
In genetic algorithm, Yang (2015), we basically perform four steps to optimize a function these steps are defined as follows; Selection: In selection process we select population of finite size randomly from the search boundary. There are various methods of selecting the population; Enlisted selection, Roulette Wheel, Tournament Selection, Rank Selection, Boltzmann Selection etc. In our case we have performed elitist selection method to choose our population. Crossover:The crossover operation is usually depending on the problem to optimize. Due to the continuous objective function of real decision variables we consider the convex combination of two chromosomes. Mutation:The purpose of mutation operation is to retard the rate of convergence. The probability of mutation is usually small and included in the algorithm to protect the evolution of solution from being trapped in local optimum. In our GA coding we consider the mutation operation as a minute change in the decision variable. Stopping Criteria: In this step we provide some valid condition to terminate the iteration process.
There are many stopping criteria available although in our GA coding we terminate the algorithm after a certain number of iteration and accept the solution if the variance of the set of global optima is less than 10 −5 .

Procedure of Genetic Algorithm
The individuals are selected for crossover on the basis of fitness probability of each individual and the crossover probability. A set of individuals are selected by elitist selection method. We define crossover probability and mutation probability. Whenever the fitness probability is more than the crossover probability we select the individual for crossover. Similarly, if the fitness probability is less than the mutation probability then we apply the mutation operation on that individual. We repeat the steps until we get a set of global optima in the given search boundary whose variance is less than 10 −5 To perform the soft computing, we assume the following values of the parameters of GA. Population Size = 100 Crossover Probability = 0.5 Mutation Probability = 0.1 Maximum Iteration = 3000

WPSO Algorithm
In weighted PSO we attach a weight to the initial velocity of the particle. This weight is often termed as inertial weight which is defined as constant and the value usually ranges between [0.5, 0.9]Eberhart& Shi (2000) and Yang, (2015).

= −
Where, is maximum number of iteration, is maximum inertial weight, is minimum inertial weight and is the iteration number. Let be the ℎ particle at ℎ position with initial velocity . Then the velocity equation for the ( + 1) ℎ position together with a inertial weight , Where and are acceleration coefficients and , ∈ (0,1) are random numbers The ℎ particle at ( + 1) ℎ position is given by; +1 = + +1 Table 16: Pseudo-Algorithm of weighted PSO Initialized the population of size Compute fitness of objective function for each particle Initially set * = * = ℎ Compute the fitness of objective function for each particle at + 1 Obtain the maximum fitness and assign it to * .

CPSO Algorithm
In CPSO we assign a constant weight to the velocity equationEberhart and Shi (2000). It is an effective measure to control the motion of search particle in the swarm and helps in optimum search of the search boundary. Let be the ℎ particle at ℎ position with initial velocity .
Then the velocity equation together with constriction factor for the ( + 1) ℎ position Where, and is known as acceleration coefficient and , ℎ ∈ (0,1) are random numbers.

Numerical Verification
In the lack of real data, we assume the following values of the parameters.

Problem (Case I):
A vendor selling 0 = 1000 at the rate = $ 250which deteriorating at a rate = 0.001 and the rate at which the product emits carbon gases is = 0.5 and the effect of carbon emission on demand is = 0.7. The carbon emission and deterioration can be controlled by a proper green investment and preservation technology with sensitivity = 0.15 and = 0.13. There is a constant replenishment of = 500 per unit time throughout the planning horizon. The initial demand of the product is = 150 and the price sensitivity is = 0.2. The shelf life of the product is = 150 . The ordering cost 0 = $[10000,11000], the holding cost of the product is = $ [30,35], the purchase cost of the product from the source is = $ [250,255] and the carbon tax is = $[55, 60]    Managerial Implications: In the light of above results, the profit is maximum if the amount of carbon emission follows pattern II, although the vendor has to sell the product within 0.64 days which seems difficult except for the case when the vendor is selling products to the industries in bulk. Similar situation can be observed for pattern III. For pattern I, it can be observed that, the planning horizon is more for selling the product is almost 79 days but it reduces the profit in a significant amount which quite obvious because the seller has bear more holding, deterioration and carbon emission cost is the selling period is more. Further the demand of the product is more if the amount of carbon emission is less. Comparing the results obtained for three different search method it is observed that CPSO produce the better results subject to the statistical analysis.           Managerial Implications:In Case II we assumed to be probabilistic and observed that,  When follows uniform and triangular distribution then the supplier must consider the products whose carbon emission follows pattern II and III as for pattern I the average profit is negative. Comparing the profit earn for pattern II and pattern III under uniform and triangular distribution, the manager should select the materials, taking under triangular distribution because the preservation cost and green investment is less compare to follows uniform distribution  For the products whose rate if carbon emission follows pattern I, it is suggested that, the manager should consider the beta distribution for analysing and decision making purposes. Further the manager could select the product whose carbon emission follows pattern II, because the average profit is maximum for pattern II under beta distribution.

