OPTIMIZATION OF AN INVENTORY MODEL FOR DETERIORATING ITEMS ASSUMING DETERIORATION DURING CARRYING WITH TWO-WAREHOUSE FACILITY Текст научной статьи по специальности «Математика»

OPTIMIZATION OF AN INVENTORY MODEL FOR DETERIORATING ITEMS ASSUMING DETERIORATION DURING CARRYING WITH TWO-WAREHOUSE FACILITY

Krishan Kumar Yadav1, Ajay Singh Yadav1'*, Shikha Bansal1

•

department of Mathematics, SRM Institute of Science and Technology, Delhi-NCR Campus, Ghaziabad, India, 201204 krishankumaryadav222@gmail.com, ajaysiny@srmist.edu.in, shikhab@srmist.edu.in

Corresponding author

Abstract

A common topic in the context of its application in today's business contexts is inventory modelling and management. It is well-known that deterioration has a big impact on inventory management. One of the most frequent supply chain concerns is the deterioration of items during transit from a supplier's storehouse to a retailer's storehouse. In light of this, a two-level supply chain inventory model for decaying goods is developed with two warehouse (storehouse) facilities for retailers, namely Owned Warehouse (OW) and Rented Warehouse (RW), assuming deterioration both during carrying from a supplier's storehouse to a retailer's storehouses and in the retailer's storehouses themselves. Also, we are assuming the selling price and time sensitive demand. We are developed this model under inflation. Shortages are not allowed. The main objective of this study is to determine the optimal ordering policy in order to maximizes the retailer's profit per unit of time. The applicability of our suggested model is investigated using a numerical example and with the support of MATLAB programming software (version: R2021b). Sensitivity analysis is used to examine the effects of changing the values of system parameters. Graphical representations are also shown in this paper.

Keywords: Two-warehouse, Demand based on timing and selling price, Inflation, Deterioration during carrying and Optimization.

1. Introduction and literature survey

Design and production operations plays an important role in supply chain inventory management. Two-warehouse inventory management is a useful for optimising discrete item design and production operations. It enables producers to customise their manufacturing and distribution processes based on unique product quality, demand patterns, and lead times, resulting in increased operational efficiency and customer satisfaction. Deterioration is a key factor in both deterministic and probabilistic inventory models of the classical type. Profit changes anti-proportionally to the decline rate, meaning that if the deterioration rate rises, the retailer's profit falls, and if the deterioration rate falls, the retailer's profit rises. In the current analysis of an inventory model, the rate of deterioration cannot be disregarded. Deterioration is defined as the loss of the initial product's marginal values as well as damage, decay, disappearance, obsolescence and harm to utility. Poswal, P., et al. [47] and Mahata, S. and Debnath, B.K. [51] are also constructing an

inventory model based on certain novel assumptions. Kumar, A., et al. [49], Kumar, K., et al. [50], Kundu, T. and Islam, S. [52], Yusuf T.I., et al. [53] and Kumar, P., et al. [54] have also implemented optimisation approaches in various domains.

Inventory management is essential for preventing waste, maintaining product quality, and guaranteeing timely component delivery in the mechanical and electrical industries when it comes to perishable or deteriorating products. Here are a few specific applications for deteriorating goods in these industries: Temperature-controlled storage, management of humidity, First In First Out (FIFO), tracking of expiration dates, real-time monitoring, frequent quality checks, appropriate packaging, cooperation with suppliers, shortened storage durations, customised storage solutions, emergency response plan, waste reduction techniques, and continuous improvement. A study on the application of manufacturing in the Malaysian electrical and electronics industries was conducted by Wong, Y.C., et al. [55]. Basdere, B., et al. [56] Electronic and electrical product disassembly factories to recover resources in material and product cycles. Colledani, M., et al. [57] Manufacturing system design and management for superior product quality. Yusuf T., et al. [53] studied about analysis of the parameters relating to manufacturing flexibility and efficient performance.

Several kinds of realistic assumptions are taken into account when developing this work. Deterioration during carrying is one such sensible supposition. A portion of the entire order spoils during carrying for a variety of causes. Long distances travelled by the carrying vehicle (such kind of justification is appropriate for medicine, blood, radioactive elements, vaccine, fruits, vegetables, etc.), weather conditions while carrying (such kind of justification is appropriate for sugar, salt, vegetables, fruit, fish, meat, eggs, etc.), carelessness while loading and unloading (such kind of justification is the primary cause of an untrained labour force, and such kind of justification is appropriate for any kind of product), etc. are some possible reasons. A lot of study has already been done in inventory control and inventory management systems that take deterioration into consideration as a crucial factor. Many researchers in the past, including Ghare and Schrader [1] and Aggrawal and Jaggi [5], accepted that once things are received, they begin to deteriorate. The analyses of the development of the deteriorating inventory literature were given by some researchers, including Bakker et al. [12] and Yadav et al. [10].

