Optimizing profit in a controlled environment: Assessing the synergy between preservation technology and cap-and-trade policy

1.1 Literature review

In inventory management, the presence of carbon emission is inevitable, and over the years, the industries have adopted numerous strategies to limit emissions. For better maintenance of greener and cleaner environment for a comfortable life, some precise practical footsteps to reduce the carbon emissions, for instance, climate change tax (CCT), capture and storage (CCS), carbon offset (COS), climate change agreements (CCA), Green technology investment (GTI), carbon price support (CPS) and cap-and-trade (CAT) have to turn out to be indispensable to some extent company or for betterments of people of any country (Zhang et al., 2018). All these strategies will intensify the practitioners' participation in a clean and emission-free sustainable environment. It could also work as an inspiration for non-participants because they will turn away traditional coordination to numerous emission diminishing rehearses through a minimum carbon production, a minimum-carbon supply chain, and remanufacturing (Feng et al., 2019). A generous review of literature of numerous regulations of emissions and their consequence to the environment has been anticipated by Waltho et al. (2019). According to Skiba (2020) carbon capture and storage (CCS) is fruitful for an industry where petroleum products are burned for energy while demands a plant of at least 800 m more profound in the soil to store it safely. Sometimes this sort of plant expenses enormous investment, which is difficult for middle-scale industries. So, bearing this perplexity in implementing the technique, business owners always hunt for some new approach to curb the carbons in minimum investments. Huang et al. (2020) established a model bearing in mind the effect of green technology and uncovered that carbon tax policy further contentedly confines carbons' emissions to the atmospheres. Green technology investment is widely used and became popular nowadays in middle-scale industries despite its limitations (Mishra et al., 2021; Datta, 2017; Mashud et al., 2020a; Aronsson and Brodin, 2006; Jumadi and Zailani, 2010). It demands some energy-efficient logistic support with modern equipment (Lou et al., 2015). The critical fact that the government had significantly less information about the amount of carbon emitted during the process of production or other activities. Because the type or the brand of energy-efficient equipment may vary from producer company to company as the production company has the right to purchase any brand of that equipment, so, a proper database is needed for this process to implement.

To avoid complications and provide better facilities to the entrepreneurs, most of the government in the modern era has decided to implement a cap-and-trade policy. According to Du et al. (2016), the European Union Emissions Trading System (EUETS) has adopted the cap-and-trade policy, and they have formed some regulations based on this strategy. The main point they proposed is the emissions have a financial value. A carbon tax is similar type of strategy nowadays used by the emission-reducing countries, which is somewhat a particular case of a cap-and-trade policy. Another similar strategy akin to cap-and-trade is carbon offset. Rout et al. (2020) anticipated a model where carbon tax, cap-and-trade, and carbon offset policy are discussed explicitly all together. They provided a clear and transparent discussion on these strategies for curbing emissions. Sana (2020) proposed a newsvendor inventory model comparing green and non-green products with government tax implementation where the government rules higher subsidies and lower taxes to green producers, as opposed to lower subsidies and higher taxes to non-green producers. Taleizadeh et al. (2018) projected an EPQ (economic production quantity) model with a carbon tax for numerous shortages. Later, the exploration of the consequence of green production together with the cap-and-trade regulation on reducing the amount of produced carbon emissions was analyzed in Taleizadeh et al. (2021). A multi-objective oriented model for a sustainable supply chain thru carbon tax policy was anticipated by Sazvar et al. (2018), while product deterioration and backorder were also considered in that model. Daryanto et al. (2019) provided an integrated supply chain with carbon emissions for a three-echelon system for deteriorating products only. Daryanto et al. (2019)'s main limitation is they did not consider the preservation sense to control the deterioration, which is also absent at Sazvar et al. (2018). This paper has tried to bridge this gap with a cap-and-trade strategy.

