Imperfect Production Model for Sensitive Demand with Shortage Текст научной статьи по специальности «Математика»

Imperfect Production Model for Sensitive Demand with

Shortage

Uttam Kumar Khedlekar and Ram Kumar Tiwari

•

Department of Mathematics and Statistics Dr. Harisingh Gour Vishwavidyalaya, Sagar M.P. India (A Central University) e-mail: uvkkcm@yahoo.co.in

Abstract

In this paper, we have presented economic production inventory model considering non-linear demand depanding on selling price. Here, all imperfect quality items are reworked after the regular production process and the reworked items are considered as similar as good quality items. Rework is important in those businesses where last product is expensive and raw materials are insuficient. Now, our objective is to find out the optimal ordering lot size, optimal selling price and shortage for which the profit of the model is maximum. A numerical example is presented to illustrate the validity of the model. Manageral implications has been presented in terms of the production and pricing of imperfect items.

Keywords: Dynamic pricing, Non linear price sensitive demand, Optimal price settings, Imperfect item, Rework, Partial backlogging.

AMS Subject Classification: 90B05, 90B30, 90B50

1 Introduction

Inventory control is an important part of business because it ensure quality control in business. Inventory management secure the business and help to smooth runing of business affair. Today, pricing and production strategies are two fundamental components of the daily operations for manufacturers, particularly in the presence of imperfect production system. Production system is one of the most important ascepts of company's business strategy. To avoid the of overeges and shortages of products, firm should carefully design the production process to enrich any business.

As a consequence of this paper, the topic of pricing with production system has recently been the focus of acadmic research increase diverse as economics, marketing and operation managemant. There has been several studies analysing condition under which different pricing strategys optimize the compnies profitability Bose et al. (1995) desinged an economoc order quantity inventory model for deteriorating products with demand followed linear and positive trend under allowable shortage and backlogging. Chakrabarti and chauduri (1997) presented an inventory model for perishable items. In this model the demend was taken as linear function and shortage in each cycle. Wee (1999) explained an inventory model for deteriorating items. In which, shortage was partially backlogged at constant rate and demand was taken to be linear function on selling price.

The production is an essential part of inventory system and not produced hundred persent perfect items . Many researchers designed a production inventory model under backlogging situation such as Chern et al. (2008); Dye et al. (2007); Lodree (2007); Leung (2008) Goyal and Imran (2008); Thannyam and Uthayakumar (2008); Cardenas and Berson (2009); Taleizadeh (2011) Roy et al. (2011). Das et al. (2011) presented an economic order quantity model foe imperfect quality items with partialy backlogging. In this model, they also considered the cost of lost sale. Taleizadeh et al.

(2012) proposed an EOQ model in which they considered a special sale price along with partial backlogging, in that model customer can take the advantage of discount price. Lee and Dye (2012) formulated an economic order quantity model with shortage, in that model demand was taken stock dependent. They also considered the optimal ordering and preservation policies for maximize the total profit.

Several inventory moderl considered demand dependence on other factors such as product selling price quality. Datta (2013) investigated an inventory model assuming that the demand depends on both the selling price and quality. Kumar et al. (2013) proposed EOQ model under the consideration of price-dependent demand, where the carrying cost is a time function of the trade credit for deteriorating products. Sana (2010) designed an economic order quantity (EOQ) model in that model, the demand wsa considered as function of selling price and they also assumed the deterioration rate of defective item is time proportional. They studied about that the shortage followed by an inventory of replesnisment is also followed. They developed this model over an infinite time horizon for defective products. Sana (2011) suggested an inventory model in which, they taken the demand function is quadratic function and the selling price increases in each cycle, but demand decreases quadraticly with selling price. They studied the lot of change in the rate of demand.

By use of item preservation concept for deteriorating items, Khedlekar et al. (2016) conceptualized an EOQ in that model the demand wsa considered as function of selling price and linearly decreases. They considered as the profit is the concave function of the optimal selling price, they also stadied that the optimal selling price, the length of the replenishment cycle and the optimal preservation concept investment simultaneously. Mishra (2016) proposed a single-manufacturer single-retailer inventory model by incorporating preservation technology cost for defective items and determined optimal retail price, replenishment cycle and the cost of preservation lechnology.

