Mathematical Model for Forecasting and Estimating of Market Demand

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1 Mathematical Model for Forecastig ad Estimatig of Market Demad Da Nicolae, Valeti Pau, Mihaela Jaradat, Mugurel Iout Adreica, Vasile Deac To cite this versio: Da Nicolae, Valeti Pau, Mihaela Jaradat, Mugurel Iout Adreica, Vasile Deac. Mathematical Model for Forecastig ad Estimatig of Market Demad. Recet Advaces i Applied Mathematics (ISBN: / ISSN: ) [ Proceedigs of the America Coferece o Applied Mathematics (AMERICAN-MATH) ], Ja 00, Cambridge, Uited States. pp <hal > HAL Id: hal Submitted o 3 Dec 0 HAL is a multi-discipliary ope access archive for the deposit ad dissemiatio of scietific research documets, whether they are published or ot. The documets may come from teachig ad research istitutios i Frace or abroad, or from public or private research ceters. L archive ouverte pluridiscipliaire HAL, est destiée au dépôt et à la diffusio de documets scietifiques de iveau recherche, publiés ou o, émaat des établissemets d eseigemet et de recherche fraçais ou étragers, des laboratoires publics ou privés.

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3 RECENT ADVANCES i APPLIED MATHEMATICS Mathematical model for forecastig * ad estimatig of market demad DAN NICOLAE α, VALENTIN PAU β, MIHAELA JARADAT χ, MUGUREL IONUT ANDREICA δ, VASILE DEAC δ Titu Maiorescu Uiversity α, β Bucharest, Bogda Vodă Uiversity χ Clu Napoca, Polytechic Uiversity δ of Bucharest, Academy of Ecoomic Studies δ Bucharest ROMANIA topmath009@yahoo.com *Nobel prize, 97, Joh R.Hicks, Keeth J.Arrow : for their pioeerig cotributios to geeral ecoomic equilibrium theory ad welfare theory. Abstract : The scietific study article(a moograph), presets a forecast ad estimate the evolutio of the market demad. Key-Words: fuzzy sets, forecastig, estimatig, statistical extrapolatio, market demad. Problem formulatio Methods for forecastig. Probabilistic Model [0] The methods operate with certai statistical parameters. The most frequetly used is the mea square deviatio, amely: ( x ) i x σ = () We ca use the statistical tables, if we kow a certai distributio. Kowig the sequece of the likelihood coefficiets k, we determie a probability P ( x x+ kσ ), respectively P ( x x+ kσ ), if x is maximized ad a probability P( x x kσ ), respectively P( x x kσ ) if x is miimized. If the distributio fuctio is ot kow, the we apply Cebâşev's formula: P( x kσ x x+ kσ ) () k where k = the likelihood coefficiet.. Fuzzy Model Used to Mea Expadig [3], [3] There are cases whe mea expadig is useful. For istace, it is possible that the mea ot to be foud i the excellece domai of the variable uder research. This case, after the mea value x is determied, the membership degree [5], [6], µ ( x) = is assiged to it. I order to fid a expaded mea, we proceed to "expadig" figure, for istace, o iterval ' [0.95;]. We calculate a value x with the property that: µ ( x ' ) = 0,95 (3) If this value is sigle, the the expaded mea is the iterval x ', x or ' x, x. If two values, ' x ad would exist, such that: ' '' µ x = µ x = 0,95 (4) ( ) ( ) the, the expaded mea is give by the iterval ' '' ' '' x, x x, x = x, x.3 Liguistic Model For a liguistic procedure to be applied, it is ecessary that the cotiuous parameter x (or the discrete parameter, but with a large umber of restricted values) to be subect to a graulatio operatio. We cosider the maximum ad the miimum levels ad amog them, certai levels which are deomiated by usig words (verbs, adverbs) etc., as for istace: if five levels must be itroduced betwee a miimum weight piece ad a maximum weight oe, we shall use the followig gradatios : very light - for miimum; light - for immediately ext level; average - for immediately ext level; heavy - for immediately ext level; '' x ISSN: ISBN:

4 RECENT ADVANCES i APPLIED MATHEMATICS very heavy - for immediately ext level. I order to fid the average weight, if the frequecy of gradatios is symmetrical, we allow for the weights: X mi ad X max, ad we divide the level X, X ito equal parts (four itervals exist [ ] mi max amog the five levels). A iterval I will have the legth: xmax xmi I = (5) 4 I this case, x is determied by the relatio: x= x + I = x I (6) mi max Practically, x is lower rouded, ad we obtai for mea x a iterval uder the form: ( r r, x r ) + (7) where r = the roudig value. It is possible that this roudig ot to be symmetrical, the obtaied iterval is: ' '' x r, x+ r 8 ( ) Where: r'= roudig to left; r''= roudig to right. If the umber of gradatios is eve, the, the umber of itervals is odd, therefore the mea is already represeted i a iterval. If the frequecy of gradatios is ot symmetrical, the each gradatio is weighted with the iterval cetre (attached to each gradatio). Thus, a average gradatio represetig a iterval is obtaied..4 Mea Expadig Method.4. The Trucatio Method The steps of the method are: Step. The value x i is trucated ad a miimum - level trucated value is obtaied, accordig to the m -level of trucatio: y = T x i i Step. A uit for -level is added, amely b (where b is the umeratio basis) ad the maximum level is obtaied. M m y = y + b (for iteger umber) ad i y = y + b (for -rak fractios) M m i = decimals umber..4. The Roudig Method The method cosists i two steps: Step. The umber is lower rouded for sequece, yieldig: m x Step. The umber is upper rouded for the level, M x yieldig: The differeces are calculated i the same way: M m = x x with the property that: lim = 0 x Models for estimatig the evolutio of market demad. Theory of demad Quatitative theory of demad is based o the followig assumptios: I ) If a stable icome, the demad for certai goods decreases with icreasig price, ad vice versa. This assumptio allows the formulatio of umerical methods with which to predict the icrease (decrease) demad for the reductio (icrease) the price i a defied proportio. Graphic illustratio where oted: q i = the demad volume for the product I; p i = uit price for product i, (see fig. o. a, b, c). The sesitivity of demad to price chages is illustrated by the elasticity of demad coefficiet (c), of the price (p), which shows the percetage chage (i reverse) applicatio of a good, if its price chages by %. The expressio for calculatio is: C p Ec / p = : (9) C p C, p = demad growth; chage (+/-)referece price i two periods. q i q i Figure (a) p i Figure (b) (a) demad-price fuctio curve (q i = (p i )); (b) a decrease i price causes a rapid icrease i demad (luxury items); (c) demad is less sesitive to price chages q i Figure (c) Fig. o..- Possible evolutio of the price p i p i ISSN: ISBN:

5 RECENT ADVANCES i APPLIED MATHEMATICS I ) If a variable icome, demad for a good icreases with icome growth ad decreases with icreasig price. If we assume for each level of icome other f v demad fuctio: q i = f v (p i ), (0) where possible chages i demad ca be represeted by several successive demad curves as show i fig.o..: q i p i Fig.o..- Evolutios of demad If the price is kept costat, the applicatio may be described as a fuctio of icome: q i = f(v) () The coefficiet of elasticity of demad (c) to icome (v), shows the percetage icrease i demad whe icome icreases by %. That is: C l E = () c / v : C V Chagig prices determis travel demad curves. I the literature, curves expressig the depedece betwee demad ad icome are kow as Egel curves. Depedece is liear, the demad for differet products or product groups kow saturatio poit ad is iflueced by the iterchageability of products.demad-icome depedece ca be expressed primarily through the followig types of fuctios: a) C = m V (for strictly ecessary goods) (3) V+ b) C = m (V-p )(for everyday cosumer goods)(4) V+ c) C 3 = m 3 V(V-p 3 ) (for luxury goods) (5) V+ 3 d) C 4 = m 4 V- 4 V (for goods that are out of use whe a certai level of reveue) (6) The sigificace of otatios are: C = demad for the product (or product group) cosidered; V = icome; m,, p = ecoometric parameters. The iterpretatio of the four fuctios, is as follows: a) C, first fuctio, kow as Törquist fuctio I, show that with icreasig icome, demad icreases at a rate lower, ad teds to be capped. The graph is show i fig.4a.: C m Fig.o.4a. -Törquist I Fuctio Whe V limit is m, so the level toward which C is M. The first derivative of the fuctio is decreasig, tedig to 0, whe V. C m C = = (7) V ( V + ) The graph is show i fig. o.4 b.: C m / Fig. o. 4b.-Törquist I Derivative fuctio b).c fuctio (Törquist II) is similar to C i terms of demad-icome depedece, because it has a threshold of saturatio (at m ), which meas that the growth of icreasigly high o V, C ted to m, which ca be oticed by: m( V p) lim C = lim = m (8) V V V + The demad for this group of products or services begis to maifest oly after V > p, meaig that p parameter is egative, after calculatios. For V = 0: m p C = (9) C value for V = 0 is ot egative, as p < 0. The graph is show i fig. o.4c.: C -m p Fig. o.4c.-törquist II Fuctio First derivative fuctio is also decreasig (fig.o.4d.): ISSN: ISBN:

6 RECENT ADVANCES i APPLIED MATHEMATICS C m ( - p ) / Fig.o.4d.-Törquist II Derivative fuctio ' C m( + p) C = = (0) V ( V + ) c) C 3 fuctio (Törquist III) is specified for the products that required a cotiuous icrease with icreasig icome. The curve presets a oblique asymptote to the brach to +. C3 m= lim = m () 3 V V = lim( C m V ) = m ( p + ) () V 3 3 x 3 3 Therefore, C = m 3 V - m 3 (p ) is the asymptotically wated. For 3 < 0 demad occurs whe - 3 < p 3. For graphic represetatio, we determie the itersectio with the abscissa: C = 0 V = p For the asymptote, ad V p3 C3+ 0 m3v = 0 V = p (3) 3 V + 3 for the fuctio. As 3 < 0 p < p 3 ad C 3 fuctio graph is show i fig.o. 4e.: C 3 < 0 0 +p 0 3 V Fig.o.4e.-Törquist III Fuctio First derivative of the fuctio C 3 is icreasig ad has the expressio: ' C + = 3 m = 3 3 p3 (4) C3 m3 V ( V + 3) It may be oted that C = m 3 is a horizotal asymptote, ad V = - 3 is the vertical asymptote of C I 3 fuctio. Give the fact that 3 < 0-3 > 0, ad the derivative graph is show i fig.o. 4 f: C 3 -m 3 p 3 3 m 3 Fig.o.4 f.-törquist III Derivative fuctio For: m p V = 0 C = m p ; > 0 (5) ' as 3 <0 ad m 3 < - (m 3 p 3 ) / 3, give that - 3 < p 3. d) C 4 fuctio is is typical for products or groups of product, which are out of use after a certai level of reveue. So, the demad for such products are iitially icreases, reaches a maximum for V = m 4 / 4, the start to decrease reachig zero, which meas that for higher icomes tha this limit, there is o demad for such products. Problem Solutio Cosumer demad is treated cotiuously i coectio with the supply of goods, as amog them there is a complex iteractio. A approach to the processes occurrig o the market require prior research ad a proper uderstadig of the real complexity of the categories of idividual ad social demad, potetial ad actual demad. By research o a type of behavior, we ca determie future structure of the supply / demad actually recorded i previous years (t=,,...- N). We ote: m i = demad of article of family i; i= α (fixed), umber of families;,,...,, umber of articles of family i (may vary from year to year). Assortmet structure is give by the probability of producig a applicatio of article i m α quatity, the quatity demaded would be (6), (7): m = α m y Pα = mα ( t ) mα ( t ) ISSN: ISBN:

7 RECENT ADVANCES i APPLIED MATHEMATICS Assume that requests for items form a complete system of evets. For t =,,..., N years, it is defied a stochastic matrix: P = Pα ( t) ; Pα ( t) = (8) t N As : m α (t) > 0 α (P i ) - max (mi P α (t)>0)(9) t =,..., N ; =,...,. the matrix is Markov type. We cosider cosecutive years: = P, t+ - P,t, (P,t = P, (t)) (30) If > 0, we ote : = + for =,,...,m < 0, = - = m+,..., m + + = = = (3) m+ Startig from the defiitio of matrix P i, x, it is defied a trasitio matrix: Tg = Ti, Ti = Pi / P (3) x y Same for the followig years t =,,...,N, we obtai (N=) matrixes type T t. Matrix is built: T Σ N = T, Σ Σ g = T i t= T (33) from which to obtai the matrix: (34) M = T i Σ T i from which it ca determie the structure of the vector elemets of sales ext year. We ote (35): K = i i = K i i= m+ i = It builds P i, x with elemets defied as: - if i= put P i, = mi (P i,t ; P i,t+ ) For i, for which + i put P i,i = mi (...,...), ad P i, = 0, for =. For i, for which - i to do so: if comes from elemets for which: - put P i, = K - - if comes from elemets for which: + put P i, = K + + The practical applicatio questios a umber of years which is cosidered to obtai satisfactory results. For a few years, there is a stochastic matrix (Markov). From oe year to aother: Σ N i = T t i= T (36) The calculatio for year N of matrix T N is doe by applyig the recurret relatioship: T = T + T N N is determied based o the matrix T, the do calculatios to determie the structure vector (N+) to determie the coefficiets K - ad K + ad determie differeces, buildig o practical applicatio requiremets. Observatios must cover a sufficiet umber of years, somethig that is determied by simulatio. 3 Coclusios Give the dyamic lik betwee demad ad icome, ad its progress, the ormal tred is diversificatio. Philosophical coectios Ubi Cocordia, ibi Victoria! Refereces: [] Albu, L.I, Model to Estimate the Composite Idex of Ecoomic Activity i Romaia, Romaia Joural of Ecoomic Forecastig, Bucharest, Romaia, o.., vol.9, 008. [] Alua, J. Gil, Ivestmet Selectio Base o Diversified Criteria, Aals of Royal Academy of Ecoomic Ad Fiacial Scieces, Academic Year 993/994, pp [3] Bellma, R.L., Zadeh, L.A., Decisio-makig i a Fuzzy Eviromet,Maagemet Sciece, No.7, 970. [4] Dubois, D.M, Sabatier, For a Naturalist Approach to Aticipatio from Catastrophe Theory to Hypericursive Modellig, i CASYS Iteratioal Joural of Computig Aticipatory System (IJICAS ),Liege, Belgium,D.M Dubois Ed., 998. [5] Kauffma, A., Alua, J.G., Las matematicas del azary de la icertidumbre, Editorial Ceura, Madrid, Spai,990. [6] NegoiŃă, C.V. Vag (Vague), Paralela 45 Publishig House, Pitesti, Romaia,003. [7] Odoblea, St., Psychologie cosoatiste, Meloie Publishig House, Paris, Frace,938. [8] Ruxada, Gh., Botezatu, A., Spurious Regressio ad Coitegratio, Numerical Example Romaia's M Moey Demad, Romaia Joural of Ecoomic Forecastig, o. 3, Bucharest, Romaia,008. [9] Stoica, M., Adreica, M., Nicolae, D., Adreica, R., Subtle Sets ad their Applicatios, Ciberetica Publishig House, Bucharest, Romaia,008 ISSN: ISBN:

8 RECENT ADVANCES i APPLIED MATHEMATICS [0] Stoica, M., Adreica, M., Nicolae, D., Catău, Methods ad Models of Ecoomic Forecastig, Uiversitara Publishig House, Bucharest, Romaia, 006. [] Stoica, M., Nicolae, D., Ugureau, M.A.,Adreica, A., Adreica, M., Fuzzy Sets ad their Aplicatios, Proceedigs of the 9 th WSEAS Iteratioal Coferece o Mathematics & Computers i Busiess & Ecoomics ( MCBE'08 ), Bucharest, Romaia, 008. [] Zadeh, L.A., Fuzzy Logic ad its Applicatios to Approximate Reasoig,, Iformatio Processig 3, IFIP Cogress, Stockolm, Swede, 974. [3] Zadeh, L.A., Fuzzy Sets, Ifo&Cth., Vol. I, 965, pp [4] ** Oxford - Dictioary of Ecoomics. [5] ** Ecoomic Glossary. [6] ** Ecyclopedia Britaica. ISSN: ISBN: