Inventory Model with Truncated Weibull Decay Under Permissible Delay in Payments and Inflation Having Selling Price Dependent Demand Текст научной статьи по специальности «Биологические науки»

Inventory Model with Truncated Weibull Decay Under Permissible Delay in Payments and Inflation Having Selling Price Dependent Demand

K Srinivasa Rao1, M Amulya2 and K Nirupama Devi3

1,2,3Department of Statistics, Andhra University, Visakhapatnam, India

[email protected], [email protected], [email protected]

Abstract

For optimal utilization of resources, the inventory models are required in several places such as market yards, production processes, warehouses, oil exploration industries and food vegetable markets. Huge work has been produced by several researchers in inventory models for obtaining optimal ordering quantity and pricing policies. This paper addresses an EOQ model for deteriorating items having Weibull decay under inflation and permissible delay in payments. It is considered that the demand of items is a function of selling price. It is further assumed that the decay of items starts after certain period of time which can be well characterized by truncated Weibull probability model for the life time of the commodity. The optimal ordering and pricing policies of this system are derived and analyzed in the light of the input parameters and costs. Through sensitivity analysis it is demonstrated that the delay in the payments and rate of inflation have significant effect on the optimal policies. This model is very useful in the analyzing market yards where sea foods, vegetables, fruits, edible oils are stored and distributed.

Keywords: EOQ model, selling price depended demand, truncated decay.

1. Introduction

Decay is the major consideration for planning inventory and scheduling orders. The decay is in general random due to various factors such as environmental conditions, type of commodity, storage facility and natural life time. Considering the life time of commodity as random several authors developed various inventory model for deteriorating items with various plausible assumptions. The review on inventory models with deteriorating items is given by [1], [2], [3], [4]. Recently [5], [6], [7], [8], [9], and [10] have developed several inventory models with the assumption that the life time of a commodity is random and follows a specified distribution depending on the nature of commodity. In all these papers they assumed that the decay starts immediately after the procurement. But in many practical situations the deterioration of items in the stock starts only after certain period of time. This type of delay in decay can be characterized by truncated Weibull life time distribution which is often known as three parameter Weibull distribution.

Another basic assumption made by all these authors is that the payments must be made to the supplier immediately after receiving the items. However, it is a common phenomenon that the supplier allows a certain fixed period for finalizing the accounts and does not charge any interest during that period from the retailer. In [11] studied an EOQ model with assumption of permissible delay in payments. His work was extended to deteriorating items by [12]. Later [13], [14], [15] and others have developed EOQ models with permissible delay in payments.

K Srinivasa Rao, M Amulya, K Nirupama Devi RT&A, No 4 (71)

INVENTORY MODEL WITH SELLING PRICE DEPENDENT DEMAND_Volume 17, December 2022

In today's business transaction, the supplier will offer a cash discount to encourage the retailer in addition to allowing a fixed period for settlement of account. In addition to this there is a change in money value over time. Ignoring inflation may leads falsification in the model. Recently [16] has studied Inventory Model with Generalized Pareto life time under permissible delay in payments while deriving the optimal pricing and ordering policies. Considering the inflation several authors have studied various inventory model with permissible delay in payments. However, they assumed the decay is constant or independent of time, but in many practical situations the deteriorating rate is time dependent. An EOQ model with time quadratic demand by [17]. They considered the inflation while determining the optimal policies.

Little work has been reported regarding EOQ models under permissible delay in payments having inflation and selling price dependent demand, which are very useful for analyzing many practical situations arising at market yards, warehouse etc. Hence in this paper we develop and analyze the Economic Order Quantity model with truncated Weibull decay under permissible delay in payments and inflation having selling price dependent demand.

Section (2) of this paper deals with the assumptions of the model and notation. Section (3) is to develop the instantaneous inventory level at any given time t. The optimal ordering and pricing policies of the model are derived in Section (4). Section (5) considers Numerical illustration of the model. The sensitivity analysis is presented in Section (6). Section (7) deals with conclusions.

2. Assumptions

For developing the Economic Order Quantity model, the following assumptions are made

• Deterioration start time is y.

• Weibull distribution is the life time distribution of the commodity. Its p.d.f is

f(t) = a/3(t — y)!-1e

Where a is the scale parameter, p is the shape parameter and y is the location parameter The instantaneous deterioration rate is

h(t) = a(t — y)! , t>y

• Demand function is

R(p(t)) = a — bp(t) = a — bpert Which is selling price dependent demand. Where, a is the fixed demand, a > 0, b is the demand parameter, b > 0, and a > b, p(t) is the selling price of an item at time t and p is the selling price of the item at time t = 0.

• Rate of inflation is r, 0 < r < 1

• Shortages are not allowed.

• Zero lead time.

• During the permissible delay period (M), the account is not settled, the generated sales revenue is deposited in an interest-bearing account. At the end of the trade credit period, the customer pays off for all the units ordered.

• There is no repair or replacement of the deteriorated units during the cycle time.

Notation

H : Finite horizon length.