Product deterioration is a common phenomenon for any product and a very known problem for daily retailing business. Mashud et al. (2019) tried to anticipate a model considering this issue and suggest an efficient preservation technology to curb the deterioration of products together trade-credit policy. This paper lags to describe the essential factor of carbon emission. Preservation technology means energy-efficient equipment that lengthened products' lifetime, e.g., cold storage to preserve potatoes, refrigerators to preserve medicine, etc. Hasan et al. (2020b) provided an inventory model for agricultural products in deteriorating nature with perfect and imperfect items. Some discount approach has been adopted in that paper but not consider any emissions and emission regulatory strategy, which is a limitation of that paper. Besides, Mishra et al. (2020) anticipated a non-instantaneous deteriorating inventory model that simultaneously implements preservation technology and green technology. Mashud et al. (2020c) predicted a model with product deterioration and preservation technology considering carbons' emissions. In that paper, an efficient green technology had been suggested by the authors to limit emissions. Few studies addressed these two issues simultaneously, and prior research shows a gap in simultaneous investment in preservation and cap-and-trade policy. Considering stochastic demand, Dolai and Mondal (2021) presented an imperfect production inventory system where the screening process was used to isolate defective items and this defective rate was controlled by investing in preservation technology. Since it was a production model it would have been the best combination if carbon emissions and emission control measures had been considered in that study. Based on the effects of carbon emissions at the breeding stage for growing items, an EOQ model was created by Rana et al. (2021) in which retention and deterioration were also found to be responsible for excessive carbon emissions.

Some recent study addresses preservation technology with the price of the products while pricing is another vital factor that needs to be maintained dynamically. Mashud et al. (2020a) developed an inventory model with price-sensitive demand and preservation technology. Wei et al. (2020) emphasized the difference between business time and non-business time in his inventory model and represented demand by a piece-wise function of price and product freshness. Considering the demand varying with green quality and sales prices, Sana (2021) demonstrated two situations in two models where the investment in green product development was shared by both the manufacturer and the retailer. Barman et al. (2021) set up a framework to control the connection between the government and a two-level duopoly green supply chain. In that study, both green and non-green products were also considered and market demand was a linear function of the product sales price and greening level. Mashud et al. (2021a) projected an inventory model with pricing strategies and carbon emissions for a greenhouse. Mashud et al. (2021c) anticipated a model with price-dependent demand and a hybrid payment system under the effect of COVID-19. In contrast, Ruidas et al. (2021) projected a model considering price-sensitive demand and interval-valued carbon emissions. They discussed four different policies on restricting the carbons to the environment, and they solved these cases with soft computing algorithms. In this model, we have implemented carbon limiting regulations cap-and-trade policy—moreover, the associated problem we have solved by a classical optimization technique. The synergy between a cap-and-trade policy and preservation technology has been synthesized in this proposed model. Table 1 presents a brief overview of some of the related studies.

2.1 Model interpretation and assumptions

The entire cycle from the beginning to the end of the retailer's inventory $L\left(t\right)$ is delineated by Fig. 1 accompanying the differential equation below.

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Fig. 1

Graphical representation of the inventory model.

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With given conditions: $$L\left(0\right)=QandL\left(K\right)=0$$

Equation (1) describes the inventory fluctuations over time $t$ due to the effects of degradation $\beta$ and demand rate $D_s$ . Here $\omega \left(\gamma \right)=e^{-\psi \gamma }$ (He and Huang, 2013, Mishra et al., 2018, Mashud et al., 2019) addresses the preservation technology, that is utilized to decrease the pace of decay of the item. The cycle of inventory begins through the acquisition of $Q$ quantity of products by the retailer and closes with the finishing of the stock at the $t=K$ time.

By solving equation (1) and using given conditions, we get

and

Retailer’s associated costs and earnings of the inventory during the total cycle $K$ are calculated as follows:

Ordering cost:

Purchase cost:

Holding Cost:

Preservation technology investment:

Transportation cost:

Total amount of carbon emissions due to holding and transportation is

Mandatory carbon emission tax:

Sale’s revenue:

Then the total profit per cycle is

2.2 Carbon cap and trade policy

In cap-and-trade policies, the government determines an objective level for contamination, generally identified as cap, and permits the businessmen to decide how to face the cap. Carbon dioxide and other gases that drive a dangerous atmospheric deviation are fundamental focuses of such caps. The cap consists of allowance that licenses the holder for a certain measure of emissions. There should be an allowance equivalent to the measure of emissions produced. Regulated companies are permitted to purchase and sell allowances. Penalties are imposed for cap infringement due to excess carbon emissions.