Taleizadeh and Noori-daryan (2016) studid a production inventory model with a three-level decentralised supply chain with price sensitive demand. Haider et. al (2016) proposed an economic production quantity (EPQ) model in which they studied if we give the discount in defective item and apply rework process then we get maximum profit. Teksan and Geunes (2016) reported an economic order quantity model for finished goods. In this model they they assumed that the demand rate was more price sensitive for supplier and customer both. Taleizadeh et.al (2017) outlined an imperfect production inventory model without shortages. Pal and Adhikari (2017) conceptualized an imperfect production, inventory model with exponential partially backlogging with rework, in that model they assumed that all imperfect quality products are reworked after the regular production process and demand rate was price sensitive and it was monotonic decreasing function selling price. Among other researcher in the exposure, the notworthy contribution of Sarkar, Sana and Chaudhuri (2011); Yu, and Chen (2007); Wee, and Kuo

(2013); Pal, Sana, and Chaudhuri (2014); Sarkar (2012, 2013); Haider, Salameh, and Nasr (2016); Tyyab and Sarkar (2016) should be mentioned.

We have considered an imperfect production model which depend on the selling price. We assume, all the defective products are reworked just after the regular production process and no any scarp product is produced during production as well as reworking run time. Shortage occur at the beginning of the cycle and production starts after backorder time and backlogging rate is variable and consideration of impatient behaviour of customer. The price of goods is definitely shown to the customer at the beginning of time cycle in many situations. So it is very difficult to

take the different price within same inventory cycle. In this paper we deal with the three issues: first, what will be selling price for the items, second one how much inventory should be produced and third one what time period shortage would be allowed in order to optimum frofit.

3 Assumptions & Notations

3.1 Assumptions

The model is designed for infinite time horizon, This model is developed for single item, Production rate if perfect item p is constant and production rate of defective items is pd = xp, where x is continuous randam variable, In this model the shortages occur at the beginning of the cycle and during the shortage time interval a fraction of the demand varying with waiting time is backlogged for the clients, who have patience to wait, assume that customers impatient function by

b(t) = e aT, a > 0, After the continue production process all imperfect quality items are reworked, The holding cost for both type (perfect and imperfect) items is the same, Every constant costs as inspection cost and purchasing cost are included within the production cost of the items, The demand function of the product is D(s) = ps-r; t] > 0.

3.2 Notations

[D(s)j - Demand function for good products,

[I (t)j - On-hand inventory of product at time t in jth cycle,

[p] - Production rate for perfect item per units per unit time, [ pd ] - Production rate for imperfect quantity items unit per unit time, [x] - Percentage of produced imperfect quality items which is randam variable, [f(x)] - Probability density function of x, [r] - Rework rate of imperfect quality item per unit per unit time, [w] - Backorder level, [b(t)] - Customers impatient function, where t is the waiting time for customer, [ch] - Holdig cost per item per unit time, [ chl ] - Holdig cost of reworked item per item per unit time, [cp]- Production cost per unit of item, [cb]-Backorder cost per item, [ck ]- Per production set-up cost, [cL ]- Lost sale cost per item, [s]- Selling price per item, [ cp]- Stock dependent parameter, [ n]- The total profit,

1. - Average total profit,

2. - Excepted average total profit.

4 The Mathematical Model

Suppose a business start with shortage of products which are partially backlogged. The backlogging rate is a function of customer waiting time as b(t) = e-aT, a > 0, where t is waiting time x = ti — t. Suppose the production start at time and it continue up to time t3. Due to production run, all the products which are backlogged, during time period [0, t2] are provide at the time t2. The production rate is considered constant. The qx amount of defective item is produced by the total production. The rework rate of defective products is , and these are reworked after the regular production process. ^ is the amount of time required for reworking of defective items, where qx is total items produced and r is rework rate. There is the same price of good products and reworked product and demand rate is depend on selling price and defined as,

D(s) = ps-r (1)

We take Ti = ti- ti-1.