Problem (Case III):
A vendor selling 0 = 1000 with selling price probabilistic which is deteriorating at a rate = 0.001 and the rate at which the product emits carbon gases is = 0.5 and the effect of carbon emission on demand is = 0.7. The carbon emission and deterioration can be controlled by a proper green investment and preservation technology with sensitivity = 0.15 and = 0.13. There is a constant replenishment of = 500 per unit time throughout the planning horizon. The initial demand of the product is = 150 and the price sensitivity is = 0.2. The shelf life of the product is = 150 . The ordering cost 0 = $[10000,11000], the holding cost of the product is = $ [30,35], the purchase cost of the product from the source is = $ [250,255] and the carbon tax is = $[55, 60] For Uniform Distribution: 1 = 250 and 2 = 375. The rate of carbon emission is;            For all the probabilistic distributions, the products with carbon emission pattern II can produce maximum average profit and simultaneously the preservation cost and green investment is less than pattern I and pattern II.  For probabilistic price under uniform distribution shows maximum profit.

Sensitivity Analysis
We perform sensitivity of analysis of the cost parameters of in all the cases. Let be any parameter then the change in the parameter is considered by the formula → + * where, ∈ {−0.2, −0.1, 0.1,0.2}. Since profit is maximum in Case I for pattern II. So we perform the sensitivity analysis of the parameters under pattern II, Moreover, , and increases with the increase in . 7) Rate of Deterioration ( ): A mild decrease in the profit is observed with the increase in the rate of deterioration further, preservation cost increases with the increase in the rate of carbon emission.

Conclusion
This is a single product inventory model to provide a decision making process for selecting right product to sale in an open market. The products considered in this model are perishable in nature and has different rate/ amount of carbon emission. Further the production industries select those products for manufacturing that has low rate/amount of carbon emission so that the burden of carbon tax on overall production can be minimize. Thus the amount of carbon emission has an effect on the demand of the product. Moreover, the selling price reduces with time and eventually diminish so the product must be sold within the finite shelf-life. Therefore, both shelf-life and carbon emission affects the demand of a product. In our present paper we consider three patterns of carbon emission and analyse the profit earns by the seller, also we have computed optimum preservation cost and green investments to reduce deterioration and carbon emission. We divide the problem in three different cases; In Case I, we have studied the effective preservation cost and green investment to maximize the profit. It is observed that, the product under pattern II emission can generate maximum profit for the supplier.
In Case II, we assume the rate of carbon emission probabilistic and follow uniform, triangular and beta distribution. We observed that, when rate of carbon emission follows beta distribution then supplier can earn maximum profit.
In Case III, we consider the selling price is probabilistic and follow uniform, triangular and gamma distribution. We observed that, supplier can generate maximum profit under pattern II when price follow uniform distribution. We perform sensitivity analysis and noted that, holding cost, carbon emission cost, purchase cost, rate of carbon emission, shelf-life are sensitive parameters and the supplier should select the product for sale subject to the effect of parameters on the profit function as given in Table 29. In our present paper we propose a solution procedure and apply Genetic Algorithm (GA), Weighted PSO (WPSO) and Constriction PSO (CPSO) to solve the profit function for optimization. Form the results and statistical analysis given in Table 1-28 we conclude that constriction PSO provide better results as compare to the GA and WPSO. This is an extension in the literature on inventory modelling that studies the effect of emission and shelf-life on the demand of a product and the profit earn by the supplier under both deterministic and stochastic environments which is not reported in the literature earlier as per the authors knowledge is concern. Present model can be further extended to study the effect trade credit on the demand and profit, and it would be an important study as large vendors and suppliers start their business in credit system.

Compliance with Ethical Standards
Conflict of Interest: All the authors declare that they have no conflict of interest.
Ethical approval: There is no involvement of animal Funding details: Not applicable.