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International Journal of Engineering and Technology (IJMET), ISSN 0976 INTERNATIONALMechanical Volume 4, Issue 1, January - February (2013) © IAEME– JOURNAL OF MECHANICAL ENGINEERING 6340(Print), ISSN 0976 – 6359(Online) AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4 Issue 1 January- February (2013), pp. 44-53 IJMET © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2012): 3.8071 (Calculated by GISI) www.jifactor.com ©IAEME A GENERALISED ALGORITHM FOR THE DEMAND PREDICTION OF A SHORT LIFE CYLCLE PRODUCT SUPPLY CHAIN AND ITS IMPLEMENTAION IN A BAKED PRODUCT Bijesh Paul1 Dr Jayadas.N.H.2 1 Research scholar, 2Associate Professor Division 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 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 algorithm for optimizing demand which will act as a benchmark for future production and will lead to huge annual savings for each product.For very short life cycle products or perishable products such as baked products and newspapers with maximum shelf life of one day it’s very difficult to predict 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 life cycle products with very low shelf life. Moreover little literature is available to predict the demand of short life cycle or very perishable product. Hence a generalized algorithm for optimal future demand was developed by using Markov chain.This is very relevant in Indian scenario where small firms are going to face stiff competition from multinational and indigenous retail chains. Thealgorithm was implemented for a baking product and the optimal demand forecast wasdetermined. This paper offers a novel optimization technique for optimizing demand forecast of short life cycle supply chain products by using Markov chains.The algorithm can be also implemented for novel products as it requires only demand data of any two consecutive time periods. Keywords: Demand, Markov Chain, Algorithm, Optimization, Short life cycle, Supply chain. 44 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME 1. INTRODUCTION Short Life cycle product is a kind of product with a comparatively short and fixed selling time, such as baked products, fashion clothes, books, magazines, electronics merchandise, festival adornment etc [1]. In this product supply chain the indetermination degree of downstream demand is very high, and the indetermination degree of the upper stream will enlarge further. The seller’s average out of stock rate is even up to 10-40%. The demand 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. This phenomenon will play more in the commodity market of fierce competition. The main reasons are 1. Speed of technology refreshing is more and more 2. The consumption is more and more of short life characteristic [3] Many researchers in the field of logistics analyzed the returning goods problem of short life cycle product. Most investigated impact of supply chain management on logistical performance indicators in food supply chains especially in the case of baked products. [4]. Lee found that product that has not been sold at the end of the season 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 the optimal demand as under stocking will result in the switching of loyalty of the customers and loss of possible profit and overstocking will result in obsolete products which will enhance the financial burden. For small retailers it’s expensive to use software tools to predict demand in advance. Hence an attempt is made to develop an algorithm by using Markov chain with past sales data of any two successive months. This technique is to be applied when demand is treated as a random variable.ie trend;seasonality and cycleness associated with demand data are negligible. Agrawal and Smith used negative binomial distribution (NBD) for the demand model and suggested that NBD model provides a better fit than the normal or Poisson distributed data.[6]Cachon used the negative binomial distribution model to analyze the demand of the fashion goods where it is assumed that the demand process follows the Poisson distribution and demand rate varies according to a gamma distributed model [7].Hammond studied the Quick Response policy with ski apparel (ski suits, ski pants, parkas, etc), and showed that forecast accuracy can be substantially improved by adopting QR policy [11].A Markov chain model is a stochastic process, with discrete states and continuous time in which modeling is done 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 the following states, since the current state is known [19]. In 2001, Zhang and He have developed a Grey–Markov forecasting model for forecasting the total power requirement of agricultural machinery in Shangxi Province [21]. In 2007, Akay and Atak have formulated a Grey prediction 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. The Markov-chain forecasting model is applicable to problems with random variation, which could improve the GM forecasting model [23].Markov models are used in many disciplines for many different applications, from thermodynamic modeling in physics to the population modeling in biology.Theycan be used to model almost any dynamical system whose evolution over time involves uncertainty" [24]. Due to the uncertainty and randomness of the data it’s appropriateto use a Markov chain to predict the demand. 45 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME 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 a system 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 described in terms of its transition probabilities, pij , which can berepresented in a transition probability matrix Pij = P11 P12 P13 ………P1n P21 P22 P23……….P2n ……………………………. …………………………….. Pn1 Pn2Pn3....…..Pnn The columns of P are stochastic, meaning the entries are non-negative and sum toone. At each time step, k, the state of the chain, xk, is determined by the previousstate and the transition probabilities associated with that state. The assumptions of markov chain analysis are that 1. 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 previous statevector, which is a stochastic vector representing the probabilities of the system beingin any one of the given states. The stationary characteristics of the Markov chain reveal that same output will be produced irrespective of the input. This Property is utilized for generating the optimal demand forecast for a short life cycle product with minimum shelf life because options of stocking beyond one day is not possible and the minimization of financial burden that overstocked products brings to the firm is crucial in these days of intense competition. Moreover as time passes the probability of a particular state increases and reaches a steady state probability and the demand corresponding to this state is taken as the optimal demand forecast. An Algorithm is developed by incorporating the above mentioned features 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 two successive months. 2. Implement the generalized algorithm for the selected demand data. 3. Deduce the initial probability matrix and the Transition probability matrix for the different states of demand. 4. By utilizing the above two matrices the probability of different states of demand for any future period can be determined.The evolution ofthe system is determined by multiplying the transition matrix by the previous statevector(probability matrix), which is a stochastic vector representing the probabilities of the system beingin any one of the given states 5. Choose the state with maximum probability from the obtained current probability vector (initial probability matrix). 6. Determine the annual savings by adopting the demand of the state with maximum probability. 46 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME 3. ALGORITHM 1) Collect the observed data for sales of a particular product with minimum shelf life for any two consecutive or successive months, say t and t+1 2) Determine the upper limit and lower limit of the collected sales data for the tth month. Determine the range or band width of the collected data as the difference between upper limit and lower limit for the tth month 3) Discretise the obtained range into states or class intervals with minimum possible no of sample size. Let us denote these states as X1, X2, X3………..Xn. 4) Determine the initial probability vector P0 for the month t. This matrix gives the initial probability of all states say X1, X2, X3………..Xnin month t. 4, a) List out all the days (m) in a month in the month t as the first column, in the ascending order of the table 4, b) In second column enter the state of the observed sales data for all the days of tthmonth listed in the first column 4, c) Count the no of occurrence of each state in tth month. (For eg say state Xi is occurring j times in the month t of m days, then initial probability of Xi= j/m) 4, d) Determine the initial probability of all states by using the formulae Xi= J/M where J is the occurrence of ith state in tthmonth 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 P0. 5) Construct state occurrence table for tth month and t+1th month. 5, a) List out all the days of tthand t+1th month in the ascending order as the first column of the table. Assume the number of working days in both months as same 5, b) In the second column of the table enter the state corresponding to sales data for all the days listed in tthmonth. 5, c) In column three enter the state corresponding to sales data for all days listed in the t+1th month. 6) Deduce transition probability matrix from the event occurrence table. 6,a ) Any current state Xi in a particular day of tth month can transform into states X1, X2, X3………..Xn during the same day of t+1th month. Hence there exits n probabilities which results from the probable transformation of current state Xi to other possible states X1, X2, X3………..Xn. Represent these probabilities as P11,P12………..P1n 6, b) Form the Transition probability matrix by representing all the current states as rows and next states as columns. Now enter the probabilities as P11,P12………..P1n in 1st row and repeat the same procedure for other rows. Any entry say Pij= No of transformations of current state i of tth month in a particular day to next state j of t+1 th month in the same day/ Total no of occurrence of current state i in the t th month 7) Deduce the current probability vector for the succeeding months t+2, t+3 as P1=P0* TPM P2=P1* TPM ……… ……… Pm= Pm-1 *TPM 8) Choose the state with maximum probability from the obtained current probability vector for say the mthmonth which is a row matrix with probability of each state during say mthmonth. 9) Determine the possible profit to firm by the adoption of this state of production as indicated by the step 8 47 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME 4. IMPLEMENTATION OF THE ALGORITHM FOR A BAKED PRODUCT The data collected from a reputed baking firm is furnished below.The firm has been producing 1300 items per day and selling this item @ Rs 12. Any leftover item is sold at a rate of Rs 3 and there by occurring a possible loss of profit of Rs 9 per product for left over item.Cost of each item is Rs7/-. Step1 The table below shows the demand data gathered for two successive months of a baked product with a short life cycle of one day. Table-1 Date Total Demand during each day Total Demand during each of April 2012 day of May 2012 01-04-2012 1260 1260 02-04-2012 1267 1260 03-04-2012 1260 1268 04-04-2012 1252 1255 05-04-2012 1248 1249 06-04-2012 1266 1267 07-04-2012 1271 1267 08-04-2012 1260 1265 09-04-2012 1259 1265 10-04-2012 1264 1268 11-04-2012 1256 1259 12-04-2012 1265 1267 13-04-2012 1260 1264 14-04-2012 1271 1269 15-04-2012 1265 1271 16-04-2012 1271 1265 17-04-2012 1259 1262 18-04-2012 1266 1268 19-04-2012 1270 1269 20-04-2012 1260 1263 21-04-2012 1262 1268 22-04-2012 1268 1265 23-04-2012 1265 1260 24-04-2012 1271 1269 25-04-2012 1263 1265 48 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME Figure-1 Daily Demand Pattern 1275 1270 1265 Daily Demand 1260 1255 1250 Series1 1245 1240 1235 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 The above pattern shows that demand is Random. Hence we treat the demand as random variable and apply Markov chain based algorithm for predicting demand. Step 2 The Range = Highest demand – Lowest Demand =1272 -1248 = 24. Also when demand was highest volume of discount sales was (1300 – 1272) 28and when demand was lowest volume of discount sales was (1300 – 1248) 52. Range 2 = 52- 28 = 25 Step 3 Discretize the range into class intervals or states with a minimum sample size of 3. I.e. Say 8 states denoted by X1, X2, X3………..X8. Step 4 Determine the initial probability vector P0 for the month April. This matrix gives the initial probability of all states say X1, X2, X3………..X8in the month of April. Table-2 Class interval of discounted State No of occurrence Probability sales 52, 51, 50 X1 1 .04 49, 48, 47 X2 1 .04 46, 45, 44 X3 1 .04 43, 42, 41 X4 2 .08 40, 39, 38 X5 6 .24 37, 36, 35 X6 5 .20 34, 33, 32 X7 4 .16 31, 30, 29,28 X8 5 .20 The Fourth row of the above table gives initial probability vector P0 for the month April. This matrix gives the initial probability of all states say X1, X2, X3………..X8 in the month of April. P0 = [.04, .04, .04, .08, .24, .2, .16, .2] 49 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME Step 5 – State transition table Table-3 Day Current state (April) Subsequent state (May) 1 5 5 2 7 5 3 5 7 4 2 3 5 1 1 6 7 7 7 8 7 8 5 6 9 4 6 10 6 7 11 3 4 12 6 7 13 5 6 14 8 8 15 6 8 16 8 6 17 4 5 18 7 7 19 8 8 20 5 6 21 5 7 22 7 6 23 6 5 24 8 8 25 6 6 Step 6 Deduction of Transition Probability Matrix (T P M) a. Any current state Xi in a particular day of tth month can transform into states X1, X2, X3………..Xn during the same day of t+1th month. Hence there exits n probabilities which results from the probable transformation of current state Xi to other possible states X1, X2, X3………..Xn. Represent these probabilities as P11,P12………..P1n b. Form the Transition probability matrix by representing all the current states as rows and next states as columns. Now enter the probabilities as P11, P12………..P1n in 1st row and repeat the same procedure for other rows. c. Any entry say Pij= No of transformations of current state i of t th month in a particular day to next state j of t+1th month in the same day/ Total no of occurrence of current state i in the tth month d. Transition Probability Matrix, 50 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME Table-4 1 2 3 4 5 6 7 8 1 1 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 3 0 0 0 1 0 0 0 0 4 0 0 0 0 .5 .5 0 0 5 .167 0 0 0 0 .5 .333 0 6 .2 0 0 0 .2 0 .4 .2 7 .5 0 0 0 .25 .25 0 0 8 .6 0 0 0 0 .2 .2 0 Step 7 Deduce the current probability vector for the succeeding months t+2, t+3 ….t+9 as P1=P0* TPM P2=P1* TPM ……… Pm= Pm-1 *TPM P1=P0* TPM = [.3201, 0, .04, .04, .12, .24, .1994, .04] P2 = [.5121, 0, .0, .04, .118, .138, .144, .048] P3 = [.6602, 0, .0, .0, .0836, .1246, .1041, .0276] P4 = [.7676, 0, .0, .0,.0509, .0733,.0832 .024] P5= [.8473, 0, .0, .0, .0355, .0512, .0513, .0147] P6= [.8980, 0,.0, .0, .0231, .0335, .0352, .0102] P7= [.9323, 0, .0, .0, .0155, .0224, .0231, .0067] P8= [.9549, 0, .0, .0, .0103, .0149, .0155, .0045] P9= [.97, 0, .0, .0, .0068, .0099, .0103, .0030] Step 8 The maximum Probability is for state 1 and as time passes we can see that probability of demand for state 1 is approaching steady state probability. The Probability V/s Time curve for state one is shown below Figure-2 1.2 1 0.8 0.6 Series1 0.4 0.2 0 0 2 4 6 8 10 Step 9 The firm should produce the mean of the class interval corresponding to state 1 namely 1249 items .Thus the predicted or forecasted value of demand is Df = 1249. 51 International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 1, January - February (2013) © IAEME 5. RESULT The maximum Probability is for state 1 and as time passes we can see that probability of demand for state 1 is approaching steady state probability. Hence the firm should produce the mean of the class interval corresponding to state 1 namely 1249 items which will enhance the profit of the firm.Annual Possible Profit for a single product = {(P-Df) (E-S)*25- ∑ (Dn- Df) C}*12= {(1300- 1249) (7-3) 25 – 354*5}12 = Rs 41000, where P= Production rate per day, Df is the predicted value of demand by using algorithm, E is the cost of making unit quantity, S is the price of the product after discount to be sold after 8pm, ∑ (Dn-Df) represents the error in forecast for one month and R represents the profit of selling of unit quantity.The above concept can be extended for many products and huge annual savings can be achieved. 6. CONCLUSION AND FUTURE SCOPE A generalized algorithm for optimizing the demand data was developed by using Markov chain for a short life cycle supply chain with least possible shelf life. Little literature exits in the demand forecasting of short life cycle of baked product. The algorithm has been validated by implementing it in a baking firm and by the huge annual savings Rs 41000/for one product. The above concept can be extended for many products and huge annual savings can be achieved. It acts as a tool for recovering the lost possible profit.This is very relevant in today’s world when these small firms are going to face stiff competition from multinational and indigenous retail chains.Further validation can be carried out by comparing the results obtained by implementing this algorithm with that obtained by other statistical methods.It can be further validated by applying in different type of short life products such as Newspaper 7. REFERENCES [1] Pasternack B A. Optimal pricing and returns policies for perishablecommodities.Marketing Science, vol 4(4), pp166-67, 1985. [2] T. Davis, Effective “Supply Chain Management”. Sloan managementReview Summer, pp 35-46, 1993. 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