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This paper deals with optimization of demand prediction of a short life product with a minimum shelf life. A generalized algorithm for optimizing the future demand was developed by using markov chain. The purpose of this paper is to develop an

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International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME
44
A GENERALISED ALGORITHM FOR THE DEMAND PREDICTIONOF A SHORT LIFE CYLCLE PRODUCT SUPPLY CHAIN AND ITSIMPLEMENTAION IN A BAKED PRODUCT
Bijesh Paul
1
Dr Jayadas.N.H.
21
Research scholar,
2
Associate ProfessorDivision of Mechanical Engineering, School of Engineering,Cochin University of Science and Technology (CUSAT),Cochin - 682 022, Kerala, India.
E-mail:bijeshpaul@hotmail.com ABSTRACT
This paper deals with optimization of demand prediction of a short life product witha minimum shelf life. A generalized algorithm for optimizing the future demand wasdeveloped by using markov chain. The purpose of this paper is to develop an algorithm foroptimizing demand which will act as a benchmark for future production and will lead to hugeannual savings for each product.For very short life cycle products or perishable products suchas baked products and newspapers with maximum shelf life of one day it’s very difficult topredict the future demand. For such products demand forecast erroris found in between 40 –100%.This gives an opportunity to identify the problem of demand forecast error in short lifecycle products with very low shelf life. Moreover little literature is available to predict thedemand of short life cycle or very perishable product. Hence a generalized algorithm foroptimal future demand was developed by using Markov chain.This is very relevant in Indianscenario where small firms are going to face stiff competition from multinational andindigenous retail chains. Thealgorithm was implemented for a baking product and the optimaldemand forecast wasdetermined. This paper offers a novel optimization technique foroptimizing demand forecast of short life cycle supply chain products by using Markovchains.The algorithm can be also implemented for novel products as it requires only demanddata of any two consecutive time periods.
Keywords
: Demand, Markov Chain, Algorithm, Optimization, Short life cycle, Supply chain.
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET
)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)Volume 4 Issue 1 January- February (2013), pp. 44-53
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2012): 3.8071 (Calculated by GISI)www.jifactor.com
IJMET
© I A E M E
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME
45
1. INTRODUCTION
Short Life cycle product is a kind of product with a comparatively short and fixedselling time, such as baked products, fashion clothes, books, magazines, electronicsmerchandise, festival adornment etc [1]. In this product supply chain the indeterminationdegree of downstream demand is very high, and the indetermination degree of the upperstream will enlarge further. The seller’s average out of stock rate is even up to 10-40%. Thedemand forecast error of supplier or manufacture is generally between 40 – 100% [2].Actually, with the quick development of science and technology and with continuous rise of peoples demand more and more products will have the characteristic of short life cycle. Thisphenomenon will play more in the commodity market of fierce competition. The mainreasons are 1. Speed of technology refreshing is more and more 2. The consumption is moreand more of short life characteristic [3] Many researchers in the field of logistics analyzed thereturning goods problem of short life cycle product. Most investigated impact of supplychain management on logistical performance indicators in food supply chains especially inthe case of baked products. [4]. Lee found that product that has not been sold at the end of theseason may be either returned to the manufacturer or processed at the discount shop [5].It’s important for a retailer of a short life product with minimum shelf life to predict theoptimal demand as under stocking will result in the switching of loyalty of the customers andloss of possible profit and overstocking will result in obsolete products which will enhancethe financial burden. For small retailers it’s expensive to use software tools to predict demandin advance. Hence an attempt is made to develop an algorithm by using Markov chain withpast sales data of any two successive months. This technique is to be applied when demand istreated as a random variable.ie trend;seasonality and cycleness associated with demand dataare negligible.Agrawal and Smith used negative binomial distribution (NBD) for the demand model andsuggested that NBD model provides a better fit than the normal or Poisson distributeddata.