If $C_p$ denotes the carbon cap, and $A_e$ denotes the total amount of carbon emissions then two cases arise:

Case I: When Carbon cap is less than total emissions i.e. ${\mathit{A}}_{\mathit{e}}>{\mathit{C}}_{\mathit{p}}$

Suppose the amount of carbon cap imposed by any regulatory authority, e.g., government or any agency, is less than the retailer’s produced emission quantity. According to the cap-and-trade policy, the retailer needs to compensate per unit penalty charges for the excess amount. In mathematically, the cost for excess carbon emission is:

Thus, changes profit is:

Case II: When Carbon cap is greater than total emissions i.e. ${\mathit{A}}_{\mathit{e}}<{\mathit{C}}_{\mathit{p}}$

In this case, the retailer’s produced emission quantity is less than the cap forced by any regulatory authority, e.g., government or agency. Thus, agreeing to a cap-and-trade policy, the retailer can choose to sell its rights to any industry and earn some revenues. In mathematically, the earning from the sale of extra permits is:

Then the profit becomes

2.3 Concavity nature of profit function

The cycle's total profit of a retailer from equation (16) provides a concave nature, shown later thru some propositions. Thus, the concavity of total profit can be easily extracted through Proposition 1 with the implementation of some constraints. The retailer will take decisions based on the conditions of the following situations on the cap illustrated in Proposition 1.

Proposition 1

For any stable non-negative $\gamma$ , total carbon emissions $A_e,$ and $\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{a}\mathrm{p}{C}_p$ , the attained profit function of the retailer’s inventory always presents the concave nature at the optimum point $K^{\ast }$ , which is characterize by the following piecewise function: $$K^{\ast }=\left\{\begin{array}{c}\begin{array}{cc}\sqrt{2{\propto }_1},& \mathrm{n}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{p}\\ \sqrt{2{\propto }_2},& C_p{<A}_e\\ \sqrt{2{\propto }_3,}& C_p{>A}_e\end{array}\end{array}\right.$$

Proof. Simplifying the profit function in equation (12) through the series expansion (considering the first three terms) of the exponential terms, we get

Let us take the 1st order derivative of equation (17) with regard to $K$ ,

Now apply the necessary condition $\frac{\partial \phi }{\partial K}=0$ in order to make the solution for $K$ , which yields the following result

Here ${\propto }_1=\frac{\left(2E_1{F}_c{T}_d{t}_x+{T_{d}v}_2+f\right)z+M}{\left(\beta c\omega +t_{x}uw+h_d\right)D_s+T_d{v}_1\beta \omega z}$

Then performing the derivative of equation (18) with regard to $K$ ,

Since the all parameters contained in ${\propto }_1$ and equation (20) assume positive values, so obviously $\frac{{\partial }^2\phi }{\partial K^2}$ will yield negative result at $K=\sqrt{2{\propto }_1}$ and presents the concave nature of the profit function.

For ${\mathit{C}}_{\mathit{p}}{<\mathit{A}}_{\mathit{e}}$ :

Simplifying the profit function in equation (14) through the series expansion (considering the first three terms) of the exponential terms, we get

Let us take the 1st order derivative of equation (21) with regard to $K$ ,

Now apply the necessary condition $\frac{\partial {\phi }_1}{\partial K}=0$ in order to make the solution for $K$ , which yields the following result

Here ${\propto }_2=\frac{\left(2E_1{F}_c{T}_d{t}_x+{T_{d}v}_2+f\right)z+\left(A_e-C_p\right)x_1+M}{\left(\beta c\omega +t_{x}uw+h_d\right)D_s+T_d{v}_1\beta \omega z}$

Then performing the derivative of equation (22) with regard to $K$ ,

Since the all parameters contained in ${\propto }_2$ and equation (24) assume positive values, so obviously $\frac{{\partial }^2{\phi }_1}{\partial K^2}$ will yield negative result at $K=\sqrt{2{\propto }_2}$ and presents the concave nature of the profit function.

For ${\mathit{C}}_{\mathit{p}}{>\mathit{A}}_{\mathit{e}}$ :

Simplifying the profit function in equation (16) through the series expansion (considering the first three terms) of the exponential terms, we get

Let us take the 1st order derivative of equation (25) with regard to $K$ ,

Now apply the necessary condition $\frac{\partial {\phi }_2}{\partial K}=0$ in order to make the solution for $K$ , which yields the following result

Here ${\propto }_3=\frac{\left(2E_1{F}_c{T}_d{t}_x+{T_{d}v}_2+f\right)z-\left(C_p{-A}_e\right)x_2+M}{\left(\beta c\omega +t_{x}uw+h_d\right)D_s+T_d{v}_1\beta \omega z}$

Then performing the derivative of equation (26) with regard to $K$ ,

Since the all parameters contained in ${\propto }_3$ and equation (28) assume positive values, so obviously $\frac{{\partial }^2{\phi }_2}{\partial K^2}$ will yield negative result at $K=\sqrt{2{\propto }_3}$ and presents the concave nature of the profit function.