For the Time period 0 < t < t1, the differential equation governing the inventory level is

d-t = -D(s)B(r) (2)

with the boundary condition/(0) = 0 and I(t1) = —w where t = t1 — t.

The solution of above differential equation by using the boundary condition is

K 0 = ^--(3)

and using the boundary condition I (t= —w, we get The backorder cost during 0 < t < t± is

ц = D^1-e-atl) (4)

cbC(Kt))d t = (5)

The demand rate is D(s), out of this only D(s)e-a(tl-t) is fullfilld during [0, tj and D(s) — D(s)e-a(tl-t) wich not fullfilld. Then the cost of lost sale is given by

Cl £ D(s)[l — e-aCi-'^dt = ciD(s)(atl-1+e-atl) (6)

For the time interval ti<t< t2, the governing differential equation of inventory level is

%=p — pd—D(s) (7)

with boundary condition Ifa) = — w, I (t2) = 0

Then the solution of above differential equation is

I( t) = [(1 — X)p — D(s)}(t — t2) (8)

using the condition I(t) = — w, we have

w = [(1 — x)p — D(s)}T2, (9)

where T2 = t2 — ti

The cost of backorder in time interval ti<t<t2 is

rt1 flft-W^l- — CbMT2 2

Eq. (9) & Eq. (10) leads the back order cost during ti<t<t2

2

cbr04Kt))d t = c-^ (10)

(11)

cbM

2{(l-x)p-D(s)}

For the time interval t2<t< t3, the governing differential equation of inventory level is

%=p-pd-D(s) (12)

with boundary condition I(t2) = 0, I(t3) = z3 where z3, is inventory level of good product. Then the solution of above differential equation is

I( t) = {(l-x)p-D(s))(t-t2) (13)

using I(t3) = z3, we get

z3={(l-x)p-D(s)]T3 (14)

The holding cost for good items in time period 2 < < 3 is

ch£(I(t))d t = c-^ (15)

Now T2 + T3= -, using the Eq. (9) & Eq. (14) the holding cost is

p

22

= Cd{(1 — x)p — D(s)}q- — C^ +-^--(16)

2 v Ji p2 p 2{(1-x)p-D(s)} V '

The differential equation for time period t3 < t < t4, is

Tt = r — D(s) (17) with boundary condition I(t3) = z3, I(t4) = z4, where z4 is the highest inventory level of good items

!( t) = Z3+{r — D(s)}(t—t3) (18)

by using the condition ( 4) = 4

4 — 3 = { — D( )} T4 (19) After some simplification and putting T4 = —, we get

z4=q{1 — DD^-} — w (20)

Holding cost for good poducts for the time interval t3 < t < t4 is given by

ch£(I(t))d t = c-f(z3+z4)T4 (21)

Putting the value from Eq. (19) then holding cost

= -^^4[z3+z3+[r-D(s)}T4}

_ т^ I -h[r-P(s)]T42 = chl4z3 +

= Ch{(1-x)p-D(s)]T3T4 .2cmbyEq.(3.14) (22)

22

= ch{(1 - x)p - D(s)}fi----)^ + {r-D(s)}l^r

hLK V JiKp {(l-x)p-D(s)} r L v Ji r2

2 2 2 = ChW-xfr-DW^-^ + bir-DW^

Now it can be seen that the difective products produced during the time interval t1<t< t3 at rate pd. The defective products are reworked perfectly during the time interval [t3, t4] by the rework rate r. In this system there is no defective items after time t = t4.

