3.1.2. Notation

The decision variables and the parameters used in the proposed mathematical modeling are denoted by the notations enlisted comprehensively in Appendix A.

#### *3.2. Model Formulation*

The inventory diagram of the imperfect agri-food processing firm with vendor/supplier is shown as in Figure 1, where the cycle time of production is given on the x-axis and inventory is given on the y-axis. The upper portion is showing the inventory of the agri-product processing firm, while the lower portion is associated with the vendor inventory. The objective of the research is to minimize the total cost of Agri-SCM, and the formulation of the cycle time of the processing firm is prerequisite to calculate the total cost (TC). Cycle time is taken as a decision variable in the production model, which is dependent upon the production rates of processing firm. The production rate of the processing firm, i.e., *Pja* and *Pjc* for first and final stage is relying on the production rate of the machines (<sup>ε</sup>*ja* and <sup>ε</sup>*jc*). In order to meet the customer demand and no shortages in the processing firm, the production rates are considered as a variable (i.e., ε∗*ja* and <sup>ε</sup><sup>∗</sup>*jc*) to take an advantage of flexible production, where, ε∗*ja* ∈ [ε<sup>∗</sup>*ja*−*min*, <sup>ε</sup><sup>∗</sup>*ja*−*max*]

and ε∗*ja* ∈ [ε<sup>∗</sup>*jc*−*min*, <sup>ε</sup><sup>∗</sup>*jc*−*max*]. The total cost of production is the sum of Agri-processing firm and vendor cost as given in Equation (1) and their formulations are further represented as following.

$$\text{Total cost of production} = \text{Agri-processing cost} + \text{Vender cost} \tag{1}$$

#### 3.2.1. Agri-Processing Cost

The total cost associated with the manufacturer include the cost related to the first stage and final stage of the processing firm, where the basic cost includes setup, production, labor, holding, carbon emission, and stress, as expressed in Equation (2).

$$\begin{aligned} \text{Total cost of aggi-processing} &= \text{Setup cost} + \text{processing cost} + \text{Labor cost} \\ &+ \text{Holding cost} + \text{Carbon emissions cost} \end{aligned} \tag{2}$$

The breakup of the agri-processing cost is necessary to understand each cost clearly. That is the reason all the costs are described mathematically and theoretically in Equations (3)–(10) as follows.

#### 3.2.2. Setup Cost

The incurred cost is subject to the initial cost required to operate the production setup. Generally, it may contain setup cost, substitution, and tool setting cost. Further, this is fixed nature cost, independent of quantities, though time dependent. The expression is given in Equation (3).

$$SC = \sum\_{j=1}^{J} A\_j \tag{3}$$

#### 3.2.3. Production Cost

Explicitly, cost incurred in the processing, which is largely dependent on quantity and variable in nature. The cost is accumulation of all costs utilized on resources that are required to manufacture a product. It contains the processing and machining cost, utilities needed for machines, and labor cost. Further, it is linked with the manufacturing processes of initial stage manufacturing. and are the production rates of the initial stage processing plant and machines respectively. Lastly, the production cost is the summation of all the costs linked with the raw materials, production, and tool-die cost [50]. The expression is given as in Equation (4).

$$\text{MC} = \sum\_{j=1}^{J} (\text{C}\_{rm} + \text{TD}\_{\text{maj}}P\_{\text{ja}} + \frac{\text{g}\_{\text{maj}}}{P\_{\text{ja}}})P\_{\text{ja}}t\_{1j} + (\text{TD}\_{\text{maj}}P\_{\text{ja}} + \frac{\text{g}\_{\text{maj}}}{P\_{\text{ja}}})Q\_{\text{ja}} \tag{4}$$

#### 3.2.4. Holding Cost

Holding cost is the basic cost need to understand the basic production model, which is variable and dependent upon the variable inventory at any instant of time. This is the carrying cost of holding inventory. Holding cost depends on production of semi-finished and finished goods used as stock. It is incurred on production and crashing quantities in the proposed model and includes costs such as wedges, warehouse rent, and insurance. It also depends on the holding time of the product in inventory. The holding cost will be applied on the inventory supported by manufacturing firm, outsourcing operations. The average inventory is calculated as the ratio of the sum of inventories in the form of area under the curve to the cycle time of the production, which is expressed as in Equation (7). The cycle time and inventory levels of the processing firm are calculated by a step-by-step procedure gives in Appendix B.

$$TotalInventory = Area\_{123} + Area\_{10,11,12} + Area\_{11,12,13} \tag{5}$$

$$=\frac{Q\_j^2}{2P\_{j\mathfrak{a}}} + \frac{(Q\_j - u)^2}{2P\_{j\mathfrak{c}}}(1 - \frac{D\_j}{P\_{j\mathfrak{c}}}) + \frac{(Q\_j + u)^2}{2D\_j}(1 - \frac{D\_j}{P\_{j\mathfrak{c}}})^2\tag{6}$$

$$HC\_{mj} = \sum\_{j=1}^{J} h\_{mj} [\frac{Q\_j^2}{2P\_{j\mathfrak{a}}} + \frac{\left(Q\_j - u\right)^2}{2P\_{j\mathfrak{c}}} (1 - \frac{D\_j}{P\_{j\mathfrak{c}}}) + \frac{\left(Q\_j + u\right)^2}{2D\_j} (1 - \frac{D\_j}{P\_{j\mathfrak{c}}})^2 \, ]. \tag{7}$$

#### 3.2.5. Carbon Emission Costs

The model optimized the carbon emission cost as a function of production rate to develop an eco-friendly and eco-e ffective agri-product SCM. The carbon emission in the agri-processing firm depends upon the source of energy consumed by the equipment/machines. The cost of carbon is incurred by the state governmen<sup>t</sup> to support the global warming issue. The cost of carbon emission is generated during the life cycle production, which is given as in Equation (9), where *A* is the emissions function parameter (ton.year/unit), *B* is the emissions function parameter (ton year/unit), and *C* is the emissions function parameter (ton/unit) [51].

