EPQ MODELS WITH MIXTURE OF WEIBULL PRODUCTION EXPONENTIAL DECAY AND CONSTANT DEMAND Текст научной статьи по специальности «Промышленные биотехнологии»

EPQ MODELS WITH MIXTURE OF WEIBULL PRODUCTION EXPONENTIAL DECAY AND CONSTANT DEMAND

V. Sai Jyothsna Devi1, K. Srinivasa Rao2

Department of Statistics1, 2, Andhra University, Visakhapatnam, India [email protected], [email protected]

Abstract

This paper deals with an economic production quantity (EPQ) model in which production is random and having heterogeneous units of production. The production process is characterized by mixture of Weibull distribution. It is assumed that the demand is constant and the lifetime of the commodity is random and follows an exponential distribution. Assuming that the shortages are allowed and fully backlogged the instantaneous state of inventory in the production unit is derived. The minimizing the expected total production cost, the optimal production quantity, the production uptime and downtime are derived. Through sensitivity analysis it is observed that the random production with mixture distribution have a significant influence on the optimal production schedules and production quantity. It is also observed that the rate of deterioration can tremendously influence the optimal operating policies of the system. This model also includes some of the earlier models as particular cases. The model is extended to the case of without shortages. A comparison of the two models reveals that allowing shortages will reduce expected total cost of the model.

Keywords: Stochastic production, Mixture of Weibull Distribution, Exponential decay, Production Schedules, Sensitivity analysis.

I. Introduction

In production quantity models much emphasis is given for the lifetime of the commodity. In many production processes the lifetime of the commodity is random and can be characterized by a probability distribution. The literature on inventory models for deteriorating items are reviewed by Pentico and Drake (2011), Ruxian Lie et al (2010), Goyal and Giri (2001), Raafat (1991) and Nahmias (1982). The exponential decay models of inventory are studied by Ghare and Schrader (1963), Shah and Jaiswal (1977), Cohen (1977), Aggarwal (1978), Dave and Shah (1982), Pal (1990), Kalpakam and Sapna (1996), Giri and Chaudhari (1999). The exponential decay is used when the rate of deterioration is constant which coincide with the deterioration of several perishable items such as medicine, sea foods, vegetable oils, cement and paints. Hence it is reasonable to assume exponential decay of the product.

Another important consideration in EPQ models is the rate of production and it is studied by several authors Perumal and Arivarignan (2002), Pal and Mandal (1997), Sen and Chakrabarthy (2007), Lin and Gong(2006), Maity et al(2007), Hu and Liu(2010), Uma Maheswararao et al (2010), Venkata Subbaiah et al (2011), Essey and Srinivasa Rao (2012), Ardak and Borade (2017), Anindya Mandal, Brojeswar Pal and Kripasindhu Chaudhuri (2020), Sunit Kumar, Sushil Kumar and

Rachna Kumari (2021). In all these papers they assumed that the production is deterministic and having finite rate. However, in many production processes the production is not deterministic and random.

Stochastic production is a reality in the modern technological industrial developments. One of the major consideration for scheduling the production and determining the optimal production quantity lies on several factors such as availability of raw material, power supply, man power skill level, machine tool wear which are governed by laws of chance and become stochastic. Because of the stochastic factors the production process in many industries is random and can be characterized by a probability distribution.

Recently Sridevi et al. (2010), Srinivasa Rao et al. (2010), Laxmana Rao et al. (2015), Srinivasa Rao et al. (2017), Madhulatha et al. (2017), Punyavathi et al. (2020) have developed and analyzed production quantity models with random production. In all these papers they assumed that the production is homogeneous even though governed by stochastic nature i.e., all the production is done in one unit or in a single machine. But in practice several of the products are produced by different machines or in different units which are operated under different conditions. Hence these heterogeneous production processes can be characterized by mixes of probability distributions. It is also observed that in each unit the production rate may be increasing/decreasing/remains constant. This type of variable rate of production can be represented by Weibull probability distribution. Hence in this paper we develop and analyze stochastic production quantity models assuming that the production is random and follows a two component Weibull mixture distribution. It is also further assumed that the demand is constant and in the production backorders are allowed and fully backlogged.

Using the differential equations the production quantity at a given time is derived. With suitable costs the total expected production cost is derived. By minimizing the total expected production cost the optimal production schedules, the production quantities are derived. Through sensitivity analysis the effect of the change in parameters and cost on optimal production schedules and production quantity is discussed. This model is extended to the case of without shortages.

II. Assumptions For developing the model the following assumptions are made:

• The demand rate is constant say k. (1)

• The production is random and follows a mixture of two-parameter Weibull distribution. The instantaneous rate of production is:

pa1B1t"1~1e -ait"1 + (1 -p)a2B2t"#-1e~a#t"2

R(t)=^-±-"---F 2^2 "-;ava2 >0,p1,p2 >0,0<p<1 (2)

pe-ait"1 + (1 - v)e-a2%"2 ; 1 2 2 '

Lead time is zero.

Cycle length is 7TIt is known and fixed. Shortages are allowed and fully backlogged. A deteriorated unit is lost.