Proposition 2

For any stable non-negative $K$ , total carbon emissions $A_e,$ and $\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{b}\mathrm{o}\mathrm{n}\mathrm{c}\mathrm{a}\mathrm{p}{C}_p$ , the attained profit function of the retailer’s inventory always presents the concave nature at the optimum ${\gamma }^{\ast }$ .

Proof. The process is similar to that of Proposition 1.

Sometimes it is difficult for the retailer to make decisions based on only variable replenishment time or variable preservation technology investment. Moreover, this technique alleviates mathematical perplexity to make decisions, but exact calculations of profit are sometimes becoming critical. So, to provide some better constraints to the retailer on profit maximization, which should be maintained through the chain, is illustrated through Proposition 3.

Proposition 3

The attained profit function of the retailer’s inventory always presents the concave nature at the optimal point $\left({\gamma }^{\ast },K^{\ast }\right)$ .

Proof. Firstly, let us make the series expansion (considering the first three terms) of all the exponential terms in profit function (12) and simplify it, which gives the following

Taking the first derivative of equation (29) with regard to $\gamma$ ,

Now apply the necessary condition (i.e., first derivative equal to zero) in equation (30), which yields the following result.

Substituting this value of $\gamma$ in equation (29), we get the profit function in terms of variable $K$ .

Let us take the 1st and 2nd order derivatives of equation (32) with regard to $K$ .

Now place $\frac{\partial \phi \left(K\right)}{\partial K}=0$ and solves for $K$ . This implies

Here, $${∊}_1=\beta \psi \left(T_d{v}_{1}z+c{D}_s\right)\left\{\left(2t_{x}uw+c\beta +2h_d\right)D_s+T_d{v}_1\beta z\right\}$$ $${∊}_2=\left\{\beta \left(T_d{v}_{1}z+c{D}_s\right)+2D_s\left(t_{x}uw+h_d\right)\right\}\left(T_d{v}_{1}z+c{D}_s\right)$$ $${∊}_3=\left\{-1+\left(T_d{v}_{1}z+c{D}_s\right)\left\{\left(\left(2E_1{F}_c{t}_x+v_2\right)T_d+f\right)z+M\right\}\beta {\psi }^2\right\}\beta$$

To examine the optimality at $K=K^{\ast }$ , let us place this value in equation (34).

If ${\left[\frac{{\partial }^2\phi \left(K\right)}{\partial K^2}\right]}_{K=K^{\ast }}<0$ , then the profit function presents the concave nature at the optimal point $\left({\gamma }^{\ast },K^{\ast }\right)$ .

Similarly, the concavity of the profit function at $\left({\gamma }^{\ast },K^{\ast }\right)$ can be proved for carbon cap and trade policies.

Lemma. The sufficient condition ${\left[\frac{{\partial }^2\phi \left(K\right)}{\partial K^2}\right]}_{K=K^{\ast }}<0$ for the concavity of the retailer’s profit function $\phi \left(\gamma ,K\right)$ is satisfied when $\begin{array}{c}\left[4{\beta \psi }^2\left(T_d{v}_{1}z+c{D}_s\right)\left[\left\{\left(E_1{F}_c{t}_x+\frac{v_2}{2}\right)T_d+\frac{f}{2}\right\}z+\frac{M}{2}\right]\right]>2\end{array}$ .

Proof. It is seen that the mentioned term $\left[4{\beta \psi }^2\left(T_d{v}_{1}z+c{D}_s\right)\left[\left\{\left(E_1{F}_c{t}_x+\frac{v_2}{2}\right)T_d+\frac{f}{2}\right\}z+\frac{M}{2}\right]\right]$ involves summation and multiplication of some positive parameters and also there is no negative term present. Thus, it refers to a larger value than 2 and ${\left[\frac{{\partial }^2\phi \left(K\right)}{\partial K^2}\right]}_{K=K^{\ast }}<0$ is satisfied.

3.1 Algorithm

• Step 1.

  Start

• Step 2.