$$\text{Carbon emission cost} \text{(CEC}\_{\text{j}}) = \text{Costof carbon emissionimum factoring} \text{(CEM)} \tag{8}$$

$$\dot{\lambda} = \sum\_{j=1}^{l} \gamma\_2 \left[ (A P\_{j\text{a}}^2 - B P\_{j\text{a}} + \mathbb{C}) P\_{j\text{a}} t\_{1j} + (A P\_{j\text{c}}^2 - B P\_{j\text{c}} + \mathbb{C}) Q\_{j\text{c}} \right] \tag{9}$$

#### 3.2.6. Labor Cost

The cost is associated with the utilization of the workforce in the agri-SCM. The wages paid to the workers on the basis of the level of the skilled. Here, the cost is incurred to reflect the importance of the unskilled workers to understand the importance of the human factor in the processing firm. The labor cost is calculated on the basis of the machines required in the agri-product processing firm, which is expressed in Equation (10).

$$Laborcost(LC) = \sum\_{j=1}^{J} L\_j \mathcal{W}\_j \tag{10}$$

The number of machines required and the amount of labor required in the processing firm are given in Equations (11) and (13), where laj, and lcj are the labor rate or the number of labors working on each machine.

$$\mathcal{K}\_{\dot{j}} = \mathcal{K}\_{\dot{j}\dot{a}} + \mathcal{K}\_{\dot{j}\dot{c}} \tag{11}$$

*Numbero flabors* = *Laborrate* × *Numbero f machines* × *Productiontime* (12)

$$L\_j = \frac{l\_{aj}}{\rho} \mathcal{K}\_{ja} + \frac{l\_{cj}}{\rho} \mathcal{K}\_{jc} \tag{13}$$

#### 3.2.7. Stress and Workers' E fficiency

The e fficiency of the processing firm depends upon the e fficiency of the workers to fulfill the customer demand before deadline. The labor-machine coordination in the processing firm is very important for e ffective and e fficient processing firm. The relationship between the average worker's stress and e fficiency is expressed as given in Equation (15), where *m* is scale factor, ρ0 is the workers' efficiency a ffected by stress, and ρ1 is the e fficiency due to the e ffect of another factor. Therefore, the total e fficiency of the worker will be ρ [52,53].

$$
\rho\_0 = e^{-s/m},
\tag{14}
$$

$$
\rho = \rho\_1 \rho\_0. \tag{15}
$$

#### 3.2.8. Stress Level and Defective Rate

The relationship is developed between the defective rate and averages stress among worker, where the defective rate is considered as a function of stress having a significant impact on the processing firm. The increasing rate of average stress among workers increases the defective rate. That is the reason, the expression for defective is the sum of initial and variable defective rate as expressed in the Equation (16), where variable defective rate is a function of stress among workers, α0 is the initial defective rate, τ and, are the scaling factors, and *s* is average stress among worker [53]. The analysis of workers' stress on the production system has already been analysed by the research work of Omair et al. [53]. Therefore, the formulation has been incorporated in this study. However, the analysis of workplace stress on the production system is not highlighted.

$$
\alpha\_j = \alpha\_0 + \pi(s)^{\mathfrak{s}}.\tag{16}
$$

Now, the mathematical form of the total cost of the agri-processing firm is expressed as in Equation (17).

$$\begin{split} TC\_{aj} &= \sum\_{j=1}^{l} \left[ A\_{j} + (C\_{rm} + TD\_{muj}P\_{j\bar{x}} + \frac{\mathcal{G}\_{muj}}{P\_{j\bar{x}}}) P\_{j\bar{x}} \mathbf{1}\_{1j} + (TD\_{m\bar{x}j}P\_{j\bar{x}} + \frac{\mathcal{G}\_{m\bar{x}j}}{P\_{j\bar{x}}}) Q\_{j\bar{x}} + L\_{j} \mathcal{W}\_{j} \\ &+ h\_{\text{mj}} \big] \frac{Q\_{j}^{2}}{2P\_{j\bar{x}}} + \frac{(Q\_{j} - u)^{2}}{2P\_{j\bar{x}}} (1 - \frac{D\_{j}}{P\_{j\bar{x}}}) + \frac{(Q\_{j} + u)^{2}}{2D\_{j}} (1 - \frac{D\_{j}}{P\_{j\bar{x}}}) \big] \\ &+ \gamma\_{2} [(AP\_{j\bar{x}}^{2} - BP\_{j\bar{x}} + \mathcal{C}) P\_{j\bar{x}} \mathbf{1}\_{1} + (AP\_{j\bar{x}}^{2} - BP\_{j\bar{x}} + \mathcal{C}) Q\_{j\bar{x}}] + s \mathcal{S} \mathcal{C}\_{j}]. \end{split} \tag{17}$$

The costs related to the manufacturer in the first and final stages of production are calculated, and the costs associated with the vendor are given in the next section.

#### *3.3. Vendor Cost*

The semi-finished agri-products are delivered to the vendor for processing few operations. The cost of vendor is the sum of the costs associated with the production, holding, inspection, and recycling of the processes is given in Equation (18). These costs are expressed in the Equations (19)–(24).

$$\begin{aligned} \text{Vender cost} &= \text{Production cost} + \text{Holding cost} + \text{Inception cost} + \text{Reovorking cost} \\ &+ \text{Scrap recyccle cost} + \text{Buffer cost} \end{aligned} \tag{18}$$

#### 3.3.1. Production Cost of Outsourcing

The expression for production cost is utilized from the research work done by [50] except the cost of raw material, because of receiving the semi-finished products from the manufacturer. The expression is given in Equation (19).

$$\text{MCCO} = \sum\_{j=1}^{l} [(TD\_{oj}P\_{jb}(1-\alpha\_{j}) + \frac{\mathcal{G}o^{j}}{P\_{jb}(1-\alpha\_{j})})P\_{jb}(1-\alpha\_{j})t\_{2j}].\tag{19}$$