Another significant factor related to an inventory system is the item's demand. One of the modelling community's major concerns has been it. In order to reflect practical scenarios, a variety of inventory models have been built and explored over time for various item types, taking into account various demand patterns. The demand is influenced by a wide range of variables, including quality, stock, various promotional deals, service quality, etc. One of these crucial factors that greatly influences customer's demand is selling price. The majority of the products are evidently price dependent. Some goods are extremely sensitive, while others are not. As a consequence, when an item's selling price increases, demand for that item declines, and when it decreases, demand for that item increases. Demand obviously declines as the selling price rises. On the other hand, a cheap selling price for some goods might make consumers wonder about their freshness and quality. The demand is also influenced by time. According to some researchers, demand can be a time- based function, while others contend that a quadratic function of time would be more suitable. The market demand in the earlier instance varies dramatically over time, while the market demand in the later case changes gradually over time. In especially for products like vegetables, fruits, sweets, etc., these situations rarely correspond to actual market scenarios. Thus, it appears that a demand function with a linear time dependence is more accurate and a better representation of the changing market needs over time. (An in-depth analysis of the time-dependent linear demand rate pattern is provided in [45]). A large number of inventory management models are informed by the realistic feature that are lower selling prices result in higher sales for many decaying products. Mondal et al. [7] established a selling price-dependent inventory model based on customer's demand. You [8] optimized the product's selling price to maximize the average profit of a manufacturing company. Maihami and Kamalabadi [11] investigated the impact of time and selling price on customer demands in an inventory system. Sarkar et al. [17] considered price and time- based demand in their manufacturing inventory

system under reliability and inflation. Manna et al. [29] conducted additional research on the impact of advertising and selling price on the rate of demand in a manufacturing inventory model. In the past few years, Kumar et al. [36], Yadav et al. [32], Yadav and Swami [33], Aditi and Jaggi [37], Gautam et al. [34] and Yadav and Swami [26] are used price sensitive demand or time sensitive demand in their research.

Another crucial factor of an inventory system is inflation. The total price of products and services rises as a result of inflation over time. When prices rise overall, each unit of currency may purchase fewer products and services. Therefore, inflation denotes a decline in the buying power of money, or a loss of real value in the internal medium of exchange and unit of account of the economy. In 1975, Buzacott [2] constructed the first Economic Order Quantity (EOQ) model that took inflationary effects into account, and it was at this time that he first introduced inventory models with inflation. The inventory model with inflation is then suggested by Harold and Thomas [4]. In the past few years, Inventory models Kausar et al. [38], Tiwari et al. [25], Yadav et al. [23], Yadav and Swami [27] and the Yadav et al. [24] have been proposed in an inflationary context.

Another crucial component of inventory management is deciding where to store the goods. The development of traditional inventory models took into account only one storage facility with infinite capacity. Typically, this storing space is referred to as an owned warehouse. (OW). In actual market situations, however, a merchant may choose to buy more goods than his storage capacity at once due to a price reduction offered for bulk purchases, a high reordering cost, high demand for a product, or seasonal products. Consequently, a second storage space is leased in order to store extra items. The owned warehouse (i.e., OW) is nearby and is known as the rented warehouse (i.e., RW), which we typically presume to have unlimited capacity. Typically, the carrying cost in RW is greater than that in OW. So, in order to lower inventory expenses, RW items are released first, followed by OW items. The two-warehouse inventory approach was first presented by Hartley [3]. A two-storage stock model for degradation goods with time-sensitive demand was then put forth by Bhunia and Maiti [6]. Yang [9] provided some consideration to the two-storage model with incomplete backlogs for deteriorating products and a constant demand rate under inflation. In the recent few years, Swati et al. [13], Chaman and Singh [14], Yadav et al. [21], Hatibaruah and Saha [46], Yadav and Swami [30], Nath and Sen [39], Yadav et al. [43], Vandana and Das [44] and Aarya, D.D., et al. [48] are developed inventory models under two storage facility.

Table 1: The comparison of our current work with previously published work

Deterioration Warehouses Preservation Technology Inflation Deterioration during carrying

Ghiami et al. [15] Stock dependent Yes Single No No No

Rizwanullah et al. [35] Stock dependent Yes Two No Yes No

Tayal et al. [16] Selling price & time dependent Yes Single Yes Yes No

Tiwari et al. [41] Constant Yes Two No No No

Saha and Chakrabarti [28] Stock and advertisement dependent Yes Single No No No

Momeni et al. [31] Stock and selling price dependent Yes Single No No No

Mahata and Debnath [42] Selling price dependent Yes Single Yes No Yes

Bhunia et al. [19] Time, selling price and advertisement dependent Yes Two No No No

Palanivel et al. [20] Stock dependent Yes Two No Yes No

Jiangtao et al. [18] Stock dependent Yes Single No No No

Huang et al. [40] Selling price & stock dependent Yes Single No Yes No

Akhtar et al. [45] Selling price & time dependent Yes Single No No No

Present paper Selling price & time dependent Yes Two No Yes Yes

Demand

2. Presumptions and notations

2.1. Presumptions

The following presumptions were used in the formulation of the mathematical model.

1. We know that selling price and time have an impact on market demand. As a result of this finding, we share Akhtar, et al. [45] and Aarya, D.D., et al. [48] view that demand is a function of both time and selling price-dependent.

2. Demand rate function is f (p, t) = a — bp + ct, where a, b, c > 0 are constants.

3. We consider the deterioration during carrying same as Mahata and Debnath [42].

4. We are assuming constant rate of deterioration, which is 0(0 < 0 << 1) in supplier's storehouse and y(0 < y << 1) in retailer's storehouses..

5. Planning horizon is infinite.

6. There is no lead time.

7. In retailer's storehouses, OW capacity is limited but RW capacity is deemed boundless same as Vandana and Das [44] and Aarya, D.D., et al. [48].

8. We are assuming constant holding cost in both the storehouses.

9. Both the time and the expense of transportation are insignificant.

10. We consider the inflation for developed this model same as Palanivel et al. [20] and Huang et al. [40].

11. There is no item replacement or repair.

12. Stock out are not allowed.

2.2. Notations

Table 2 is provided a description of the notations utilised for the constructed mathematical model.