R(p(t)) : Demand per unit time as a function of selling price.

h : Holding cost of inventory per unit time after excluding interest.

r : Rate of inflation.

p(t) = pert : Per unit selling price.

g(t) = gert : Purchase cost of a unit at time t.

K Srinivasa Rao, M Amulya, K Nirupama Devi RT&A, No 4 (71)

INVENTORY MODEL WITH SELLING PRICE DEPENDENT DEMAND_Volume 17, December 2022

A(t) = Aert : Per order cost at time t.

lc : Interest charged per Rs. INR in stock per a year by the supplier.

le : Interest earned in Rs. INR per a year.

M : Permissible delay period which is allowed in settling the account.

Q : Order quantity per a cycle.

T : Cycle length

I(t) : On-hand inventory at time t,0 < t < T.

TC(p, T) : Total cost over (0, H).

NP(p, T) : Net profit rate function over planning period.

3. Inventory Model

Let Q be the inventory level of the system at time t = 0. During (0, y) inventory will decrease due to demand and during (y , T) inventory will decrease due to demand and deterioration. Since no shortages are allowed, at time T the inventory level reaches zero, the stock is replenished instantaneously. The schematic diagram representing the inventory level is shown in Figure-3.1.

Figure -1: Schematic diagram representing the inventory level of selling price dependent demand model

Let I(t) be the on-hand inventory at time t. The differential equations governing the on-hand inventory at time t are

I(t) = -R(p(tj) 0<t<y (1)

I(t) + h(t)I(t) = -R(p(t)) y<t<T (2)

where h(t) = aft (t- y)!~# y <t<T

and R(p(t)) = a — bp(t) = a — bpert with initial conditions 1(0) = Q and I(T) = 0. Solving equation (1) and using the initial condition 1(0) = Q, we get bp

I(t) = Q — at + — (ert — 1) 0<t<y (3)

r

Solving equation (2) and using the initial condition I(T) = 0, we get

I(t) = e

-a(t-')!

,J ea(u-y)!du-bpj eru+a(u-Y)! du

Y <t<T

Equating equations (3) and (4) when t = y, we get

T T

Q = a.Y-y (erY -1) + aj ea(u-Y)!du - bp J eru+a(u-Y)!du

YY

Substituting Q in equation (3), we get

TT

I(t) = a(y-t) + 1y (ert - erY) + aJ ea(u-Y)! du - bp J e ru+a(u-Y)!du 0<t<Y

YY

Since the length of time intervals are all the same, we have I(JT + t)

TT

bp

a(y -t)+~y (ert -erY ) + aJ ea(u-Y)!du - bp J eru+a(u-Y)!

du

-a(t-')!

I J

, J ea(u-Y)!du-bpJ eru+a(u-Y)! du

(4)

(5)

(6)

0 <t <y

Y <t <T

(7)

4. The Optimal Ordering and Pricing Policies

Total cost function is the sum of Ordering Cost (OC), Cost Deterioration (CD), Inventory Carrying Cost (ICC), Interest Charged (IC#) and Interest Earned (IE#). Each cost component is computed as follows: Ordering Cost, OC is

\P)H - !

(8)

erH -1

erT - 1

OC = ¿(0) + A(T) + A(2T)+... +A(n - 1)T = A Cost Deterioration, CD is

n-1 T

CD = 1 °e"T

J=0 0

where, Q is as given in equation (5). On simplification, we get

TT

CD =g ay-aT (erY - erT) + aJ ea(u-')! du -bp J eru+a(u-Y)! di

erH -1

erT - 1

Inventory Carrying Cost, ICC is

n-1 T

icc = h^g(JT) JIdT + t)dt

j=o

cly 6 bp

= hg Jr + J6 [erY (1-rY)-1] + Y 2 r6

T

,J ea(u-Y)! du-bpJ eru+a(u-Y)! di

+

dt

erH - 1

erT - 1

(9)

(10)

J e-a(t-Y)! aJ ea(u-Y)!du - bp J eru+a(u-Y)!di

Y t t

For computing interest charged and earned, there are two possibilities based on the customer's choice. Interest Charges (IC) for unsold items at the initial time or after the permissible delay period M and interest Earned (IE) from the sales revenue during the permissible delay period. Case (i): Optimum cycle length T is larger than or equal to M i.e., T > M Interest Charged in (0, H), IC# is

Tl — ± '

IC# = ic asO'T) ji(jT + t)dt

j=o \m

= ¡0a [|(r2 +M2 - 2My) + ^ [e#$(l - r(y - M)) - erM] + (y - M)

l + i J ea(u-$)!du-bpJ eru+a(u-$)!di

11 i

t

dt

erH -1

erT - 1

(11)

Interest Earned in (0, H), IE- is

n-l 8

IE# = Ie a P(JT) f (a- bpert)tdt

j=o

= h P

aM2 bp 2 r2

[erM(rM -1) + 1]

erH -1

erT - 1

The total cost over (0, H) is TC(p, T) and is given by TC(p, T) = OC + CD + ICC + IC# - IE#

