Analysis of a production system of green products considering single-level trade credit financing via a parametric approach of intervals and meta-heuristic algorithms

Abstract

The use of chemicals and preservatives during the production of most daily-used products has increased rapidly over the last few decades. In turn, manufacturers, to improve the sustainability of a manufacturing unit in the current highly competitive market, are obliged to provide various facilities to their customers. However, estimating the best optimal decision for manufacturing companies seems like a daunting task amid highly uncertain market conditions. By shedding more light on these facts and considering the gaps in the existing literature on production inventory, this work aims to incorporate a production model which is concerned with the factors mentioned above, so that it can be beneficial for a manufacturer. This work involves the demonstration of an imperfect green production system under single-layer full-trade credit financing and partial backlog shortages using interval flexibility. Based on the credit period lengths provided by the manufacturer to the consumers, three distinct cases have arisen and for each case, the average profit of the system appears to be an interval valued optimal control problem, and several theorems are evaluated to obtain analytical forms of respective objective functions of each case. Additionally, to test the effect of different credit lengths on optimal policy and to examine the feasibility of each case, a numerical example is solved using the Equilibrium Optimizer (EO) algorithm. Finally, a sensitivity experiment is performed and the results obtained from the simulation are shown graphically to gain some managerial insights.

Appendix

1.1 Algebra of intervals (Ramezanadeh et al. [75]).

Let, ${I}_c=\left\{\left[{a}_L,{a}_U\right]:{a}_L,{a}_U\in \mathbb{R}\ \mathsf{and}{a}_L\le {a}_U\right\}.$

Now the parametric forms of an element [a_L, a_U] ∈ I_care defined as follows:

• (i) Increasing parametric form (IPF): [a_L, a_U] = {a(ξ) = a_L + ξ(a_U − a_L) : ξ ∈ [0, 1] }

• (ii) Decreasing parametric form (DPF): [a_L, a_U] = {a(ξ) = a_U + ξ(a_L − a_U) : ξ ∈ [0, 1] }.

Therefore, the set of all compact intervals in parametric form is denoted by I_P and it is defined by

$$ {I}_P=\left\{a\left(\xi \right):a\left(\xi \right)\;\mathsf{is}\ \mathsf{IPF}\;\left(\ \mathsf{or}\ \mathsf{DPF}\right)\ \mathsf{of}\ \left[{a}_L,{a}_U\right],\forall \xi \in \left[0,1\right]\;\mathsf{and}\forall \left[{a}_L,{a}_U\right]\in {I}_c\right\} $$

Clearly, the sets I_c and I_P are equivalent.

Definition A.1

(Ramezanadeh et al. [75]). Let I₁ = {a(ξ) : ξ ∈ [0, 1]}, I₂ = {b(ξ) : ξ ∈ [0, 1]} ∈ I_p, and let μ ∈ ℝ. Different arithmetic operations on I_Pare defined as follows:

• (i) Addition: I₁ + I₂ = {a(ξ₁) + b(ξ₂) : ξ₁, ξ₂ ∈ [0, 1]}

• (ii) Subtraction: I₁ − I₂ = {a(ξ₁) − b(ξ₂) : ξ₁, ξ₂ ∈ [0, 1]}

• (iii) Multiplication :I₁I₂ = {a(ξ₁)b(ξ₂) : ξ₁, ξ₂ ∈ [0, 1]}

• (iv) Division: $\raisebox{1ex}{${I}_1$}\!\left/ \!\raisebox{-1ex}{${I}_2$}\right.=\left\{\frac{a\left({\xi}_1\right)}{b\left({\xi}_2\right)}:{\xi}_1,{\xi}_2\in \left[0,1\right]\right\},$ provided 0 ∉ I₂.

• (v) Parametric difference: I₁−_pI₂ = {a(ξ) − b(ξ) : ξ ∈ [0, 1]}

• (vi) Scalar Multiplication: μI₁ = {μa(ξ) : ξ ∈ [0, 1]}

• (vii) Equality: ${I}_1={I}_2\kern0.24em \mathsf{if}\kern0.17em \mathsf{and}\kern0.17em \mathsf{only}\kern0.17em \mathsf{if}\;a\left(\xi \right)=b\left(\xi \right),\mathsf{for}\;\;\mathsf{all}\;\;\xi \in \left[0,1\right].$

Definition A.2

(Ramezanadeh et al. [75]). Let I₁ = [a_L, a_U], I₂ = [b_L, b_U] ∈ I_c, and their parametric representations I₁ = {a(ξ) : ξ ∈ [0, 1]}, I₂ = {b(ξ) : ξ ∈ [0, 1]} ∈ I_p and let μ ∈ ℝ. The different arithmetic operations on I_care defined as follows:

• (i) ${I}_1+{I}_2=\left[\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\min}\left(a\left({\xi}_1\right)+b\left({\xi}_2\right)\right),\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\max}\left(a\left({\xi}_1\right)+b\left({\xi}_2\right)\right)\right]$

• (ii) ${I}_1-{I}_2=\left[\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\min}\left(a\left({\xi}_1\right)-b\left({\xi}_2\right)\right),\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\max}\left(a\left({\xi}_1\right)-b\left({\xi}_2\right)\right)\right]$

• (iii) ${I}_1{I}_2=\left[\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\min}\left(a\left({\xi}_1\right)b\left({\xi}_2\right)\right),\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\max}\left(a\left({\xi}_1\right)b\left({\xi}_2\right)\right)\right]$

• (iv) $\raisebox{1ex}{${I}_1$}\!\left/ \!\raisebox{-1ex}{${I}_2$}\right.=\left[\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\min}\left(\frac{a\left({\xi}_1\right)}{b\left({\xi}_2\right)}\right),\underset{\xi_1,{\xi}_2\in \left[0,1\right]}{\max}\left(\frac{a\left({\xi}_1\right)}{b\left({\xi}_2\right)}\right)\right]$ provided 0 ∉ I₂.

