Optimal production schedule in declining market for an imperfect production system

ORIGINAL ARTICLE

Optimal production schedule in declining market for an imperfectproduction system

Nita H. Shah • Kunal T. Shukla

Received: 14 March 2010 / Accepted: 7 September 2010 / Published online: 22 September 2010

� Springer-Verlag 2010

Abstract This is a study about economic production in an

imperfect production system. The imperfect production

system means is a system in which the machine shifts from

an ‘in-control’ state to an ‘out-of-control’ state. This may

happen at any random time during the production period.

The classical economic production lot size model is based

on the assumption that the produced items are of perfect

quality. However, this is not observed during the produc-

tion phase. In the long run, the process shifts from the

‘in-control’ state to the ‘out-of-control’ state after certain

time due to continuous usage of the machine. The proposed

model is developed to study the optimal production when a

certain percent of total product is of imperfect quality.

These items are reworked to maintain the quality of the

products. The production cost per unit item is convex

function of production rate. The demand is assumed to be

decreasing function of time. The total cost of inventory

system per time unit is minimized. The development of the

model is validated by numerical examples and its sensi-

tivity analysis is carried out too.

Keywords Imperfect production process �Defective items � In-control-state � Out-of-control state �Declining demand

1 Introduction

The flexible manufacturing system helps the manufacturing

industries to reduce the rate of production to avoid piling of

inventories and defective items. The continuous usage of

machine for production decreases the quality of items. It is

observed that the percentage of defective items increases

with increase of production rate and production-run time.

At the start of production, system is ‘in-control’ state and

items produced are of 100% quality. After some time, it

may shift to an ‘out-of control’ state producing defective

items.

Rosenblatt and Lee [18] considered the time of shift

from ‘in-control’ state to ‘out-of-control’ state and this

follows an exponential distribution with a mean 1l, assum-

ing l to be very small. They derived closed form solution

for Economic Manufacture quantity using second order of

Maclaurin series expansion of the exponential function.

Lee and Rosenblatt [12, 13] derived an optimal production

run-time and optimal inspection policy simultaneously to

observe the production process. Cheng [2] has derived a

closed form expression for the optimal demand to satisfy

order quantity and process reliability while the demand

exceeds supply and the production process is imperfect.

Khouja and Mehrez [5] assumed that the elapsed time until

the production process shifts to an ‘out-of-control’ state to

be an exponentially distributed random variable. Hariga

and Ben-Daya [4] and Kim and Hong [6] have extended

Rosenblatt and Lee [18] model by considering the general

time required to shift distribution and an optimal produc-

tion run time shown to be unique. Makis [16] derived

several properties of the optimal production and inspection

policies in an imperfect production process. Some of the

above cited references assumed that defective items are

N. H. Shah (&)

Department of Mathematics, Gujarat University,

Ahmedabad, India

e-mail: [email protected]

K. T. Shukla

JG College of Computer Application, Drive-in road,

Ahmedabad, Gujarat, India

123

Int. J. Mach. Learn. & Cyber. (2010) 1:89–99

DOI 10.1007/s13042-010-0005-9

reworked instantaneously. Wang [23] discussed an imper-

fect Economic Manufacturing Quantity model for produc-

tion which is required and sold under free-repair warranty

policy. Sheu and Chen [22] derived a lot size model to

determine the level preventive maintenance for an imper-

fect production control. Lee [7–9] has extended the model

to increase the service level and reduce the defective items

in imperfect production system. Chen and Lo [1] formu-

lated an imperfect production system with allowable

shortage and items are sold with free minimal repair war-

ranty. The probabilities of defective items in both the states

(out-of-control and in-control) are considered to be dif-

ferent. Lee [10] investigated the investment model with

respect to repetitive inspections and measurement equip-

ment in imperfect production process. Sana et al. [20]

analyzed Economic Production Lot-size model for pro-

duction system producing items of perfect as well as

imperfect quality. The probability of imperfect quality

items increases with increase of production run time

because of continuous usage of machines. They assumed

that the demand rate of perfect quality item is constant

whereas the demand rate of defective items which is not

repaired is a function of reduction rate. Sana et al. [21]