Substituting equations (8), (9), (10), (11) and (12) in (13), we get TC(p,T) =

(12)

(13)

A + g

ay-aT-y (erY - erT) + af ea(u-Y)! du - bp f eru+a(u-Y)! di

+hg

av2 bp

Jr + J2 [er'(1-rY)-1]+Y

2 r2

+

T

fe-a

(t-y)!

f ea(u-Y)!du-bpf eru+a(u-Y)!dv

y

f ea(u-Y)!du-bpf eru+a(u-Y)!dv

+!od

Q. bv

2 (y2 +M2 - 2My) + -¡2 [e#$(l - r(y - M)) - erM] + (y - M)

l + tfe-O-r^tpfe^-r»*

T T T

+ fe-a(t-Y,! afea(u-r,! dU-bpfe™0a<u-Yl> dl

dt

-Iep

aM2 bp 2 r2

- ^ [erM (rM -1) + 1]

erH -1

erT - 1

(14)

The net profit is the difference of gross revenue and total cost. The gross revenue is (perT - gerT)(a - bperT)

Hence, the net profit is NP(p, T) = (perT - gerT)(a - bperT) - TC(p, T) where, TC(p, T) is as given in (14)

K Srinivasa Rao, M Amulya, K Nirupama Devi RT&A, No 4 (71)

INVENTORY MODEL WITH SELLING PRICE DEPENDENT DEMAND_Volume 17, December 2022

For obtaining the optimal policies of the model, maximize NP(p, T) with respect to T and p. The conditions for obtaining optimality are

dNP(p,T) dNP(p,T)

= 0,---= 0 and D =

dT

dp

d2NP(p,T) d2NP(p,T)

dp2 dTdp

d2NP(p,T) d2NP(p,T)

dTdp dT2

< 0

where D is the determinant of Hessian matrix

—= 0 implies,

(P — g)[arerT — 2pbre2rT]

erH —1

erT — 1

[g [—a + bp

erT + ae$(T-y)! — hnprT+a(T

bperT+a(T-Y)']

+hg

+h g

+

+hg

+

T

V [aea(T-y)! — bperT+a(T-y)!] + J e-a(t-')! [aea(T-Y)' — bperT+a(T-^i>] dt

y

T

(y — M) — ^e^"] + J e-a(t-r» ^^ — ^«-„e ] „ t

Y

TT

A + g ay — aT — ^ (er' — erT) + aJ ea(u-Y,!du — bp J eru+a(u-Y)'d\

y y

TT

J ea(u-Y,!du — bpJ eru+a(u-Y,!d y

J ea(u-Y,!du — bp J eru+a(u-Y,!d

av2 bp

Jr + J2 [erY(1 — ry) — 1]+y 2 r2

Je-a

(t-Y)!

+icg

Q, bv

-(y2 +M" - 2My) + [err(l - r(y - M)) - erM] + (y - M)

t +

aJ ea(u-r)!du - bp J eru+a(u-r)!di r

+

T

J e-a(t-Y,! aJ ea(u-Y,!du — bp J eru+a(u-Y,!di

dt

—IeP

aM2 bp 2 r2

— -2 [erM (rM — 1) + 1]

■ erH —1

(erT "1)2J

rerT l = 0

(16)

dNP(p,T,

= 0 implies,

erT (a + bgerT — 2pberT)

t

\s

b c

--(err - erT) - b j eru,a(u-r)!di

—b

Je-a"-"' I'™"*-"'du

dt

+ hg

+ h g

;[err(1-ry)-1]-by

I

J eru+a(u-r)!dl

— [erY(l — r(y — M)) — erM]

" T T T

-b(y-M) J eru+a(u-r)!du r -b J e-a(t-r)! r J eru+a(u-r)!du t d t

-I,

aM2 2bp

r r -rM

--f [erM(rM -1) + 1]

erH -1

erT - 1

= 0

(17)

For given values of the parameters and costs, equations (16) and (17) are solved using MATHCAD to get the optimal cycle length T = T* and selling price p = p*. Substituting the optimal values T* and p* in equation (14) we get the minimum total cost. Substituting this minimum total cost, T* and p* in equation (15), we get the maximum profit as

NP*(p1,T1) = (p1erTl - gerT")(a - bp1erT")

A + g

ay-aT* (err - erT") + aJ ea(u-r)!du - bp* J eru+a(u-r)!di

+hg

av2 bp*

J- + J2- [err(1-ry)-1]+y 2 r2

r T

r T

,+

' 1

Je-a

a J ea(u-r)!du - bp* J eru+a(u-r)!dv

r

J ea(u-r)! du-bp* J eru+a(u-r)! di

+!cd

2(y2 + M2 - 2My) + ^1 [e#$(l - r(y - M)) - erM] + (y - M)

* " * " tfe-C-r^tn j

+ je-

$

dt

-'e Pi

aM2 bp-,

—---Si [erM(rM -1) + 1]

2 r2

erH -1

erT" - 1

(18)