• (v) ${I}_1{-}_p{I}_2=\left[\underset{\xi \in \left[0,1\right]}{\min}\left(a\left(\xi \right)-b\left(\xi \right)\right),\underset{\xi \in \left[0,1\right]}{\max}\left(a\left(\xi \right)-b\left(\xi \right)\right)\right]$

• (vi) $\mu {I}_1=\left[\underset{\xi_1\in \left[0,1\right]}{\min}\left(\mu a\left({\xi}_1\right)\right),\underset{\xi_1\in \left[0,1\right]}{\max}\left(\mu a\left({\xi}_1\right)\right)\right].$

Definition A.3

(Stefanini and Bede [76]). Let an interval-valued function G : D ⊆ ℝ → I_cbe defined asG(t) = [G_L(t), G_U(t)], where G_L, G_U : D ⊆ ℝ → ℝ withG_L(t) ≤ G_U(t), forallt ∈ D.

The parameterized form (IPF) or p-interval-valued function of G(t)is defined as G : D ⊆ ℝⁿ → I_p and it is defined by $G(t)=\left\{\overset{\sim }{G}\left(t,\xi \right)={G}_L(t)+\xi \left({G}_U(t)-{G}_L(t)\right):\xi \in \left[0,1\right]\right\}$, forallt ∈ D.

DefinitionA.4

(Stefanini and Bede [76]). The function G(t) = [G_L(t), G_U(t)] is said to be p-differentiable at t₀ if ${\lim}_{h\to 0}\frac{G\left({t}_0+h\right){-}_pG\left({t}_0\right)}{h}$ exists finitely. The p-derivative of G at t₀ is denoted by G^′(t₀),

Definition A.5

Let G : [a, b] → K_p be a p-interval-valued function. Then G is integrable over [a, b] if for every fixedξ ∈ [0, 1], the parameterized function $\overset{\sim }{G}\left(t,\xi \right)$ is integrable over [a, b] as usual sense and$\int_a^bG(t) dt=\left\{\int_a^b\overset{\sim }{G}\left(t,\xi \right)\; dt:\xi \in \left[0,1\right]\right\}.$

Definition A.6

(Rahman et al. [49]). Let Y : [a, b] → I_c be a p-differentiable interval valued function of a real single variable and let F : [a, b] × I_c → I_c be a continuous interval valued function. Then an interval differential equation is defined as follows:

$$ \frac{dY}{dt}=F\left(t,Y\right),t\in \left[a,b\right] $$

where

$$ Y(t)=\left[{Y}_L(t),{Y}_U(t)\right]\kern0.24em \;\mathrm{and}\;\;\;F\left(t,Y\right)=\left[{F}_L\left(t,Y\right),{F}_U\left(t,Y\right)\right]. $$

(A.1)

Now the parametric representation of (A.1) is given below:

$$ {\displaystyle \begin{array}{c}\frac{d\overset{\sim }{Y}\left(t,{\xi}_1\right)}{dt}=\overset{\sim }{F}\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right),{\xi}_2\right),t\in \left[a,b\right]\kern0.24em \mathrm{and}\;\;{\xi}_1,{\xi}_2\in \left[0,1\right]\\ {}\begin{array}{c}\mathrm{where}\kern0.24em Y(t)=\left\{\overset{\sim }{Y}\left(t,{\xi}_1\right)={Y}_L(t)+{\xi}_1\left({Y}_U(t)-{Y}_L(t)\right):{\xi}_1\in \left[0,1\right]\right\}\kern0.36em \\ {}\mathrm{and}\kern0.36em F\left(t,Y\right)=\left\{\overset{\sim }{F}\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right),{\xi}_2\right)={F}_L\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right)\right)+{\lambda}_2\left({F}_U\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right)\right)-{F}_L\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right)\right)\right):{\xi}_1,{\xi}_2\in \left[0,1\right]\right\}.\end{array}\end{array}} $$

(A.2)

Proposition A.1

(Rahman et al. [49]). The Eqs. (A.1) and (A.2) are equivalent.

Proof

Now, the parametric representation of Eq. (A.1) is given by

$$ \left\{\frac{d\overset{\sim }{Y}\left(t,{\xi}_1\right)}{dt}:{\xi}_1\in \left[0,1\right]\right\}=\left\{\overset{\sim }{F}\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right),{\xi}_2\right):{\xi}_1,{\xi}_2\in \left[0,1\right]\right\} $$

By Definition A.2, it gives

$\frac{d\overset{\sim }{Y}\left(t,{\xi}_1\right)}{dt}=\overset{\sim }{F}\left(t,\overset{\sim }{Y}\left(t,{\xi}_1\right),{\xi}_2\right),{\xi}_1,{\xi}_2\in \left[0,1\right]$which is the Eq. (A.2).

The converse part is straightforward.