#### 3.3.2. Holding Cost of Vendor

The holding cost of outsourcing operations is obtained by the sum of inventories calculated from Appendix B. The expression for calculating the total cost of production is expressed in Equation (22).

$$Total\ Inventory = Area 456 + Area 5678 + Area 689\tag{20}$$

$$=\frac{Q\_j^2(1-\alpha\_j)}{2P\_{j\flat}} + \frac{\alpha\_j(1-\alpha\_j)Q\_j^2}{P\_{j\flat}} + \frac{\alpha\_j^2Q\_j^2(1-\alpha\_j\beta\_j)}{2P\_{j\flat}},\tag{21}$$

$$\text{HCC} = \sum\_{j=1}^{l} [h\_{oj}] \frac{Q\_j^2 (1 - \alpha\_j)}{2P\_{jb}} + \frac{\alpha\_j (1 - \alpha\_j) Q\_j^2}{P\_{jb}} + \frac{\alpha\_j^2 Q\_j^2 (1 - \alpha\_j \beta\_j)}{2P\_{jb}}] \,\text{}.\tag{22}$$

#### 3.3.3. Inspection Cost

Inspection of the agri-product is carried out by the vendor, where the products are checked according to the quality control dimensions. The parts are categorized into good and rejected. The total inspection cost of the production is the sum of the fixed and variable inspection cost in the processing, as expressed in Equation (23).

$$IC\_{\dot{j}} = \sum\_{j=1}^{I} [\theta\_{\dot{j}} + \psi\_{\dot{j}a} P\_{\dot{j}a} t\_{1\dot{j}} + \psi\_{\dot{j}b} P\_{\dot{j}b} t\_{2\dot{j}}] \tag{23}$$

#### 3.3.4. Recycling Cost/Disposal Cost

Recycling cost is considered in the agri-processing firm. It is not concerned with the recycling of the agri-product after deteriorated but it is the cost incurred on rejected/defective or imperfect agri-product. The customer/user is not the part of the proposed agricultural supply chain managemen<sup>t</sup> (Agri-SCM) that is the reason, the perishability factor is not considered. Here, recycling cost is the cost incurred on disposing the imperfect/defective not due to deterioration or perishability factor. These products are bio-waste and further utilized into other byproducts, i.e., fertilizers, bio-fuel, feeds, etc., in the processing expressed as in the equation given below.

$$\mathcal{RC}\_j = \gamma\_1 P\_{jb} t\_{2j} \alpha\_j \beta\_j$$

The mathematical expression to sum all the costs equations is represented as given in Equation (24).

$$\begin{split} TC\_{vj} &= \sum\_{j=1}^{l} \left[ (TD\_{oj}P\_{jb}(1-\alpha\_{j}) + \frac{\mathcal{G}v\_{j}}{P\_{jb}(1-\alpha\_{j})})P\_{jb}(1-\alpha\_{j})t\_{2j} + \frac{Q\_{j}^{2}(1-\alpha\_{j})}{2P\_{jb}} + \frac{a\_{j}(1-a\_{j})Q\_{j}^{2}}{P\_{jb}} \\ &+ \frac{a\_{j}^{2}Q\_{j}^{2}(1-a\_{j}\theta\_{j})}{2P\_{jb}} + \theta\_{j} + \psi\_{j\mu}P\_{j\mu}t\_{1j} + \psi\_{j\bar{\eta}}P\_{j\bar{\eta}}t\_{2j} + \gamma\_{1}P\_{j\bar{\eta}}t\_{2j}x\_{\bar{\eta}}\theta\_{\bar{\eta}} \right] \end{split} \tag{24}$$

The production system of agri-product processing firm is analyzed by the formulation of mathematical model. The mathematically model is based on the cycle time of production. The objective of the proposed model is to minimize the total cost (*TCsj*) of processing firm. The total cost per cycle is given in Equation (26).

$$Totalcost = Agri-processing cost + MR(Vendercost)\tag{25}$$

*TCj* = *J j*=1 1 *Tj* [*Aj* + (*Crm* + *TDmajPja* + *gmaj Pja* )*Pjat*1*j* + (*TDmcjPjc* + *gmcj Pjc* )*Qjc* + *LjWj* +*hmj*[ *Q*2*j* 2*Pja* + (*DjTj*)<sup>2</sup> 2*Pjc* (1 − *Dj Pjc* ) + (*Qj* + *u*)<sup>2</sup> 2*Dj* (1 − *Dj Pjc* )2] + <sup>γ</sup>2[(*AP*2*ja* − *BPja* + *<sup>C</sup>*)*Pjat*1*j* +(*AP*2*jc* − *BPjc* + *C*)*Qjc*] + *<sup>s</sup>*.*SCj* + *MR*[(*TDojPjb*(1 − <sup>α</sup>*j*) + *goj Pjb*(1 − <sup>α</sup>*j*))*Pjb*(<sup>1</sup> − <sup>α</sup>*j*)*<sup>t</sup>*2*j* + *<sup>Q</sup>*<sup>2</sup>*j*(<sup>1</sup> − <sup>α</sup>*j*) 2*Pjb* + <sup>α</sup>*j*(<sup>1</sup> − <sup>α</sup>*j*)*Q*<sup>2</sup>*j Pjb* + <sup>α</sup>2*jQ*<sup>2</sup>*j*(<sup>1</sup> − <sup>α</sup>*j*β*j*) 2*Pjb* + θ*j* + ψ*jaPjat*1*j* + ψ*jbPjbt*2*j* <sup>+</sup>*Rj*α*jQj*(<sup>1</sup> − <sup>α</sup>*j*β*j*) + <sup>γ</sup>1*Pjbt*2*j*<sup>α</sup>*j*β*j*]] (26)

where,

$$L\_{\hat{j}} = L\_{j\hat{x}} + L\_{j\hat{x}}$$

$$Q\_{\hat{j}} = \frac{T\_{\hat{j}}D\_{\hat{j}}(\Omega)}{1 - \alpha\_{\hat{j}}^2 \beta\_{\hat{j}}}$$

