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

Initially, most of the inventory models were based on the assumption that every manufacturer has its warehouse with unlimited capacity but in a realistic world, it is not possible. Every warehouse has limited capacity due to some real reasons. When manufacturer starts production, he accumulates all well in his warehouse. But when manufacturer got attractive price discount on raw material for bulk purchase or maybe the order cost is too higher than manufacturer have to use RW for the storage purpose. Inventory managers usually attracted to hold more quantity of items that can be stored in an owned warehouse. Rented ware house (RW) may also useful due to its good condition, like low-risk factors, low deterioration rate, and better storage facility, etc. But the manufacturer has to pay some rent or cost which is much greater than own ware house (OW). So that manufacturer uses the RW but tries to utilize the inventory of RW first and then OW inventory. In reality, various factors was induced by the decision-maker to order more items. Recently warehouse situation generally arises when the acquisition is higher than maintaining an RW which have better preservation technology. Hartely

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.

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 (

References | Parameters | ||||

Two warehouse | Recycling | Product Stewardship | Green design | Environmental Management capabilities | |

1 | No | No | No | No | No |

2 | No | No | No | Yes | No |

3 | No | Yes | No | No | Yes |

4 | No | No | No | No | Yes |

5 | No | No | Yes | No | Yes |

6 | No | No | Yes | No | Yes |

7 | No | No | Yes | No | Yes |

8 | No | Yes | No | Yes | No |

9 | No | No | No | Yes | Yes |

10 | Yes | No | No | No | No |

11 | No | Yes | No | No | Yes |

12 | No | No | No | Yes | No |

13 | No | No | No | Yes | Yes |

14 | No | No | No | No | Yes |

15 | No | No | No | No | Yes |

16 | No | No | No | No | Yes |

17 | No | No | No | Yes | No |

18 | Yes | No | No | No | Yes |

19 | No | No | No | No | Yes |

20 | No | No | No | Yes | Yes |

21 | Yes | No | No | No | No |

22 | Yes | No | No | No | No |

23 | Yes | No | No | No | No |

24 | Yes | No | No | No | No |

25 | No | Yes | No | Yes | No |

26 | Yes | No | No | No | No |

27 | Yes | No | No | No | No |

28 | Yes | No | No | No | No |

29 | Yes | No | No | No | No |

30 | Yes | No | No | No | No |

31 | Yes | No | No | No | No |

32 | Yes | No | No | No | No |

33 | Yes | No | No | No | No |

34 | Yes | No | No | No | No |

35 | Yes | No | No | No | No |

Proposed Model | Yes | Yes | Yes | Yes | Yes |

P (t) Production rate is demand dependent (KD(t)).

D(t) Demand rate which is (a-bp)t (a, b>0), p is the selling price.

G_{d} Green design life cycle cost.

G_{dv} Fixed product life cycle design cost ratio for green design.

N Number of life cycles before a product is disposed of or recycles.

RW Rented ware house.

OW Own ware house.

R_{DV}_{ }Variable product life design cost ratio for green design.

R_{j}_{ }Reliability of the function.

PS Product stewardship.

TC Total cost.

Demand is price and time dependent in this model. D(t)=(a-bp)t, where a,b are constant, p is selling price of the commodity and t is time. (for example Zhou

The OW has a limited capacity of w units and the RW has unlimited capacity.

The inventory cost of RW is greater than OW.

Inventory decreased due to demand and deterioration only.

Production rate depends on time and rate of production is KD(t), where K is constant.

The deteriorating rate depends on time and deteriorating units cannot be replaced or repaired. Deterioration rate for the model is

The RW is located near the OW so the transportation cost is negligible.

Deterioration rate of RW is lower than the OW.

The lead time is considered as negligible.

The shortage is allowed and completely backlogged.

In the proposed mathematical model, price and time-dependent demand are considered for different inventory levels. Here the production starts at zero time.

Throughout the time interval, 0 to t_{1} inventory level levitate at the rate of

With the boundary conditions

Throughout the time interval, t_{1} to t_{2} inventory level levitate at the rate of

With the boundary conditions

Throughout the time interval, t_{2} to t_{3} inventory level diminished at the rate of

With the boundary conditions

Throughout the time interval, t_{3} to t_{4} inventory level diminished at the rate of

With the boundary conditions

Throughout the time interval, t_{1} to t_{3} inventory level diminished at the rate of _{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

Throughout the time interval, t_{4} to t_{5} shortage occurs at the rate of _{6}(t) inventory is calculated with the help of differential equation

With the boundary conditions

Throughout the time interval t_{5} to t_{6} production starts again and inventory level increases at the rate

With the boundary conditions

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:

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.

TC=Setup cost+ Holding cost+ Deterioration cost+ Shortage cost+ Green design cost

According to the proposed model, there are three independent variables in the total cost equation

Step. 1

Calculate the first-order partial derivatives w.r.t all the independent variable

Step 2.

Equate the first-order partial derivatives to zero and solve for the value of

Step 3.

Now, calculate the second-order partial derivative w.r.t. all the independent variables like

Step 4.