The differential equation for time period t4 <t < t5, that show inventory level is

Tt = -D(s) (23)

with boundary conditions I( t4) = z4 and I( t5) = 0

Then the solution of this differential equation

I(t) = D(s)(ts - t) (24)

Byusing .2cml(t) = z4, .5cmz4 = D(s)T5 (25)

Holding cost for the time interval t4 < t < t5 is given by

Chft4 (I(t))dt= c-hzT

2

_ -hz42

= IDoo (26)

2D(s) И 1 pr J 1

The inventory of defective products is given figure(2) then the differential equation for time period ti<t<t3

~^ = Pd, .2cmwithboudarycondition .2cmld(t1) = 0, .2cmld(t3) = qx (27) Then the solution is

¡d( t) = Pd(t-h) (28)

Holding cost for the defective products is

ChItt3Q d( t))d t= ^ (29)

For time interval t3 <t <t4 the governing differential equation inventory level of the defective item, is given by

= —r, .2cmwithboudarycondition .2cmld(t3) = qx, .2cmld(t4) = 0 (30)

Then the solution is

¡d( t)=r( t4-t) (31)

The holding cost of reworked items

22

Chr £ (Id(t))dt = ^ (32)

The total profit = Revenue - total cost

= Revenue - (backorder cost + cost of lost sale + holding cost for good and defective products + holding cost for reworked items + purchase cost + repairing cost for defective items + set-up cost)

n(q, t^ s) = sq —

chD(s){1-at1e-ati-e-ati] clD(s)(at1-1+e-ati'>

cbV

2{(1-x)p-D(s)}

ci(1 — X)2_ + fD^ + :

p

q x Cfrtàqx

— ch{(1 — x)p — D(s)]^ + - r

2

2{(1-x)p-D(s)}

— ^ — D^ — ^Ml—^} — "]

chq2x chlq2x2

2 p

cpq — crqx — k

The total average profit of the model

D(s)

na tp= Df)n(q,t1,s)

D(s) r cbD(s){1-at1e-ati-e-ati] clD(s)(at1-1+e-ati)

— Is q--2---

q a2 a

cbV

2{(1-x)p-D(s)}

CJh(1 — x)q- + YDPßr + '-

p

chV

qx chVqx

— ch{(1 — x)p — D(s)}qpprx + - r

2

2{(1-x)p-D(s)}

— ïlr — Dm^ — ^ïqll — ^} — »]

chq2x ch1q2x2

2 p

cpq — crqx — k]

The total expected average profit of the model

na tp =

E(Ü\Sa - cbD(s){1-atie-atl-e-at1} _

q a2

clD(s)(at1-1+e-atl)

, cbV2__chh — m) Si ch D(s)q2 chqv

2{(1-m)p-D(s)} 2 ( ) p 2 p2 p

chV

2{(1-m)p-D(s)}

Ch{(1 — m)p

cf{r — D(s)}

q2(m2 + rr2)

r2

ch1q2(m2 + G2)

ch 2D(s)

D(s)}qm +

p r ß(r+m),

chvqm

Eq. (4) & Eq. (35) leads to

2 p

— cpq — crqm — k]

leatp

= A(q,s,t1) = Uo(s)+U1(s,t1) +

U2(s,ti) W(s)q

where

Uo(s) = U1(S, tj U2(s, tj V(s) = V1(s) = V2(s) = V3(s) =

V4(S) = V5(S) = W1(S) =

W2(s) = ¿11 = ¿22 = ¿31 = ¿51 = ¿53 =

x01 = x02 = x12 =

xoo + x01D(s) + x02D(s)2 w1(s) + w2(s)e-atl

V1(s)e-2atl + {v2 (s) + t1V3(s)}e-atl + V4(s)t1 + v5(s)

2a2{(1 — m)p — D(s)}

¿11 D( )2 + ¿12 D( )3

¿21D(S) + ¿22D(S)2 + ¿22D(S)3

¿31d(s)2 + ¿32d(s)3

¿41d(s)2 + ¿42d(s)3

¿5lD(s) + ¿52D(s)2+¿5зD(s)3

xuD(s) + x12D(s)2 x21D(s) + x22D(s)2

— /

ChP(1 — m); ¿12 = —cb; ¿21 = 0 2cbp(1 — m) + 2{ChP(1 — m) — Cip(1 — m)a}-^23 = 2cta 2cbp(1 — m)a; ¿32 = —2cba; ¿41 = —2clp(1 — m)a2 2kp(1 — m)a2-^52 = 2^p(1 —m) + {—Chp(1 — m) + ctp(1