$$t\_{1j} = \frac{T\_{\hat{j}}D\_{\hat{j}}}{K\_{j\hat{x}}\varepsilon\_{j\hat{x}}(1 - \alpha\_{\hat{j}}^2 \beta\_{\hat{j}})}$$

$$t\_{2\hat{j}} = \frac{T\_{\hat{j}}D\_{\hat{j}}}{P\_{j\hat{x}}(1 - \alpha\_{\hat{j}}^2 \beta\_{\hat{j}})}$$

$$t\_{4\hat{j}} = \frac{T\_{\hat{j}}D\_{\hat{j}}}{K\_{\hat{j}}\varepsilon\_{j\hat{x}}}$$

$$t\_{5\hat{j}} = (\frac{T\_{\hat{j}}D\_{\hat{j}}}{1 - \alpha\_{\hat{j}}^2 \beta\_{\hat{j}}} - \mu)(\frac{K\_{\hat{j}}\varepsilon\_{j\hat{x}} - D\_{\hat{j}}}{D\_{\hat{j}}K\_{\hat{j}\hat{x}}\varepsilon\_{j\hat{x}}})$$

The SCM mathematical model is non-linear by minimizing total cost of SCM, where the decision variables are (*Tj*, *Lja*, *Ljc*, *Kja*, *Kjc*, *Pja*, and *Pjc*).

#### *3.4. Solution Algorithm*

The variability in the proposed Agri-SCM model make the model non-linear in nature. The decision variables considered are relying on the decisions of the production planning. Analytically, the proposed model is optimized with the help of improved methodology called algebraic function, which is based on quadratic equation. There are four decision variables, i.e., cycle time (*Tj*), machines (*Kja*, *Kjc*), production rate (<sup>ε</sup>*ja*, <sup>ε</sup>*jc*), and labors (*Lja*, *Ljc*) to optimize the non-linear imperfect production model. The algebraic method consists of a positive expression type, and can be rewritten as:

$$f(\mathbf{x}) = a\_1 \mathbf{x} + a\_2 / \mathbf{x} + a\_3 = \frac{a\_1}{\mathbf{x}} (\mathbf{x}^2 + a\_2 / a\_1 - 2\mathbf{x}\sqrt{a\_2/a\_1} + 2\mathbf{x}\sqrt{a\_2/a\_1}) + a\_3 \tag{27}$$

$$f(\mathbf{x}) = \frac{a\_1}{\mathbf{x}} (\mathbf{x}^2 + a\_2/a\_1 - 2\mathbf{x}\sqrt{a\_2/a\_1}) + 2a\_1\sqrt{a\_2/a\_1} + a\_3 \tag{28}$$

$$f(\mathbf{x}) = \frac{a\_1}{\mathbf{x}} (\mathbf{x} - \sqrt{a\_2/a\_1})^2 + 2\sqrt{a\_2 a\_1} + a\_3 \tag{29}$$

Since the quadratic expression is non-negative and *a*1 is positive, it is always minimized for *x* = √*a*2/*<sup>a</sup>*1, which reaches the minimum at *f*(*x*) = 2 √*a*2/*<sup>a</sup>*1 + *a*3.

In the first step of solution algorithm, by using algebraic function methodology, the form of decision variable *Tj* can be written as given in Equation (30).

$$\begin{split} \mathrm{TC\_{ij}(}T\_{j},K\_{j\mu},L\_{j\nu}L\_{j\nu}L\_{j\nu},\varepsilon\_{j\mu},\varepsilon\_{j\kappa}\rangle &= \frac{1}{T\_{j}}[A\_{j}+(\mathrm{C\_{rn}}+\mathrm{TD\_{mj}}P\_{j\mu}+\frac{\mathcal{G}\_{\mu\dot{n}}}{P\_{j\mu}})P\_{\mu}t\_{1}] \\ + (\mathrm{TD\_{mj}}P\_{j\bar{\kappa}}+\frac{\mathcal{G}\_{\mu\dot{n}}}{P\_{j\bar{\kappa}}})Q\_{\bar{\kappa}}+L\_{j}W\_{\bar{\jmath}}+h\_{\bar{m}j}[\frac{\Omega\_{j}^{2}}{2P\_{j\mu}}+\frac{(D\_{j}T\_{j})^{2}}{2P\_{j\bar{\kappa}}}(1-\frac{D\_{j}}{P\_{j\bar{\kappa}}})+\frac{(Q\_{j}+u)^{2}}{2D\_{j}}(1-\frac{D\_{j}}{P\_{j\bar{\kappa}}})^{2} \\ + \gamma\_{2}[(AP\_{j\mu}^{2}-BP\_{j\mu}+C)P\_{j\bar{\kappa}}t\_{1j}+(AP\_{j\mu}^{2}-BP\_{j\bar{\kappa}}+C)Q\_{\bar{\kappa}}]+s.S\bar{C}\_{j}+MR[(TD\_{oj}P\_{j\bar{\kappa}})(1-a\_{j}) \\ + \frac{\mathcal{G}\_{\bar{\jmath}}}{P\_{j\bar{\kappa}}(1-a\_{j})})P\_{j\bar{\kappa}}(1-a\_{j})t\_{2j}+\frac{Q\_{j}^{2}(1-a\_{j})\mathcal{G}\_{j}^{2}}{2P\_{j\bar{\kappa}}}+\frac{a\_{j}^{2}Q\_{j}^{2}(1-a\_{j}\beta\_{j})}{2P\_{j\bar{\kappa}}}+\theta\_{j} \\ + \psi\_{\bar{\mu}}P\_{j\bar{\kappa}}t\_{1j}+\psi\_{\bar{\mu}$$