Table 2: Notations

Notation Units

a Constant

b Constant

c Constant

e Constant

Y Constant

Q Units

M Units

W Units

S - W Units

tl Weeks

I(t) Units

h $/Unit

ß $/Units

r Constants

a $/Units

TAIPF $/Cycle

Description

Coefficient of demand function Coefficient of demand function Coefficient of demand function Rate of deterioration during carrying. Rate of deterioration in retailer's warehouses.

The quantity of orders made each cycle Deteriorating items quantity due to carrying Retailer's Owned Warehouse capacity Retailer's Rented Warehouse The stock arrived in retailer's warehouses at this time. Inventory level at time t. Holding cost per unit.. Deterioration cost per unit.

Inflation rate. Cost of purchasing per unit. The total average inventory profit function

Table 3: Decision-makingparameters

Notation Units Description

p $/Units Selling price of each product, where p > a.

T Weeks Length of the cycle..

3. Mathematical Model Formulation

In the starting, Q units of deteriorating goods were ordered by the retailer from supplier. Thus, Q represents the inventory quantity at time zero. The inventory level steadily drops M at the time t = t\ due to the deterioration rate 9 while during carrying from the supplier's storehouse to retailer's storehouses (i.e., RW and OW). At the time interval t € [ti, t2], the joint impact of deterioration and demand decreases the inventory/stock level in RW of the retailer's storehouse to drop until it reaches zero. Also, at the time interval t € [ti, t2], only the effect of deterioration decreases the inventory level in OW of the retailer's storehouse. Again, at the time interval t € [t2, T], the joint impact of deterioration and demand decreases the inventory/stock level in OW of the retailer's storehouse to drop until it reaches zero (See fig.l).

Figure 1: A graphical representation of a deteriorated inventory model with two warehouses

The stock level at t = 0 to t = T is characterised in the differential equations as follows:

dIi(t)

dt

+ 9Ix(t) = -f (p, t); t G [0, ti]

with the boundary conditions (B.C.) Ii(ti) = S and Ii(0) = Q.

dI2 (t)

dt

+ Y Il (t) = - f ( p, t); t g [ti, t2]

with the boundary conditions (B.C.) I2(t2) = 0 and I2(ti) = S — W.

dh(t)

dt

+ yI3 (t)= 0; t G [ti, ti]

with the boundary conditions (B.C.) I3(ti) = W.

dI4 (t)

dt

+ Yh(t) = — f (p, t); t g [ti, T]

(i)

(2)

(3)

(4)

with the boundary conditions (B.C.) I4(T) = 0.

The equations (5), (6), (7) and (8) are the solutions of equations (i), (2), (3) and (4), respectively:

i

Ii(t) = 9i

Ii(t) = y2

c + 9ie(ti —t)9( S +(a — bp + cti ) — 9

c + y2e(tl—t)y ( (a — bp + ct2)

Y

Y2

(a — bp + ct)

~9

(a — bp + ct) Y

(5)

(6)

I4 (t) = -2

Y

h(t) = We(ti-t)Y

c + 72e(T-th((a — bp + cT) - 4 V Y Y2

(a — bp + ct)

Y

From the equations (7) and (8), using the continuity at t = t2 , we get

W = e(t2—ti ^j-1

c + y2 e(T—t2 )y( (a — bp + cT) — i_ Y Y2

(a — bp + ct2)

Y

From the equations (7) and (9), we get

Í3 (t) = e^^Y,

c + sep—Vif(a — bp + cT ) — < V Y Y2

(a — bp + ct2 )[0(tl—t)y

7

Using I2(t) = S — W in equation (6), we get

S = W +

Y

c + y2e(t2—h)i( (a — bp + ct2) —

V Y Y2

(a — bp + ct1)

7

From the equations (9) and (11), we get

(7)

(8)

(9) (10) (11)

S = e(f2—ti^ [c + Y2e(T—t2(a — hp + cT> —

(a — bp + ct2 ) 1 7 J

Y2

c +12e(t2—ti)i^(a — bpY + C¿2) —

(a-bp + ct1)

7

Using h(0) = Q in equation (5), we get

Q = 02

c + 92efid (S +(a — bp + ct1 ) — c

0

02

( a - b p )

Since M = Q - S , So from equations (12) and (13), we get

M = 02

c + d2eti9 (S +(a — bp + ct1 ) — c

( a - b p )

Je(t2—ti)y{ [c + y2e(T—*2h((a — bp + cT) — 4

(aV 7

Y

(a -bp + ct2)

7

Y2

c + y2e(t2—ti)i( (a — bp + ct2) —

\ Y Y2

(a — bp + ct1)

7

(12)

(13)

(14)

Next, we compute the associated costs and profit as follows: 1. The Ordering cost:

OC = A

2. The Holding cost:

HC = h

rT

I h(t)e—rtdt + I3(t)e—rtdt + / I4(t)e—rtdt _Jt1 Jt1 Jt2

(15)

HC

M i

Y

(1 + rT) (1 + rt2 ) r2ert2

■2 a rT

r2g'

+

ry\ erT ert2 J ry

erT eH2

ae

tye-t(r+y) - aetye-t2(r+Yh T / ce-t(r+y) - ce-1 y(r + y) ) e V Y2 (r + y)

-h(r+y)

pbe rT — pbe rt2 ry

- hi 1 c Y

+

bpeyT

Y(r + y) V et(r+y) e*2(r+y)

TceyT

1

1

(1 + rti ) (1 + rt2)

r2ert1 r2 ert2

aet2ye-t1(r+y) aet2ye-t2(r+y)