derived a volume flexible inventory model with an

imperfect production system where demand rate of good

quality items is a random variable and the demand rate of

defective items is a function of a random variable and

reduction rate. Giri and Dohi [3] developed the inspection

scheduling in an imperfect production process when

machine shifts from ‘in-control’ state to an ‘out-of-control’

state. They assumed that the shift time follows an arbitrary

probability distribution with increasing failure rate and the

products are sold with a free minimal repair warranty. The

inspection during production period reduces the number of

imperfect quality products. Liao [14] analyzed an imper-

fect production processes that requires production correc-

tions and imperfect maintenance. Lo et al. [15] extended a

production-inventory model for a varying rate of deterio-

ration, partial back-ordering, inflation, imperfect produc-

tion process and multiple deliveries. Panda et al. [17]

formulated an economic production lot size model for

imperfect items in which production rate is constant and

the demand rate is probabilistic under certain budget and

shortage constraints. They have assumed that the percent-

age of defective items is stochastic and the nature of

uncertainty in the constraints is stochastic or fuzzy. Here,

the percentage of defective item is independent of pro-

duction rate and production-run-time. Lee [11] developed a

maintenance model in multi-level multi-stage system. In

this model, the investment in preventive maintenance is to

reduce the variance and the deviation of the mean from

the target value of the quality that reduce the proportion

of defectives and increase reliability of the product.

The proportion of defectives can be linked to the cost of

manufacturing, cost of inventory and loss of profit. The

total cost comprises of the cost of manufacturing, set-up

cost, holding cost, loss of profit and warranty cost. Sana

[19] derived model with assumptions that the percentage of

defective items varies with production rate and production-

run-time, and demand to be deterministic and constant.

The most prevailing scenario of recession is studied in

this paper. It is observed that the demand had decreased

during the recession period. The model assumes that the

percentage of defective items varies non-linearly with

production rate and production-run-time. The production

starts with a variable production rate up to an optimal time.

During production period, the machine may shift to an

‘out-of-control’ state after certain time that follows expo-

nential distribution. During this ‘out-of-control’ state, the

defective items are produced which are to be reworked

immediately. The demand is considered to be decreasing

function of time. The profit of an inventory system is

maximized with respect to production rate and maximum

produced items. The proposed model is validated by

numerical examples. The sensitivity analysis is carried out

to determine critical model parameters.

2 Assumptions and notations

The following assumptions and notations are considered to

develop the model.

2.1 Assumptions

1. The system under consideration deals with a single

item.

2. The rate of production is a decision variable.

3. The demand rate is decreasing function of time.

4. Lead time is zero.

5. Shortages are not allowed.

6. The production process shifts from the ‘in-control’

state to an ‘out-of-control’ state. During this shift, the

defective items are produced which require rework at a

cost immediately.

7. The planning horizon is infinite.

2.2 Notations

P Production rate in units per year (a decision

variable)

Im Production lot size in units

T (t1 ? t2) cycle time in years (a decision

variable)

t1 Production time period in year (a decision

variable)

90 Int. J. Mach. Learn. & Cyber. (2010) 1:89–99

123

t2 Non-production time period in year (a decision

variable)

A Set-up cost per set up

h Inventory holding cost per unit per year

Cr Cost of rework of one unit

R(t) (=a(1 - bt)), demand rate a [ 0 is scale

parameter of the demand and 0 \ b \ 1 is

the rate of change of demand

s An exponential random variable that depends

on P and denotes the elapsed time until the

process shifts to the ‘out-of-control’ state

k(t, s, P) Percentage of defective items produced at any

instant of time t, When the machine is in the

out-of-control state

C(P) Unit production cost as a function of the

production rate1

f ðPÞ The mean of the s where f(P) is an increasing

function of P

N The number of defective items produced

during the production period

EK Integrated expected total cost per unit time

3 Mathematical model

The production starts with the rate P at time t = 0 and

continues up to time t = t1, the maximum inventory Im is

reached. During the time [0, t2], the inventory depletes due

to the demand and reaches to zero at time t2 (See Fig. 1)

The differential equations governing inventory levels

are

dI1ðtÞdt¼ P� RðtÞ; 0� t� t1 ð1Þ

and

dI2ðtÞdt¼ �RðtÞ; 0� t� t2 ð2Þ

with initial condition I1(0) = 0 and boundary I1(t1) =

I2(0) = Im and I2(t2) = 0.