  Declare and assign the following variables

• Step 3.

  Write the basic assignment statements [equations (3)-(28)] and stabilize $\gamma$ .

• Step 4.

  Set the value of $C_p$ (cap)

  if $\left(C_p==\mathrm{n}\mathrm{u}\mathrm{l}\mathrm{l}\right)$

    calculate roots for $K$ (say $K$  =  $K_1)$ from statement in equation (19)

  else if $\left(C_p{<A}_e\right)$

    calculate roots for $K$ (say $K$  =  $K_2)$ from statement in equation (23)

  else if $\left(C_p{>A}_e\right)$

  calculate roots for $K$ (say $K$  =  $K_3)$ from statement in equation (27)

• Step 5.

  Check the sufficient conditions for each $K$ = $K_i$ , $i=1,2,3$ from the statements in equations (20), (24) and (28).

• Step 6.

  If the sufficient conditions are satisfied for $K$ = ${K_i}^{\ast }$ , then evaluate the feasible profit from the statements in equations (12), (14) and (16) by using the concavity marker Max= $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{i}\mathrm{t}$ ( $\gamma ,{K_i}^{\ast }$ ).

• Step 7.

  Stop

3.2 Examples

3.2.1

3.2.1 Example 1

This explains the fact when there is no cap strategy to control carbon emissions just the mandatory taxes are levied on the estimation of emissions. Consider the following parametric values: $$M=\$3500/\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r},c=\$80/\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t},\beta =0.2,h_d=\$5/\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t},D_s=76,f=\$2/\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p},v_1=\$0.02/\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{n}/\mathrm{k}\mathrm{m},v_2=\$0.3/\mathrm{k}\mathrm{m},z=4,T_d=100\mathrm{k}\mathrm{m},F_c=2\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{e}/\mathrm{k}\mathrm{m},t_x=\$1.7/\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{O}2,E_1=0.4\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{O}2/\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}\mathrm{u}\mathrm{e}\mathrm{l},u=0.8\frac{\mathrm{k}\mathrm{W}\mathrm{h}}{\mathrm{g}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{n}}/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r},w=3\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{O}2/\mathrm{k}\mathrm{W}\mathrm{h},\mathrm{\psi }=0.7$$

In order to obtain a feasible solution, it is necessary to verify the feasibility of the corresponding mathematical terms (constraints) of the profit function (12). To this end, with the help of Lingo 18 software and following the algorithm discussed in the previous section 3.1, we get the following feasible solutions: $${\mathrm{K}}^{\ast }=3.695\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r},{\mathrm{\gamma }}^{\ast }=\$10.822/\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}{\phi }^{\ast }=\$1388.507/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}$$ which satisfy the associated constraints (2)-(11), and the objective (profit) function (12) reaches its maximum.

Now to check the concavity, utilizing the mentioned parametric values in equation (20), we get

${\left[\frac{{\partial }^2\phi }{\partial K^2}\right]}_{K=K^{\ast }}=-186.965<0$ , which yields negative result. So, it proves the concavity of equation (12) and thus converges to a specific value $K^{\ast }$ at which the maximum profit is attained (Fig. 2).

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Fig. 2

Concavity of total profit function $\phi$ with respect to $\gamma$ and $K$ .

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3.2.2

3.2.2 Example 2

When the quantity of emissions is higher than carbon cap, for this situation let us consider the same parametric values of Example 1 and additional values $C_p=1200\mathrm{t}\mathrm{o}\mathrm{n}$ , $x_1=\$1.5/\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{O}2$ . In order to obtain a feasible solution, it is necessary to verify the feasibility of the corresponding mathematical terms (constraints) of the profit function (14). To this end, with the help of Lingo 18 software and following the algorithm discussed in the previous section 3.1, we get the following feasible solutions:

$K^{\ast }=2.835\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r},{\gamma }^{\ast }=\$10.455/\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}{{\phi }_1}^{\ast }=\$1206.904/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}$ which satisfy the associated constraints (2)-(11), (13) and the objective (profit) function (14) reaches its maximum.

Now to check the concavity, utilizing the mentioned parametric values in equation (24), we get

${\left[\frac{{\partial }^2\phi }{\partial K^2}\right]}_{K=K^{\ast }}=-10992.734<0$ , which yields negative result. So, it proves the concavity of equation (14) and thus converges to a specific value $K^{\ast }$ at which the maximum profit is attained (Fig. 3).

