Green design and product stewardship approach for two-warehouse inventory model

Background/Objectives: To trim down the recycling cost of any manufactured goods with the help of green design and product stewardship. Methods/Statistical analysis: For the planned EPQ (economic production quantity model) model, all costs are calculated to find total cost and this total cost is optimized with the help of the Hessian matrix. Sensitivity analysis is also carried w.r.t. different parameters, to illustrate the impact of these parameters on the proposedmodel. The convexity of the total cost function is also checked with the help of mathematical softwareMathematica 9.0. Findings:Major finding of the proposedmodel are as follows: (i) Increase in the number of recycles results in the reduction of the total cost. (ii) Product stewardship parameter has a negative effect on total cost as the PS increases from 1 to 4 units, total cost decreases from 5926.00 to 5918.96 units (see Table 9 ) (similar findings can be written for numeric example 1 after correcting it). (iii) Green design costs have a positive effect on total cost, as the green design cost increases from 3 to 6 units, total cost also increases from 5918.49 to 5920.37 units (see Table 10 ). (iv) increase in the number of recycles results in the reduction of the total cost, as the number of recycles increases from 20 to 50 units total cost decreases from 5922.87 to 5919.12 units (see Table 11 ). Novelty/Applications: The Study of the effects of recycling by this green design and product stewardship approach makes the proposed model distinctive from the existing methods. The proposed model applies to eco-friendly manufacturing items with green design and product stewardship.


Introduction
In the current inventory modeling, green design and product stewardship are quite emerging issues. To resolve environmental stumbling blocks, inventory modeling can play a significant role in terms of green design and product stewardship. Manufacturer designs products considering it to be a profitable task, but most https://www.indjst.org/ also presented different approaches for different models.
In this study, a green design and product stewardship approach is used in the model for deteriorating items with shortages. Product stewardship is a predominant factor in green design. It is assumed that the demand rate is a function of price and time. The model is developed with a two-warehouse storage facility. To illustrate the utility of the model, two numerical examples are expounded; convexity and sensitivity analysis is also illuminating the constructive path for the proposed model.
The proposed study is further dealt in the following way: In section 2 deals with research gap analysis regarding the utility of the proposed model, in section 3, notations, and assumptions are provided which is used for the development of the proposed model. In section 4, the problem's definition is presented in the form of a flow chart. In section 5, a mathematical model is derived. In sections 6 and 7, an algorithm to solve the mathematical model and numerical examples are shown respectively.

Research gap
In the existing literature, different kinds of production inventory models are introduced and studied yet a lot of work had been done by the researcher in the field of green production. In a two warehouses inventory model green design and product stewardship is not used yet. In the proposed paper production depends on demand and still manufacturer needs a rented ware house due to sudden fluctuation in market. To resolve these fluctuation manufacturer should store goods in rented ware house. In the proposed work green design and product stewardship are applicable in the form of cost, which may be responsible to make a product green. This cost may increase the total cost, but it can reduce the recycling cost of inventory and it may reduce the salvage of the system. This concept become the main focal point of this study and that is achieved in it. To show the research gap the previous reports are tabulated as follows ( Table 1 ).

Mathematical model
In the proposed mathematical model, price and time-dependent demand are considered for different inventory levels. Here the production starts at zero time. Figure 1 shows the different inventory levels during the production cycle when two different warehouses are used OW and RW. Throughout the time interval, 0 to t 1 inventory level levitate at the rate of P − D (t) for manufactured own warehouse, where P is the rate of production, and D(t) is the demand rate. This situation solved with the help of differential equation With the boundary conditions 0 ≤ t ≤ t 1 , I 1 (0) = 0 Throughout the time interval, t 1 to t 2 inventory level levitate at the rate of P − D (t) for the rented warehouse, where P is the rate of production, and D(t) is the demand rate. This situation solved with the help of differential equation With the boundary conditions t 1 ≤ t ≤ t 2 , I 2 (t 1 ) = 0 Throughout the time interval, t 2 to t 3 inventory level diminished at the rate of D (t) for the rented warehouse, where D(t) is the demand rate. This situation is solved with the help of differential equation With the boundary conditions t 2 ≤ t ≤ t 3 , I 3 (t 3 ) = 0 Throughout the time interval, t 3 to t 4 inventory level diminished at the rate of D (t) for manufactured own warehouse, where D(t) is the demand rate. This situation is solved with the help of differential equation With the boundary conditions t 3 ≤ t ≤ t 4 , I 4 (t 4 ) = 0 Throughout the time interval, t 1 to t 3 inventory level diminished at the rate of α (t) from manufactured own warehouse, where α (t) is the deterioration rate of own warehouse. In time interval t 1 to t 3 inventory stored in the manufactured own warehouse deteriorate. This deteriorated inventory is calculated with the help of differential equations given below, where 'w' is the capacity of manufactured own warehouse.
With the boundary conditions t 1 ≤ t ≤ t 3 , I 5 (t 1 ) = w Throughout the time interval, t 4 to t 5 shortage occurs at the rate of D (t), where D(t) is the demand rate. This I 6 (t) inventory is calculated with the help of differential equation With the boundary conditions t 4 ≤ t ≤ t 5 , I 6 (t 4 ) = 0 Throughout the time interval t 5 to t 6 production starts again and inventory level increases at the rate P − D (t) , where P and D(t) are the rates of production the rate of demand respectively. The levitated inventory is calculated with the help of differential equation With the boundary conditions t 5 ≤ t ≤ t 6 , I 7 (t 6 ) = 0 https://www.indjst.org/ On solving above differential equation (1), subject to the boundary condition On solving above differential equation (2), subject to the boundary condition On solving above differential equation (3), subject to the boundary condition On solving above differential equation (4), subject to the boundary condition On solving above differential equation (5), subject to the boundary condition On solving above differential equation (6), subject to the boundary condition On solving above differential equation (7), subject to the boundary condition The different inventory levels obtained as follows: To find out the relation between t 2 , t 3, and t 5 the equation of continuity is solved Inventory stored at the manufacturer own warehouse is calculated by integrating different inventory level at a different time period The inventory stored at the rented warehouse when the capacity of own warehouse is filled can be calculated as given below: https://www.indjst.org/ The number of deteriorating stocks in both own and rented warehouse during the production cycle: Total holding cost for rented and own warehouse: Shortage cost: When a shortage occurs in any production system manufacturer lose some sale. This cost is called a shortage cost.
Green design cost: When a green design product is manufactured there is some extra expenditure that occurs in this process, that expenditure is called green design cost.  ) (

Algorithm to solve the mathematical model
According to the proposed model, there are three independent variables in the total cost equationt 1 ,t 2 andt 5 . To optimize the total cost equation and to find the value of all the independent parameters, the following steps are pursued.
Step. 1 Calculate the first-order partial derivatives w.r.t all the independent variable ∂ f Step 2.
Equate the first-order partial derivatives to zero and solve for the value of t 1 ,t 2 ,t 3 Step 3. Now, calculate the second-order partial derivative w.r.t. all the independent variables like ∂ 2 f Step 4. Now, form a Hessian matrix as follows  Step 5.

Numerical analysis
Two numerical examples are solved to show the reliability of the model. Random data is used to solve the following numerical.

Observation for the above sensitivity analysis
1. Firstly, when the value of 'a' increases then there is a decrease in the value of TC, but t 1, t 2 and t 5 becomes constant regularly. In any real market situation when demand increases then total cost decreases. Parameter 'a' is a factor of the demand function. Hence when the value of 'a' increases the value of demand function also increases, which may reduce total cost. 2. In the next step if we increase the value of 'b' then TC increases, but t 1 , t 2 and t 5 will remain constant. Parameter 'b' is a negative impact on demand function. On increasing the value of 'b' demand decreases and if demand decrease then the total cost goes up. 3. On increasing the value of parameter 'PS' then TC increases regularly but, the value of t 5 , t 1 and t 2 remains constant.
Product stewardship is a concept which helps in making a product green. In the proposed research green design and product stewardship collaborated, by increasing the value of product stewardship, the total cost increases. 4. When the 'G d ' increases then TC increases on the other hand t 1 , t 2, and t 5 remains constant. 'G d ' is a green design life cycle cost if 'G d ' increases then it also increases the value of total cost. 5. On increasing the number or recycling the product 'N' then the value of t 1 , t 2, and t 5 remains constant, but the value of TC continuously reduces. It is the main objective of the paper that recycling of a commodity will reduce the total cost.

Convexity of TC
The Figures 8, 9 and 10 show the convexity of TC function w.r.t different independent variables.

Numerical example 2
Following parameters are used to demonstrate the numerical: θ 2 =$0.

Conclusion
This study has collaborated on the concepts of green design, product stewardship, and two warehouses with the recycling of the used items. Numerical examples and sensitivity analysis illustrate that number of recycling of the items had a reverse effect on the total cost. i.e., an increase in the number of recycles results in the reduction of the total cost. Green design and product stewardship help the manufacturers to decrease the recycling cost. This study may be very useful for the manufacturer to deal with inventory management as well as environmental issues. Future research can be unfolded as integral research model by which of inflation, trade credit policy, different demand patterns, etc. can be included.