JL( _L

ry\ ert1

+e

hy

y(r + Y)\ et(r+y) et2(r+y) _ N f _1___1_

ce-t1(r+y) — ce-t2 (r+y)

y(r + y) J V Y2(r + y)

pbe-rt1 - pbe-rt2 \ bpeyt2 f 1 1 \ t2ceyt2 f 1 1

y(r + y)\ et1(r+y) et2(r+y) J y(r + y)\ eh (r+y) et2(r+y)

rY

+

h

Y2 (r + Y)

1

1

[c(eTy - et2y) - ay(eTy - et2Y)

et2 (r+y) et1(r+y)

+ bpy(eTy - et2y) + (cyt2et2Y - TcyeTy)

(16)

3. The Deterioration cost:

DC = yß

t2 t2 T

I I2(t)e-rtdt + I3(t)e-rtdt + / I4(t)e-rtdt _Jt1 Jt1 Jt2

c

DC

yß[Y

(1 + rT) (1 + rt2)

r2 erT

r2ert2

j^i _1___1_

+ ry{¿rT

rY2 e

rT

ert2

ae

tye-t(r+y) - aetye-t2(r+y) N . Ty ( ce-t(r+y) - ce-t2(r+y)

Y(r + Y) pbe-rT - pbe-rt2^ bpeyT

+ eT

rY

- Yßi Yy

V Y2 (r + y)

TceyT

Y(r + Y) eT(r+y) - et2(r+y)

1

1

(1 + rt\) (1 + rt2 )

r2ert1 - r2ert2 aet2ye-t1(r+y) aet2ye-t2(r+y)

1

Y(r + Y) eT(r+y) - et2(r+y)

1 N c f 1 1

a

rY ert1 ert2 rY2 ert1 ert2

+e

t2y

ce-t1(r+y) — ce-t2 (r+y)

y(r + y) J V Y2(r + y)

pbe-rt1 - pbe-rt2N bpeyt2 ( _J___\_ t2 ceyt2 f ___

y(r + y)\ et1 (r+y) et2(r+y) ) y(r + y)\ et1 (r+y) et2(r+y)

rY

+

ß

Y2 (r + y)

1

1

et2 (r+y) et1(r+y)

c(eTY - et2Y) - aY(eTY - et2Y)

+ bpY(eTY - et2Y) + (cYt2et2Y - TcYeTY)

(17)

4. The Purchasing cost:

PC = a ■ Q = a ■ -77

^ e2

c + e^f s +(a - bpe + ct1 ) - e2

(a - bp)

(18)

5. The Sales Revenue:

SR = p /Tf (p, t)dt = p(2a + Tc - 2bp + ct1)(T - t1) J tl 2

1

c

c

1

1

Thus, we compute the total profit per unit time by the following equation:

TAIPF(p, T) = T [SR — OC — HC — DC — PC]

TAIPF( p, T) = 11 p(2a + TC — 2bp + Ctl)(T — tl) — A

— <(h + YP){ -

(1 + rT) (1 + rt2) N c' " r2eH2 J

r2erT

ry \ erT ert2 J ry2\ erT

ae

tye—t(r+y) — aetye—^(r+y)\ t / ce—t(r+y) — ce—t2(r+y)

Y(r + y) ) ^ Y2(r + y)

1 1 ' TceyT ( 1

i pbe rT — pbe rtz\ bpeyT V ry ) + y(r + y)\ et(r+y) et2(r+y)

— (h + yp){ y

(1 + rh) (1 + rt2)

r2ert1 r2ert2

ae*2ye—t1(r+y) - ae*2ye—t2(r+y)

a

ry V ert1 ert2

Y(r + Y) pbe—rt1 — pbe—rt2' bpeyt2 ( Y(r + Y) V

+e

t2y

ry

+

1

1

et1(r+y) et2(r+y)

h + p Y2 (r + Y)

1

1

ert2

1

y(r + y)\ et(r+y) et2 (r+y) 1 1 ' c ( 1 1 ' — ry^\ e^ — ~e^r2

ce—h(r+y) — ce—h (r+y)

1

Y2(r + Y)

t2 ceyt2 (_1

Y(r + y)\ et1(r+y) ef2 (r+y)

et2(r+y) eh(r+y)

^ (c(eTy — et2y) — ay(eTy — ehY)

+ bpy(eTY — et2Y) + (cyt2 et2 Y — TcyeTy)

— " ' 02

c + 02eh0 ( S +

(a —bp + ct1) c 0 02

( a b p )

(20)

Let ti = kt2, 0 < k < 1, then we get equation (21) from equation (20).

TAIPF( p, T) = 11 p(2a + Tc — ckt2 )(T — kt2) — A— |(h + yP){1

(1 + rT) (1 + rt2 ) r2ert2

r2erT

c

1

1

+ — ( —___1 , ,

ry\ erT ert2 J ry2\ erT ert2

aety e—t(r+y) — aety e—^ (r+y)

Y(r + Y) 1 1 ' TceyT

+e

Ty

ce—t(r+y) — ce—h (r+y)

Y2(r + Y)

1

1

f pbe rT — pbe rt2\ bpeyT

V ry J + y(r + y)\ et(r+y) et2(r+y) J y(r + y)\et(r+y) ef2(r+y)

— (h + YP)\ 1

(1 + rkt2) (1 + rt2)

r2 erkt2

r2ert2

J^iJ____ J^f J___

+ ry\ erkt2 eH2) ry2\ ¿rkt2 erf2

fae^y e—kt2(r+y) — ae^ y e—t2(r+y)

