Sustainable production inventory management through bi-level greening performance in a three-echelon supply chain

Abstract

Recently, different greening activities in various steps of a supply chain are being used to keep a system sustainable. As a result, it creates a terrific business opportunity for a business concern. In such a direction, the present study develops a supplier-manufacturer-retailer green supply chain model under a bi-level greening performance. The first level of a greening performance is in the raw-materials delivered by a supplier, while the second level is in the production process of the manufacturer’s production house. The novelty of this model is that stocking decision has been taken on the basis of levels of these greening activities. Also, here the supplier offers a credit period to the manufacturer based on the quantity and quality of the raw-materials ordered by him/her, whereas the manufacturer forwards a credit period, which has been assumed as type-2 fuzzy in nature to tackle a non-random uncertainty, to the retailer. Under such a fuzzy uncertain environment, a new defuzzification process has been proposed to get the optimal solution of the model. Comparing different scenarios of the proposed model, our findings reveal that although bi-level greening performance is beneficial from the perspective of the whole supply chain, it is less profitable from the manufacturer’s point of view. It also reveals that bi-level greening activity positively impacts the credit policy of supply chain members. On the sustainability issue, the imperfect production has a negative impact on economic and environmental performances. Finally, several practical managerial implications have been provided to make a decision on bi-level greening performance.

Appendices

Appendix A

Manufacturer’s holding cost for the raw-materials:

As discussed in Section 4.2.1, here we have derived the total raw-materials holding cost by considering two different cycle structures: Raw-materials holding cost for the first $k_1$ cycles and raw-materials holding cost for remaining $(k-k_1)$ cycles. We have considered the scenarios because of the variation in inventory patterns. As we can observe from Fig. 2, during the first $k_1$ cycles, at the start of every cycle, the manufacturer has raw-materials stock carried out from the previous cycle along with the ordered amount from the supplier. On the other side, in the remaining $(k-k_1)$ cycles, there is no replenishment from the supplier; only the unused raw-materials of the first $k_1$ cycles are used for production. Now, we will formulate the holding costs for these two cycle structures and then finally, taking the sum of the obtained holding costs, we get the manufacturer’s total holding cost for raw-materials.

Raw-materials holding cost for first $k_1$ cycles:

Let us denote $I_r(t)$ as the manufacturer’s raw-materials inventory level at any instant t $(0 \le t \le kT)$. For the first $k_1$ cycles, the governing differential equation for $I_r (t)$ is given by

$$\begin{aligned}{} & {} \frac{dI_r (t)}{dt} = - fP, \quad (i-1)T \le t \le iT,\quad \nonumber \\{} & {} \quad \text{ under } \text{ the } \text{ boundary } \text{ condition } I_r((i-1)T) = iQ_m - (i-1) Q_p, \nonumber \\{} & {} \quad \text{ where } i= 1, 2, ..., k_1 \end{aligned}$$

(A.1)

The solution of the differential equation (A.1) is (using $Q_p = fPT$)

$$\begin{aligned} I_r (t) = iQ_m - fPt \quad (i-1)T \le t \le iT, \text{ where } i = 1,2, ...., k_1 \end{aligned}$$

Therefore the present value of holding cost for the i-th cycle ($i = 1, 2,...., k_1$) is

$$\begin{aligned} MRHC_i= & {} C_{mr} \Big \{\frac{iQ_m}{R} [e^{-(i-1)RT} - e^{-iRT}] - \frac{fP}{R}[(i-1)T e^ {-(i-1)RT}- iT e^{-iRT}]\nonumber \\{} & {} -\frac{fP}{R^2}[e^{-(i-1)RT} - e^ {-iRT}]\Big \} \end{aligned}$$

(A.2)

Raw-materials holding cost for remaining $k - k_1$ cycles:

The governing differential equation that delineate the structure of manufacturer’s remaining in-stock raw-materials is expressed as the following

$$\begin{aligned}{} & {} \frac{dI_r (t)}{dt} = - fP, \quad iT \le t \le (i+1)T,\quad \nonumber \\{} & {} \quad \text{ under } \text{ the } \text{ boundary } \text{ condition } I_r(iT) =(k-i) Q_p,\nonumber \\{} & {} \quad \text{ where } i= k_1, k_1 + 1,....., k-1 \end{aligned}$$