[6]Cachon used the negative binomial distribution model to analyze the demand of thefashion goods where it is assumed that the demand process follows the Poisson distributionand demand rate varies according to a gamma distributed model [7].Hammond studied theQuick Response policy with ski apparel (ski suits, ski pants, parkas, etc), and showed thatforecast accuracy can be substantially improved by adopting QR policy [11].A Markov chainmodel is a stochastic process, with discrete states and continuous time in which modeling isdone on observable parameters. This model can be utilized to evaluate the probability of different states with respect to time. The earlier states are irrelevant for predicting thefollowing states, since the current state is known [19]. In 2001, Zhang and He have developeda Grey–Markov forecasting model for forecasting the total power requirement of agriculturalmachinery in Shangxi Province [21]. In 2007, Akay and Atak have formulated a Greyprediction model with rolling mechanism for electricity demand forecasting of Turkey [22].A Grey–Markov forecasting model has been developed by Huang, He and Cen in 2007. TheMarkov-chain forecasting model is applicable to problems with random variation, whichcould improve the GM forecasting model [23].Markov models are used in many disciplinesfor many different applications, from thermodynamic modeling in physics to the populationmodeling in biology.Theycan be used to model almost any dynamical system whoseevolution over time involves uncertainty" [24]. Due to the uncertainty and randomness of thedata it’s appropriateto use a Markov chain to predict the demand.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME
46
Markov chains are dynamic systems that describe the evolution of a probability distribution.Since this analysis is concerned with demand prediction based on a finite time interval,discrete time stationary Markov chains with a fixed number of states are used. A state of asystem is where the system is at a point of time. Transition probability is the probability of transforming from one state to another in a specific time period.AMarkov model is describedin terms of its transition probabilities, p
ij
, which can berepresented in a transition probabilitymatrix
P
ij
= P
11
P
12
P
13 ………
P
1n
P
21
P
22
P
23……….
P
2n…………………………….……………………………..
P
n1
P
n2
P
n3
....…..P
nn
The columns of P are stochastic, meaning the entries are non-negative and sum toone. Ateach time step, k, the state of the chain, x
k
, is determined by the previousstate and thetransition probabilities associated with that state. The assumptions of markov chain analysisare that1.
Theprobabilities of travelling from one state to all other sates add to one.2.
The states are independent of time.The evolution ofthe system is determined by multiplying the transition matrix by the previousstatevector, which is a stochastic vector representing the probabilities of the system beinginany one of the given states. The stationary characteristics of the Markov chain reveal thatsame output will be produced irrespective of the input. This Property is utilized forgenerating the optimal demand forecast for a short life cycle product with minimum shelf lifebecause options of stocking beyond one day is not possible and the minimization of financialburden that overstocked products brings to the firm is crucial in these days of intensecompetition. Moreover as time passes the probability of a particular state increases andreaches a steady state probability and the demand corresponding to this state is taken as theoptimal demand forecast. An Algorithm is developed by incorporating the above mentionedfeatures of a Markov chain to predict the demand forecast.
2. METHODOLOGY
1.
Observed demand data for a short life cycle product is collected for any twosuccessive months.2.
Implement the generalized algorithm for the selected demand data.3.
Deduce the initial probability matrix and the Transition probability matrix for thedifferent states of demand.4.
By utilizing the above two matrices the probability of different states of demand forany future period can be determined.The evolution ofthe system is determined bymultiplying the transition matrix by the previous statevector(probability matrix),which is a stochastic vector representing the probabilities of the system beingin anyone of the given states5.
Choose the state with maximum probability from the obtained current probabilityvector (initial probability matrix).6.
Determine the annual savings by adopting the demand of the state with maximumprobability.