The Equation (30) can be written as in Equation (31)

$$\begin{split} &= \frac{1}{T\_{j}} [A\_{j} + s.SC\_{j} + \theta\_{j} + \text{MR} \times TD\_{oj}P\_{jb}(1-a\_{j})] + T\_{j} [h\_{mj}] \frac{D\_{j}^{2}}{2P\_{j\bar{k}}(1-a\_{j}\theta\_{j})^{2}} \\ &+ \frac{1}{2P\_{j\bar{k}}} (\frac{D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}} - \frac{\alpha\_{j}\beta\_{j}D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}})^{2} (1-\frac{D\_{j}}{P\_{j\bar{k}}}) + \left(\frac{D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}} - \frac{\alpha\_{j}\beta\_{j}D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}}\right) (1-\frac{D\_{j}}{P\_{j\bar{k}}})^{2} \\ &+ \text{MR}[(\frac{D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}})^{2} \frac{1-\alpha\_{j}}{2P\_{j\bar{k}}} + \frac{\alpha\_{j}(1-\alpha\_{j})}{P\_{j\bar{k}}} (\frac{D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}})^{2} + \frac{\alpha\_{j}^{2}(1-a\_{j}\beta\_{j})}{2P\_{j\bar{k}}} (\frac{D\_{j}}{1-\alpha\_{j}^{2}\beta\_{j}})^{2} \end{split} \tag{31}$$

Our assumptions are given as in Equations (32) and (33).

$$A\_1 = \sum\_{j=1}^{l} \left[ A\_j + s.S\mathbf{C}\_j + \theta\_j + MR \times TD\_{oj}P\_{jb}(1 - \alpha\_j) \right] \tag{32}$$

$$\begin{split} A\_{2} = \sum\_{j=1}^{l} \left[ h\_{\rm inj} \right] \frac{D\_{j}^{2}}{2P\_{j\rm in}(1-a\_{j}\theta\_{j})^{2}} + \frac{1}{2P\_{j\rm}} (\frac{D\_{j}}{1-a\_{j}^{2}\beta\_{j}} - \frac{a\_{j}\beta\_{j}D\_{j}}{1-a\_{j}^{2}\beta\_{j}})^{2} (1 - \frac{D\_{j}}{P\_{j\rm c}}) + (\frac{D\_{j}}{1-a\_{j}^{2}\beta\_{j}} - \frac{a\_{j}\beta\_{j}D\_{j}}{1-a\_{j}^{2}\beta\_{j}})^{2} \\ (1 - \frac{D\_{j}}{P\_{j\rm c}})^{2} \left[ 1 + \text{MR} [ (\frac{D\_{j}}{1-a\_{j}^{2}\beta\_{j}})^{2} \frac{1-a\_{j}}{2P\_{j\rm b}} + \frac{a\_{j}(1-a\_{j})}{P\_{j\rm b}} (\frac{D\_{j}}{1-a\_{j}^{2}\beta\_{j}})^{2} \\ & + \frac{a\_{j}^{2}(1-a\_{j}\beta\_{j})}{2P\_{j\rm b}} (\frac{D\_{j}}{1-a\_{j}^{2}\beta\_{j}})^{2} ] \right], \end{split} \tag{33}$$

In the algebraic function approach, the constant term becomes neglected and equal to zero. Therefore, the cost function can be given as in Equation (34).

$$T\mathcal{C}\_{\text{s}\bar{\jmath}}(T\_{\text{j}\prime}^{\star}\mathcal{K}\_{\text{ja}\prime}\mathcal{K}\_{\text{j}\prime}L\_{\text{ja}\prime}L\_{\text{ja}\prime}\varepsilon\_{\text{ja}\prime}\varepsilon\_{\text{ja}}) = \frac{A\_1}{T\_{\bar{\jmath}}} + A\_2T\_{\bar{\jmath}}\tag{34}$$

$$= (\sqrt{\frac{A\_1}{T\_j}})^2 + (\sqrt{A\_2 T\_j})^2,\tag{35}$$

$$\hat{\lambda} = (\sqrt{\frac{A\_1}{T\_j}} - \sqrt{A\_2 T\_j})^2 + \sqrt{2A\_1 A\_2}.\tag{36}$$

By the algebraic approach, in Equation (36), having the square term as a maximum value, the square will be zero, i.e.,

$$
\sqrt{\frac{A\_1}{T\_j}} - \sqrt{A\_2 T\_j} \stackrel{2}{\ } = 0,\tag{37}
$$

$$
\sqrt{\frac{A\_1}{T\_j}} - \sqrt{A\_2 T\_j} = 0,\tag{38}
$$

$$
\sqrt{\frac{A\_1}{T\_j}} = \sqrt{A\_2 T\_{j\nu}}\tag{39}
$$

$$T\_j^\* = \sqrt{\frac{A\_1}{A\_2}}\tag{40}$$

By putting the value of in Equation (34). The Equation (41) is obtained as.

(

$$T\mathbb{C}\_{\mathbf{s}\mathbf{j}}(T^\*\_{\mathbf{j}\prime}K\_{\mathbf{j}\prime\prime}K\_{\mathbf{j}\prime\prime}L\_{\mathbf{j}\prime\prime}L\_{\mathbf{j}\prime\prime}\varepsilon\_{\mathbf{j}\prime\prime}\varepsilon\_{\mathbf{j}\prime}) = \sum\_{j=1}^{J} \frac{A\_1}{T^\*\_{\mathbf{j}}} + A\_2 T^\*\_{\mathbf{j}}.\tag{41}$$

In the second step, the production rate of each machine in the first stage (<sup>ε</sup>*ja*) and final stage (<sup>ε</sup>*jc*) of the production system are also calculated by using algebraic approach. First of all, to find optimal (<sup>ε</sup>*jc*), the *TCsj* can be converted into the form given as in the following equation:

$$\begin{split} & \text{TC}\_{ij}(T\_j^\*, K\_{j\text{in}} K\_{j\text{c}}, L\_{j\text{c}}, L\_{j\text{c}}, \varepsilon\_{j\text{c}}, \varepsilon\_{j\text{c}}) = \varepsilon\_{j\text{c}} [\text{TD}\_{m:j} K\_{j\text{c}} D\_j - B K\_{j\text{c}} \gamma\_2 D\_j] \\ & + \frac{1}{\varepsilon\_{j\text{c}}} [\frac{g\_{m:j} D\_j}{K\_{j\text{c}}} + \frac{l\_{i\text{j}} D\_j W\_j}{\rho} + \frac{h\_{mj} (Q\_j - u)^2}{2 K\_{j\text{c}} T\_j} - \frac{(Q\_j + u)^2 h\_{mj}}{K\_{j\text{c}} T\_j^\*}] \\ & + [-\frac{h\_{mj}}{T\_j^\*} \left\{ \frac{(Q\_j - u)^2}{2 K\_{j\text{c}}} \frac{D\_j}{(\varepsilon\_{j\text{c}})^2} + \frac{(Q\_j + u)^2}{2 D\_j} \frac{D\_j}{\rho K\_{j\text{c}} \varepsilon\_{j\text{c}}} \right\} + \gamma\_2 T\_j^\* D\_T A (K\_{j\text{c}} \varepsilon\_{j\text{c}})^2] \\ & + [\gamma\_2 D\_j \varepsilon\_{j\text{c}} + \frac{(Q\_j + u)^2}{2 D\_j T\_j^\*} h\_{mj} + \frac{l\_{i\text{j}} D\_j W\_j}{\rho \varepsilon\_{j\text{a}} (1 - \alpha\_j^2 \beta j)}] \end{split} \tag{42}$$

$$R\_1 = \varepsilon\_{j\bar{c}} [TD\_{m\bar{c}j} K\_{\bar{j}c} D\_{\bar{j}} - BK\_{\bar{j}c} \gamma\_2 D\_{\bar{j}}] \tag{43}$$

$$R\_2 = \frac{1}{\varepsilon\_{j\varepsilon}} [\frac{g\_{m\varepsilon j} D\_j}{K\_{j\varepsilon}} + \frac{l\_{\varepsilon j} D\_j \mathcal{W}\_j}{\rho} + \frac{h\_{m\dot{\jmath}} (Q\_j - u)^2}{2K\_{j\varepsilon} T\_j^\*} - \frac{(Q\_j + u)^2 h\_{m\dot{\jmath}}}{K\_{j\varepsilon} T\_j^\*}] \tag{44}$$

$$R\_3 = \left[ -\frac{\hbar\_{mj}}{T\_j^\*} \left\{ \frac{\left(Q\_j - u\right)^2}{2\mathcal{K}\_{j\varepsilon}} \frac{D\_j}{\left(\varepsilon\_{j\varepsilon}\right)^2} + \frac{\left(Q\_j + u\right)^2}{2D\_j} \frac{D\_j}{\rho \mathcal{K}\_{j\varepsilon}\varepsilon\_{j\varepsilon}} \right\} + \gamma\_2 T\_j^\* D\_{\hat{\boldsymbol{\beta}}} A(\mathcal{K}\_{j\varepsilon}\varepsilon\_{j\varepsilon})^2 \right] \tag{45}$$

$$R\_4 = \left[\gamma\_2 D\_{\dot{j}} \varepsilon\_{\dot{j}\dot{a}} + \frac{\left(Q\_{\dot{j}} + u\right)^2}{2D\_{\dot{j}}T\_{\dot{j}}^\*} h\_{\text{mf}} + \frac{l\_{\dot{a}\dot{j}} D\_{\dot{j}} \mathcal{W}\_{\dot{j}}}{\rho \varepsilon\_{\dot{j}\dot{a}} \left(1 - \alpha\_{\dot{j}}^2 \beta \dot{j}\right)}\right] \tag{46}$$

$$T\mathbb{C}\_{\rm sj}(T^\*\_{\rm j}, K\_{\rm j\alpha}, K\_{\rm j\alpha}, L\_{\rm j\alpha}, L\_{\rm j\varepsilon}, \varepsilon\_{\rm j\varepsilon}, \varepsilon\_{\rm j\varepsilon}) = R\_1 \varepsilon\_{\rm j\varepsilon} + \frac{R\_2}{\varepsilon\_{\rm j\varepsilon}} + R\_3 + R\_4 \tag{47}$$

*R*3 consists of squared <sup>ε</sup>*jc* and also in the denominator, whereas *R*4 is constant term. Therefore, both are neglected and considered as zero.

$$T\mathbb{C}\_{\text{sj}}(T\_{j\text{\text{\textdegree}}}^{\text{\textdegree}}K\_{\text{j\text{\textdegree}}}K\_{\text{j\text{\textdegree}}}L\_{\text{j\text{\textdegree}}}\mathbb{L}\_{\text{j\text{\textdegree}}}\varepsilon\_{\text{j\text{\textdegree}}}\varepsilon\_{\text{j\text{\textdegree}}}) = (\sqrt{R\_1\varepsilon\_{j\text{\textdegree}}})^2 + (\sqrt{\frac{R\_2}{\varepsilon\_{j\text{\textdegree}}}})^2 \tag{48}$$

$$\dot{\varepsilon} = (\sqrt{R\_1 \varepsilon\_{j\bar{c}}} - \sqrt{\frac{R\_2}{\varepsilon\_{j\bar{c}}}})^2 + 2\sqrt{R\_1 R\_2} \varepsilon\_{j\bar{c}} \tag{49}$$

By the algebraic approach, Equation (49), having the square term as a maximum value, the squared expression will be zero, i.e.,

$$(\sqrt{\mathbf{R}\_1 \varepsilon\_{j\mathbf{c}}} - \sqrt{\frac{\mathbf{R}\_2}{\varepsilon\_{j\mathbf{c}}}}) = 0 \tag{50}$$

$$
\varepsilon\_{jc}^{\*} = \sqrt{\frac{R\_2}{R\_1}}\tag{51}
$$

In step 2, the optimal <sup>ε</sup>*ja* is obtained by using algebraic approach, i.e., the *TCsj* can be given as in Equation (52).