(A.3)

Similarly, using the solution of differential equation (A.3) we get the present value of holding cost for the i-th cycle ($i = k_1, k_1 + 1, ....., k-1$) as

$$\begin{aligned} {\text{MRHC}}_{i} = & C_{{mr}} \{ \frac{{kQ_{p} }}{R}[e^{{ - iRT}} - e^{{ - (i + 1)RT}} ] - \frac{{fP}}{R}[iTe^{{ - iRT}} - (i + 1)Te^{{ - (i + 1)RT}} ] \\ - & \frac{{fP}}{{R^{2} }}[e^{{ - iRT}} - e^{{ - (i + 1)RT}} ]\} \\ \end{aligned}$$

(A.4)

Using (A.2) and (A.4), we get the present value of total holding cost for raw-materials as

$$\begin{aligned} {\text{MRHC}} = & \sum\limits_{{i = 1}}^{{k_{1} }} {{\text{MRHC}}_{i} } + \sum\limits_{{i = k_{1} }}^{{k - 1}} {{\text{MRHC}}_{i} } \\ = & C_{{mr}} \{ \frac{{Q_{m} }}{{R(1 - e^{{ - RT}} )}}(1 - e^{{ - k_{1} RT}} - k_{1} e^{{ - k_{1} RT}} + k_{1} e^{{ - (k_{1} + 1)RT}} ) \\ + & \frac{{kQ_{p} }}{R}(e^{{ - k_{1} RT}} - e^{{ - kRT}} ) + \frac{{fP}}{{R^{2} }}(kRTe^{{ - kRT}} + e^{{ - kRT}} - 1)\} \\ \end{aligned}$$

(A.5)

Manufacturer’s paid interest to the supplier:

Present value of the total paid interest to the supplier is given by

$$\begin{aligned} MIP= & {} I_p S (T-M) \Big [ \frac{Q_m }{(1-e^{-RT})^2} \big ( 1 - (k_1 + 1) e^{-k_1 RT} + k_1 e^{-(k_1 + 1)RT}\big )\nonumber \\{} & {} - \frac{fP (T+M)}{2(1- e^{-RT})} (1-e^{-k_1 RT})\Big ] \end{aligned}$$

(A.6)

Appendix B

Manufacturer’s holding cost for the good finished products:

Here we have formulated the total holding cost for perfect quality items by considering two different cycle structures: Holding cost of good finished products for the first k cycles and holding cost of good finished products for the remaining $(n-k)$ cycles. During the first k cycles, production continues in the manufacturing system and at the end of the $i-$th cycle ($i=1,2,......,k$), the retailer replenishes $Q_r$ amount of perfect quality items from the manufacturer. After fulfilling the retailer’s requirement, the manufacturer stocks the remaining items carried out into the next cycle. This process continues until the time kT, i.e., the end of the k-th cycle. On the other side, there is no production for the remaining $(n-k)$ cycles. Only the items kept in stock during the last k cycles are used to fulfil the retailer’s requirement.

Holding cost of good finished products for first k cycles:

Let ${(I_f)}_{i} (t)$ be the good green products’ inventory level at time $t \in [(i-1)T, iT]$, where $i = 1,2,3,...,k$. Then the differential equation for ${(I_f)}_{i} (t)$ can be expressed as

$$\begin{aligned} \frac{d{(I_f)}_{i} (t)}{dt} = \beta P, \quad (i-1)T \le t \le iT, i = 1,2,...,k \end{aligned}$$

(A.7)

with the initial condition ${(I_f)}_{1} (0) = 0$ and ${(I_f)}_{i} (iT) = {(I_f)}_{i} ((i-1)T) - Q_r, i = 2,3,...,k$

Therefore the manufacturer’s good green products’ inventory level at time $t \in [(i-1)T, iT]$ is given by

$$\begin{aligned} {(I_f)}_{i} (t)= & {} (\beta _0 - \beta _1 P)Pt - \beta _2 P \frac{t^2}{2} - (i-1) Q_r, \nonumber \\{} & {} \quad (i-1)T \le t \le iT, i= 1,2,....,k \end{aligned}$$

(A.8)

So, using (A.8), we get the present value of the holding cost for good green products for the i-th cycle ($i = 1,2,...., k$) as

$$\begin{aligned} MFHC_{i}= & {} (\beta _0 - \beta _1 P)P \bigg \{\frac{T}{R}\Big [(i-1)e^{-(i-1)RT} - ie^{-iRT} \Big ] + \frac{1}{R^2}\Big [ e^{-(i-1)RT} - e^{-iRT}\Big ]\bigg \}\nonumber \\{} & {} - \frac{\beta _2 P}{2} \bigg \{ \frac{T^2}{R}\Big [ (i-1)^2 e^{-(i-1)RT} - i^2 e^{-iRT}\Big ] \nonumber \\{} & {} +\frac{2T}{R^2}\Big [(i-1)e^{-(i-1)RT} - ie^{-iRT} \Big ] + \frac{2}{R^3}\Big [ e^{-(i-1)RT} - e^{-iRT}\Big ]\bigg \}\nonumber \\{} & {} -\frac{(i-1)Q_r}{R}\Big \{ e^{-(i-1)RT} - e^{-iRT}\Big \} \end{aligned}$$

(A.9)

Holding cost of good finished products for remaining $(n -k)$ cycles:

The present value of the holding cost for good green products for the remaining i-th cycle ($i = k, k + 1, ....., n-1$) is given by

$$\begin{aligned} MFHC_{i}= & {} \frac{C_{mh} Q_r}{R}(n-i) (e^{-iRT} - e^{-(i+1)RT}) \end{aligned}$$

(A.10)

Using (A.9) and (A.10), we get the present value of total holding cost for good finished items as

$$\begin{aligned} MFHC= & {} \sum _{i=1} ^{k} MFHC_{i} + \sum _{i=k} ^{n-1} MFHC_{i}\quad \nonumber \\= & {} C_{mh} \bigg \{ \frac{P}{R^2}(\beta _0 - \beta _1 P) \big ( {1-(1+kRT) e^{-kRT}}\big )\nonumber \\{} & {} - \frac{\beta _2 P}{2R^3} \Big ( {2-2(1+kRT)e^{-kRT} - k^2 T^2 R^2 e^{-kRT}}\Big )\nonumber \\{} & {} + \frac{Q_r}{R(1-e^{-RT})} \Big ( {n\big ( e^{-kRT} - e^{-(k+1)RT}\big ) + e^{-(n+1)RT} - e^{-RT}}{}\Big )\bigg \} \end{aligned}$$

(A.11)

Appendix C

Derivation of $RNP_n$ and RIEB:

First, we derive the net profit in total n cycles ($RNP_n$). To do that, the net profit in each of the ith-cycle ($RNP_i, i=1,2,...,n$)has to be calculated. Then by adding all these profits, we can get the net profit in n-cycles.

Let ${(J_r)}_{i}(t)$ be the inventory level of the retailer at time $t\in [iT, (i+1)T], i = 1,2,...,n$. Then from the inventory pattern of the retailer (see Level 4) for the i-th cycle we can derive the differential equation of ${(J_r)}_{i}(t)$ as:

$$\begin{aligned}{} & {} \frac{d {(J_r)}_{i}(t)}{dt} = -D \quad \text{ when } iT \le t \le (i+1) T\text{, } i = 1,2,......,n \quad \nonumber \\{} & {} \text{ with } \text{ the } \text{ boundary } \text{ condition } J_r ((i+1)T) = 0 \end{aligned}$$

(A.12)

Then the solution of the differential equation (A.12) is $J_r (t) = (i+1)DT -Dt, iT \le t \le (i+1) T$ and $i = 1,2,......,n$.

Using the solution, we get the present value of the retailer’s holding cost for the i-th cycle is $RHC_i = \frac{C_{rh} D}{R^2} {\big (e^{-(i+1)RT} + (RT-1) e^{-iRT}\big )}$, where $i = 1, 2, ...., n$.