- ch

cb — 2cia;xoo = ~

ch(1-m) chm

2 p 2 p ch 1 chm ch(m2 + g2)

2 p2 p r 2 r 2

chm chm _ _ ch . „ _ _ chm , chm

; x21 = ; x22 = + r a p ra a r a pra

2

ch(1-m)m ch(m2 + a2)

2 r

ch(m+r)2 2 p2 r2

; x11 =

p+

p r

crm + —

(33)

(34)

(35)

(36)

m) a}

chi(m2 + (T2) ch(m+r)

2

a

a

a

r

2

2

h q2 m

a

Proposition. The profit function ft (q, s, is concave if the corresponding Hessian matrix H of expected profit function is negative definite. where

H=

'd2fl d2fi d2fi \

dq2 ds dq dqdtl

d2fi d2fi d2fi

ds dq ds2 dti ds

d2fi d2fi d2fi ,

^dq dti dti ds dtl2 )

Proof: We have

n ëàtp = flfa s, h) = U0(s) + u1(s, tj + ^^

fq= xm + X01D(s) + XmD(s)2 - ^I(s)e-2a^1+{v2(s) + t^e-a^1+v4<.s)t1+v5<.s) w1(s) + w'(s)e-atl + q[x0iD'(s) + 2x02D'(s)D(s)}

{vi(s)e-2atl+{v2(s) + tiv3(sy}e-atl+v4(s)ti+v5(sy}'¥'(s)

dfi ds

+

qW(s)2

v'1(s)e-2atl+[v'(s) + tiv'(s)}e-atl+vi(s)ti + v'5(s) qW(s)

dfi ^--at, , -2avi(s)e-2atl + e-atlv3(s)-{v2(s) + tiv3(s)}ae-atl+vi(s)

— = —aw? (s )e l +--

d tl 2W qV(s)

Solve above equations by puting

ajL=0>ajL=0tajL=0

dq ds dti

and get the values of variable q, s, ^

■V J.-V n _L v n/^2 vi(s)e 2ati+{v2(S) + tiv3(S)}e ati+vi(S)ti + v5(S) _

x00 + x01D(b) + X02D(i) q2W(s) =

Then

_ Vl(s)e-2atl+{v2(s) + tlv3(sy}e-atl + v4(s)tl+v5(s) . .

q V {X00 + X0lD(s) + X02D(s)2}W(s) ( )

Substituting the value of q in the Eq. = 0 & jfl = 0 and solving them, we get the solution of

decision variable q, s, ti of the model.

If the second order condition of of optimization method will be satisfied then above solution will be optimal.

Now the second order derivatives

d2fl _ 2[vl(s)e-2atl+{v2(s) + tlv3(s)}e-atl+vi(s)tl+v5(s)]

dq2 q2W(s) d2fi _ -2vl(s)e-2ati+v3(s)e-ati+v4(s)-{v2(s) + tlv3(sy}ae-ati

(38)

(39)

dqdti q2W(s)

d2fi _ 2{v'1(s)e-2ati+{v!z(s) + tiv!3(s)}e-ati+v'^(s)ti+v'^(s)}W'(s) ds2 q^(s)2

2{vi(s)e-2ati+{v2(s) + tiv3(s)}e-ati+v4(s)ti+v5(s)}{W'(s)}2

qW(s)3 (40)

vï(s)e-2ati+{v2'(s) + tiv3'(s)}e-ati+v4'(s)ti+v'5'(s)

+ qV(s)

+w1'(s) + w2'(s)e-2ati + q[xo1D"(s) + 2X02D''(s)D(s) + 2xo2{D'(s)}2]

d2fi _ {v'2(s)e-2atl+{v2(.s) + tlv3(s)}e-atl+v4(s)tl+v2¡(s)} ds dq q2W(s)

{vi(s)e-2atl+{v2(s) + tiv3(sy}e-atl+v4(s)ti+v5(sy}{'¥'(sy} (41)

qW(s)2

+2x01x02D'(s)+D'(s)D(s)