Case (ii): Cycle Length T is smaller than M i.e., T < M Interest Earned, IE2 is

n-l T

IE2 =Ie R(p(t))tdt + R(p(T))[T(M-T)]

j=o ^ 0

= „,e\j(a-bpe-)tdt + (a-bpe'T)[T(M-T)]

erH -1

erT - 1

= P^e

aT2 bp

---2 [erT(rT - 1) - 1] + (a - bperT)[T(M - T)]

2 r2

erH -1

erT - 1

Thus, the total cost over (0, H) is TC(p, T) TC(p, T) = OC + CD + ICC - IE2

Substituting equations (8), (9), (10) and (19) in (20), we get

(19)

TC(p,T) =

A + g

ay - aT -

^L (err - erT ) + aJ e a(u-y)! du -bp J eru+a(u-r)! dv

+hg

ay2 bp

Jr + J6 [erY(1 — ry) — 1]+y 2 r2

-y)!

+ J e-a(t-Y)

y

J ea[u-Y)!du —bp J eru+a(u-Y)!dt

Y

j ea(u-Y)!du — bp J eru+a(u-Y)!di

-pJe

œT 2 bn

---2 [erT(rT — 1) — 1] + (a — bperT)[T(M — T)]

2 r2

erH — 1

erT — 1

(21)

The net profit is the difference of gross revenue and total cost. The gross revenue is (perT — gerT)(a — bperT )

Hence, the net profit is NP(p, T) = (perT — gerT)(a — bperT) — TC(p, T)

(22)

where, TC(p, T) is as given in equation (21)

For obtaining the optimal policies of the model we maximize NP(p, T) with respect to T and p. The conditions for obtaining optimality are

d2NP(p,T) d2NP(p,T)

dNP(p, T) dNP(p, T)

= 0,---= 0 and D =

dT

dp

dp2 dTdp

d2NP(p,T) d2NP(p,T)

dTdp

dT2

< 0

where D is the determinant of Hessian matrix

dNP(p,T)

dT

= 0,,implies,

(p — g)[arerT — 2brpe2

{■eru — 1]

erT — 1 jg j—a + bperT + aea(T-Y)! — bperT+a(T-Y)!]

+hg

T

■ jaea(T-Y)! — bperT+a(T-Y)!] + J e-a(t-Y)! jaea(T-Y)! — bperT+a(T-Y)!] dt

—Ie p[aT — bpTerT + (a — bperT )(M — 2T) + (MT — T2)(—bprerT )]

TT

+

A + g

+hg

+

T

Je-a

ay — aT — y (erY — erT) + aj ea(u-Y)!du — bp J eru+a(u-Y)!dt

YY

TT

J ea(u-Y)!du —bp J eru+a(u-Y)!dt Y

J ea(u-Y)!du —bp J eru+a(u-Y)!dt

ay2 bp

Jr + J2 [erY(1 — ry) — 1]+y 2 r2

(t-Y)!

— JeP

aT2 2bp

---f [erT (rT — 1)] + (a — bperT)[T(M — T)]

2 r2

■ erH —1

(e rT "1)2J

rerT I = 0

dNP(p,T)

= 0 implies,

erT (a + bgerT — 2pberT ) —

j

b f

— (erY — erT) — b J eru+a(u-Y)!dt

e" ) — b J e'

Y

420

(23)

g

+hg

A [e(1 - ry) - l]-byj e^Wdu -b j e-W j d

dt

-I,

aT2 2pb

—---Ç [erT(rT - 1)] + (a - 2pberT)[T(M - T)]

erH - 1

erT - 1

= 0

(24)

For given values of the parameters and costs, equations (23) and (24) are solved using MATHCAD to get the optimal cycle length T = T2 and selling price p = p2. Substituting the optimal values of T2 and p2 in equation (21), we get the minimum total cost. Substituting this minimum total cost, T2 and p2 in equation (22), we get the maximum profit as

NP*(p2, T2) = (P2erT# - gerT#)(a - bP2erT#)

bp

A + g - -- P

+hg

+

ay2 bp2

+ [err(1-ry)-1]+y 2 r2

ay-aT2 (err - erT#) + aj ea(u-r)!du-bp2 j eru+a(u-r)!d

r r

t# t#

j ea(u-r)!du-bp2 j eru+a(u-r)!du r

j e-a(t-r)! aj ea(u-r)!du-bp2 j eru+a(u-r)!&

-P2le

aT2 bp2

[erT# (rT2 -i)-i] + (fl - bp2erT#)[T2(M - T2)]

erH -1

erT# - 1

(25)

5. Numerical Illustration

The optimal values of selling price (p) and cycle length (T) are obtained by using the equation (16) and (17) or (23) and (24). The optimal values of T are taken as T = T# if T# >M and T = T2 if T2<M.

To illustrate the developed model of Case (i) i.e, if T# > M, a numerical example with the following parameter values is considered. The deteriorating parameters a,p and y vary from 0.020 to 0.024, 0.06 to 0.72 and 0.06 to 0.72 respectively. The values of the other parameters and costs are considered as follows: a = 1000 to1200 ,b = 0.010 to 0.012 units, A = Rs. 250.0 to 300.0,g = Rs. 0.20 to 0.24 = Rs. 0.100 to 0.120 Ic = Rs. 0.150 to 0.180, Ie = Rs. 0.120 to 0.144, M = 15 days = — = 0.500 to 0.600, r = 0.010 to 0.012, H = 12.0 to 14.4 months.