The lifetime of the item is random and follows a exponential distribution with probability density function:

f(t) = ee-6t; e>0,t>0 Therefore the instantaneous rate of deterioration is

h(t) = e; e > 0 (3)

The following notations are used for developing the model. Q is the production quantity A i s setup cost C is cost per unit

h Inventory holding cost per unit per unit time n Shortages cost per unit per unit time

III. EPQ Model with Shortages

Consider a production system in which the stock level is zero at time t = 0. The stock level increases during the period (0, tx), due to production after fulfilling the demand and deterioration. The production stops at time tx when stock level reaches S. The inventory decreases gradually due to demand and deterioration in the interval (t^ t2). At time t2 the inventory reaches zero and backorders accumulate during the period (t2, t3). At time t3 the production again starts and fulfills the backlog after satisfying the demand. During (t3, T) the backorders are fulfilled and inventory level reaches zero at the end of cycle T. The Schematic diagram representing the inventory level is given in Figure 1.

Figure 1: Schematic Diagram representing the inventory level

Let I(t) be the inventory level of the system at time 't' (0 < t < T). The differential equations governing the instantaneous state of I(t) over the cycle of length are:

d pa1ß1t"i-1e-aitßl + (1 - p)a2ß2t"#~1e~a#tß2

—I(t) + h(t)I(t) -ß-^-" --k;0<t<t1

dt w w pe~aitßl + (1 - p)e-a2tß2 ' 1

d

— I(t) + h(t)I(t) = -k' t1 < t < t2 dt

d

—/(t) = -k; t& <t< t3

d paißitß1-1e-aitß1 + (1-p)a2ß2t"2-1e-a2tß2

tuO =-ß-ß--k' t3 < t <T

dt pe-aitß1 + (1 - p)e-a2tß2 3

( )

(5)

(6) (7)

Where, h(t) is as given in equation (3), with the initial conditions 1(0) = 0,1(tx) = S, I(t2) = 0 and I(T) = 0

Solving the differential equations, the on hand inventory at time T is obtained as:

rti

I(t) = Se6(t!~t) - e—t' f

I(t) = See(t!~t) - ke—te f

t

pa1p1u"1—1e—aiuPi + (1 - p)a2p2u"'2—1e

"2-lp-a2u"2

—a1u"i

pe

eu' du; t1 <t<t2

+ (1 -p)e~

a2

u"2

■-k

I(t) = k(t& -t);t& <t< t3

rT paipit"i—1e—ait"1 + (1 - p)a2p2t"2—1e—a2t"2

eu'du; 0 < t < t1(8)

(9) (10)

/(t) = i

pe—ait"1 + (1 - p)e—a2t"2

dt + k(T -t);t3 <t<T

Production quantity Q in the cycle of length T 4 s :

Q=f R(t)dt + f R(t)dt

Jo 't$

i

=J!

pa1p1t"i—1e—ait"1 + (1 -p)a2p2t"2—1e—a2t"2

+

l J

pe—ait"1 + (1 -p)e—a2tP2

o

pa1p1t"i—1e—ait"1 + (1 - p)a2p2t"2—1e—a2t"2

dt

dt

pe-ait"1 + (1 - p)e-a2%"2 From equation (8) and using the initial condition I(0) = 0, we obtain the value of'S' as:

5 = "—et

i fti (P^1P1u"1—1e—aiuPi + (1-p)a2P2u"2—1e—a2u"2S^ k _ )

Jo V pe—aiu"1 + (1 - p)e—a2u"2 ) eT U

When t = t3, then equations (10) and (11) become:

I(t3) = k(t2 - t3) and

I(t() = k(T - t() -

T pa1p1u"i—1e—aiu"1 + (1 - p)a2p2u"2—1e—a2u"2

pe—aiu"1 + (1 -p)e~

du

Equating the equations (14) and (15) and on simplification one can get:

=1f

k L

pa1p1u"i—1e—aiu"1 + (1 - p)a2p2u"2—1e—a2u"2

pe

a!u

Pi

+ (1 - p)e

—a2up2

du + T = x(t3) say

(11)

(12)

(13)

(1 )

(15)

(16)

Let KTt1t2, t3) be the total production cost per unit time. Since the total production cost is the sum of the set up cost, cost of the units, the inventory holding cost. Hence the total production cost per unit time become:

K{ti,t2, t()= 0 + T + T [Jo1 №dt + £ I(t)dt] + 4 [j£ -I(t)dt + J, -I(t)dt\

(17)

Substituting the values of I(t) given in equations (8), (9), (10) and (11) and given in equation (12) in equation (17) one can obtain KTt1t2, t3U as:

t

2

A C

K(ti,t2,t3)=- + -

M

K

paipit"1~1e~$lt"1 + (1 - p)a2p2t"#~1e~a#t"2 pe-ait"1 + (1 - p)e-a2t"2

paipit"i~1e~ait"1 + (1 - p)a2p2t"2-1e-a2t"2 pe-ait"1 + (1 - p)e-a#t"2

/pa1fi1u"i-1e-aiu"1 + (1 - p)a2p2u"2-1e-a2+"2

h + T

K

Se'(ti-t) - e-te

+

l

K

t$

ti K

-dt

dt

pe

-a 1 u" 1

+ (1 - p)e~

-a2u"2

-k \euddu

dt

+

2 K2

Se'(ti-t) - ke-te I eud du

K

dt

t r / t

k I(K

t-3 [_ \t

K

k I(t2 - t)dt

pa1p1u"i-1e-aiu"1 + (1 - p)a2p2u"2-1e-a2u"2

pe-aiu"1 + (1 -p)e~

a2

u"2

due dt

t

+ k j(T-t)