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME
47
3. ALGORITHM
1) Collect the observed data for sales of a particular product with minimum shelf life for any twoconsecutive or successive months, say t and t+12) Determine the upper limit and lower limit of the collected sales data for the t
th
month.Determine the range or band width of the collected data as the difference between upper limit andlower limit for the t
th
month3) Discretise the obtained range into states or class intervals with minimum possible no of samplesize. Let us denote these states as X
1,
X
2,
X
3
………..X
n.
4) Determine the initial probability vector P
0
for the month t. This matrix gives the initialprobability of all states say X
1,
X
2,
X
3
………..X
n
in month t.4, a) List out all the days (m) in a month in the month t as the first column, in the ascending orderof the table4, b) In second column enter the state of the observed sales data for all the days of t
th
month listedin the first column4, c) Count the no of occurrence of each state in t
th
month. (For eg say state X
i
is occurring j timesin the month t of m days, then initial probability of X
i
= j/m)4, d) Determine the initial probability of all states by using the formulae X
i
= J/M where J is theoccurrence of i
th
state in t
th
month of M days and i= 1, 2,3…….n.4, e) Represent the initialprobabilities obtained from step 8 as a row vector (1*n) with n no of entries and is called as initial probability vector denoted by P
0
.5) Construct state occurrence table for t
th
month and t+1
th
month.5, a) List out all the days of t
th
and t+1
th
month in the ascending order as the first column of thetable. Assume the number of working days in both months as same5, b) In the second column of the table enter the state corresponding to sales data for all the dayslisted in t
th
month.5, c) In column three enter the state corresponding to sales data for all days listed in the t+1
th
month.6) Deduce transition probability matrix from the event occurrence table.6,a ) Any current state X
i
in a particular day of t
th
month can transform into states X
1,
X
2,
X
3
………..X
n
during the same day of t+1
th
month. Hence there exits n probabilities which resultsfrom the probable transformation of current state X
i
to other possible states X
1,
X
2,
X
3
………..X
n.
Represent these probabilities as P
11
,P
12………..
P
1n
6, b) Form the Transition probability matrix by representing all the current states as rows and nextstates as columns. Now enter the probabilities as P
11
,P
12………..
P
1n
in 1st row and repeat the sameprocedure for other rows.Any entry say P
ij
= No of transformations of current state i of t
th
month in a particular day to nextstate j of t+1 th month in the same day/ Total no of occurrence of current state i in the t th month7) Deduce the current probability vector for the succeeding months t+2, t+3 asP
1
=P
0
* TPMP
2
=P
1
* TPM………………P
m
= P
m-1
*TPM8) Choose the state with maximum probability from the obtained current probability vector forsay the m
th
month which is a row matrix with probability of each state during say m
th
month.9) Determine the possible profit to firm by the adoption of this state of production as indicated bythe step 8
International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME
48
4. IMPLEMENTATION OF THE ALGORITHM FOR A BAKED PRODUCT
The data collected from a reputed baking firm is furnished below.The firm has beenproducing 1300 items per day and selling this item @ Rs 12. Any leftover item is sold at arate of Rs 3 and there by occurring a possible loss of profit of Rs 9 per product for left overitem.Cost of each item is Rs7/-.Step1The table below shows the demand data gathered for two successive months of a bakedproduct with a short life cycle of one day.
Table-1
Date Total Demand during each dayof April 2012Total Demand during eachday of May 201201-04-2012 1260 126002-04-2012 1267 126003-04-2012 1260 126804-04-2012 1252 125505-04-2012 1248 124906-04-2012 1266 126707-04-2012 1271 126708-04-2012 1260 126509-04-2012 1259 126510-04-2012 1264 126811-04-2012 1256 125912-04-2012 1265 126713-04-2012 1260 126414-04-2012 1271 126915-04-2012 1265 127116-04-2012 1271 126517-04-2012 1259 126218-04-2012 1266 126819-04-2012 1270 126920-04-2012 1260 126321-04-2012 1262 126822-04-2012 1268 126523-04-2012 1265 126024-04-2012 1271 126925-04-2012 1263 1265

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