30

By substituting the above values in equations (16) and (17) and solving, the optimal values of cycle length T and selling price p are obtained. Substituting the optimal values of cycle length T and selling price p in equations (5) and (15), the optimal values of Order quantity Q and net profit NP are obtained and presented in Table-1.

From Table-1, it is observed that when the parameter 'a' is increasing from 1000 to 1200 units, the optimal ordering quantity 'Q\ the cycle length 'T' and the net profit 'NP' are increasing from 1250.845 to 1585.738 units, 1.245 to 1.314 and Rs.1977.152 to Rs.2050.474 respectively and the unit selling price 'p' is decreasing from Rs. 4.275 to Rs. 3.625, when other parameters and costs are fixed.

When the parameter 'b' is increasing from 0.010 to 0.012 units, the optimal ordering quantity 'Q' increasing from 1250.845 to 1250.849, cycle length 'T, selling price 'p' are remains constant at 1.245, Rs.4.275 and the net profit 'NP' is decreasing from Rs.1977.151 to Rs.1977.150 respectively, when other parameters and costs are fixed.

As the deterioration parameter a is increasing from 0.020 to 0.024, the optimal ordering

quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1375.107 units, 1.245 to 1.365 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. Rs.4.275 to Rs. 4.140 and Rs.1977.152 to Rs.1953.603 respectively, when other parameters and costs are fixed.

Table-1: Optimal values of Q, NP, T and p for different values of parameters and costs

For h=0.1, Ic=0.15, Ie=0.12, M=0.5, r=0.01, H=12

a b a P Y A a Q T V NP

1000 0.01 0.02 0.6 0.6 250 0.2 1250.845 1.245 4.275 1977.152

1050 1328.542 1.259 4.090 1994.236

1100 1410.353 1.275 3.921 2012.050

1150 1496.130 1.294 3.766 2030.755

1200 1585.738 1.314 3.625 2050.474

0.0105 1250.847 1.245 4.275 1977.151

0.0110 1250.847 1.245 4.275 1977.151

0.0115 1250.849 1.245 4.275 1977.150

0.0120 1250.849 1.245 4.275 1977.150

0.021 1281.746 1.275 4.239 1971.233

0.022 1312.768 1.305 4.204 1965.332

0.023 1343.894 1.335 4.171 1959.454

0.024 1375.107 1.365 4.140 1953.603

0.63 1290.509 1.284 4.227 1970.017

0.66 1332.079 1.325 4.181 1962.661

0.69 1375.594 1.368 4.135 1955.097

0.72 1421.082 1.413 4.090 1947.340

0.63 1252.355 1.247 4.272 1976.975

0.66 1253.806 1.248 4.269 1976.811

0.69 1255.199 1.250 4.266 1976.658

0.72 1256.534 1.252 4.263 1976.517

262.5 1170.346 1.165 4.489 1984.103

275.0 1167.304 1.162 4.498 1984.386

287.5 1161.260 1.156 4.516 1984.953

300.0 1160.959 1.156 4.517 1984.982

0.21 1255.235 1.249 4.277 1961.926

0.22 1259.979 1.254 4.279 1946.543

0.23 1265.073 1.259 4.280 1930.992

0.24 1270.516 1.264 4.281 1915.263

For a=1000, b=0.01, a=0.02, |3=0.6, y=0.6, A=250, g=0.2

h lr I, M r H Q T p NP

0.105 1252.897 1.247 4.275 1971.381

0.110 1255.002 1.249 4.274 1965.584

0.115 1257.161 1.251 4.274 1959.762

0.120 1259.374 1.253 4.273 1953.913

0.1575 1251.530 1.245 4.272 1970.434

0.1650 1252.233 1.246 4.270 1964.532

0.1725 1253.034 1.247 4.267 1958.606

0.1800 1254.068 1.248 4.264 1951.804

0.126 1268.103 1.262 4.228 1975.626

0.132 1285.617 1.279 4.183 1974.133

INVENTORY MODEL WITH SELLING PRICE DEPENDENT DEMAND_Volume 17, December 2022

h lr I, M r H Q T P NP

0.138 1303.390 1.296 4.139 1972.676

0.144 1321.422 1.314 4.097 1971.258

0.525 1282.482 1.276 4.182 1976.588

0.550 1316.396 1.309 4.091 1976.187

0.575 1352.685 1.345 4.002 1976.004

0.600 1391.447 1.383 3.916 1976.099

0.0105 1242.297 1.236 4.292 1979.547

0.0110 1242.297 1.236 4.292 1979.547

0.0115 1233.796 1.228 4.309 1981.935

0.0120 1233.796 1.228 4.309 1981.935

12.6 1228.266 1.222 4.330 1974.585

13.2 1206.405 1.201 4.386 1972.158

13.8 1185.239 1.180 4.443 1969.864

14.4 1164.742 1.160 4.501 1967.694

When the parameter p is increasing from 0.60 to 0.72 the optimal ordering quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1421.082 units, 1.245 to 1.413 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.090 and Rs.1977.152 to Rs.1947.340 respectively, when other parameters and costs are fixed.