+e

t2Y

+

ce—kt2(r+Y) — ce—t2(r+Y)

Y2(r + Y) t2ceyt2 ( 1

+

Y(r + y)

bpeyt2

y(r + y)\ ekt2 (r+y) et2(r+y) J y(r + y)\ ekt2(r+y) et2 (r+y) h + p Y2(r + Y)

pbe rkt2 pbe

ry

1

1

et2 (r+y) ekt2 (r+y)

^ (c(eTy — e*2Y) — ay(eTY — e^)

+ bpy(eTY — et2Y) + (cyt2et2Y — TcyeTy)

— "' 02

c + 62ekt2° S +

(a —bp + ckt2) c 0 02

( a b p )

1

1

c

c

2

1

Krishan Kumar Yadav, Ajay Singh Yadav and Shikha Bansal

OPTIMIZATION OF AN INVENTORY MODEL FOR RT&A, No 3 (79) DETERIORATING ITEMS ASSUMING DDC WITH TWF_Volume 19, September 2024

4. Solution procedure

In this section, we explore the concavity of the objective function. Tiwari et al. [22] is also used the following technique for optimization in their research article. The necessary criteria must be satisfied in order to attain a maximum total profit:

dTAIPF(p, T) = 0 dTAIPF(p, T) = 0

dp ' dT 0 ()

Two equations are derived from equation (22). The optimal values of p and T (namely, p* and T* ) are determined by solving these two equations, which include two unknown variables, p and T , the subsequent sufficient conditions are also satisfied by it.

The conditions as mention below that must be satisfied in order to maximize TAIPF(p, T) using the hessian matrix HM, a matrix of 2nd order partial derivatives:

U

d2 TAIPF( p,T) d2 TAIPF(p,T) dp2 dpdT

d2 TAIPF( p,T) d2 TAIPF(p,T) dTdp dT2

D11 = " ^ ,[F/1) < 0, D22 = det(HM) > 0.

d2TAIPF(p, T) ~df2

Here, D11 and D22 are the minors of the hessian matrix HM. Figure 2 displays the whole solution strategy for our proposed model, which was derived using MATLAB software (version: R2021b).

5. Numerical illustration

To maximize the total average profit function TAIPF(p, T), the present model seeks to identify the p and T optimal values. Since TA IPF( p, T) generated in equation (21) is to find the best values for the decision variables p and T, it is extremely difficult to calculate a complex function analytically. The model is solved with the help of the following algorithm.

5.1. Algorithm

1. Fix the values of the parameters a, b, c, 9, t2,7, S, a, fi, h, r, A, k.

2. Build the function TAIPF(p, T) given by equation (21).

3. Maximize TAIPF(p, T) subject to the constraints 0 < t1 < t2 < T and 0 < a < p.

4. Calculate the optimum values p*, T*, t*, Q* and TAIPF*.

5.2. Example

A numerical example is given in this section to show how the model works. The following parameters values are used as input:

a = 75, b = 1.5, c = 2.1, 9 = 0.46, t2 = 4.0 weeks, 7 = 0.44, S = 900 units,

a = $1.1 /unit, fi = $1.8/unit, h = $3.6/unit, r = 0.85, A = $110, k = 0.55. The optimal solution obtained is as below:

p* = $32.4827/unit, T* = 10.6011 weeks, t* = 2.2000 weeks,

Q* = 2586.3 units, TAIPF* = $550.0893/cycle.

Figure 2: A flowchart depicting our established model solving process.

6. Sensitivity analysis

Sensitivity analysis is used in this part to investigate how changes in the parameter's values affect the optimum values. In order to do these studies, one parameter was changed by ±5% and ±10% at a time while remaining at its original value for the other parameters. The following table 3 is demonstrate the outcomes of the sensitivity analysis.

Table 4: Sensitivity analysis with respect to above example

Parameters 0/ % % change in optimal value

Change p* T* t* H Q* TAIPF*

a -10% 29.9669 10.5885 2.2000 2572.1 400.1475

-5% 31.2235 10.5938 2.2000 2579.2 473.2606

+5% 33.7443 10.6103 2.2000 2593.3 630.6352

+10% 35.0079 10.6211 2.2000 2600.4 714.9006

b -10% 35.9273 10.7299 2.2000 2587.1 664.1233

-5% 34.1133 10.6628 2.2000 2586.7 603.9984

+5% 31.0093 10.5443 2.2000 2585.8 501.4878

+10% 29.6713 10.4918 2.2000 2585.4 457.4537

c -10% 31.9951 10.5684 2.2000 2588.0 523.7440

-5% 32.2385 10.5847 2.2000 2587.1 536.8524

+5% 32.7277 10.6176 2.2000 2585.4 563.4554

+10% 32.9733 10.6341 2.2000 2584.5 576.9513

Table 5: Sensitivity analysis with respect to above example (Continue)