(18)

Substituting the value of S given in equation (13) in the total production cost equation (18), we obtain:

f paipit"i-1e-a1 + (1 - p)a2p2t"#-1e~a#t"2

A C

K(t1,t2,t3)=- + -

K

pe

-ait"1 + (1-p)e-a2t"2

■dt

l

K

pa1p1t"i-1e-ait"1 + (1 - p)a2p2t"2-1e-a2t"2

+ ' pe-ait"1 + (1 - p)e-a2t"2 dt

h + T

1 - e 't2 r pa1p1u"i-1e-aiu"1 + (1 - p)a2p2u"2-1e-a2u

K

"2

pe-aiu"1 + (1 - p)e-a2u'

-a2u"2

-euddu + — (1 - e-t28U

k -i

e

^[(T-t3)2 -(t2 -t3)2] +

K e-et ( K pa1P1U"1-1e-aiu"1 + (1- p)a2P2u"2-1e-a2u"2 ^ \

J6 (J pe-aiu"1 + (1 - p)e-a2u"2 6 Uj

t / T

K (K

dt

■ pa1p1u"i-1e-aiu"1 + (1 - p)a2p2u"2-1e-a2u"2

pe-aiu"1 + (1 - p)e-a2u

du I dt

(19)

Substituting the value of 't2' given in equation (16) in the total production cost equation (19), we obtain:

paipit"1-1e-$lt"1 + (1 - p)a2p2t"2-!e-a2t"2

A C

K(t1,t2,t3)=- + -

K

pe

-$i%"1 + (1 - p)e a2

t"2

-dt

+

l

K

pa1p1t"i-1e-ait"1 + (1 -p)a2p2t"2-1e-a2t"2

pe

-$i%"1 + (1 - p)e-a2

t"2

dt

h + T

1 - e-ex(t$) Ç%i pa1p1U"i-1e-aiu"1 + (1 - p)a2p2u"2-1e-a2u

j

Jo

u"2

0ud

pe-a1u"1 + (1 - p)e-a2u"2

du + e2 (1-e-dx(t$))

k

--x(t() -

e

f1 e—dt (f1 pa^u"1^—aiu"1 + (1- p)a2P2u"2—1e—a2u"2 ^^ ^

pe—aiuPi + (1 - p)e—a2u

^[(T-t()2 - (x(t() - t()2]

TT

f (f

t$ \t

' pa1p1u"i—1e—aiu"1 + (1 - p)a2p2u"2—1e—a2u"2 + 111 -s-s-du | dt

pe—aiuPi + (1 -p)e—a2

—a2up2

|

.ij I

(20)

IV. Optimal Production Schedules of the Model

In this section we obtain the optimal policies of the system under study. To find the optimal values of ti and fe, we obtain the first order partial derivatives of K(ti,t3) given in equation with respect to ti and fe and equate them to zero. The condition for minimization of Kftify))

Where D i s the Hessian matrix

D =

d2K(t1,t() d2K(t1,t()

dt1dt3

d2K(t1,t() d2K(t1,t()

dt,dt3

dt(2

> 0

Differentiating K(t1, t3) given in equation (20) with respect to 1 and equating to zero, we get

pa1p1t1"i—1e—aiti"1 + (1 - p)a2p2t1"2—1e—a2ti

P2

pe—aiti"1 + (1 -p)e—a2ti

P2

h

+ Y

1 - e—x(t$)e

pa1p1t1"i—1e—aitiPi + (1 - p)a2p2t1"2—1e—a2ti

P2

pe—aitiPi + (1 -p)e—a2ti

P2

= 0 (21)

I

ti

Differentiating K(t1, t3) given in equation (20) with respect to 3 and equating to zero, we get

pa1p1t3"i—1e—ait$Pi + (1 - p)a2p2t3"2—1e—a2t$

P2

h + T

pe—ait$Pi + (1 -p)e—a2t3P2

pa1p1t3"i—1e—ait$Pi + (1 - p)a2p2t3"2—1e—a2t$P2

pe ait3Pi + (1 -p)e—a2t$

P2

(1 - e—ex(t3))

e-0x(t3)

pa1"1t3Pi-1e-ait3Pi + (1—p)a2"2t$P2~1e~a2t3

P2

pe-ai)3Pi + (1—p)e-a2t3P2

J%!

o

pa1"1uPi-1e-aiuPi + (1—p)a2"2uP2-1e-a2uP2

pe-aiuPi + (1—p)e-a2uP2

eud du

k(t3 -T) + (x(t3) - t3)

pa1p1t3"i—1e—ait3Pi + (1 - p)a2p2t3"2—1e—a2t3

P2

pe—ait3Pi + (1 - p)e—a2t3

P2

+ k

V.Sai Jyothsna Devi and K.Srinivasa Rao

epq models with mixture of weibull production RT&A, No 4 (65) exponential decay and constant demand_Volume 16, December 2021

i

I

pa1ß1u"i-1e-ai+ßl + (1 - p)a2ß2u"2-1e-a2+ß2

pe

-aiu

ß!