0

d2fi „2.., , 4a2vi(s)e-2atl-2e-atlv3(s)+{v2(s) + tiv3(s)]a2e-atl

—t — a w? ( s )e 1 +--

d 112 2W q¥(s)

(42)

d2fi ds dt-\

-2avf1(s)e-2ati+v3(s)e-ati+v4(s)-{v2(s) + t1v3(s)}ae-ati

qW(s)

aw'2(s)e~

[-2av1(s)e-2ati + e-ativ3(s)-{v2(s) + t1v3(s)}ae-ati+vi(s)]W'(s) qW(s)2

(43)

putting all values of second derivatives in Hessian matrix

H=

'd2f1 d2fi

dq2 ds dq

d2fi d2fi

ds dq ds2

d2fi d2fi

^dq dti dti ds

d2fi d2fi \ Ar* Ar. Ar. Ai- '

dq dti d2fi dti ds d2fi I d U2 )

If all eigen values are nagetive i.e Hessian matrix H of expected profit function is negative definite, then the profit function is concave. 5 Numerical Example & Sensitivity Analysis

Consider a numerical example taking the demand function as given in Eq. (1) 5.1 Example

We consider the demand function D(s) as D(s) = ys-r and the value of the parameter in appropriate units as follows r] = 1.2, ct = 2 per unit per unit time, cb = 1.5 per unit per unit time, k = 500, ch = 1 per unit per unit time, chl = 1 per unit per unit time, cr = 1.5 per unit, cp = 4 per unit, y = 3000, r = 1200 units per unit time, a = 1.6, m = 0.05, a2 P = 800 units per unit

time, and randam variable follows uniform distribution in the interval (0,0.1). Then the optimal values for the model are f1 = 1107.4 s* = 39.15 q* = 206 if = 0.69. These values are optimal as the

eigen value of the Hessian matrix

/d2fi dq2 d2fi d2fi

ds dq dq dti

d2fi d2fi d2fi

ds dq ds2 dti ds

\ d2fi (dqdt1 d2fi d2fi

dti ds d ti2

are negative. i.e -1.510, -0.12, -0.00042.

So the profit function is concave. 5.2 Sensitive Analysis

We observed the sensitive of the key parameters which help the decision makers to take appropriate decision on their marketing strategy.

From Table 1, we observed that, with the increasing values of holding cost of products there is a minor change in the optimal lot size and selling price, but the expected average profit decreases shortlly and there is negligible changes in the period of shortage. It is clear that higher holding cost reduce the lot size. So smaller commodity causes the increas in shortage period. In this situation the expected average total profit in decreasing order.

From Table 2, we noticed that, the optomal lot size, shortage period and selling price are increasing with increasing production cost and we also fund that expected profit decreases with increasing the production cost.

From Table 3, we observed that, with the increasing values of backorder cost there is a minor changes in the optimal lot size and selling price, and there is negligible changes in the expected profit and shortage period.

We observed that, with the increasing values of parameter q there is a major change in the optimal lot size and selling price, the expected average profit decreases and there is negligible changes in the period of shortage (table 5). With the changes of parameter a, ther are minor change in optimal lot size, selling price and expected average profit with the increasing values of parameter a shortage period decreases. (table 4). If the demand function parameter y increases, the expected average profit, and lot size increases highly while the selling price and shortage period decreases (from table 6).

Now we have followed graphical analysis method three-dimensional (3D) plots for the

i

profit function nsatp Figure 1 and 2 present the piecewise 3D plots for the profit function, ngatp, versus the two corresponding variables subsequently out of the three variables, , q and 1 . In each Figure 1 and 2, 3D plot of function, 1 using the other two variables, and q at a fixed shortage time period t1 and 3D plot of function, ngatp, using the other two variables, s and t1 at a fixed lot-size q.

Table 1: Changges in h

Ch s k q A

1 39.15 0.69 206 1107.4

1.1 39.49 0.72 195 1099.26

.2 37.30 0.72 197 1090.8

.3 25.69 0.70 275 1052.24

Table 2: Changges in cp

cp s k q f1

3 31.30 0.64 240 1144.37

4 39.15 0.69 106 1107.4

46.83 0.74 183 1074.41

54.39 0.78 166 1047.31

Table 3: Changges in cb

Cb s h q f1

.5 19.88 0.73 446 1027.81

1 39.06 0.72 207 1107.82

.5 39.15 0.64 206 1107.4

39.24 0.66 206 1107.01

Table 4: Changges in a

a s k q f1

1.3 38.90 0.82 209 1109.22

1.4 31.78 0.73 245 1103.37

.5 32.62 0.70 240 1103.95

.6 39.15 0.69 206 1107.4

Table 5: Changges in ]

t s h q f1

1.1 77.56.15 0.78 167 1687.86

1.2 39.15 0.69 206 1107.4

.3 22.29 0.68 289 744.67

.4 22.71 0.69 210 519.93

Table 6: Changges in cp

cp s k q f1

3000 39.15 0.69 206 1107.4

3500 37.77 0.65 235 1308.24

36.69 0.62 254 1510.38

35.83 0.59 277 1713.58

40

Fig.1. Expected average total profit versus quantity and price

Fig.2. Expected average total profit versus shortage time and price

6 Conclusion

Several manufacturers have to call back their items after use and rework on them to make protect. satisfy the demands with new ones in recent years. This type of remanufacturing system may prevent disposal cost and reduce environment dilemmas. To overcome this problem, an economic production quantity model has been portrayed for imperfect items with rework and production.

We have presented an imperfect production inventory model by considering demand as nagetive power function of selling price. The shortage occurs in begning bears the more cost for inventory manager, but it helps to project the product and optimize the selling price also. We have also illustrated the model numerically for demand depending on selling price. In the sensitivity of parameters of the model, we observed that the optimal expected average profit decreases with higher holding cost of items and optimal expected average profit increases with higher value of parameter p .

References

[1] Bose, S., Goswami, A., and Chaudhuri, K. S. (1995). An EOQ model for deteriorating items with linear time-dependent demand rate and shortage under inflation and time discount. The Journal of Operational Research Society, 46(6),771-782.

[2] Chakrabarti, T., Chaudhuri, K. S. (1997). An inventory model for deteriorating items with

alinear trend International Journal of Production Economics, 49(3), 205-213.

[3] Chern, M.S., Yang, H.L., Teng, J.T., and Papachristos, S. (2008). Partial backlogging inventory lot-size models for deteriorating items with fluctuating demand under inflation, European Journal of Operational Research, 191(1), 127-141.

[4] Cárdenas Barrón, L.E. (2009). Economic production quantity with rework process at a singlestage manufacturing system with planned backorders. Computers and Industrial Engineering, 57(3), 1105-1113.

[5] Datta, T.K. (2013). An inventory model with price and quality dependent demand where some items produced are defective. Advances in Operations Research.

[6] Dye, C.Y., Hsieh, T.P., and ouyang, L.Y. (2007). Determining optimal selling price and lot-size with a varying rate of deteriorating and exponential partial backlogging. European Journal of Operational Research, 181(2), 668-678.

[7] Das Roy, M., Sana, S.S., and Chauduri, K.S. (2011). An economic order quantitymodel of imperfect quality items with partial backlogging. International Journal of Systems Science, 42(8), 1409-1419.

[8] Haider, M.L., Salameh, M., and Nasr, W. (2016). Production lot sizing with quality screening and rework. Applied Mathematical Modelling, 40(4), 3242-3256.

[9] Jaber, M.Y., Goyal, S.K.,and Imran, M. (2008). Economic production quantitymodel for items with imperfect quality subject to learning effects. International Journal of Production Economics, 115(1), 143-150.

[10] Khedlekar, U. K., Shukla, D. and Namdeo, A. (2016). Pricing policy for declining demand using item preservation technology. SpringerPlus, 5(1):1957.

[11] Konstantaras I., Goyal, S.K., Papachristos, S. (2007). Economic ordering policy for an item with imperfect quality subject to the in-house inspection. International Journal of Systems Science, 38(6), 473-482.

