Stability of the solution of standardization and unification problems with respect to accuracy of approximation

V.S,SR. Cornput. Maths. Math. Phys. Vol. 20, No. 3, pp. lOl-108,198O Printedin Great Britain

Owl-5553/80/030101-08$07.50/O 0 1981. PergamonPressLtd.

STABILITY OF THE SOLUTION OF STANDARDIZATION AND UNIFICATION PROBLEMS WITH RESPEm TO ACCURACY OF APPROXIMATION*

Moscow

(Received 6 Febnwy

1979; rwised 4 July 1979)

IN ORDER to check the stability of the solution of ~~~~tion and action problems, their solutions as the accuracy of approximation of the criterion function increases are considered. The mathematical properties of these problems are investigated, on the basis of which rules are established whereby versions can be rejected more eff%iently than is possible with the usual branch and bound methods. 1. A wide class of ~~~~tion and action problems can be foisted as discrete programming problems. It is only in the simplest cases, when, e.g. parts of the initial series have homogeneous properties and can be ordered with respect to the values of a principal parameter, that these problems can be solved by using an efficient agony based pan [ 1,2] . In general, when the parts of the initial series do not have the properties mentioned, or when the dynamic nature of the development of the series is considered, in the context of an indeterminate demand and extra ~0~~~ of an integral type, the problems have to be solved by more general methods of mathematical programming and combinatorics. A possible approach to the solution of the general problem, utilizing the branch and bound algorithm and the multiplicative algorithm of the simplex method, is described in f3] ; it is based on the fact that, by ~trodu~g auxihary variables and connecting conditions, piecewise linear approximation of the non4inear criterion function can be realized, and the initial problem reduced to a partially integer-valued Iinear prig (1 .p_) problem. The coarser this piecewise linear appro~tion, the less complete is the account taken of the non-linear and multi-extremal properties of the original problem, and the less trustworthy is the solution obtained: practical examples show that, if the approximation is too inaccurate, the solution obtained may be q~~tively as well as qu~titatively different from the optimal solution. In this co~ection, it is of practical as well as theoretical interest to check the stability of the solution as the accuracy of piecewise linear approximation of the criterion function is gradually increased. But it has to be borne in mind that, as the number of linear pieces increases, so does the number of otter-bed variables; hence, since the volume of ~rnpu~tio~ in the branch and bound method depends exponentially on the dimensionahty of the problem, we are faced with a very rapid rise in the volume of computations as the accuracy of approximation improves. And apart from this, there is serious ~~n~~en~ in increasing the number of auxihary variables and connecting conditions, since it becomes more difficult to solve the ordinary continuous 1.p. problems that arise in the branching process.

*Zh. vj&ist

Mat. mat. Fiz., 20,3,640-647,1980.

101

I. B. Vapnyamkii

102

In view of these contradictory aspects, it is worth seeking better computing schemes with as little sensitivity as possible to increasing accuracy of approximation of the criterion function, and also to look for mathematical properties of the standardization and unification problem, which might provide the basis for more efficient rejection of versions as compared with the ordinary algorithm of the branch and bound method. These are the topics to be discussed below. 2. It was shown in [3] that a wide class of standardization and unification problems can be stated as follows: to minimize

(1) under the constraints

9.

j-1,2

GJ'ijl~Gjr'p

Y i-i

i-4,2,

,...,

n,

E--l,2 ,..., q,

q, ....m. l--1,2,...,

(2)

j-1

6 f-

0, 2,

1

NW4 Niz>O,

i-i,2,...,m,

Here,i=l,2 ,..., m is the initial series of alternative types of technical system (parts), n is the set of different types of work, I = 1,2, . . . , q are the possiile versions of i=l,2,..., demand. The following data are assumed to be given: the cost C;u of initial development and preproduction work on the fth part; the coefficient C; of series and service expenses; the coeffrdent pi of fall of series and seMce expenses, -i
Solution of standardizationand unification problems

103

By solving problem (1) - (5) we can find the total number Nix of parts and their distribution with respect to types of work in different versions of demand Nil. The systems for which Nix > 0 form the optimal series, the search for which is the primary content of the standardization and unit?cation problem. 3. Let us now describe the method of solving problem (1) - (5). The total expenses corresponding to the i-th part,

represent, since -1 < fli < 0, a concave (upwards convex) monotonically increasing function of Ni,,definedforA$x >Oandhavingajumpof-co asA& ++O. We divide eachNix axis by means ofr - 1base points OtN,< . . .
2,. . . ,s},

ri={O,

1).

(7)

The variables Jli indicate the number of the approximating piece of step-line, while ri establish prohibition (at value 0) and allowance (at value 1) on the use of the i-th part for satisfying the demand. Using (7), the piecewise linear approximation of the criterion function (1) can be written as

where

(9) (10) M is a “large” number, M * Ci,.