Parameters 0/ % % change in optimal value

Change p* T* t* h Q* TAIPF*

9 -10% 32.3843 10.5333 2.2000 2342.1 575.5644

-5% 32.4326 10.5667 2.2000 2461.1 563.1258

+5% 32.5348 10.6366 2.2000 2717.9 536.4270

+10% 32.5889 10.6731 2.2000 2856.2 522.1102

t2 -10% 32.2278 9.9178 1.9800 2331.5 542.8833

-5% 32.3541 10.2592 2.0900 2455.7 546.6066

+5% 32.6138 10.9435 2.3100 2723.6 553.3205

+10% 553.3205 11.2863 2.4200 2868.0 556.2894

Y -10% 32.7035 11.4222 2.2000 2585.0 606.0342

-5% 32.5851 10.9900 2.2000 2585.7 577.2419

+5% 32.3940 10.2492 2.2000 2586.8 524.3797

+10% 32.3168 9.9289 2.2000 2587.2 499.9510

a -10% 32.3676 10.5285 2.2000 2586.9 577.0201

-5% 32.4255 10.5652 2.2000 2586.6 563.5307

+5% 32.5394 10.6363 2.2000 2585.9 536.6945

+10% 32.5955 10.6708 2.2000 2585.6 523.3453

ß -10% 32.4765 10.6361 2.2000 2586.3 553.6050

-5% 32.4796 10.6185 2.2000 2586.3 551.8408

+5% 32.4859 10.5839 2.2000 2586.2 548.3502

+10% 32.4891 10.5668 2.2000 2586.2 546.6234

h -10% 32.4551 10.7659 2.2000 2586.4 566.4975

-5% 32.4687 10.6816 2.2000 2586.3 558.1543

+5% 32.4973 10.5241 2.2000 2586.2 542.2818

+10% 32.5122 10.4503 2.2000 2586.1 534.7136

r -10% 32.5857 10.1147 2.2000 2585.7 499.6508

-5% 32.5293 10.3579 2.2000 2586.0 525.3019

+5% 32.4454 10.8445 2.2000 2586.5 574.0624

+10% 32.4164 11.0879 2.2000 2586.6 597.2693

A -10% 32.4794 10.5985 2.2000 2586.3 551.1270

-5% 32.4811 10.5998 2.2000 2586.3 550.6081

+5% 32.4844 10.6025 2.2000 2586.3 549.5705

+10% 32.4861 10.6038 2.2000 2586.2 549.0518

k -10% 32.2278 9.9178 1.9800 2331.5 542.8833

-5% 32.3541 10.2592 2.0900 2455.7 546.6066

+5% 32.6138 10.9435 2.3100 2723.6 553.3205

+10% 32.7473 11.2863 2.4200 2868.0 556.2894

Following are a few insights drawn from the sensitivity analysis's observations.

1. If we increase in a, then the total average inventory profit TAIPF is increases, because demand is increases. Simultaneously selling price, total cycle length T and total quantity Q are increases (See figure 7).

2. If we increase in b , then the total average inventory profit TA IPF is decreases, because demand is decreases. Simultaneously selling price, total cycle length T and total quantity Q

are decreases.

3. If we increase in c , then the total average inventory profit TAIPF is increases, because demand is increases. Simultaneously selling price and total cycle length T are increases, and total quantity Q is decreases.

4. If we increase in 9 (deterioration during carrying), then the total average inventory profit TAIPF is decreases, because the associated cost is increases. Simultaneously selling price, total cycle length T and total quantity Q are increases (See figure 4).

5. If we increase in 7, then the total average inventory profit TAIPF is decreases, because the deterioration in retailer's warehouses are increases, so that associated inventory cost is increases. Simultaneously selling price, total cycle length T are decreases, and total quantity Q is increases.

6. If we increase in r, then the total average inventory profit TAIPF is increases, because selling price will be decreases, so that demand will be increases. If demand will be increase, then the total average inventory profit is increases. Simultaneously the total cycle length T and total quantity Q are increases.

7. If we increase in a, p and h , then the total average inventory profit TAIPF is decreases, because associated inventory costs are increases. Simultaneously, selling price p is increases and the total quantity Q is decreases.

0

1 it

T

p

Figure 3: Concavity of TAIPF(p, T) with respect to p and T

$

Figure 4: Variation between Profit vs. Deterioration during carrying (d)

10 1 5 20 25

P

Figure 5: Variation between Profit vs. Selling price (p)

Figure 6: Variation between Profit vs. Purchasing cost (a)

7. Conclusion and future directions

In this study, the carrying of decaying goods from the supplier™s storehouse to the retailer™s storehouses and deterioration in the retailer™s storehouses with time and selling price incumbent demand under inflation have both been addressed in a two-level supply chain inventory model. We are presuming that retailers have two warehouses, named as OW and RW. We are taking a steady rate of decline into account. In such a scenario, shortages are not permitted. This study aims to find the optimal selling price and cycle length by implementing an algorithm that maximizes the total average inventory profit per unit of time. Considering deterioration during carrying is the most important part of this article. Many researchers have not yet taken into account that part of this article. Finally, the applicability of the proposed model is demonstrated

with a numerical example and pictorial representation. A sensitivity analysis of important parameters is provided with the help of MATLAB software (version: R2021b). The proposed model can also be modified to take into consideration different types of variable demands. It is also possible to recommend future research, such as investigating payment policies and preservation technologies.

References

[1] Ghare, P. M., and G. F. Schrader. "An inventory model for exponentially deteriorating items". Journal of Industrial Engineering 14.2 (1963): 238-243.

[2] Buzacott, J. A. "Economic order quantities with inflation." Journal of the Operational Research Society 26.3 (1975): 553-558.

[3] Hartley, R. V. "Operations research: a managerial emphasis (Vol. 976)." (1976).

[4] Bierman Jr, Harold, and Joseph Thomas. "Inventory decisions under inflationary conditions." Decision Sciences 8.1 (1977): 151-155.

[5] Aggarwal, S. P., and C. K. Jaggi. "Ordering policies of deteriorating items under permissible delay in payments." Journal of the operational Research Society 46.5 (1995): 658-662.

[6] Bhunia, Asoke Kumar, and Manoranjan Maiti. "A two-warehouse inventory model for deteriorating items with a linear trend in demand and shortages." Journal of the Operational Research Society 49.3 (1998): 287-292.