The solutions of differential equations are,

I1ðtÞ ¼ ðP� aÞt þ abt2

20� t� t1 ð3Þ

and

I2ðtÞ ¼ aðt2 � tÞ � ab

2t22 � t2� �

0� t� t2: ð4Þ

Clearly, Inventory level at any instant of time t can be

defined as,

IðtÞ ¼ I1ðtÞ; 0� t� t1I2ðtÞ; 0� t� t2

� �:

Using I1 (t1) = Im, we get the maximum produced units

are,

Im ¼ ðP� aÞt1 þabt2

1

2: ð5Þ

Since, I(t) is continuous function of t, we have

I1(t1) = I2(0) which gives the relation between time

periods t1 and t2 as

t1 �at2

P� a� abt2

2

2ðP� aÞ ð6Þ

and hence

T ¼ t1 þ t2 ¼Pt2

P� a� abt2

2

2ðP� aÞ: ð7Þ

The produced units are of perfect quality in the

beginning of the production process. The system remains

‘in-control’ state up to a certain time s, after which the

production process shift to an ‘out-of-control’ state. The

production rate of defective items is k(t, s, P) percent of

production rate P.

Define k t; s;Pð Þ ¼ aPb t � sð Þc ð8Þ

where b, c C 0 and t C s. Due to long usage of machine,

the percentage of defective items increases with increase of

production rate and production time. So k(t, s, P) is well

justified. The total defective items during [0, s] is zero and

during [s, t1] is,

N ¼ P

Zt1

s

kðt; s;PÞdt ¼ acþ 1

Pbþ1ðt1 � sÞcþ1: ð9Þ

Therefore, the total defective items during [0, t1] is,

N ¼ 0 if t1� sa

cþ1Pbþ1ðt1 � sÞcþ1

if t1� s

�ð10Þ

Consider, the distribution function of ‘out-of-control’

state to be GðsÞ ¼ 1� e�f ðPÞs such thatR1

0dGðsÞ ¼

f ðPÞR1

0e�f ðPÞsds ¼ 1:

In general, the exponential distribution is used to

describe the elapsed time to failure of machineFig. 1 Time-Inventory status

Int. J. Mach. Learn. & Cyber. (2010) 1:89–99 91

123

components. The mean time to failure, 1f ðPÞ is a decreasing

function of production rate, P. Therefore, the expected

number of defective items in a maximum production lot

size, Im is

EðNÞ ¼ acþ 1

Pbþ1

Zt1

0

ðt1 � sÞcþ1f ðPÞe�f ðPÞsds

¼ acþ 1

Pbþ1f ðPÞe�f ðPÞt1X1

n¼0

tnþcþ21 ðf ðPÞÞn

n!ðnþ cþ 2Þ: ð11Þ

The rework cost, CRW, of unit item is,

CRW ¼ Cr � EðNÞ

¼ Cra

cþ 1Pbþ1f ðPÞe�f ðPÞt1

X1

n¼0

tnþcþ21 ðf ðPÞÞn

n!ðnþ cþ 2Þ: ð12Þ

The production cost of unit item is [19]

CðPÞ ¼ Cm þg

Pþ gPd ð13Þ

where Cm is the material cost per unit item, g is the total

labor/energy cost per unit time of a production system

which is equally distributed over the unit time, so, the

second cost component decreases with increase in the

production rate. The third cost component in C(P) is due to

component failure of a machine which is proportional to

the positive powers of the size of production rate; P. Here,

many external and internal factors are responsible for

increasing cost of production. It is obvious that propor-

tional changes of fixed factors with variable factors are not

possible. This heterogeneous combination between fixed

and variable factors may increases the cost of production.

