Price Floating Range Model and Linear General Embedded Optimization Problem

!Copyright © (FAC (lth Triennial Wo rld Congress, T allinn , Estonia, USS R, 1'1'10

PRICE FLOATING RANGE MODEL AND LINEAR GENERAL EMBEDDED OPTIMIZATION PROBLEM Le Weiliang and Hu Baosheng SnlplIls 1:: lIgill a ri llg IlISlilulr, Xi 'a ll jiaululIg (i llil'frs it)" Xiall, PR C

Abstract. How to cont.rol the upper and lower limits of the floating pri L-es is one of the important problems in the price system reform in China. In this paper ,an optimization model of the price floating range is developed, which is a new class of optimization problems --- General Embedded Optimization Problem (GEOP). The linear general embedlied optjmization problem is studied ami a two le vel algorithm with only one iteration is propos ed for the linear GEOP in the pape r. fin a lly, a numerical example is given. Keywords. Optimization; general embedded problem; price control.

INTRODUCTION

pr ogramming me thods . In th e third part , the linear GEOPs are studied and a quite simple two l e v e l a lgorithm with only on e iterati o n is de veloped for them. Finally, a numerical example is give n. Thus a s cientific bas e is provided for solving the problem of finding the price floating r easonable range quantit.atively in this paper.

The price system reform i s one of th e key problems in the economic system readjustment in China. In reforming price sys tem, the government is reorganizing the ways to control and manage the price syst e m, i.e. from the t.r aditional centrally-planned price sys tem to the multiple channel management one, which includs the free market price channel, the floating price channel and the direct control price channel e tc .. In these management channels, the floating price is playing an increasingly important role, because the floating price not only can bring the government's functions of controling prices into full play, but also gives enterprises and salesmen some freedom to set or f100t their prices within the upper and lower bounds, according to the market situation and production conditions . Thus how to work out the upper and lower limits of the floating prices is one of the unsolved important problems in the price system reform in China.

THE PRICE FEASIBLE SET The price s y s tem is a ve ry complex one. The prices of various products affect and restrict each other. It is ne<.~""SIlry to study and find 011 t the law dominating the effect and restriction. In this se ction , this problem i s inves tigated from the relationship between prk-es , cost and profit. Let p represent n- dimension price row vector, PJ the pri ce of the jth product, A the (n Xn) input- output matrix and AJ the jth column of A. We can defin e the cost- price ratio. zJ = pAJ / PJ

In the first part of this paper, the price feasible set is proposed and defined, which describes the feasible range in which the prices can guarantee all economic sectors involved again s t loss es under the given profit level. It is proved that the set is a simplex in (n- J) dimension space. In the second part, the optimization model of the price floating range is developed based on the price feasible set. But the model is involved in a new class of optimization problems --- General Embedded Optimization Problem (GEOP), which can not be solved by the existing mathematical

j=I,2,···,n,

(J)

which represe nt s th e proportion of the cost in the price of the jth unit product. Obviously, ZJ < I, j= I,2,···,n in ge neral cases. If the profit (or profit and wage) requirements

of verious products are given to enterprises and salesme n by the government, then the cost-price ratio will be restricted, Le. j=I,2,···,n, where

359

~J

(2)

represents the largest proportion the

cost can make up in the price of the jth unit producL Frolll

idea is S<;;;;;F,

(5)

j= I,2,···,n,

z,=pA, / p,
where S=(plp;>p.>p~, i= 2,3,···,n, P. = Po)' which is a price floating ge n e ralized rectangular parallelepipe
we can see that , when ~"j =I,2,···,n are given, the value of the price vector I' is restricted in a certain range and c an not be taken arbitrarily. In order to remove the effed of the price level, we let p.=po' where Po is given. Then we can define price feasible set in which prices can guarantee all economic sectors involved against losses under the given profit level.

This gives us the way tD fDrmulate the floating price range mDde ), i.e. to find DUt the maximal pric e flDating generalized rectangular parallelepiped in the pri ce feasibl e set. Thus the following optimization prDble m can be pr~--ented

Price Feasible Set is liefined as:

DEFINITION

F, =(~, =pA, / p, <{,.,

p. = p.,

(}<:~, < I ,

j=I,2,···,n) s.t. S<;;;;;F,

(3)

(6)

p7 ::;;·· p~

THEOREM I Price fea s ible set is a s imple x in (n- I) dimension s pace ami can be formulated as:

where w; and w~ are weights USett to control the floating bou/Ilts of some prices to be larger or s maller, and the fir s t part Df the Dbjective function, n~:.( p;-p~) , is the vDlume Df th e price floating generalized r ectangular parallelepiped.

[~JpT_ATpT >~

«()

P.=p. P. > (},

i= 2,3,···,n

i=2,3,···,n ~l

where

[~J=

~

By solving above o ptimization problem (6), the price f1Dating range can the r efore be quant.itatively work ed Dut. But (ID is different from the traltitional optimization problems, because of its right se t- inclusion co ns traint (5). It can not be handled by the ex isti ng mathematical programming methods and it i s a ne w class of Dptimization probl e m - - - General Embedded Optimization PrDbl e m (GEOP)' In th e n ex t se ctiDn , its algorithm in th e lin ear case is studied for solving (6).

Proof: It can be directly obtained from Defini tion I. The concepts of the price feasible set gives the following new ide.as: I. In the planned price s ys t.em, in order to keep every economic sedors operating nonnally, the price of various products can not be worked out arbitrarily, they must be given in the corresponding feasible set. It negat es th e traditional iliea that prices couhi be giv e n arbitrarily.

THE ALGORITHM FOR LINEAR GENERAL EMBEDDED OPTIMIZATION PROBLEM The general form of

(6)

can be described as

max f(x)

2. The higher the given profit level, the samller the corresponding price feasible set. It i s possible that the set could be empty in the case of extreme high profit level.

s.t. xEX S,<;;;;;y

(7)

where SA={yl<1>(x,y)(x,y)ER', VC R" and XC R". By comparing (7) with other optimization problems, the di s tinc t feature of (7) is the right set-inclus ion constraint S,<;;;;;Y.

1 It provides the e.:onomics base for establishing the mathematical model to find out the range of the floating pril.:e quantitatively. OPTIMIZATION MODEL OF PRICE FLOATING RANGE

Only linear GEOPs are s tudied in this paper, in which (x,y) alllt 'I'(y) which de fined SA and Y are linear. In a special linear ca se, (7) can be describeti as

By the meanings of the price feasible set, it is clear that, when working out the upper and lower bounds of floating prices, in order to keep every economic sectors involved operating normally under the given profit level, the range of the floating prices must be in the prke feasible set. The mathematical formulation of this

max f(x ) s.L g(xKn s.<;;;;;y

360

(8)

where

SA= {yi'1>
(6)

is a s pecial case of

take S . a s th e con s traint se t and under hypothe s i s L th e optimal ex tr e me point of a linear programming will be alway s appe ar e d on same e xtreme point despite how .x changes. If we con s ider H, y a s th e obj ec tive functi o n of the line ar programming with x fix e d (wh e r e H, be the ith row of H, s o we hav e r o bjedive fun c tion s ), th e n th e o ptimal ex tr e me point s o f the s e r lin e ar programming s ubproblem s ar e obvious ly unc hange d with the variation of x . If y' (x ) is the optimal ex treme point corres ponding to the max imizett objec tive func tion H.y, the n we have H.y"( x );;" H,y' ( x ), j :;t:k, j= 1,2,.. ·,T. Therefore , we could e ns ur e H,y' (x )< d,., j= 1,2, .. ·,T, by only holding H, y'( x )<.d, . Thus we have prove d le mma

(8).

LEMMA I The s uffi c ient and nece s sary l' ondition s for S .={yIBy
2. LEMMA 2 For the co ns tra ined etluations of (9), th e r e is only on e cons traint , i.e. H.y· ( x )< d., is active within eac h group of the constraints H, y' (x )< d., j= I,2,· .. ,T, y' (x ) is the optima l ex tre me point of th e following linear programming subllroblems

S. = {yIBy
0<.:-", < L i= L2, .. ·, Tl. And if

= 1,2,.. ·,T, allll H(L;"A,y') < L;" A,d = ,1. Al s o it i s hold in th e opposite, so the le mma is tru e.

max H, y DEFINITION 2 (E x tr e me Point Cons titut ion) If e xtr e me point s y', 1= 1,2, .. ·,T, of th e s et S. ar e c on s tru c t e d by int e r se ctin g m e quation s ordered [!,g,.. ·,I: in equation By= x, the n we call th e inde x se t O: ,g, .. ·, I:} a s th e ex tr e me point L"Onstitution of y' .

s.t. By< x ',

Aft e r s olving above optimi zati on subprobl ems (10) , if the optimal s olution exis ts , we could find the ac ti ve cons traints in By
Under hypothe s is I, S A ha s only on e se t of e xtr e me points, and thi s set of points uniquely descr ibe s th e ge ometrical shape of the S •. The hypothes is I has al s o its prac ti cal background, s uch a s S. be a gen e ralized r e ctangular parallele piped or a closed simplex cons tructed by m+1 con s train e d e quation s, or in anyca ses the hypothes is I can be he ld whe n the variati o n of x in X is small.