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Fig. 3

Concavity of total profit function ${\phi }_1$ with respect to $\gamma$ and $K.$

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3.2.3

3.2.3 Example 3

When the quantity of emissions is lower than carbon cap, then consider the same parametric values of Example 1 and additional values $C_p=1500\mathrm{t}\mathrm{o}\mathrm{n}$ , $x_2=\$2/\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{C}\mathrm{O}2$ . In order to obtain a feasible solution, it is necessary to verify the feasibility of the corresponding mathematical terms (constraints) of the profit function (16). To this end, with the help of Lingo 18 software and following the algorithm discussed in the previous section 3.1, we get the following feasible solutions:

$K^{\ast }=2.382\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r},{\gamma }^{\ast }=\$10.193/\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}{{\phi }_2}^{\ast }=\$1426.250/\mathrm{y}\mathrm{e}\mathrm{a}\mathrm{r}$ which satisfy the associated constraints (2)-(11), (15) and the objective (profit) function (16) reaches its maximum.

Now to check the concavity, utilizing the mentioned parametric values in equation (28), we get

${\left[\frac{{\partial }^2\phi }{\partial K^2}\right]}_{K=K^{\ast }}=-647.186<0$ , which yields negative result. So, it proves the concavity of equation (16) and thus converges to a specific value $K^{\ast }$ at which the maximum profit is attained (Fig. 4).

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Fig. 4

Concavity of total profit function ${\phi }_2$ with respect to $\gamma$ and $K$ .

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3.3 Checking the robustness of the model

An analysis to check the robustness of the model has been performed on some important parameters by making varieties of a single parameter from −20% to +20% while others remain consistent to show their significant impacts on the retailer's profit (Table 2). The analysis has been conducted based on two main cases: when the cap is absent and the implementation of cap-and-trade policy. In cap-and-trade policy, two subcases are considered when the cap is greater or less than total emissions.

From the above analysis, we come to the following perceptions

• In light of the analysis, it is seen that if the retailer needs to purchase the item at a more significant cost $(c)$ than expected, however selling price remains steady, the profit earned at that time may not be able to achieve a strong net profit. On the other hand, the reduction in per unit purchase cost $(c)$ leads to a relative expansion in profit. This is because the retailer buys the product at a lower price than before but still sells at the previous price, which contributing to higher profits.

• At the point when the retailer has to go through more cost to hold the item per unit, he does not keep the product in stock for a long time so as not to incur huge losses, which prompts less preservation investment $(\gamma )$ and a slight decrease in profit.

• The cost of transporting the goods depends mainly on the distance ${(T}_d)$ covered. The shorter the distance, the lower the cost and the higher the profit. Again, if the number of trips $(z)$ is increased, the total covered distance increases and so does the cost of transportation, which allays the profit. That is, the number of trips ${(T}_d)$ , and the distance traveled $(z)$ are inversely proportional to the profit.

• When the rate of deterioration $(\beta )$ of the product expands, the retailer starts investing $(\gamma )$ more to prevent it, and during this period the profit diminishes.

• An increment in emissions standards by electricity during holding $\left(w\right),$ and by vehicles during transportation ${(E}_1),$ altogether significantly increase the amount of tax paid by the seller, bringing about lower profits.

• In the event that the carbon cap ${(C}_p)$ expands, that is, when the disparity among emissions, and caps is more noteworthy, the retailer makes higher profits by selling increased allowances. Also, a significant profit is seen, when the selling price of carbon per unit ${(x}_2)$ is increased. On the other hand, profit declines because of the increment in mandatory unit tax for each unit of carbon emissions ${(t}_x)$ and unit penalty charges for overabundance emissions ${(x}_1)$ .

3.4 Managerial insights

A sustainable inventory always proposes some significant recommendations for a retailer to implement in real life. Implementing a cap-and-trade policy always provides a better experience for a manager, while preservation technology added extra features of facilities to the proposed model. Some specific insights which any managers can easily practice in real business arena are:

• A roadmap on proper utilization of transport system is suggested with its associated attributes, e.g., distance, number of trips, carbon emissions etc., for industry managers to calculate how to optimize the industries profits in efficient way.

• This study synthesizes the shortcomings of cap-and-trade policy together with preservation technology investment.

• A proper scenario for preservation technology is suggested here when to stop the investments and how much optimal investment is beneficial for the manager.