+ (1 -p)e

-a2uß2

du

= 0

(22)

Solving the equations (21) and (22) simultaneously, we obtain the optimal time at which production is stopped t* of t1 and the optimal time t( of t3 at which the production is restarted after accumulation of backorders.

The optimum production quantity Q* of Q in the cycle of length T is obtained by substituting the optimal values of t*, t** in equation (12).

1

V. Numerical Illustration

In this section we discuss the solution procedure of the model through a numerical illustration by obtaining the production uptime, production downtime, optimum production quantity and the total production cost of an inventory system. Here, it is assumed that the production is of deteriorating nature and shortages are allowed and fully backlogged. For demonstrating the solution procedure of the model the parameters are considered as A = Rs.300/-, C = Rs.10\-, h = Re.0.2\-, n = Rs.3.3\-, T = 12 months. For the assigned values of production parameters (a, a2, Pi, P2, pp = (11, 15, 0.55, 2, 0.5), deterioration parameter e = 3, demand rate k = 3.3. The values of parameters above are varied further to observe the trend in optimal policies and the results are obtained are shown in Table 1. Substituting these values the optimal production quantity Q*, the production uptime, production downtime and total production cost are computed and presented in Table 1.

From Table 1 it is observed that the deterioration parameter and production parameters have a tremendous influence on the optimal values of production times, production quantity and total production cost.

When the ordering cost 'A ' increases from 300 to 345, the optimal production quantity Q* decreases from 33.867 to 33.863, the optimal production down time 11* remains constant, the optimum production uptime t3* increases from 3.685 to 3,686, the total production cost per unit time K* increases from 80.793 to 84.529. As the cost parameter ' C ' increases from 10 to 11.5, the optimal production quantity Q* increases from 33.867 to 33.872, the optimal production down time t^* and optimal production uptime t3* remains constant, the total production cost per unit time K* increases from 80.793 to 82.451. As the holding cost 'h ' increases from 0.2 to 0.23, the optimal production quantity Q*, the optimal production down time 11*, the optimal production uptime t3* remains constant, the total production cost per unit time K* decreases from 80.793 to 80.755. As the shortage cost ' n ' increases from 3.3 to 3.795, the optimal production quantity Q* increases from 33.867 to 33.966, the optimal production down time 11* remains constant, the optimal production uptime t3* decreases from 3.685 to 3.655, the total production cost per unit time K* increases from 80.793 to 87.753.

As the production parameter 'ai' varies from 11 to 12.65, the optimal production quantity Q* increases from 33.867 to 39.086, the optimal production down time 11* increases from 1.274 to 1.277, the optimal production uptime t3* decreases from 3.685 to 3.628, the total production cost per unit time K* increases from 80.793 to 93.146.As the production parameter 'a 'varies from 15 to 17.25, the optimal production quantity Q*, the optimal production down time t1*, the optimal production uptime t3*, the total production cost per unit time K* remains constant.

Table 1: Numerical Illustration

A C h n T a1 a2 ßi ß2 e k P Î3 Q* K

300 10 0.2 3.3 12 11 15 0.55 2 3 3.3 0.5 1.274 3.685 33.867 80.793

315 1.274 3.685 33.866 82.039

330 1.274 3.686 33.865 83.284

345 1.274 3.686 33.863 84.529

10.5 1.274 3.685 33.869 81.346

11 1.274 3.685 33.87 81.898

11.5 1.274 3.685 33.872 82.451

0.21 1.274 3.685 33.867 80.781

0.22 1.274 3.685 33.867 80.768

0.23 1.274 3.685 33.867 80.755

3.465 1.274 3.675 33.9 83.1

3.63 1.274 3.665 33.933 85.42

3.795 1.274 3.655 33.966 87.753

11.55 1.275 3.666 35.599 84.778

12.1 1.276 3.647 37.338 88.895

12.65 1.277 3.628 39.086 93.146

15.75 1.274 3.685 33.867 80.793

16.5 1.274 3.685 33.867 80.793

17.25 1.274 3.685 33.867 80.793

0.578 1.275 3.648 36.366 88.423

0.605 1.276 3.609 38.996 97.002

0.633 1.277 3.565 41.973 107.394

2.1 1.274 3.685 33.867 80.793

2.2 1.274 3.685 33.867 80.793

2.3 1.274 3.685 33.867 80.793

3.15 1.274 3.685 33.867 80.805

3.3 1.274 3.685 33.867 80.816

3.45 1.274 3.685 33.867 80.826

3.465 1.274 3.689 33.853 79.886

3.63 1.274 3.693 33.841 79.062

3.795 1.274 3.696 33.83 78.31

0.525 1.274 3.685 33.818 80.753

0.55 1.274 3.685 33.772 80.714

0.575 1.274 3.685 33.728 80.677

As the production parameter 2 ' varies from 0.55 to 0.633, the optimal production quantity Q* increases from 33.867 to 41.973, the optimal production down time 11* increases from 1.274 to 1.277, the optimal production uptime t3* decreases from 3.685 to 3.565, the total production cost per unit time K* increases from 80.793 to 107.394. As the production parameter '2 varies from 2 to 2.3, the optimal production quantity Q*, the optimal production down time t±*, the optimal production uptime t3* and the total production cost per unit time K* remains constant. As the production parameter p ' varies from 0.5 to 0.575, the optimal production quantity Q* decreases from 33.867 to 33.728, the optimal production down time t!* and the optimal production uptime t3* remains constant, the total production cost per unit time K* decreases from 80.793 to 80.677.