Wh e n th e number of th e acti ve con s traint s in the kth subproblem in (10) is m, the n (ID

where D. is the square matri .x con s is ted of m active r ow e leme nts in B, D;:', the inve r se of 0., is obvi o us ly ex is te d. F r om le mma 2, we kn o w that only the con s tra i nt H. y' ( x )< d. is active within the cons traints H, y' ( x )< d., j= I,2,· ..,T. So we only ne ed to introdu ce the co nstr a int H, D;: ' x < d, into th e pr o bl e m (9) t o r e pla ce all cons traints H,y'(xKd., j= I,2,.. ·,T.

Unde r hypoth e si s I and by us ing le mma I, th e following theorem 2 can be obtainetl.

ma)(

(8)

is

Wh e n th e numbe r of th e a cti ve con s traint s of th e kth s ubprobl e m is I, ob v ious ly th e active cons traint B,y= x; in By
f( x )

SoL g(xKO

Hy'(xKd

(10)

whe r e x'EX give n.

HYPOTHES IS I In th e following di scussion we s uppose that , when xE X={xlg( x )';:::.o} c hanges, the e xtreme point cons titution s of S. are fixed.

THEOREM 2 Und e r hypo thes is I, problem equivalent to the following proble m ( 9 )

k= I,2, .. ·,r

(9) i= I,2,.. ·,T

Although (9) i s a ge n e ral math e matical programming problem , but larg e amount of c on s traint s are in c ud e ,t in (9), and many constraints in whi c h are non - a c tive. On the other hand , even x i s gi ven, it i s al s o very diffimit to find out y' (x ), i= I,2,.. ·,T.

into (9) to s ubstitut e all the cons traints H,y' < d" j= I,2,···,T. Whe n the number of the active cons traints is 1< q
From linear programming we know that if we

% 1

Step 2: Solve all subproblems in (10) where the invers e of Then

~

max H,y

i s obviously existed.

s.t By
we could get

y"(X)=[ Y. l=r (~)-'x..-(~Y'JJ;"'x,..... ; l ~'

l x.._

(13)

Ste p 3: Solve the problem

J

Thu s we only ne e d constraint H,y'(x)-< d. into

to introdu c e

k

mBx

the

(14)

re,,)

(9),

s.t.. g(x KO H,y"(x )-<.:::d"

By s olving r s ubproblems in (10) , we could find out all r active constraints in (9), thu s problem (9) can be transformed to

k = ),2,.. ·,r

th e n we o btain the optimal s olution of th e proble m (8).

max f( x ) S tep (: End. s.t. g(x K O H,y'(x )< d"

(14) k= I,2, .. ·,r

Obviou s ly the algorithm s ugges te d he re is quite s imple.

is a ge ne ral nonline ar programming probl e m with r lin e ar c ons traint s and the con s traint s g( x ) < O. Th e optimal s olution of problem <10 en s ure s that all e x tr e me point s of S. are s ubjected to Y. Thus according to theorem 2, its optimal s olution is jus t th e optimal s olution of proble m (8). And problem (8) can be s olved by a two level algorithm with only one iteration, at the lower l e v e l, r lin e ar programming subproblems (10) are solveti to introduce r active c ons tra i nt s to th e upp e r le ve l, at th e upp e r le ve l, the main probl e m (14) is solveo.i to get the optimal s olution, whi ch is a s imllle mathe matical programming problem. (14)

Up to now , t h e linear g e neral emb e d,te,t optimization problem (S) ha s be en s olv e d c ompletely und e r hypothe s i s I. Using the knowledge of linear algebra ,the res ults and the algorithm could al s o be e .x te lllte d eas ily to the mor e ge ne ral lin e ar GEOP when S.={yIB, x+B. y< d umi e r hypothe sis I. But for linear general e mbedded optimization problems without hypothesis I, s ome of above res ults and the algorithm are ina,tquat e . Now we are de ve loping the algorithm for that case. EXAMPLE

We furth e r di sc uss the cases that (8) has no s olution or dive rge s to infinity. If one of r linear programming subproble ms is dive rgent or (It) has no s olution, th e n ( 8) has no s olution . The form e r corresponds to the situation that the boundary of the set S. is unclo s ed and the laller corres ponds to t he s i tuation that there ex i s ts no xE X={ x lgCx )< O} to make S. <;;; Y. If problem (14) i s dive rge nt , th e n s o is (8). Thi s correspolllts to the s i tua tion tlult the boullltnry of Y i s unclo se d.Thu s we hav e the following theorem.

He r e only a s implified prke floating range problem in volving 3 price variables is give n as an example. Let A be an input- output matrix

°

I

0.2 0.1 A= 0.2 OJ 0.3 [ 0.1 0.1

°

~ =[0.8

and THEOREM 3 Under hypothesis I, the optimization problem ( 8 ) c an be s olv e d by a tw o l e vel algorithm with only one ite ration as me ntion e d above , i. e . aft e r s ol ving r lin e ar programming s ubprobl e ms (10) and one nonlinear programming (I(), the optimal s olution can be obtained, or no solution or dive rgence of (8) can be proved.

O.S 0.8J

p, = I.

We can get the price fe,a s ible s imple x :

.8.S ] .8 p,=1.

ALGORITHM

J~ ~ ~Il r~) LO .3 .1 ,p"

then we have

Step I: Select appr opriate x from

o..2p. < .0.6 - 0.4P. +0.1 p,< - 0.1 o..3p.- O.7p,< 0

xEX= {xlg(x )<.Ol

362

;>{)

And the prk--e feasible set is

constrained nonlinear programming problem, which can be solved easily. Its optimal solutions (P;, ~, P;, P;) are:

F.=(plp.<3, -4p.+p,<1, 3p.-7p,
2.165, 7.662, 1.28) <3., 2.151, 7.210, 1.28) n, 2.770, 10.09, 1.28) (~,

The price floating generalized rectangular parallelepiped set is S=(pI P~
respectively corres ponding to given The optimization model of this price floating range problem can be given as:

(w;,

w~,

w;,

w;) :

I, 1, D (2, 2, I, D (I, I, 2, 2) (I,

max Sot.

(P; - ~XP;- P;)+
P;>~

CONCLUSION

p;>p; S<;;;;F.

In t.his paper , the definition and concept of the price feasible set is proposed, whi ch reveals the relationship among prit' es as well as the relationship between price , cost and profit, ami provides some new ideas for t.he t.heoretical study of price. Based on the price feasible set, tJte optimization model of the price float.ing range is developed. But its optimization problem is a new class of opt.imization called General EmbetMed Optimization Problem. For solving t.he optimization model, linear GEOPs are studied, some theoretical results ami a simple two level algorit.hm are s uggested. So the problem to work out. the upper and lower limHs of floating prices quantitatively has been solved complete ly. The model ami the algorit.hm can be provilied as the ,iec\sion support tools for the price management depart.ment. Meanwhile the idea of the model can be parallelly extended to the production planning problem and other socio-economic problems.

where the variables to be solved are p;, p;, p; and p;, whidt are t.he upper and lower limits of the floating prk--es. We should solve the subproblems in (I~) first. In the following subproblms , the variables of the primal problem, p~, p;, p; and p;, are given appropriately. Solving subproblem 1: max p. s.t.

~
p;
P;<3.

Solving subproblem 2:

For t.he general embelided optimizat.ion problem, the authors have s tudied t.he non - linear case and obtained some results and algorHhm, which will be published in other paper. But t.he study of this class of optimization is just at. the very beginning, many open problems should be solved, ami t.he applicat.ions of this kind of optimization should be explored. We believe that GEOP will find its wide applications in engineering design, socio-economic and other various fields.

max -4p.+p, s.t.

~
P;
-4~+P;<- 1.

Solving subproblem 3: max 3p.-7p, Sot.

ACKNOWLEIx.EMENT

~
p;< p,
The authors are grateful for t.he financial support received from the Youth Reward Fund of SINICA and the Youth Scientific Resear c h Fund of Xian Jiaotong University.

3P;-7p;
Then we obt.ain the main problem (14) as follows:

REFERENCES Sot.

Le, Weiliang (J986). Price System Analysis, Modelling and Control. Ph.D Thesis, Xian Jiaotong University

P;>~ >O

p;>p;>,O P;<3 -'~+P;<-I

Le , Weiliang and B.S. Hu
3P;-7p;< .o Here the main problem is a simple linear

363

Report 86-598 and 86-599, Xian Jiaotong University. Le, Weiliang, Q.K. Peng, and 8.S. Hu (1989). General Embedded Optimization Problem. System Science and Mathematics, Vol.9, NoJ.

364