As the deterioration parameter y is increasing from 0.60 to 0.72, the optimal ordering quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1256.534 units, 1.245 to 1.252 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.263 and Rs.1977.152 to Rs.1976.517 respectively, when other parameters and costs are fixed.

If the ordering cost 'A' increases from Rs.250 to 300, the optimal ordering quantity 'Q' and the cycle length 'T' are decreasing from 1250.845 to 1160.959 units, 1.245 to 1.156 respectively and the unit selling price 'p' and the net profit 'NP' are increasing from Rs. 4.275 to Rs. 4.517 and Rs.1977.152 to Rs.1984.982 respectively, when other parameters and costs are fixed.

When the unit cost 'g' is increasing from Rs.0.20 to 0.24, the optimal ordering quantity 'Q\ the cycle length 'T' and the unit selling price 'p' are increasing from 1250.845 to 1270.516 units, 1.245 to 1.264 and Rs. 4.275 to Rs. 4.281 respectively and the net profit 'NP' is decreasing from Rs.1977.152 to Rs.1915.263 respectively, when other parameters and costs are fixed.

When holding cost 'h' is increasing from Rs.0.100 to 0.120, the optimal ordering quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1259.374 units, 1.245 to 1.253 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.273 and Rs.1977.152 to Rs.1953.913 respectively, when other parameters and costs are fixed.

When interest charged 7c' increases from Rs.0.150 to 0.180, the optimal ordering quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1254.068 units, 1.245 to 1.248 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.264 and Rs.1977.152 to Rs.1951.804 respectively, when other parameters and costs are fixed.

If interest charged 7e' increases from Rs.0.120 to 0.144, the optimal ordering quantity 'Q' the the cycle length 'T' are increasing from 1250.845 to 1321.422 units, 1.245 to 1.314 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.097 and Rs.1977.152 to Rs.1971.258 respectively, when other parameters and costs are fixed.

If the permissible delay period 'M' increases from 0.5 months to 0.6 months, the optimal ordering quantity 'Q' and the cycle length 'T' are increasing from 1250.845 to 1391.447 units, 1.245 to 1.383 respectively and the unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 3.916 and Rs.1977.152 to Rs.1976.099 respectively, when other parameters and costs are fixed.

The inflation rate V increases from 0.010 to 0.0120 the optimal ordering quantity 'Q' and the cycle length 'T' are decreasing from 1250.845 to 1233.796 units, 1.245 to 1.228 respectively and the

unit selling price 'p' and the net profit 'NP' are decreasing from Rs. 4.275 to Rs. 4.309 and Rs.1977.152 to Rs.1981.935 respectively, when other parameters and costs are fixed.

When the time horizon 'H' increases from 12 months to 13.8 then the optimal ordering quantity 'Q', the cycle length 'T' and the net profit 'NP' are decreasing from 1250.845 to 1164.742 units, 1.245 to 1.16 and Rs.1977.152 to Rs.1967.694 respectively and the unit selling price 'p' is increasing from Rs. 4.275 to Rs. 4.501, when other parameters and costs are fixed.

6. Sensitivity Analysis

To study the effect of changes in the model parameters and costs on the optimal values of the order quantity, cycle length, selling price and net profit, the sensitivity analysis is carried by considering a = 1000, b = 0.01 units, a = 0.02, 0 = 0.60, y = 0.60, A = Rs. 250, g = Rs. 0.20, h = Rs. 0.100, Ic = Rs. 0.150, Ie = Rs. 0.120, M = 0.500,r = 0.01, H = 12 months. Table-2 summarizes these results for variations of -15%, -10%, -5%, 0, 5%, 10%, 15% of the parameters and costs.

As the parameter a increases from -15% to +15%, the optimal order quantity Q is increases from 1044.252 to 1496.13, cycle length 'T' increases from 1.223 to 1.294, selling price 'p' decreases from Rs.4.940 to Rs.3.766 and the net profit increases from Rs.1927.777 to Rs.2030.755.

When the total demand during the cycle period b increases from -15% to +15%, the optimal order quantity 'Q' increases from 1250.843 to 1250.849, cycle length 'T' and selling price 'p' remains constant 1.245 and Rs.4.275 and the net profit 'NP' decreases from Rs.1977.153 to Rs.1977.150.

As the deterioration parameter a increases from -15% to +15%, the optimal order quantity 'Q' increases from 1159.039 to 1343.894, cycle length 'T' increases from 1.155 to 1.335, selling price 'p' decreases from Rs.4.394 to Rs.4.171 and the net profit 'NP' decreases from Rs.1994.984 to Rs.1959.454.