As the deterioration parameter 'e' varies from 3 to 3.45, the optimal production quantity Q*, the optimal production down time t!* and the optimal production uptime t3* remains constant, the total production cost per unit time K* increases from 80.793 to 80.826.

As the demand rate parameter 'k ' increases from 3.3 to 3.795 the optimal production quantity Q* decreases from 33.867 to 33.83, the optimal production down time t!* remains constant, the optimal production uptime t3* increases from 3.685 to 3.696, the total production cost per unit time K* decreases from 80.793 to 78.31.

VI. Sensitivity Analysis of the Model

Sensitivity analysis is carried to explore the effect of changes in model parameters and costs on the optimal policies, by varying each parameter (-15%, -10%, -5%, 0%, 5%, 10%, 15%) at a time for the model under study. The results are presented in Table 2. The relationships between the parameters and the optimal values of the production schedule are shown in Figure 2.

Table 2: Sensitivity Analysis of the Model - With Shortages

Variation Parameters Optimal Policies -15% -10% -5% 0% 5% 10% 15%

A ii* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.684 3.684 3.685 3.685 3.685 3.686 3.686

Q* 33.871 33.869 33.868 33.867 33.866 33.865 33.863

K* 77.058 78.303 79.548 80.793 82.039 83.284 84.529

C ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.684 3.685 3.685 3.685 3.685 3.685 3.685

Q* 33.862 33.864 33.866 33.867 33.869 33.87 33.872

K* 79.138 79.689 80.241 80.793 81.346 81.898 82.451

h ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.685 3.685 3.685 3.685 3.685 3.685 3.685

Q* 33.867 33.867 33.867 33.867 33.867 33.867 33.867

K* 80.832 80.819 80.806 80.793 80.781 80.768 80.755

n ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.715 3.705 3.695 3.685 3.675 3.665 3.655

Q* 33.766 33.8 33.834 33.867 33.9 33.933 33.966

K* 73.95 76.218 78.499 80.793 83.1 85.42 87.753

a1 ti* 1.271 1.272 1.273 1.274 1.275 1.276 1.277

£3* 3.74 3.722 3.704 3.685 3.666 3.647 3.628

Q* 28.718 30.427 32.143 33.867 35.599 37.338 39.086

K* 69.61 73.212 76.939 80.793 84.778 88.895 93.146

Variation Parameters Optimal Policies -15% -10% -5% 0% 5% 10% 15%

a2 ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.685 3.685 3.685 3.685 3.685 3.685 3.685

Q* 33.867 33.867 33.867 33.867 33.867 33.867 33.867

K* 80.793 80.793 80.793 80.793 80.793 80.793 80.793

Pi ti* 1.27 1.272 1.273 1.274 1.275 1.276 1.277

£3* 3.776 3.748 3.719 3.685 3.648 3.609 3.565

Q* 27.639 29.56 31.58 33.867 36.366 38.996 41.973

K* 63.978 68.832 74.257 80.793 88.423 97.002 107.394

Pi ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.685 3.685 3.685 3.685 3.685 3.685 3.685

Q* 33.867 33.867 33.867 33.867 33.867 33.867 33.867

K* 80.793 80.793 80.793 80.793 80.793 80.793 80.793

e ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.685 3.685 3.685 3.685 3.685 3.685 3.685

Q* 33.867 33.867 33.867 33.867 33.867 33.867 33.867

K* 80.749 80.766 80.78 80.793 80.805 80.816 80.826

k ti* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.67 3.676 3.68 3.685 3.689 3.693 3.696

Q* 33.917 33.899 33.882 33.867 33.853 33.841 33.83

K* 84.168 82.916 81.798 80.793 79.886 79.062 78.31

p h* 1.274 1.274 1.274 1.274 1.274 1.274 1.274

£3* 3.685 3.685 3.685 3.685 3.685 3.685 3.685

Q* 34.029 33.972 33.918 33.867 33.818 33.772 33.728

K* 80.929 80.881 80.836 80.793 80.753 80.714 80.677

VII. Observations

The major observations drawn from the numerical study are:

• ti*and t?* are less sensitive, Q*is slightly sensitive and Kis moderately sensitive to changes of ordering cost 'A'.

• ti*and t?* are less sensitive, QQ*is slightly sensitive and Kis moderately sensitive to changes of cost per unit 'C'.

• ti, t?* and Q* are less sensitive, A* is slightly sensitive to changes of holding cost 'h'.

• ti* is less sensitive, t3* and Q are slightly sensitive and K* is highly sensitive to change in parameter 'n'.

• ti* and tt* are slightly sensitive, Q* and Kare highly sensitive to change in the production parameter' al.

• ti, t3*, Q* and A* are less sensitive to change in the production parameter 'a2.

• ti*and t?* are slightly sensitive, Q*and Kare highly sensitive to change in the production parameter' pi.