Similar to the case of the manufacturer, here also for each cycle, the retailer’s purchasing price will be evaluated at time iT, $i=1,2,....,n$. Therefore the present value of the retailer’s purchasing cost for the i-th order ($i = 1, 2, ...., n$) is $RPC_{i} = S_m Q_r e^{-iRT}$.

The retailer’s pricing strategy is also the same as in Panja and Mondal (2019, 2020a), i.e., he/she retails each product at a price $S_r = m_2 S_m$, where $m_2 (>1)$ is a mark-up on the manufacturer’s wholesale price. Here, the retailer instantly collects revenue from the market customers by selling each product. So, the present value of the sales revenue of the retailer for the i-th cycle is $RSR_i = \frac{S_r D}{R} {(e^{-iRT} - e^{-(i+1)RT})}, i = 1,2,....,n$.

Since the retailer earns his/her revenue by selling a single green product during the interval $[iT, (i+1)T]$ and pays the payment at time $(i+1)T$ for the i-th cycle, he/she can earn interest from this selling amount up to time $(i+1)T$. Thus, we get the present value of the interest earned by the retailer in each of the i-th cycle as $RIE_{i}= \frac{I_e S_rD}{R^2} \big \{e^{-(i+1)RT} + (RT-1) e^{-iRT}\big \}$, ($i = 1, 2, ...., n$).

In each cycle, the retailer has to pay interest to the manufacturer. Since the payable interest is calculated based on the purchasing price, the present value of payable interest for the i-th cycle ($i = 1, 2, ...., n$) is $RIP_{i} = I_p S_m e^{-iRT}D \frac{(T-N)^2}{2}$.

Present value of the retailer’s ordering cost for the i-th cycle ($i = 1, 2, ...., n$) is $ROC_{i} = A_r e^{-iRT}$

So, the net profit of the retailer in each of the i-th cycles under present value consideration is

$$\begin{aligned} RNP_{i}= & {} RSR_{i} + RIE_{i} - RHC_{i} - RPC_{i} - RIP_{i} - ROC_{i}, \quad i = 1,2, ..., n \end{aligned}$$

Therefore the present value of the retailer’s net profit for the total n cycles is

$$\begin{aligned} RNP_n= & {} \sum _{i=1} ^{n} RNP_{i}\nonumber \\= & {} \frac{S_r D}{R} {\big (e^{-RT} - e^{-(n+1)RT}\big )} + \frac{I_e S_rD}{R^2 (1-e^{-RT})}\nonumber \\{} & {} \bigg [ {\big ( e^{-2RT} - e^{-(n+2)RT}\big ) +(RT-1) \big ( e^{-RT} - e^{-(n+1)RT}\big )}\bigg ]\nonumber \\{} & {} - \frac{C_{rh} D}{R^2 (1-e^{-RT})} \bigg [ {\big ( e^{-2RT} - e^{-(n+2)RT}\big ) + (RT-1) \big ( e^{-RT} - e^{-(n+1)RT}\big )}{}\bigg ]\nonumber \\{} & {} - S_m Q_r \frac{e^{-RT} - e^{-(n+1)RT}}{1-e^{-RT}}\nonumber \\{} & {} - I_p S_m D\frac{(T-N)^2}{2(1-e^{-RT})} {\big (e^{-RT} - e^{-(n+1)RT}\big )}{} - A_r \frac{e^{-RT} - e^{-(n+1)RT}}{1-e^{-RT}} \end{aligned}$$

(A.13)

Next, we obtain the total interest earned from the bank (RIEB). In that policy, the retailer deposits the net profit of each cycle in the bank and earns interest on the deposited amount during the remaining time of the year. Thus, $RNP_1$, the net profit of the first cycle, is deposited in the bank for the duration of $(n-1)T$. Similarly, the net profit in the second cycle $(RNP_2)$ is deposited in the bank for the duration $(n-2)T$. Proceeding in this way, we can get the total interest earned by the retailer from the bank during his/her total business period as

$$\begin{aligned} RIEB= & {} \sum _{i=1} ^{n} RNP_{i} . \big [(n-i)TI_e\big ] = nTI_e. RNP_{n} - T I_e \sum _{i=1} ^{n} i . RNP_{i}\nonumber \\= & {} nTI_e.RNP_n - TI_e \bigg [\frac{S_r D}{R} \big \{ Tr_{1} - Tr_{2}\big \} + \frac{I_e S_r D}{R^2} \big \{ Tr_{2} + (RT-1) Tr_{1}\big \} - \frac{C_{rh} D}{R^2} \big \{ Tr_{2} \nonumber \\{} & {} + (RT-1) Tr_{1}\big \}- S_m Q_r Tr_{1} - I_p S_m D \frac{(T-N)^2}{2} Tr_{1} - A_r Tr_{1}\bigg ]\nonumber \\ \text{ where } Tr_{1}= & {} \frac{1}{(1-e^{-RT})^2} \Big (e^{-RT} - (n+1) e^{-(n+1)RT} + n e^{-(n+2) RT}\Big ) \text{ and } Tr_2=\frac{1}{(1-e^{-RT})^2} \Big (e^{-2RT}\nonumber \\{} & {} - (n+1) e^{-(n+2)RT} + n e^{-(n+3) RT}\Big ) \end{aligned}$$

(A.14)

Appendix D

Evaluation of TRS of $\widetilde{\widetilde{N}}$:

The explicit expression of $\mu _{\widetilde{N}} (x)$ has been derived in the form of following cases depending on the position of x:

Case 1. when $N_0 - \Delta _1 \le x \le N_0$:

Here, $\mu _1 (x) = 0$ and $\mu _2 (x) = \frac{\{x-(N_0-\Delta _1)\} w_2}{\Delta _1}$. So, in this domain, the membership function of TRS is given by

$$\begin{aligned} \mu _{\widetilde{N}} (x)= & {} \frac{ w_2 (\eta _1 + 2 \eta _2)}{3 \Delta _1 (\eta _1 + \eta _2)}\big \{x-(N_0-\Delta _1)\big \} \end{aligned}$$

(A.15)

Case 2. when $N_0 \le x \le N_0 + \Delta _2$:

Here, $\mu _1 (x) = \frac{(x-N_0) w_1}{\Delta _2}$ and $\mu _2 (x) = w_2$. So, in this domain, the membership function of TRS is given by

$$\begin{aligned} \mu _{\widetilde{N}} (x)= & {} \frac{ w_1 (\eta _2 + 2\eta _1)}{3\Delta _2 (\eta _1 + \eta _2)}\big \{x- (N_0 - \Delta _2 w')\big \}, \text{ where } w' = \frac{w_2 (\eta _1 + 2 \eta _2)}{w_1 (\eta _2 + 2\eta _1)} \end{aligned}$$

(A.16)

Case 3. when $N_0 + \Delta _2 \le x \le N_1 - \Delta _1$:

Here, $\mu _1 (x) = w_1$ and $\mu _2 (x) = w_2$. Therefore, in this domain, the membership function of TRS is given by

$$\begin{aligned} \mu _{\widetilde{N}} (x)= & {} \frac{1}{3(\eta _1 + \eta _2)} \big \{w_1 (\eta _2 + 2\eta _1) + w_2 (\eta _1 + 2 \eta _2)\big \} \end{aligned}$$

(A.17)

Case 4. when $N_1 - \Delta _1 \le x \le N_1$:

Here, $\mu _1 (x) = \frac{(N_1-x) w_1}{\Delta _1}$ and $\mu _2 (x) = w_2$. So, in this domain, the membership function of TRS is given by

$$\begin{aligned} \mu _{\widetilde{N}} (x)= & {} \frac{ w_1 (\eta _2 + 2\eta _1)}{3\Delta _1 (\eta _1 + \eta _2)}\big \{ (N_1 + \Delta _1 w') - x\big \} \end{aligned}$$

(A.18)

Case 5. when $N_1 \le x \le N_1 + \Delta _2$:

Here, $\mu _1 (x) = 0$ and $\mu _2 (x) = \frac{\{(N_1 +\Delta _2) -x\} w_2}{\Delta _2}$. Thus, in this domain, the membership function of TRS is given by

$$\begin{aligned} \mu _{\widetilde{N}}(x)= & {} \frac{ w_2 (\eta _1 + 2\eta _2)}{3\Delta _2 (\eta _1 + \eta _2)}\big \{(N_1 +\Delta _2) -x\big \} \end{aligned}$$

(A.19)

The explicit expression of $\mu _{\widetilde{N}} (x)$ for every $x\in [N_0-\Delta _1, N_1+\Delta _2]$ is obtained using (A.15)-(A.19) as

$$\begin{aligned} \mu _{\widetilde{N}}(x)={\left\{ \begin{array}{ll} \frac{ w_2 (\eta _1 + 2 \eta _2)}{3 \Delta _1 (\eta _1 + \eta _2)}\big \{x-(N_0-\Delta _1)\big \}, &{} \text {if} N_0 - \Delta _1 \le x \le N_0.\\ \frac{ w_1 (\eta _2 + 2\eta _1)}{3\Delta _2 (\eta _1 + \eta _2)}\big \{x- (N_0 - \Delta _2 w')\big \}, &{} \text {if }N_0 \le x \le N_0+\Delta _2.\\ \frac{1}{3(\eta _1 + \eta _2)} \big \{w_1 (\eta _2 + 2\eta _1) + w_2 (\eta _1 + 2 \eta _2)\big \}, &{} \text {if }N_0+\Delta _2 \le x \le N_1 - \Delta _1.\\ \frac{ w_1 (\eta _2 + 2\eta _1)}{3\Delta _1 (\eta _1 + \eta _2)}\big \{ (N_1 + \Delta _1 w') - x\big \}, &{} \text {if }N_1-\Delta _1 \le x \le N_1.\\ \frac{ w_2 (\eta _1 + 2\eta _2)}{3\Delta _2 (\eta _1 + \eta _2)}\big \{(N_1 +\Delta _2) -x\big \}, &{} \text {if }N_1 \le x \le N_1 + \Delta _2.\\ 0, &{} \text {elsewhere} \end{array}\right. } \end{aligned}$$

(A.20)

where $w' = \frac{w_2 (\eta _1 + 2 \eta _2)}{w_1 (\eta _2 + 2\eta _1)}$. Here, $\mu _{\widetilde{N}}(x)$ is a generalized hexagonal fuzzy number.

Evaluation of the centroid of $\widetilde{N}$:

Here, we get the TRS, $\widetilde{N}$ as a generalized hexagonal fuzzy number whose membership function has been presented in equation (A.20). The centroid of the mentioned fuzzy number can be obtained through following theorem.

Theorem 1

The centroid of the hexagonal fuzzy number $\widetilde{N}$ is given by

$$\begin{aligned} CV_{\widetilde{N}}= & {} \frac{\int \limits _{-\infty } ^{\infty } x \mu _{\widetilde{N}} (x) dx}{\int \limits _{-\infty } ^{\infty } \mu _{\widetilde{N}} (x) dx} = \frac{CV_1}{CV_2}\\ \text{ where } CV_1= & {} \frac{w_1(\eta _2 + 2\eta _1)}{6}\big [\big \{(N_1-\Delta _1)^2 + N_1 (N_1 - \Delta _1) + N_1^2\big \}\\{} & {} - \big \{(N_0+\Delta _2)^2 + N_0 (N_0 + \Delta _2) + N_0^2\big \}\big ]\\{} & {} + \frac{w_2(\eta _1 + 2\eta _2)}{6}\big [\big \{(N_1+\Delta _2)^2 + N_1 (N_1 + \Delta _2) + N_1^2\big \} \\{} & {} - \big \{(N_0-\Delta _1)^2 + N_0 (N_0 - \Delta _1) + N_0^2\big \}\big ]\\ CV_2= & {} \frac{w_1(\eta _2 + 2\eta _1)}{2}\big [\big \{ (N_1-\Delta _1) + N_1\big \} - \big \{ N_0+ (N_0+\Delta _2) \big \}\big ] \\{} & {} + \frac{w_2(\eta _1 + 2\eta _2)}{2}\big [\big \{ (N_1+\Delta _2) + N_1\big \} - \big \{ N_0+ (N_0-\Delta _1) \big \}\big ] \end{aligned}$$

Proof

We have from equation (A.20)

$$\begin{aligned} \int \limits _{-\infty } ^{\infty } x \mu _{\widetilde{N}} (x) dx= & {} \frac{w_1(\eta _2 + 2\eta _1)}{6(\eta _1 + \eta _2)} [3(N_1^2-N_0^2)-3(N_0\Delta _2+ N_1\Delta _1) - (\Delta _2^2-\Delta _1^2)] \\{} & {} + \frac{w_2(\eta _1 + 2\eta _2)}{6(\eta _1 + \eta _2)}[3(N_1^2-N_0^2)+3(N_0\Delta _1+ N_1\Delta _2) + (\Delta _2^2-\Delta _1^2)]\\= & {} \frac{w_1(\eta _2 + 2\eta _1)}{6(\eta _1 + \eta _2)} \big \{[(N_1 - \Delta _1)^2 + N_1 (N_1-\Delta _1) + N_1^2] - [(N_0 + \Delta _2)^2\\{} & {} +N_0 (N_0+\Delta _2) + N_0^2]\big \}\\{} & {} + \frac{w_2(\eta _1 + 2\eta _2)}{6(\eta _1 + \eta _2)}\big \{[(N_1 + \Delta _2)^2 + N_1 (N_1+\Delta _2) + N_1^2] - [(N_0 - \Delta _1)^2\\{} & {} +N_0 (N_0-\Delta _1) + N_0^2]\big \}\\ \text{ and } \int \limits _{-\infty } ^{\infty } \mu _{\widetilde{N}} (x) dx= & {} \frac{w_1(\eta _2 + 2\eta _1)}{2(\eta _1 + \eta _2)} [2(N_1 - N_0) - (\Delta _1 + \Delta _2)] + \frac{w_2(\eta _1 + 2\eta _2)}{2(\eta _1 + \eta _2)} [2(N_1 - N_0) + (\Delta _1 + \Delta _2)]\\= & {} \frac{w_1(\eta _2 + 2\eta _1)}{2(\eta _1 + \eta _2)} \big \{[(N_1-\Delta _1) + N_1] - [N_0 + (N_0 + \Delta _2)]\big \}\\{} & {} + \frac{w_2(\eta _1 + 2\eta _2)}{2(\eta _1 + \eta _2)} \big \{[N_1 + (N_1 + \Delta _2)] - [(N_0 - \Delta _1) + N_0]\big \} \end{aligned}$$

Theorem 2

The crisp form of the joint type-2 fuzzy profit $JTP_{(n, k_1, \eta , \theta )}(\widetilde{\widetilde{N}})$ is given by

$$\begin{aligned} {CV}_{JTP(\widetilde{N})}= & {} \frac{\frac{w_1(\eta _2+2\eta _1)}{6}\big [(t_3^2+t_3 t_4+t_4^2)- (t_2^2+t_1 t_2+t_1^2)\big ]+ \frac{w_2(\eta _1+2\eta _2)}{6}\big [(t_5^2+t_4 t_5+t_4^2)- (t_0^2+t_0 t_1+t_1^2)\big ]}{\frac{w_1(\eta _2+2\eta _1)}{2}\big [(t_3+t_4)-(t_1+t_2)\big ]+ \frac{w_2(\eta _1+2\eta _2)}{2}\big [(t_5+t_4)-(t_0+t_1)\big ]} \end{aligned}$$

where $t_0 = {JTP}_{(n,k_1,\eta ,\theta )} (N_0-\Delta _1), t_1 = {JTP}_{(n,k_1,\eta ,\theta )} (N_0), t_2 = {JTP}_{(n,k_1,\eta ,\theta )} (N_0 + \Delta _2), t_3 = {JTP}_{(n,k_1,\eta ,\theta )} (N_1 - \Delta _1), t_4 = {JTP}_{(n,k_1,\eta ,\theta )} (N_1), t_5 = {JTP}_{(n,k_1,\eta ,\theta )} (N_1 + \Delta _2)$.

Proof

The proof follows from Theorem 1.