That is why we get the cost function to be convex.

The inventory holding cost per cycle is,

IHC ¼ h

Zt1

0

I1ðtÞdt þZt2

0

I2ðtÞdt

2

4

3

5: ð14Þ

The production cost of Im-units per cycle is, PC ¼CðPÞ � Im and set up cost, OC ¼ A.

The total expected cost per unit time is,

EKðt2;PÞ ¼1

TPC þ IHC þ OC þ CRW½ � ð15Þ

Lemma 1 If oEK

ot2¼ 0 ¼ oEK

oP has a positive root t�2 and P*

(P* [ R(T*)) such that the Hessian matrix

o2EK

o2t22

o2EK

oPot2

o2EK

oPot2o2EK

oP2

��

��at

(t�2, P*) is positive definite then EK(t�2, P*) attains minimum

value at (t�2, P*).

In the next section, we demonstrate the validity of the

model and special cases using numerical values for the

model parameters.

4 Numerical examples

Example 1 Consider the values of the model parameters

in appropriate units as follows:

a = 0.05, b = 0.25, c = 1.5, d = 1, Cm = $20, g =

$2 9 105, g = $2 9 10-4, a ¼ 300 units, b = 0.05, h =

$2.0, A = $200, Cr = $100, f(P) = 0.25 ? 0.0005P. The

optimal production of P* = 610.77 units results maximum

inventory level I�m = 75.25 units, total production cost

C(P*) = $347.52 per unit and integrated expected total cost

E�K = $53438.29 per unit time.

The convexity of expected total cost per unit time is

exhibited in Fig. 2.

Example 2 Taking c = 0.0 and other parametric values as

in Example 1, the optimal production of P* = 613.05 units

results maximum inventory level I�m = 42.53 units, total

production cost C(P*) = $346.30 per unit and integrated

expected total cost E�K = $53782.04 per unit time.

Fig. 2 The convexity of the total cost for Example 1

Fig. 3 The convexity of the total cost for Example 2


123



Example 3 Here take b = 0.0 in Example 2, the optimal

production of P* = 611.91 units results maximum inven-

tory level I�m = 58.71 units, total production cost

C(P*) = $346.91 per unit and integrated expected total

cost E�K = $53579.08 per unit time.



5 Sensitivity analysis

From Tables 1, 2, 3 which exhibit the sensitivity analysis

of the Example 1, Example 2 and Example 3 respectively,

it is observed that the optimal production rate P*,Fig. 4 The convexity of the total cost for Example 3