[7] Mondal, Biswajit, Asoke Kumar Bhunia, and Manoranjan Maiti. "An inventory system of ameliorating items for price dependent demand rate." Computers & industrial engineering 45.3 (2003): 443-456.

[8] You, Peng-Sheng. "Ordering and pricing of service products in an advance sales system with price-dependent demand." European Journal of Operational Research 170.1 (2006): 57-71.

[9] Yang, Hui-Ling. "Two-warehouse partial backlogging inventory models for deteriorating items under inflation." International Journal of Production Economics 103.1 (2006): 362-370.

[10] Singh, Ajay Kumar, S. Yadav, and S. R. Singh. "A two-warehouse inventory model for a deteriorating item with exponential demand rate, partially backlogged shortages." International Transactions in Applied Sciences 2.4 (2010).

[11] Maihami, Reza, and Isa Nakhai Kamalabadi. "Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand." International Journal of Production Economics 136.1 (2012): 116-122.

[12] Bakker, Monique, Jan Riezebos, and Ruud H. Teunter. "Review of inventory systems with deterioration since 2001." European Journal of Operational Research 221.2 (2012): 275-284.

[13] Agrawal, Swati, Snigdha Banerjee, and Sotirios Papachristos. "Inventory model with deteriorating items, ramp type demand and partially backlogged shortages for a two-warehouse system." Applied Mathematical Modelling 37.20-21 (2013): 8912-8929.

[14] Singh, Chaman, and S. R. Singh. "Optimal ordering policy for deteriorating items with power-form stock dependent demand under two-warehouse storage facility." Opsearch 50

(2013): 182-196.

[15] Ghiami, Yousef, Terry Williams, and Yue Wu. "A two-echelon inventory model for a deteriorating item with stock dependent demand, partial backlogging and capacity constraints." European Journal of Operational Research 231.3 (2013): 587-597.

[16] Tayal, Shilpy, et al. "Two echelon supply chain model for deteriorating items with effective investment in preservation technology." International Journal of Mathematics in Operational Research 6.1 (2014): 84-105.

[17] Sarkar, Biswajit, Papiya Mandal, and Sumon Sarkar. "An EMQ model with price and time dependent demand under the effect of reliability and inflation." Applied Mathematics and Computation 231 (2014): 414-421.

[18] Jiangtao, Mo, et al. "Optimal ordering policies for perishable multi-item under stock-dependent demand and two-level trade credit." Applied Mathematical Modelling 38.9-10

(2014): 2522-2532.

[19

[20

[21

[22

[23

[24 [25

[26

[27

[28

[29 [30 [31

[32 [33 [34 [35

Bhunia, Asoke Kumar, et al. "A two-storage inventory model for deteriorating items with variable demand and partial backlogging." Journal of Industrial and Production Engineering 32.4 (2015): 263-272.

Palanivel, M., R. Sundararajan, and R. Uthayakumar. "Two-warehouse inventory model with non-instantaneously deteriorating items, stock-dependent demand, shortages and inflation." Journal of Management Analytics 3.2 (2016): 152-173.

Yadav, A. S., et al. "Multi objective optimization for electronic component inventory model deteriorating items with two-warehouse using genetic algorithm." International Journal of Control Theory and applications 9.2 (2016): 15-35.

Tiwari, S., et al. "Impact of trade credit and inflation on retailer™s ordering policies for non-instantaneous deteriorating items in a two-warehouse environment." International Journal of Production Economics 176 (2016): 154-169.

Yadav, Ajay Singh, et al. "Effect of inflation on a two-warehouse inventory model for deteriorating items with time varying demand and shortages." International Journal of

Procurement Management 10.6 (2017): 761-775.

Yadav, A. S., et al. "An inflationary inventory model for deteriorating items under two storage systems." International Journal of Economic Research 14.9 (2017): 29-40. Tiwari, Sunil, et al. "Two-warehouse inventory model for non-instantaneous deteriorating items with stock dependent demand and inflation using particle swarm optimization." Annals of Operations Research 254 (2017): 401-423.

Yadav, Ajay Singh, and Anupam Swami. "A partial backlogging production inventory lot-size model with time varying holding cost and weibull deterioration." International Journal of Procurement Management 11.5 (2018): 639-649.

Yadav, Ajay Singh, and Anupam Swami. "Integrated supply chain model for deteriorating

items with linear stock dependent demand under imprecise and inflationary environment."

International Journal of Procurement Management 11.6 (2018): 684-704.

Saha, Sujata, and Tripti Chakrabarti. "Two-echelon supply chain model for deteriorating

items in an imperfect production system with advertisement and stock dependent demand

under trade credit." International Journal of Supply and Operations Management 5.3 (2018):

207-217.

Manna, Amalesh Kumar, et al. "An EPQ model with promotional demand in random planning horizon: population varying genetic algorithm approach." Journal of Intelligent Manufacturing 29 (2018): 1515-1531.

Yadav, Ajay Singh, and Anupam Swami. "A volume flexible two-warehouse model with fluctuating demand and holding cost under inflation." International Journal of Procurement Management 12.4 (2019): 441-456.

Arab Momeni, Mojtaba, Saeed Yaghoubi, and Mohammad Reza Mohammad Aliha. "A cost sharing-based coordination mechanism for multiple deteriorating items in a one manufacture-one retailer supply chain." Journal of Industrial Engineering and Management Studies 6.1 (2019): 79-110.

Yadav, Ajay Singh, et al. "Supply chain inventory model for deteriorating item with warehouse & distribution centres under inflation." International Journal of Engineering and Advanced Technology 8.2 (2019): 7-13.