If the parameter p increases from -15% to +15%, the optimal order quantity 'Q' increases from 1165.449 to 1375.594, cycle length 'T' increases from 1.160 to 1.368, selling price 'p' decreases from Rs.4.388 to Rs.4.135 and the net profit 'NP' decreases from Rs.1992.891 to Rs.1955.097

When the deterioration parameter y increases from -15% to +15%, the optimal order quantity 'Q' increases from 1245.965 to 1255.199, cycle length 'T' increases from 1.238 to 1.250, selling price 'p' decreases from Rs.4.285 to Rs.4.266 and the net profit 'NP' decreases from Rs.1977.753 to Rs.1976.658.

When the ordering cost A increases from -15% to +15%, the optimal order quantity 'Q' decreases from 1636.158 to 1161.260, cycle length 'T decreases from 1.623 to 1.156, selling price 'p' increases from Rs.3.693 to Rs.4.516 and the net profit 'NP' increases from Rs.1950.119 to Rs.1984.953.

As the unit cost g increases from -15% to +15%, the optimal order quantity 'Q' increases from 1239.817 to 1265.073, cycle length 'T' increases from 1.234 to 1.259, selling price 'p' increases from Rs.4.267 to Rs.4.280 and the net profit 'NP' decreases from Rs.2021.981 to Rs.1930.992.

As the holding cost h increases from -15% to +15%, the optimal order quantity 'Q' increases from 1245.014 to 1257.161, cycle length 'T' increases from 1.239 to 1.251, selling price 'p' decreases from Rs.4.276 to Rs.4.274 and the net profit decreases from Rs.1994.320 to Rs.1959.762.

When the interest charged Ic increases from -15% to +15%, the optimal order quantity 'Q' increases from 1249.630 to 1253.034, cycle length 'T increases from 1.243 to 1.247, selling price 'p' decreases from Rs.4.281 to Rs.4.267 and the net profit decreases from Rs.1995.489 to Rs.1958.606.

If the interest earned Ie increases from -15% to +15%, the optimal order quantity 'Q' increases from 1200.594 to 1303.390, cycle length 'T increases from 1.195 to 1.296, selling price 'p' decreases from Rs.4.425 to Rs.4.139 and the net profit 'NP' decreases from Rs.1981.910 to Rs.1972.676.

When the permissible delay period M increases from -15% to +15%, the optimal order quantity 'Q' increases from 1168.658 to 1352.685, cycle length 'T' increases from 1.164 to 1.345, selling price 'p' decreases from Rs.4.562 to Rs.4.002 and the net profit 'NP' decreases from Rs.1979.374 to Rs.1976.004.

If the inflation rate r increases from -15% to +15%, the optimal order quantity 'Q' decreases from 1259.440 to 1233.796, cycle length 'T' decreases from 1.253 to 1.228, selling price 'p' increases from

Rs.4.258 to Rs.4.309 and the net profit 'NP' increases from Rs.1974.749 to Rs.1981.935.

When the time horizon H increases from -15% to +15%, the optimal order quantity 'Q' decreases from 1490.623 to 1185.239, cycle length 'T decreases from 1.480 to 1.180, selling price 'p' increases from Rs.3.820 to Rs.4.443 and the net profit 'NP' decreases from Rs.2008.083 to Rs.1969.864.

Table-2: Efect on Optimal Values with Respect to Parameters Variation

Variation Parameters Percentage change in parameter

-15 -10 -5 0 5 10 15

a Q 1044.252 1108.494 1177.432 1250.845 1328.542 1410.353 1496.130

T 1.223 1.226 1.233 1.245 1.259 1.275 1.294

p 4.940 4.699 4.477 4.275 4.090 3.921 3.766

NP 1927.777 1944.253 1960.585 1977.152 1994.236 2012.050 2030.755

b Q 1250.843 1250.843 1250.845 1250.845 1250.847 1250.847 1250.849

T 1.245 1.245 1.245 1.245 1.245 1.245 1.245

p 4.275 4.275 4.275 4.275 4.275 4.275 4.275

NP 1977.153 1977.153 1977.152 1977.152 1977.151 1977.151 1977.150

a Q 1159.039 1189.474 1220.082 1250.845 1281.746 1312.768 1343.894

T 1.155 1.185 1.215 4.275 1.275 1.305 1.335

p 4.394 4.353 4.313 1977.152 4.239 4.204 4.171

NP 1994.984 1989.032 1983.086 1.245 1971.233 1965.332 1959.454

P Q 1165.449 1177.053 1213.044 1250.845 1290.509 1332.079 1375.594

T 1.160 1.172 1.207 1.245 1.284 1.325 1.368

p 4.388 4.372 4.323 4.275 4.227 4.181 4.135

NP 1992.891 1990.722 1984.056 1977.152 1970.017 1962.661 1955.097

Y Q 1245.965 1247.650 1249.277 1250.845 1252.355 1253.806 1255.199

T 1.238 1.241 1.243 1.245 1.247 1.248 1.250

p 4.285 4.281 4.278 4.275 4.272 4.269 4.266

NP 1977.753 1977.540 1977.340 1977.152 1976.975 1976.811 1976.658

A Q 1636.158 1440.839 1340.631 1250.845 1170.346 1167.304 1161.260

T 1.623 1.432 1.333 1.245 1.165 1.162 1.156

p 3.693 3.899 4.078 4.275 4.489 4.498 4.516

NP 1950.119 1964.813 1970.615 1977.152 1984.103 1984.386 1984.953

g Q 1239.817 1243.134 1246.810 1250.845 1255.235 1259.979 1265.073

T 1.234 1.237 1.241 1.245 1.249 1.254 1.259

p 4.267 4.270 4.273 4.275 4.277 4.279 4.280

NP 2021.981 2007.169 1992.229 1977.152 1961.926 1946.543 1930.992

h Q 1245.014 1246.903 1248.847 1250.845 1252.897 1255.002 1257.161

T 1.239 1.241 1.243 1.245 1.247 1.249 1.251

p 4.276 4.275 4.275 4.275 4.275 4.274 4.274

NP 1994.320 1988.621 1982.898 1977.152 1971.381 1965.584 1959.762

Ic Q 1249.630 1249.910 1250.290 1250.845 1251.530 1252.233 1253.034

T 1.243 1.244 1.244 1.245 1.245 1.246 1.247

p 4.281 4.279 4.277 4.275 4.272 4.270 4.267

NP 1995.489 1989.675 1983.842 1977.152 1970.434 1964.532 1958.606

Ie Q 1200.594 1217.093 1233.842 1250.845 1268.103 1285.617 1303.390

T 1.195 1.211 1.228 1.245 1.262 1.279 1.296

p 4.425 4.373 4.323 4.275 4.228 4.183 4.139

NP 1981.910 1980.296 1978.709 1977.152 1975.626 1974.133 1972.676

Variation Parameters Percentage change in parameter

-15 -10 -5 0 5 10 15

M Q 1168.658 1194.025 1221.391 1250.845 1282.482 1316.396 1352.685

T 1.164 1.189 1.216 1.245 1.276 1.309 1.345

p 4.562 4.466 4.370 4.275 4.182 4.091 4.002

NP 1979.374 1978.584 1977.831 1977.152 1976.588 1976.187 1976.004

r Q 1259.440 1259.440 1250.845 1250.845 1242.297 1242.297 1233.796

T 1.253 1.253 1.245 1.245 1.236 1.236 1.228

p 4.258 4.258 4.275 4.275 4.292 4.292 4.309

NP 1974.749 1974.749 1977.152 1977.152 1979.547 1979.547 1981.935

H Q 1490.623 1402.823 1323.141 1250.845 1228.266 1206.405 1185.239

T 1.480 1.394 1.316 1.245 1.222 1.201 1.180

p 3.820 3.963 4.114 4.275 4.330 4.386 4.443

NP 2008.083 1995.997 1985.781 1977.152 1974.585 1972.158 1969.864

245QOrder Quantity (Q) I 1%™

a = 0.02

^ 2250 - p = 0.6

2050 - V = 0.6

N " A=250

5 1850 - g=0.20 1650 - h =0.1

1450 " iC = 0nf,

le = 0.12

-20% -15% -10% -5% 0% 5% 10% 15% 20% _ M=Q 5

Cycle Length (T) " j^5™

a = 0.02 (3 = 0.6 V = 0.6

* 1,9 - A=250 5 17 " g=0.20

* ' - h =0.1 1'5 - Ic = 0.15 1,3 - le = 0.12

-20% -15% -10% -5% 0% 5% 10% 15% 20% " M=0-5

2,3 2,1

-20%

-15%

-10%

Selling Price (p)

5,5 5 4,5 4 3,5 3

-5% 0% 5% 10%

15%

20% -

a=1500 b=0.01 a = 0.02 (3 = 0.6 V = 0.6 A=250 g=0.20 h=0.1 Ic = 0.15 le = 0.12 M=0.5 r=0.01 1-1=12

Net Profit (NP) " a=1500

1850 - b=0.01

a = 0.02

1800 - (3 = 0.6

N y = 0.6

i 1750 - A=250

x - g=0.20

' 1700 - h=0.1

Ic = 0.15

1650 - le = 0.12

✓ - M=0.5

1600 - r=0.01

-20% -15% -10% -5% 0% 5% 10% 15% 20% " H=12

7. Conclusion

In this paper an EOQ model for deteriorating items with permissible delay in payments having truncated Weibull distribution with inflation is proposed and analyzed. In inventory control, permissible delay in payments has significance influence in obtaining the optimal pricing and ordering policies. The truncated Weibull distribution is one of the most significant life time distributions for items such as food and vegetables markets, market yards and chemical industries, etc., where the deterioration is skewed and having long upper tail. The truncated Weibull distribution includes exponential distribution as a particular case. The sensitivity analysis of the model revealed that the pricing and ordering are highly influenced by the parameters and costs. The model with constraints on warehouse capacity and budget can also be developed with permissible delay in payment and truncated Weibull decay which will be published elsewhere.

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