Product Mix Problem with Radom Return and Preference of Production Quantity. Osaka University Japan

Size: px

Start display at page:

Download "Product Mix Problem with Radom Return and Preference of Production Quantity. Osaka University Japan"

Mervyn Scott
5 years ago
Views:

1 Product Mix Problem with Radom Retur ad Preferece of Productio Quatity Hiroaki Ishii Osaka Uiversity Japa

2 We call such fiace or idustrial assets allocatio problems portfolio selectio problems, ad various studies have bee doe till ow. As for a traditioal mathematical approach, Markowitz [] has proposed the mea-variace aalysis model, ad it is ofte used i the real fiacial field (Lueberger [2], Campbell.et al. [3], Elto ad Gruber [4], Jorio [5]. O the other had, may researchers have proposed models of portfolio selectio problems which developed Markowitz model; mea-absolute-deviatio model (Koo [6,7], semi-variace model (Bawa ad Lideberg [8], safety-first model [4], Value at Risk ad coditioal Value at Risk model (Rockfellar [9], etc..

3 [] H. Markowits, Portfolio Selectio, Wiley, New York, 952 [2] David G. Lueberger, Ivestmet Sciece, Oxford Uiv. Press, 997 [3] J.Y. Campbell, A.W. Lo. A.C. MacKilay, The Ecoometrics of Fiace Markets, Priceto Uiversity Press, Priceto, NJ, 997 [4] E.J. Elto, M.J. Gruber, Moder Portfolio Theory ad Ivestmet Aalysis, Wiley, New York, 995 [5] P. Jorio, Portfolio optimizatio i practice, Fiacial Aalysis Joural, Ja.-Feb., 68-74, 992 [6] H. Koo, Piecewise liear risk fuctios ad portfolio optimizatio, Joural of Operatios Research Society of Japa, , 990 [7] H. Koo, H. Shirakawa, H. Yamazaki, A mea-absolute deviatio-skewess portfolio optimizatio model, Aals of Operatios Research, , 993 [8] V.S. Bawa, E.B. Lideberg, Capital market equilibrium i a mea-lower partial momet framework, Joural of Fiacial Ecoomics, , 977 [9] R.T. Rockfellar, S. Uryasev, Optimizatio of coditioal value-at-risk at, Joural of Risk, 2(3-2, 2000

4 I may corporatios ad idustries, there are may decisio problems; i.e., schedulig gp problem, logistics, data miig ad allocatio problem. I these problems, it is importat to predict future total returs ad to decide a optimal asset allocatio maximizig total profits uder some costraits, particularly a moey costrait. Of course it is easy to decide the most suitable allocatio if we kow future returs a priori. However sice they always chage, there exists the case that a ucertaity from social coditios has a great ifluece o the future returs surely. Furthermore, i the real world, there may be also probabilitistic ad possibilitistic factors. Uder such situatios, we cosider how to reduce a risk, ad it becomes importat how we ear the greatest profit.

5 Portfolio Problem Fiacial Egieerig Product mix Problem Crop plaig Facility costructio problem

6 Product mix, Preferece of product quatity, Radom retur, Bi-criteria, No-domiated solutio [0] T. Hasuike ad H. Ishii: Portfolio selectio problem with two possibilities of expected retur, Noliear Aalysis ad Covex Aalysis edited by W. Takahshi ad T.Taaka, Yokohama Publishers ( [] H. Kt Katagiri, iim. Sk Sakawa ad dh. Ishii: A study o fuzzy radom portfolio selectio problems usig possibility ad ecessity measures, Mathematicae Japoicae ( , [2] S. Li ad D. Tirupati: Impact of productio mix flexibility ad allocatio policies o techology, Computers & Operatios Research (997 24, [3] P. Letmathe ad N. Balakrisha: Eviromeatl cosideratios o the optimal product mix, Europea Joural of Operatioal Research ( , [4] L. O. Morga ad R. L. daiels: Itegratig product mix ad techology adoptio decisios; a portfolio approach for evaluatig advaced techologies i the automobile idustry, Joural of Operatios Maagemet (200 9,

7 Problem Formulatio ( There exist kids of products. For each product, its future profit for each product uit is a radom variable accordig to a ormal 2 (, Nrσ Distributio. These radom variables are idepedet each other. Productio quatity for product is a decisio variable ad it is deoted by x. This variable should take a value i the fuzzy iterval I, that is, preferece of productio quatity is attached to product ad it is deoted by a membership fuctio of the value of some iterval. There exist three types of the membership fuctios with respect to products.

8 Membership fuctio with respect to products Type Type 2 (0 x u x u μ ( x = ( u x u + e e 0 ( x u + e, x < 0 =,2,..., 0 ( x x μ ( x = ( x + e e ( x + e = +,..., 2 0 ( x x ( x + e e x = + e x u x u ( u x u + k k 0 ( x u + k Type 3 μ ( ( = 2 +,...,

9 (2There exit m costraits, due to techical costrait, resource costrait, persoal costrait ad cost costrait etc. Here we assume that they are liear costraits. a is a coefficiet of product i costrait, that is, uit cosumptio k of resource k for uit productio of product. b k is a right had costat, that is, total available limit of resource k. (3Uder the above settig (, (2, we cosider the followig problem as a start. P o :Maximize = r x Maximize mi{ μ ( x =, 2,..., } = subect to a x b, k =,2,..., m, x I, =, 2,..., k k Not well defied

10 P o : Maximize F Maximize mi{ μ ( x =, 2,..., } = = subect to Pr{ rx F} α, a x b, k=, 2,..., m k k α is a probability b level that this chace ce costrait should be satisfied s ad over 0.5. rx rx F rx = = = Pr{ rx F} α Pr α = σ x σ x = = rx rx = = N(0, F rx K σ x = = σ x α =

11 . P : Maximize r x K σ x 2 α = = Maximize mi{ μ ( x =, 2,..., } subect to a x b, k =,2,...,,, m Solutio Procedure, if = k k (No-domiated dsolutio For two solutios x = ( x, x = ( x rx Kα σ x rx Kα σ x = = = = mi{ μ ( x =,2,..., } mi{ μ ( x =,2,..., } ad at least oe iequality holds without equality, the we call domiates x If there exists o solutio that domiates x x is called a o-domiated solutio x

12 β : Miimize- + α σ, = = P r x K x subect to a x b, k =,2,..., m, μ ( x β, =,2,..., = k k P R r x K x 2 β R : Miimize + α σ J= = subect to a x b, k =, 2,..., m, μ ( x β, =, 2,..., = β R : Miimize + α σ J= = P R rx K x 2 = k k subect to ax b, k=,2,..., m,0 x e( β + u, =,2,..., k k x β e +, = +,...,,, e ( β + u x β e +, = +,...,, 2 2

13 P β x( β, R = ( x( β, R Theorem. Let a optimal solutio of be. If, = σ ( β, x ( β, R R x R = Let z( β, R = R σ x( β, R ad R, the is a optimal solutio for R ( β σ ( x β = = P β for x ( β = ( x ( β a optimal solutio of. The z ( β, R > 0 R < R ( β z ( β, R < 0 R > R( β P β P β R quadratic programmig problem with hliear costrait algorithm similar to []. [] R. Helgarso, J.Keigto ad H. Hall: A polyomial bouded algoritm for a sigly costraied quadratic programmig, Mth Mathematical ti lprogrammig (980 8, R( β biary search of R.

14 A solutio procedure to fid some o-domiated solutios Step : Let ε be a suitable small positive umber. Set β = β0 (>05, ND= φ Step 2: Solve P β ad go to Step 2., fid a optimal solutio x( β = ( x( β ad calculate μβ ( = mi{ μ( x( β =, 2,..., } Set ND = ND {( x β } ad go to Step 3. Step 3: Set β = μ( β + ε If β is sufficietly close to, termiate. Otherwise retur to Step 2.

15 Coclusio R( β Efficietly to fid Exted a liear type preferece fuctio to oliear oe productio quatity of each product is discrete ad so iteger decisio i variable Coefficiet a k is usually ot fixed ad may be ambiguous ad so it should be a fuzzy umber

POSSIBILISTIC OPTIMIZATION WITH APPLICATION TO PORTFOLIO SELECTION

THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume, Number /, pp 88 9 POSSIBILISTIC OPTIMIZATION WITH APPLICATION TO PORTFOLIO SELECTION Costi-Cipria POPESCU,