$$\begin{split} \text{TC}\_{\text{sj}}(T\_{\text{j}'}^{\*}K\_{\text{j}\text{a}},\text{K}\_{\text{j}\text{a}},L\_{\text{j}\text{a}},\text{L}\_{\text{j}\text{a}},\varepsilon\_{\text{j}\text{a}},\varepsilon\_{\text{j}\text{a}}^{\*}) &= \varepsilon\_{\text{ja}}[\text{TD}\_{\text{ma}}\text{K}\_{\text{j}\text{a}}D\_{\text{j}} - \gamma\_{2}D\_{\text{j}}BK\_{\text{ja}}h\_{\text{mj}}] \\ &+ \frac{1}{\varepsilon\_{\text{ja}}}[\frac{g\_{\text{maj}}D\_{\text{j}}}{K\_{\text{ja}}} + \frac{l\_{\text{aj}}D\_{\text{j}}\mathcal{W}\_{\text{j}}}{\rho(1-\left(\alpha\_{\text{j}}\right)^{2}\mathcal{\beta}\_{\text{j}})} + \frac{h\_{\text{mj}}(Q\_{\text{j}})^{2}}{2T\_{\text{j}}^{\*}K\_{\text{ja}}}] \\ &+ [\frac{l\_{\text{c}}D\_{\text{j}}\mathcal{W}\_{\text{j}}}{\mathcal{P}\mathcal{E}\_{\text{jk}}^{\*}} + \mathcal{\gamma}\_{2}h\_{\text{mj}}A(K\_{\text{ja}}\varepsilon\_{\text{ja}})^{2}D\_{\text{j}} + \mathcal{\gamma}\_{2}\varepsilon\_{\text{j}\text{c}}^{\*}D\_{\text{j}}h\_{\text{mj}}]. \end{split} \tag{52}$$

where we can assume that

$$R\mathfrak{g} = \varepsilon\_{\dot{\mu}} [TD\_{\text{maj}}K\_{\dot{\mu}}D\_{\dot{\jmath}} - \gamma \gamma D\_{\dot{\jmath}}BK\_{\dot{\mu}}h\_{\text{mj}}] \tag{53}$$

$$R\_6 = \frac{1}{\varepsilon\_{ja}} [\frac{g\_{maj}D\_j}{K\_{ja}} + \frac{l\_{aj}D\_j \mathcal{W}\_j}{\rho \left(1 - \left(\alpha\_j\right)^2 \beta\_j\right)} + \frac{h\_{mj}(Q\_j)^2}{2T\_j^\* K\_{ja}}] \tag{54}$$

$$R\_7 = \left[\frac{l\_{\rm ij} D\_{\dot{j}} W\_{\dot{j}}}{\rho \varepsilon\_{\dot{j}\varepsilon}^\*} + \gamma\_2 l\_{m\dot{j}} A (K\_{\dot{j}\dot{n}} \varepsilon\_{\dot{j}\dot{n}})^2 D\_{\dot{j}} + \gamma\_2 \varepsilon\_{\dot{j}\dot{c}}^\* D\_{\dot{j}} l\_{m\dot{j}}\right] \tag{55}$$

Therefore, Equation (56) can be written as

$$\text{TrC}\_{\text{sj}}(T\_{\text{j}}^{\bullet}, \mathbb{K}\_{\text{ja}}, \mathbb{K}\_{\text{ja}}, L\_{\text{ja}}, L\_{\text{ja}}, \varepsilon\_{\text{ja}}, \varepsilon\_{\text{ja}}^{\*}) = R\_{\text{5}} \varepsilon\_{\text{ja}} + \frac{R\_{\text{6}}}{\varepsilon\_{\text{ja}}} + R\_{\text{7}} \tag{56}$$

$$=(\sqrt{\mathbb{R}\_5 \varepsilon\_{ja}})^2 + (\sqrt{\frac{\mathbb{R}\_6}{\varepsilon\_{ja}}})^2 + R\_7 \tag{57}$$

where *R*7 is squared <sup>ε</sup>*ja*, which can be neglected and considered as zero.

$$\text{TC}\_{\text{sf}}(T\_{\text{j}}^{\star}, \mathbb{K}\_{\text{j}\text{s}}, \mathbb{K}\_{\text{j}\text{s}}, L\_{\text{j}\text{s}}, L\_{\text{j}\text{s}}, \varepsilon\_{\text{j}\text{s}}, \varepsilon\_{\text{j}\text{s}}^{\star}) = (\sqrt{\mathbb{R}\_{\text{5}}\varepsilon\_{\text{j}\text{s}}} - \sqrt{\frac{\mathbb{R}\_{\text{6}}}{\varepsilon\_{\text{ja}}}})^{2} + 2\sqrt{\mathbb{R}\_{\text{5}}\mathbb{R}\_{\text{6}}} \tag{58}$$

$$= (\sqrt{R\_5 \varepsilon\_{ja}} - \sqrt{\frac{R\_6}{\varepsilon\_{ja}}}) + 2\sqrt{R\_5 R\_6} \tag{59}$$

The square term has a maximum value, if the squared expression will be zero, i.e.,

$$
\varepsilon^\*\_{\mu} = \sqrt{\frac{R\_6}{R\_5}}\tag{60}
$$

In the third step, the other decision variables of the production system i.e., *Kjc*, *Lja*, and *Ljc*, which are discrete and can be calculated indirectly. Therefore, the total cost from Equation (30) will be given as in Equation (61):