• ti, t3*, Q* and A* are less sensitive to change in the production parameter 'p2.

• ti* and t3* are less sensitive, Q* and K are slightly sensitive to change in the production parameter 'p'.

• ti*, t3* and Q are less sensitive, A* is slightly sensitive to change in the deterioration parameter 'e'.

• ti*is less sensitive, tt*and Qare slightly sensitive and K*is highly sensitive to change in the demand parameter 'k'.

-15%-10%-5% 0% 5% 10% 15% Percentage Change in parameters

•A •C

h ■ji al a2

PI P2 e k P

-15% 10%-5% 0% 5% 10% 15%

Percentage change in parameters

■A ■C

h

■ji ■al a2

PI P2 e k P

»

E O ï .5

re >

50 40

20 10 0

-15%10%-5% 0% 5% 10% 15% Percentage change in parameters

■A

■ C

h ■it

■ al a2

PI P2 e k P

»

E O ï .5

RJ >

-15%10%-5% 0% 5% 10% 15% Percentage change in parameters

Figure 2: Relationship betzoeen parameters and optimal values with shortages

VIII. EPQ Model without Shortages

In this section the inventory model for deteriorating items without shortages is developed and analyzed. Here, it is assumed that shortages are not allowed and the stock level is zero at time t=0. The stock level increases during the period (0, t,) due to excess production after fulfilling the demand and deterioration. The production stops at time t, when the stock level reaches S. The inventory decreases gradually due to demand and deterioration in the interval (t1; T) . At time T the inventory reaches zero. The schematic diagram representing the instantaneous state of inventory is given in Figure 3.

i(t)

Figure 3: Schematic diagram representing the inventory level

Let I(t) be the inventory level of the system at time ' t' (0 < t < T). Then the differential equations governing the instantaneous state of 1(t) over the cycle of length Tare:

d pa1ß1tß1~1e~ait + (1 - p)«,/??^ e~

—/(t) + h(t)I(t) = i—ii-i--R---HJ 2H2 .--fc; 0 < t < t±

dt pe~ait + (1 — p)e~a2t

—I(t) + h(t)I(t) = -k; t. <t<T at

(24)

Where, h(t))s as given in equation (3), with the initial conditions I(0) = 0, /(t= S and I(T) = 0. Substituting h(t) given in equation (3) in equations (23) and (24) and solving the differential equations, the on hand inventory at time 't' is obtained as:

I(t) = Se6(ti-t) - e-t6

r

pa^u

"l

+(1 - p)a2p2w

P#-ip-a2u

"2

pe-aiuPl + (1 -p)e~

a2

-fc

eu6du; 0 < t < t±

I(t) = Se6(ti-t) - ke-t6 r

eu6 du;t1 <t<T

(25)

(26)

Production quantity QQ in the cycle of length T is

Q = J R(t)dt

=J

%i paipit"i-1e-aitPl + (1 -p)a2p2t"2-!e-a2t"2

dt

pe-aith + (i-p)e-$2t"2

From equation (25) and using the initial conditions I(0) = 0, we obtain the value of' S ' a s

(27)

S = e~

-et1 f^f paiP1u"i-1e-aiuPi + (1 - p)a2p2u"#-!e-a#u"2

J0 1 ^„-a1u"i _L n _ ^\„-a2u"2

R

pe

aiu

+ (1 - p)e

-a2u"2

euedu--( 1-e-6ti U

(28)

Let K(t1) be the total production cost per unit time. Since the total production cost is the sum of the set up cost, cost of the units, the inventory holding cost. Therefore the total production cost per unit time becomes

A CO h

i

J I(t)dt + J I(t)dt

(29)

Substituting the values of I(t) and Qfrom equations ,(25), (26) and (27) in equation (29), we obtain K(ti) as

K(t.)

_A C T = T + TJ

palplt"l-1e-alt"l + (1 - p)a2p2t"2-!e-a2t"2

h + T

J

Se6(ti-t) - e-

t6

r

pe-a+ (1 - p)e-a2t"2 paipiu"i-1e-aiuPl + (1 - p)a2p2u"2-1e-a2u"2

dt

pe

-aiu

"i

+ (1 -p)e

-a2u"2

du - k

eu6 du

+

r

t

Se6(ti-t) - ke-t6 J e

u6

du

dt

dt

Substituting the value of S given in equation (28) in the total cost equation (30), we obtain

i

K(t!) =

A C r%! T + f]0

pa1ß1tßi-1e-aitßl + (1 - p)a2ß2t"#~1e~a#tß2

pe

$1%ßl + (i - p)e

dt

h + T

(l-e-dT )

1 [%i pa1ß1u"i-1e-$iußl + (1 - p)a2ß2u"2-1e-$2+ß2

if

eJ 0

pe

-aiu

ß1

+ (1 - p)e~

a#u

ß2

eu0du + — e2

k

--T -

e

f e-t' Rf1 Pa1ß1ußl-1e-$lUß + (1- P)^2ß2u"2-1e-$2uß2 eudduSdt

pe-$iußl + (1 -p)e~

(31)

IX. Optimal Production Schedules of the Model

In this section we obtain the optimal policies of the inventory system under study. To find the optimal values of it, we equate the first order partial derivatives of K(it) with respect to ti equate them to zero. The condition for minimum of K(ii))

d2K(t1)

> 0

Differentiating K(t1) with respect to t1 and equating to zero, we get

pa1ß1t1ßi-1e-ai%ißl + (1 - p)a2ß2t1ß2-1e-a2%i

"2

h + T

(1-e-8T )e

tie

pe-ai%ißl + (1 - p)e-a2%i"2 pa1ß1t1ßi-1e-aitißl + (1 - p)a2ß2t1ß2-1e-$2tißl

pe-aitißl + (1 -p)e-a2ti

"2

=0

(32)

Solving the equation (32), we obtain the optimal time 11 of t1 at which the production is to be stopped.