Table 1 Sensitivity analysis for Example 1

Parameters Variations in percentage P* I�m C(P*) RWC* E�K

c -20 0.03886 -4.22553 -0.03660 65.76346 0.02101

-10 0.01774 -1.92406 -0.01671 28.86438 0.00892

10 -0.01480 1.59957 0.01394 -22.52620 -0.00662

20 -0.02707 2.92237 0.02551 -40.06425 -0.01156

g -20 -0.00370 -0.01286 -0.00021 -0.02592 -0.00368

-10 -0.00185 -0.00643 -0.00011 -0.01296 -0.00184

10 0.00185 0.00643 0.00011 0.01296 0.00184

20 0.00370 0.01286 0.00021 0.02593 0.00368

d -20 -0.01325 -0.04614 -0.00019 -0.09323 -0.01261

-10 -0.00863 -0.03009 0.00001 -0.06087 -0.00808

10 0.01608 0.05634 -0.00069 0.11433 0.01439

20 0.04603 0.16169 -0.00311 0.32893 0.04005

a -20 -0.01314 1.43543 0.01238 -15.92528 -0.00494

-10 -0.00643 0.70355 0.00606 -7.77349 -0.00245

10 0.00617 -0.67726 -0.00581 7.42497 0.00240

20 0.01210 -1.33004 -0.01140 14.52791 0.00477

b -20 -0.01882 1.99751 0.01774 -22.25690 -0.00682

-10 -0.00991 1.05129 0.00934 -11.65293 -0.00364

10 0.01095 -1.16086 -0.01032 12.71026 0.00416

20 0.02297 -2.43443 -0.02163 26.46500 0.00889

g -20 0.66379 8.63542 -19.34224 28.69090 -18.75391

-10 0.29117 4.05333 -9.66880 13.01161 -9.37750

10 -0.23308 -3.62231 9.66476 -10.90083 9.37809

20 -0.42338 -6.88629 19.32597 -20.11418 18.75645

h -20 -0.02955 1.60256 0.02785 5.77598 -0.01413

-10 -0.01463 0.79326 0.01379 2.82997 -0.00704

10 0.01435 -0.77766 -0.01352 -2.71957 0.00698

20 0.02842 -1.54013 -0.02677 -5.33408 0.01391


123

Table 1 continued


Cr -20 -0.01314 1.43543 0.01238 -15.92528 -0.00494

-10 -0.00643 0.70355 0.00606 -7.77349 -0.00245

10 0.00617 -0.67726 -0.00581 7.42497 0.00240

20 0.01210 -1.33004 -0.01140 14.52791 0.00477

A -20 0.12607 -9.31467 -0.11862 -29.03310 -0.07983

-10 0.06101 -4.49568 -0.05744 -14.90082 -0.03893

10 -0.05757 4.22150 0.05427 15.60979 0.03725

20 -0.11218 8.20689 0.10581 31.87747 0.07305

a -20 -20.51585 -18.34531 24.31756 -25.03225 -1.17078

-10 -10.29123 -9.06012 10.80776 -12.55677 -0.58652

10 10.35848 8.83925 -8.84257 12.52782 0.58874

20 20.78484 17.46380 -16.21119 24.93768 1.17970

b -20 0.18859 6.87811 -0.17734 25.06237 -0.05151

-10 0.09158 3.30430 -0.08619 11.55426 -0.02541

10 -0.08674 -3.06188 0.08179 -9.93713 0.02476

20 -0.16917 -5.90553 0.15964 -18.52918 0.04891



g -20 -0.017855 -0.033441 -0.000332 -0.029104 -0.016992

-10 -0.017703 -0.033160 -0.000291 -0.028864 -0.016809

10 -0.017398 -0.032598 -0.000208 -0.028384 -0.016443

20 -0.017246 -0.032317 -0.000167 -0.028144 -0.016261

d -20 -0.013653 -0.025609 0.000098 -0.022330 -0.012645

-10 -0.008893 -0.016696 0.000201 -0.014574 -0.008100

10 0.016586 0.031212 -0.001048 0.027329 0.014440

20 0.047484 0.089484 -0.004151 0.078506 0.040186

a -20 -0.031247 8.445855 0.029441 -6.125451 -0.105760

-10 -0.014240 3.972625 0.013414 -2.809892 -0.051840

10 0.011980 -3.557894 -0.011282 2.409946 0.049979

20 0.022090 -6.767805 -0.020802 4.498396 0.098284

b -20 -0.057644 12.140664 0.054326 -9.033099 -0.147467

-10 -0.173552 5.724175 0.163752 -4.636057 -0.078137

10 0.027528 -5.951576 -0.025921 3.997236 0.085928

20 0.053853 -11.762542 -0.050696 7.630542 0.180844

g -20 0.700337 3.370253 -19.365241 5.225459 -18.606507