Now, form a Hessian matrix as follows

Step 5.

Find H_{1}, H_{2,}_{ }and H_{3}, where, H_{1}, H_{2,}_{ }and H_{3} denote the first principal minor, second principal minor, and third principal minor respectively. If det (H_{1})>0, det(H_{2})>0, and det(H_{3})>0, then the matrix is a positive definite matrix, and f is called convex function.

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

Following parameters are used to demonstrate the numerical:

_{d}=$5; G_{dv}_{ }=$80; N=70; R_{DV}=$0.1; r_{1}=0.1; r_{2}=0.9; PS=$4;

By using the above parameters, we minimize the total cost function with green design and product stewardship, get the following values:

TC=$5974.68;

Total cost and other parameter value without green design and product stewardship:

TC=$5972.68;

a | t1 | t2 | t5 | TC |

44 | 0.78214 | 1.22029 | 1.38124 | 5974.68 |

45 | 0.78163 | 1.22025 | 1.38136 | 5974.26 |

46 | 0.78121 | 1.22028 | 1.38144 | 5973.62 |

47 | 0.78075 | 1.22024 | 1.38155 | 5972.98 |

b
t1
t2
t5
TC
0.008
0.782051
1.22029
1.38126
5974.77
0.009
0.782083
1.22029
1.38126
5974.83
0.010
0.782140
1.22029
1.38126
5974.89
0.011
0.782150
1.22029
1.38126
5975.15

PS
t1
t2
t5
TC
4
0.78214
1.22029
1.38124
5974.68
5
0.78214
1.22029
1.38124
5974.45
6
0.78214
1.22029
1.38124
5975.05
7
0.78214
1.22029
1.38124
5975.94

Gd
t1
t2
t5
TC
5
0.78214
1.22029
1.38124
5974.68
6
0.78214
1.22029
1.38124
5975.34
7
0.78214
1.22029
1.38124
5975.78
8
0.78214
1.22029
1.38124
5976.22

N
t1
t2
t5
TC
60
0.78214
1.22029
1.38124
5975.02
65
0.78214
1.22029
1.38124
5974.95
70
0.78214
1.22029
1.38124
5974.89
75
0.78214
1.22029
1.38124
5974.82

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.

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.

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.

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.

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.

The

Following parameters are used to demonstrate the numerical:

_{d}=$5; G_{dv}=$75; N=40; R_{DV}=$0.01; r_{1}=0.01; r_{2}=0.7; PS=$3;

By using above parameters, we minimize the total cost function and get the following values:

TC=$5919.74;

a | t1 | t2 | t5 | TC |

20 | 0.712615 | 0.83245 | 2.09258 | 5961.51 |

30 | 0.708811 | 0.835243 | 2.09256 | 5940.60 |

40 | 0.70334 | 0.831332 | 2.09251 | 5919.74 |

50 | 0.70225 | 0.832285 | 2.09252 | 5898.85 |

b
t1
t2
t5
TC
0.0005
0.703323
0.831330
2.09251
5919.65
0.0010
0.70334
0.831332
2.09251
5919.74
0.0015
0.703342
0.8331329
2.09251
5919.84
0.0020
0.703358
0.831338
2.09251
5919.93

PS
t1
t2
t5
TC
1
0.70334
0.831332
2.09251
5926.00
2
0.703334
0.831332
2.09251
5921.31
3
0.703334
0.831332
2.09251
5919.74
4
0.7.3334
0.831332
2.09251
5918.96

Gd
t1
t2
t5
TC
3
0.7.3334
0.831332
2.09251
5918.49
4
0.703334
0.831332
2.09251
5919.12
5
0.703334
0.831332
2.09251
5919.74
6
0.7.3334
0.831332
2.09251
5920.37

N
t1
t2
t5
TC
20
0.703334
0.831332
2.09251
5922.87
30
0.703334
0.831332
2.09251
5920.78
40
0.703334
0.831332
2.09251
5919.74
50
0.73334
0.831332
2.09251
5919.12

Firstly, when the value of 'a' increases then there is decrease in the value of TC and t_{1}, but on the other side the values of t_{2 }fluctuate regularly. The value of t_{5 }increases first and after that decreases. On increasing the value of parameter 'a' demand function is also increasing. In any situation when demand goes up, it will decrease the total cost.

In the next step if the value of 'b' increased then TC increases, also increase, and t_{1}, t_{5 }and_{ }t_{2 }will remain constant. On increasing the value of parameter 'b' demand function decreases and when demand decrease then total cost increase.

On increasing the value of parameter 'PS'(product stewardship), TC decreased. On the other hand t_{1}, t_{5} and t_{2} become constant. Product stewardship has a sharp effect on green product design. If the parameter of product stewardship increases then total cost decreased.

When 'G_{d}' increase then TC increased, but the value oft_{2}, t_{5,} and t_{1 }will become constant. When green design cycle cost increases then green product design cost also increases, it will increase the total cost also..

n increasing the number of recycling the product ‘N’ then the value of t1_{,} t2_{, }_{a}nd t5_{ }remains constant, but the value of TC continuously reduces.

In the following

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