The optimal production quantity Q1 of Q in the cycle of length Tis obtained by substituting the optimal values of t1 in equation (27).

X. Numerical Illustration

In this section we discuss the solution procedure of the model through a numerical illustration by obtaining the production time, optimum production quantity and the total production cost of an inventory system. For demonstrating the solution procedure of the model the parameters are considered as A = Rs.310\-, C= Rs.15\-, h = Re.0.2\-, (ai, a, P, p, p) = (11, 14, 0.55, 3, 0.5) e= 3, k=3.3 and 7=12 months. The values of parameters above are varied further to observe the trend in optimal policies and the results are obtained are shown in Table 3. Substituting these values the optimal production quantity Q*, the production time and total production cost are computed and presented in Table 3

From Table 3 it is observed that the deterioration parameters and production parameters have a tremendous influence on the optimal values of the model.

1

Table 3: Numerical Illustration

A C h T a1 a2 Pi P2 e k P t\ Q* K*

310 15 0.2 12 11 14 0.55 3 3 3.3 0.5 5.495 28.771 61.871

325.5 5.495 28.771 63.163

341 5.495 28.771 64.454

356.5 5.495 28.771 65.746

15.75 5.496 28.775 63.698

16.5 5.497 28.778 65.477

17.25 5.499 28.782 67.281

0.21 5.495 28.772 61.875

0.22 5.495 28.772 61.879

0.23 5.495 28.772 61.883

11.55 5.5 30.217 63.686

12.1 5.501 31.598 65.402

12.65 5.503 33.01 67.143

14.7 5.493 28.767 61.859

15.4 5.492 28.763 61.825

16.1 5.491 28.761 61.788

0.578 5.499 30.157 63.607

0.605 5.504 31.56 65.366

0.633 5.509 33.088 67.281

3.15 5.496 28.773 61.89

3.3 5.496 28.775 61.907

3.45 5.497 28.777 61.923

3.15 5.495 28.772 61.886

3.3 5.495 28.773 61.901

3.45 5.496 28.773 61.92

3.465 5.495 28.771 61.86

3.63 5.495 28.771 61.85

3.795 5.495 28.771 61.839

0.525 5.495 28.722 61.806

0.55 5.494 28.675 61.743

0.575 5.494 28.629 61.681

When the ordering cost 'A ' increases from 310 to 356.5, the optimal production quantity Q* and the optimal production down time ti* remains constant, the total production cost per unit time K* increases from 61.871 to 65.746. As the cost parameter ' C' increases from 15 to 17.25, the optimal production quantity Q* increases from 28.771 to 28.782, the optimal production down time ti increases from 5.495 to 5.499, the total production cost per unit time K increases from 61.871 to 67.281. As the inventory holding cost 'h ' increases from 0.2 to 0.23, the optimal production quantity Q*increases from 28.771 to 28.772, the optimal production down time ti* remains constant, the total production cost per unit time A*increases from 61.871 to 61.883.

As the production parameter 'ai' varies from 11 to 12.65, the optimal production quantity Q increases from 28.771 to 33.01, the optimal production down time ti* increases from 5.495 to 5.503, the total production cost per unit time K increases from 61.871 to 67.143. As the production

parameter 'a2 ' varies from 14 to 16.1, the optimal production quantity Q* decreases from 28.771 to 28.761, the optimal production down time ti* decreases from 5.495 to 5.491, the total production cost per unit time A* decreases from 61.871 to 61.788. As the production parameter ' P' varies from 0.55 to 0.633, the optimal production quantity Q* increases from 28.771 to 33.088, the optimal production down time ti * increases from 5.495 to 5.509, the total production cost per unit time K increases from 61.871 to 67.281. As the production parameter 'P2' varies from 3 to 3.45, the optimal production quantity Q* increases from 28.771 to 28.773, the optimal production down time ti* increases from 5.495 to 5.497, the total production cost per unit time K increases from 61.871 to 61.923. As the production parameter ' p varies from 0.5 to 0.575 the total production quantity Q* decreases from 28.771 to 28.629, the optimal production down time ti* decreases from 5.495 to 5.494, the total production cost per unit time Kdecreases from 61.871 to 61.681.

As the deterioration parameter '0 ' varies from 3 to 3.45, the optimal production quantity Q* increases from 28.771 to 28.773, the optimal production down time ti* increases from 5.495 to 5.496, the total production cost per unit time A*increases from 61.871 to 61.92.

As the demand parameter 'k ' varies from 3.3 to 3.795, the total production quantity Q* remains constant, the optimal production down tim te i* remains constant, the total production cost per unit tim Ke * decreases from 61.871 to 61.839.