Yadav, Ajay Singh, and Anupam Swami. "An inventory model for noninstantaneous deteriorating items with variable holding cost under two-storage." International journal of

procurement management 12.6 (2019): 690-710.

Gautam, Prerna, Aditi Khanna, and Chandra K. Jaggi. "Preservation technology investment for an inventory system with variable deterioration rate under expiration dates and price sensitive demand." Yugoslav Journal of Operations Research 30.3 (2020): 289-305. Rizwanullah, Mohd, Sachin Kumar Verma, and Maqsood Hussain Junnaidi. "Supply-chain two-warehouse inventory model for deteriorating items on exponential time function with shortage and partial backlogging in inflationary environment." Journal of Physics: Conference Series. Vol. 1913. No. 1. IOP Publishing, 2021.

[36

[37

[38

[39

[40

[41

[42

[43

[44 [45

[46

[47 [48 [49 [50

[51

[52 [53

Kumar, Satish, et al. "An inventory model with price dependent demand rates as power law form using ant colony optimization." Journal of Management Information and Decision Sciences 24.6 (2021): 1-10.

Khanna, Aditi, and Chandra K. Jaggi. "An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment: revisited." Opsearch 58.1 (2021): 181-202.

Kausar, Amrina, et al. "Dual Warehouse Inventory Management of Deteriorating Items Under Inflationary Condition." Advances in Interdisciplinary Research in Engineering and Business Management (2021): 39-53.

Nath, Biman Kanti, and Nabendu Sen. "A Completely backlogged two warehouse inventory model for non-instantaneous deteriorating items with time and selling price dependent demand." International Journal of Applied and Computational Mathematics 7.4 (2021): 145. Huang, Xiangmeng, Shuai Yang, and Zhanyu Wang. "Optimal pricing and replenishment policy for perishable food supply chain under inflation." Computers & Industrial Engineering 158 (2021): 107433.

Tiwari, Sunil, et al. "Retailer™s credit and inventory decisions for imperfect quality and deteriorating items under two-level trade credit." Computers & Operations Research 138

(2022): 105617.

Mahata, Sourav, and Bijoy Krishna Debnath. "A profit maximization single item inventory problem considering deterioration during carrying for price dependent demand and preservation technology investment." RAIRO Operations Research 56.3 (2022): 1841-1856. Yadav, A.S., et al. "Corona virus vaccine supply chain: green inventory due to covid-19 for storage of vaccine waste items and environmental pollution removal using optimized flower pollination services." Journal of Tianjin University Science and Technology (2022): 544-561. Vandana and Das, A. K. "Two-warehouse supply chain model under preservation technology and stochastic demand with shortages." OPSEARCH (2022): 1-26.

Akhtar, Md, Amalesh Kumar Manna, and Asoke Kumar Bhunia. "Optimization of a non-instantaneous deteriorating inventory problem with time and price dependent demand over finite time horizon via hybrid DESGO algorithm." Expert Systems with Applications 211

(2023): 118676.

Hatibaruah, Ajoy, and Sumit Saha. "An inventory model for two-parameter Weibull distributed ameliorating and deteriorating items with stock and advertisement frequency dependent demand under trade credit and preservation technology." OPSEARCH (2023): 1-52.

Poswal, Preety, et al. "Investigation and analysis of fuzzy EOQ model for price sensitive and stock dependent demand under shortages." Materials Today: Proceedings 56 (2022): 542-548. Aarya, Deo Datta, et al. "Selling price, time dependent demand and variable holding cost inventory model with two storage facilities." Materials Today: Proceedings 56 (2022): 245-251. Kumar, Amit, et al. "Stochastic Petri nets modelling for performance assessment of a manufacturing unit." Materials Today: Proceedings 56 (2022): 215-219.

Kumar, Kaushal, Ajay Kumar, and Vinay Singh. "Optimization of process parameters for erosion wear in slurry pipeline." Advances in Engineering Design: Select Proceedings of FLAME 2018. Springer Singapore, 2019.

Mahata, Sourav, and Bijoy Krishna Debnath. "The impact of RD expenditures and screening in an economic production rate (EPR) inventory model for a flawed production system with imperfect screening under an interval-valued environment." Journal of Computational Science 69 (2023): 102027.

Kundu, Tanmay, and Sahidul Islam. "A new interactive approach to solve entropy based fuzzy reliability optimization model." International Journal on Interactive Design and Manufacturing (IJIDeM) 13 (2019): 137-146.

°§, Yusuf Tansel, et al. "Analysis of the manufacturing flexibility parameters with effective performance metrics: A new interactive approach based on modified TOPSIS-Taguchi

method." International Journal on Interactive Design and Manufacturing (IJIDeM) 16.1 (2022): 197-225.

[54] Kumar, Pankaj, et al. "Optimization of cycle time assembly line for mass manufacturing." International Journal on Interactive Design and Manufacturing (IJIDeM) (2023): 1-12.

[55] Wong, Yu Cheng, Kuan Yew Wong, and Anwar Ali. "Astudy on lean manufacturing implementation in the Malaysian electrical and electronics industry." European Journal of Scientific Research 38.4 (2009): 521-535.

[56] Basdere, Bahadir, and Guenther Seliger. "Disassembly factories for electrical and electronic products to recover resources in product and material cycles." Environmental science & technology 37.23 (2003): 5354-5362.

[57] Colledani, Marcello, et al. "Design and management of manufacturing systems for production quality." Cirp Annals 63.2 (2014): 773-796.