[12] Kumar, M., Chauhan, A. and Kumar, R. (2013). A Deterministic inventory model for deteriorating items with price dependent demand and time varyring holding cost under trade credit. International Journal of Soft Computing and Engineering, 2231-2307.

[13] Lee, Y.P., Dye, C.Y. (2012). An inventory model for deteriorating item under stock dependent demand and controllable deterioration rate. Computers & Industrial Engineering, 63(2), 474-82

[14] Leung, K.N.F. (2008). Using the complete squares method to analyse a lot - size model when the quantity backordered and the quantity received. European Journal of Operational Research, 187, 9-30.

[15] Lodree, E.J. (2007). Advanced supply chain planning with mixtures of backorders, lost sales, and lose contract. European Journal of Operational Research, 181(1), 168-183.

[16] Mishra, V. K. (2016). Inventory model of deteriorating item with revenue sharing on preservation technology investment under price sensitive stock dependent demand. International Journal of Mathematical Modelling & Computations, 37-48.

[17] Pal, B., Adhikari, S. (2017). Price-sensitive imperfect production inventory model withexponential partial backlogging. International Journal of System Science: Operations & Logistics, 2330-2674.

[18] Roy, T. and Chaudhuri, K. S. (2008). An EPLS model for a variable production rate with stock price sensitive demand and deterioration. Yugoslav Journal of Operational Research, 21.

[19] Sarkar, B., Sana, S.S. and Chaudhuri, K.S. (2011). An imperfect process for time varying demand with inflation and time value of money An EMQ model. Expert Systems with Applications, 38, 13543-3548.

[21] Sarkar, B. (2012). An inventory model with reliability in an imperfect production process. Applied Mathematics and Computation, 218, 4881-4891.

[21] Sarkar, B. (2013). A production-inventory model with probabilistic deterioration in two-

echelon supply chain management. Applied Mathematical Modelling, 37, 3138-3151.

[22] Sajadieh, M. S. and Akbari, M. and Jokar, R. (2009). Optimizing Shipment ordering and pricing policies in a two stage supply chain with price sensitive demand. Transportation Research Part E, 45, 564-571.

[24] Sana, S. S. (2010). Optimal selling price and lotsize with time varying deterioration and partial backlogging. Applied Mathematics and Computation, 217, 185-194.

[24] Sana, S. S. (2011). Price sensitive demand for perishable item an EOQ model, Applied Mathematics and Computation, 217, 6248-6259.

[25] Taleizadeh, A.A., Sajadi, S.J., and Niaki, S. T. A. (2011). Multiproduct EPQ model with single machine, backordering and immediate rework proces. European Journal Industrial Engineering, 5(4), 388-411.

[26] Taleizadeh, A.A., Jalali-Naini, S. Gh., Wee, H.M., and Kuo, T.C. (2013). An imperfect multi-product production system with rework. Scientia Iranica E, 20(3), 811-823.

[27] Taleizadeh, A.A., Noori-daryan, M. (2016a). Pricing, manufacturing and inventory policies for rawmaterial in a threelevel supply chain. International Journal of System Science, 47(4), 919-931.

[28] Taleizadeh, A.A., Samimi, H., Sarkar, B., Mohammadi, B. (2017). Stochastic machine breakdown and discrete delivery in an imperfect inventory production system. Journal of Industrial and Management Optimization, doi:10.3934/jimo.2017005

[29] Tayyab, M., and Sarkar, B. (2016). Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate. Journal of Cleaner Production, 139, 922-934.

[30] Teksan, Z.M., Geunes, J. (2016). An EOQ model with pricedependent supply and demand, International Journal of Production Economics, 178, 22-33.

[31] Thangam, A., and Uthayakumar, R.(2008). A two level supply chain with partial backordering and approximate Poisson demand. European Journal of Operational Research, 187, 228-242.

[32] Wee, H.M., Yu, J., and Chen, M.C. (2007). Optimal inventory model for items with imperfect quality and shortage backordering. Omega, 35(1), 7-11.