XI. Sensitivity Analysis of the Model

The sensitivity analysis is carried to explore the effect of changes in model parameters and costs on the optimal policies, by varying each parameter (-15%, -10%, -5%, 0%, 5%, 10%, 15%) at a time for the model under study. The results are presented in Table 4. The relationship between the parameters and the optimal values of the production schedule is shown in Figure 4.

Table 4: Sensitivity analysis of the model - Without Shortages

Variation Parameters Optimal Policies -15% -10% -5% 0% 5% 10% 15%

A ti* 5.495 5.495 5.495 5.495 5.495 5.495 5.495

Q* 28.772 28.772 28.771 28.771 28.771 28.771 28.771

K* 57.996 59.288 60.579 61.871 63.163 64.454 65.746

C ti* 5.491 5.492 5.494 5.495 5.496 5.497 5.499

Q* 28.761 28.765 28.768 28.771 28.775 28.778 28.782

K* 56.465 58.266 60.068 61.871 63.698 65.477 67.281

h ti* 5.495 5.495 5.495 5.495 5.495 5.495 5.495

Q* 28.771 28.771 28.771 28.771 28.772 28.772 28.772

K* 61.859 61.863 61.867 61.871 61.875 61.879 61.883

a1 ti* 5.485 5.489 5.491 5.495 5.5 5.501 5.503

Q* 24.537 26.203 27.358 28.771 30.217 31.598 33.01

K* 56.463 58.599 60.076 61.871 63.686 65.402 67.143

a2 ti* 5.496 5.496 5.496 5.495 5.493 5.492 5.491

Q* 28.775 28.775 28.774 28.771 28.767 28.763 28.761

K* 61.901 61.885 61.880 61.871 61.859 61.825 61.788

Pi ti 5.484 5.487 5.491 5.495 5.499 5.504 5.509

Q* 25.171 26.286 27.453 28.771 30.157 31.56 33.088

K* 57.359 58.756 60.219 61.871 63.607 65.366 67.281

Pi ti 5.494 5.494 5.494 5.495 5.496 5.496 5.497

Q* 28.768 28.769 28.77 28.771 28.773 28.775 28.777

K* 61.811 61.831 61.851 61.871 61.89 61.907 61.923

e ti 5.494 5.495 5.495 5.495 5.495 5.495 5.496

Q* 28.77 28.771 28.771 28.771 28.772 28.773 28.773

K* 61.832 61.845 61.857 61.871 61.886 61.901 61.92

k ti 5.495 5.495 5.495 5.495 5.495 5.495 5.495

Q* 28.771 28.771 28.771 28.771 28.771 28.771 28.771

K* 61.903 61.892 61.882 61.871 61.86 61.85 61.839

p ti 5.496 5.495 5.495 5.495 5.495 5.494 5.494

Q* 28.936 28.878 28.823 28.771 28.722 28.675 28.629

K* 62.082 62.009 61.939 61.871 61.806 61.743 61.681

XII. Observations The major observations drawn from the numerical study are:

• f/is less sensitive, Q i s slightly sensitive and A* is moderately sensitive to the changes in ordering cost 'A'.

• ti"and Q*are slightly sensitive and A*is moderately sensitive to the changes in cost per unit 'C.

• ti' i s less sensitive, Q* and A* are slightly sensitive to the changes in holding cost '//'.

• ti" is slightly sensitive, Q and K" are highly sensitive to the change in the production parameter' al.

• ti, <?* and A* are slightly sensitive to the change in the production parameter 'at.

• f/is slightly sensitive, Q and A* are moderately sensitive to the change in the production parameter' fh'.

• ti, <?* and A* are slightly sensitive to the change in the production parameter ' [ti.

• ti, <?* and A* are slightly sensitive to the change in the production parameter'//.

• ti, <?* and A* are slightly sensitive to the change in the deterioration parameter V.

• ti and Q* are less sensitive, K i s slightly sensitive to change the demand parameter 7c\

Figure 4: Relationship between parameters and optimal values without shortages

XIII. Conclusions

This paper introduces a new EPQ model with random production having mixture of two component Weibull production rate and exponential decay having constant demand. The mixture of two parameter Weibull distribution characterises the heterogeneous process more close to reality. By using the historical data we can estimate the replenishment and deterioration distribution parameters. The production manager can estimate the optimal production downtime and uptime with the distributional data of production and deterioration parameters. The Weibull rate of production can include increase/ decrease/constant rates for different values of parameters. Sensitivity analysis is used to understand the change in the parameters of Weibull rates of production and exponential deterioration. It is observed that random production and deterioration have significant influence on optimal values of the production schedule and production quantity. This model also includes some of the earlier models as particular cases. This model can be used to analyse production processes where the production is done in two different units/ machines and rate of deterioration is constant. It is possible to extend this model with other types of demand functions such as stock dependent demand, time and selling price dependent demand which will be taken up elsewhere. This paper is useful for analyzing optimal production schedules for the industries dealing with deteriorating items such as sea foods and edible oil. This model also includes some of the earlier EPQ models as particular cases for specific values of the parameters.

Funding

No funding was provided for the research.

Declaration of Conflicting Interests The Authors declare that there is no conflict of interest.

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