-10 0.309331 1.612098 -9.682018 2.543935 -9.304268

10 -0.250537 -1.495417 9.680843 -2.416853 9.305693

20 -0.457360 -2.894542 19.360553 -4.716087 18.612409

h -20 -0.009686 0.590275 0.009124 1.183907 -0.007935

-10 -0.004825 0.293823 0.004545 0.588455 -0.003962

10 0.004789 -0.291234 -0.004511 -0.581582 0.003950

20 0.009543 -0.579917 -0.008987 -1.156411 0.007888


123

Table 2 continued


Cr -20 -0.031247 8.445855 0.029441 -6.125451 -0.105760

-10 -0.014240 3.972625 0.013414 -2.809892 -0.051840

10 0.011980 -3.557894 -0.011282 2.409946 0.049979

20 0.022090 -6.767805 -0.020802 4.498396 0.098284

A -20 0.097281 -10.657695 -0.091540 -20.051804 -0.141504

-10 0.047296 -5.183826 -0.044527 -10.028601 -0.068745

10 -0.045002 4.936479 0.042406 10.033831 0.065320

20 -0.088017 9.658776 0.082976 20.072747 0.127673

a -20 -20.540854 -8.647640 24.349583 -11.063052 -1.329888

-10 -10.302718 -3.943858 10.818869 -5.043802 -0.667348

10 10.366885 3.327932 -8.847155 4.236865 0.671788

20 20.797728 6.154988 -16.215998 7.809488 1.347720

b -20 0.129412 2.300774 -0.121735 4.431372 -0.027214

-10 0.064040 1.134874 -0.060280 2.174710 -0.013561

10 -0.062757 -1.105032 0.059148 -2.096708 0.013471

20 -0.124281 -2.181353 0.117205 -4.119142 0.026855



g -20 -0.0037361 -0.0116700 -0.0001882 -0.0142017 -0.0036869

-10 -0.0018682 -0.0058356 -0.0000940 -0.0071018 -0.0018434

10 0.0018684 0.0058367 0.0000940 0.0071036 0.0018435

20 0.0037369 0.0116744 0.0001879 0.0142087 0.0036871

d -20 -0.0133635 -0.0418828 -0.0001054 -0.0510709 -0.0126247

-10 -0.0087035 -0.0273164 0.0000679 -0.0333373 -0.0080862

10 0.0162274 0.0511334 -0.0007943 0.0625540 0.0144091

20 0.0464468 0.1467506 -0.0034168 0.1798232 0.0400918

a -20 -0.0264114 3.6817106 0.0248873 -14.1230308 -0.0321126

-10 -0.0127898 1.7941768 0.0120501 -6.8061943 -0.0159178

10 0.0120242 -1.7079141 -0.0113259 6.3450690 0.0156538

20 0.0233422 -3.3359610 -0.0219843 12.2725382 0.0310558

g -20 0.6738871 7.6283845 -19.3486762 13.6202846 -18.7187970

-10 0.2962368 3.5790528 -9.6725313 6.3604290 -9.3600792

10 -0.2380083 -3.2000678 9.6693450 -5.6214733 9.3609094

20 -0.4329949 -6.0880563 19.3358575 -10.6279412 18.7223108

h -20 -0.0239357 1.4226033 0.0225539 2.8411414 -0.0124423

-10 -0.0118539 0.7038729 0.0111682 1.4008872 -0.0061992

10 0.0116336 -0.6895144 -0.0109582 -1.3630987 0.0061561

20 0.0230542 -1.3651467 -0.0217131 -2.6899056 0.0122701

Cr -20 -0.0264114 3.6817106 0.0248873 -14.1230308 -0.0321126

-10 -0.0127898 1.7941768 0.0120501 -6.8061943 -0.0159178

10 0.0120242 -1.7079141 -0.0113259 6.3450690 0.0156538

20 0.0233422 -3.3359611 -0.0219843 12.2725382 0.0310558

A -20 0.1262851 -10.4830477 -0.1188161 -19.6392020 -0.0910260

-10 0.0613815 -5.0934877 -0.0577886 -9.8023893 0.0012490

10 -0.0583764 4.8408962 0.0550254 9.7695967 0.0420761

20 -0.1141496 9.4629785 0.1076570 19.5078006 0.0822750


123

maximum produced units I�m and total expected cost per

unit time E�K are sensitive to changes in the parameters g, g,

d, a, b, c, h, Cr, A, a and b. From the Tables 1, 2, 3 and

Figs. 5, 6, 7, 8, 9, 10, 11, 12, 13, the following managerial

issues can be deducted.