*TCsj*(*T*<sup>∗</sup>*j*,*Kja*<sup>∗</sup>,*Kjc*<sup>∗</sup>, *Lja*<sup>∗</sup>, *Ljc*<sup>∗</sup>, <sup>ε</sup><sup>∗</sup>*ja*, ε∗*jc*) = 1*T*∗*j* [*Aj* + (*Crm* + *TDmaj*(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*) + *gmaj* (*<sup>K</sup>*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*))(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*)( *<sup>T</sup>*<sup>∗</sup>*jDj <sup>K</sup>*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*(<sup>1</sup> − α2*j* <sup>β</sup>*j*))+(*TDmcj*(*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*) + *gmcj* (*<sup>K</sup>*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*))(*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)( *<sup>T</sup>*<sup>∗</sup>*jDj <sup>K</sup>*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc* ) + *<sup>L</sup>*<sup>∗</sup>*jWj* +*hmj*[ ( *<sup>T</sup>*<sup>∗</sup>*jDj* <sup>1</sup>−α2*j* β*j*) 2 (2*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*) + (*DjT*<sup>∗</sup>*j*)<sup>2</sup> (2*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)(<sup>1</sup> − *Dj* (*<sup>K</sup>*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)) + (*DjT*<sup>∗</sup>*j*)<sup>2</sup> 2*Dj* (1 − *Dj* (*<sup>K</sup>*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)) 2 ] <sup>+</sup>γ2[(*A*(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*)<sup>2</sup> − *<sup>B</sup>*(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*) + *<sup>C</sup>*)(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*)( *<sup>T</sup>*<sup>∗</sup>*jDj <sup>K</sup>*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*(<sup>1</sup> − α2*j* <sup>β</sup>*j*))+(*A*(*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)<sup>2</sup> − *<sup>B</sup>*(*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*) <sup>+</sup>*<sup>C</sup>*)(*K*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc*)( *<sup>T</sup>*<sup>∗</sup>*jDj <sup>K</sup>*<sup>∗</sup>*jc*<sup>ε</sup><sup>∗</sup>*jc* )] + *<sup>s</sup>*.*SCj* + *MR*[(*TDojPjb*(1 − <sup>α</sup>*j*) + *goj Pjb*(1 − <sup>α</sup>*j*))*Pjb*(<sup>1</sup> − <sup>α</sup>*j*)( *<sup>T</sup>*<sup>∗</sup>*jDj Pjb*(1 − α2*j* β*j*)) + ( *<sup>T</sup>*<sup>∗</sup>*jDj* <sup>1</sup>−α2*j* β*j*) 2 (1 − <sup>α</sup>*j*) 2*Pjb* + <sup>α</sup>*j*(<sup>1</sup> − <sup>α</sup>*j*)( *<sup>T</sup>*<sup>∗</sup>*jDj* <sup>1</sup>−α2*j* β*j*)2 *Pjb* + α2*j*( *<sup>T</sup>*<sup>∗</sup>*jDj* <sup>1</sup>−α2*j* <sup>β</sup>*j*)<sup>2</sup>(<sup>1</sup> − <sup>α</sup>*j*β*j*) 2*Pjb* + θ*j* <sup>+</sup>ψ*ja*(*K*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*)( *<sup>T</sup>*<sup>∗</sup>*jDj <sup>K</sup>*<sup>∗</sup>*ja*<sup>ε</sup><sup>∗</sup>*ja*(<sup>1</sup> − α2*j* β*j*)) + ψ*jbPjb*( *<sup>T</sup>*<sup>∗</sup>*jDj Pjb*(1 − α2*j* β*j*)) + *Rj*<sup>α</sup>*j*( *<sup>T</sup>*<sup>∗</sup>*jDj* 1 − α2*j* <sup>β</sup>*j*)(<sup>1</sup> − <sup>α</sup>*j*β*j*) <sup>+</sup>γ1*Pjb*( *<sup>T</sup>*<sup>∗</sup>*jDj Pjb*(1 − α2*j*<sup>β</sup>*j*))<sup>α</sup>*j*β*j*]] (61)

where,

$$L\_{\rm j} = L\_{\rm ja}^\* + L\_{\rm jc}^\* \tag{62}$$

$$L\_{ja}^{\*} = \frac{l\_{aj}K\_j^{\*}}{\rho},\tag{63}$$

#### *3.5. Numerical Experiment*

The pragmatic application of the proposed Agri-SCM n model is performed by considering a sugar processing firm with vendor. The local industry is processing sugarcane as a raw material and converting it into sugar. There are various categories of the sugar obtained from the raw material depending on the quality and grades. In our case, we considered three grades of sugar, i.e., A, B, and C, respectively. The constraints of budget and resources compelled the sugar processing firm to outsource few processes of the sugar to vendors for a successful supply chain management. To avoid shortages, the managers are required to keep the production rate as a controllable to fix as per demand. The production rate of the sugar processing firm is linked with the integrated production rate of man and machine. The vendor operations are influencing the production rate of the system, which is kept constant. Therefore, it is limited to set the production rate of the operations before delivering the products to the vendor, and this should be greater than the rate of outsourcing operation. Furthermore, the rate of production operations performed after the outsourcing operation must be greater than the outsourcing rate. The numerical experiment of the research study is based on argir-SCM including sugar processing firm and vendor. The data utilized to perform the experiment is taken from the local industry of sugar processing SCM. The processing-based data for each agri-product are given in Table 2, which consists of tool-die, production, holding, and production rate, which is taken from the research work of [50]. On the basis of the capacity of the machines inside processing firm, the variable production rate, [<sup>ε</sup>*aj*−*min*, <sup>ε</sup>*aj*−*max*] is considered as [(120, 130, 140), (150, 160, 170)] for the machine to process each agri-product at the first stage where [<sup>ε</sup>*cj*−*min*, <sup>ε</sup>*cj*−*max*] is considered as [(110, 115, 125), (130, 140, 150)] to process each sugar grade processing in finishing stage of the processing.



All the data related to the imperfect production are given in Table 3, which cover inspection and recycling. Since the imperfect production is the part of normal production, this model considered few vendor operations. The inspection station is located after outsourcing to check the quality of each agri-product, and sorted the checked parts into good and rejected parts. The inspection cost is categorized as fixed including initial investment and variable cost depending upon the production quantity. The recycling cost includes the operations to recycle the rejected agri-products into other useful product. These costs have a significant impact on the total cost of processing.


**Table 3.** Vendor data to process various grades of sugar.