1. I�m and P* increase and E�K decrease with increase in c(gamma)

2. I�m, P* and E�K increase with increase in g, d, a. The

model is very sensitive to index parameter d. It is

nearly equal to 1 ([20, 21])

3. I�m and P* decrease while E�K increases with the

increase in b.

4. Increase in labor/energy cost (g) decreases I�m, P* and

E�K significantly. The production of imperfect quality

units decrease with decrease in production rate and

hence, rework cost decreases. It also lowers production

run time.

5. I�m, P* decreases with the increase in the inventory

holding cost. It is obvious that with higher inventory

holding cost reduces the production rate and increases

total cost E�K significantly.

6. P*, E�K increase and I�m decreases with the increase in

rework cost Cr. Although the production-run-time

decreases with the increase in Cr, the number of

Table 3 continued


a -20 -20.5280486 -16.0493690 24.3343898 -20.1731259 -1.2065348

-10 -10.2977932 -7.7852002 10.8148414 -9.9217730 -0.6047884

10 10.3657322 7.3273962 -8.8476869 9.5414018 0.6077508

20 20.7997994 14.2204200 -16.2199455 18.6688770 1.2183911

b -20 0.1735207 5.9825841 -0.1631811 11.7474856 -0.0447183

-10 0.0846693 2.8725110 -0.0796948 5.5647964 -0.0221006

10 -0.0809176 -2.6633577 0.0762898 -5.0339261 0.0216191

20 -0.1584504 -5.1414595 0.1495043 -9.6084985 0.0427887

-0.1

0

0.1

0.2

0.7 0.8 0.9 1 1.1

P1 IM1 EK1 P2 IM2 EK2 P3 IM3 EK3

Fig. 6 Sensitivity analysis for d

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.00014 0.00016 0.00018 0.0002 0.00022 0.00024


Fig. 5 Sensitivity analysis for g


123

-25

-5

15

250000200000150000


Fig. 8 Sensitivity analysis for g

-8

-6

-4

-2

0

2

4

6

8

10

0.039 0.044 0.049 0.054 0.059


Fig. 7 Sensitivity analysis for a

-1.75

-0.75

0.25

1.25

0.79 0.84 0.89 0.94 0.99 1.04 1.09 1.14 1.19


Fig. 9 Sensitivity analysis for h

-7.5

-3.5

0.5

4.5

8.5

01108


Fig. 10 Sensitivity analysis

for Cr


123

imperfect quality items goes higher due to higher

production rate and that results in higher rework cost.

7. P* decreases and I�m and E�K increase with the increase

in set up cost A.

8. The increase in scale demand ‘a’ encourages the

manufacturer to increase the production rate P*, I�m.

Higher production rate increases the total cost of an

inventory system.

9. P*, I�m decreases with the increase in demand rate ‘b’

and increase in total cost E�K

The Changed optimal values in the decision variables

and the objective function are shown in Figs. 5, 6, 7, 8, 9,

10, 11, 12, 13.

6 Conclusion

The prevailing scenario of recession which resulted

decrease in the demand motivated to analyze the proposed

problem. The real life problem of manufacturing unit

-25

-15

-5

5

15

25

240 270 300 330 360



for a

-7

-5

-3

-1

1

3

5

7

0.04 0.042 0.044 0.046 0.048 0.05 0.052 0.054 0.056 0.058 0.06



for b

-12

-2

8

11080



for A


123

producing imperfect quality items due to continuous usage

of machine is considered. It is observed that during ‘out-

of-control’ state, the defective items are produced. The

probability of defective items increases with increase of

production-run-time which in turn will increase rework

cost. The proposed model can be used by the decision

maker to reduce the total cost of manufacturing process

when demand is decreasing with time.

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Optimal production schedule in declining market for an imperfect production system