104

I. B. Vapnyarskii Since the variables ri will be used instead of 6i we also write the analogue of condition (3):

(11)

Those ri for which one of the two integers 0 or 1 is not fmed, will be called variable. The concept of variability is analogous to the concept of continuity of the integer-valued variables occurring during computations by the branch and bound method when solving the partially integer-valued 1.~. problem. For the variable variables we shall put $i = s. In the light of this concept we can refme the meaning of the first line in condition (9): the initial expenses are taken equal to zero, not only for prohibited 7i = 0, but also for the variable ri i.e. for all ri which have not received a fixed value of 1. The initial problem (1) - (5) is thus reduced to problem (8) - (1 l), (2), (S), (7), which is a discrete progmmming problem. It can be solved by the same mathematical procedure as was used in [3] for the partially integer-valued 1.~. problem, i.e. with the aid of the branch and bound method and the multiplicative algorithm of the simplex method. An important feature of our problem is that there is no need to introduce auxiliary variables (of type Sri) and connecting conditions (of type (8), (9), see [3], when improving the accuracy of approximation of the criterion function (1). Improvement of the accuracy of approximation is linked with an increase in the number s of approximating sections and with suitable choice of the base points IV,, r = 1, 2, . . . , s - 1, which can be made e.g. by the method of least squares, while restricting the domain of definition of the total expenses [0, N,,] , where Nmax is a sufficient large value of the total number of parts (systems), such that Ni,GNm,, i-l, 2, . . . , m. Obviously, as the approximation accuracy increases, i.e. as s increases, the optimal solution of the discrete progr amming problem (8) - (1 l), (2), (S), (7) Nix, Nir’, Ipr, yr: can change. The main interest is in the values of the integer-valued variables rip i = 1,2, . . . , m, defining the qualitative composition of the optimal series. Theoretically, determination of the stable (with respect to approximation accuracy) solution (Al,..., rm) represents a difficult, and to date, unsolved problem. Practically, to find the stable solution, we only need to solve a series of consecutive discrete programming problems (8) - (1 l), (2), (5), (7), with increasing values of s = 1,2, . . . (and with suitably chosen base points). 4. Let us describe the structure of the tree of versions. The tree vertices will be divided among m levels. The single vertex of zero level corresponds to the solution of problem (8) - (1 l), (2), (5) withvariablesyi,i= 1,2,. . . , m, for which, as we have stipulated, we take J/i = s, i = 1,2, . . . , m It follows from (9), (10) that &,=O, Ci=Ci,, i=l, 2, . . . , m, SO that the present discrete programming problem converts into an 1.p_problem.

On solving this problem, we fmd the number il corresponding to min Nir,

and consider the

s + 1 branches issuing from the zero level vertex and ending in 1st level ve&ces. Put &= (ri, $5)) then the first of these s + 1 vertices corresponds to 5;‘I = (0, s), and the rest to i=l, 2 ,**** m, All these vertices will be called continuations of the thevalues &,=(&r), r-=1,2, *. . , s. zero level vertex. Similarly, for each k-th level vertex we solve the ~o~~on~ 1.~. problem with k fixed and from the results, we fmd the number ik+r corresponding to ruin Nix, values l&r,. . . , 1, the first tree is M,=[ (St-Q”+‘-I]/& much more compact than the second, since Ml < Mz. This advantage results from taking account of the interdependence of the systems corresponding to different approximating sections of a step-line. In particular, prohibition ri = 0 is equivalent to s~~t~eous pro~~ition of all s systems, while allowance ri = 1 implies allowance of only one of the s systems corresponding to approximating sections of the i-th step line. It will be shown below that, by taking account of this interdependence of systems, we can not merely construct the turn possrble tree of versions, but we can also formulate some mathematical properties of the standardization and un&ation problem, which can provide the basis for a more efficient rejection of versions. 6. To faciliate the fo~o~ treatment, we shall introduce some definitions. A sequence of vertices, each of which is the contimration of the previous vertex, will be called a branch of the tree of versions. Each branch consists of vertices of different levels. If a vertex belongs to a given branch, we say that the branch passes through this vertex. Consider a branch passing through a given k-th level vertex with ftved values L, . . . Ci k and containing vertices of higher levels. We are interested in the nature of the variation of certain i-th components NioP of the optimal total numbers corresponding to pth level vertices, p = k, k+l , . . , , m, of the given branch We isolate the vertices ir, . . . , ikr for which the corresponding value rii = 1. The set of these indices will be denoted by 11, On the basis of the rules given in Paragraph 4 for constructing the tree of versions, the coefficients ?ik+r for the s + 1 vertexcontinuations, which are (k + 1)th level vertices, take respectively the values M, C, k, I~, *.., C.‘krl”’ Hence the coefficient 5.rk+r does not decrease on passing from the k-th level vertex, considered where it is equal to C&+r , to the s + 1 vertex~ont~~tions.

I. B. Vapnyarskii

106

Since the connecting conditions remain unchanged here, then obviously, the total number of ik+14h systems cannot increase. If it were to decrease, some types of operation would have to be performed at the expense of other systems, Le. their total numbers do not decrease. We thus have Lemma 1 below. Consider a k-th level vertex, k < m, and a branch joining it with the initial vertex of the tree of versions. To a vertex of this branch there correspond fmed values tip==(?‘iqr$iq), q=l. 3 . . ) k. Denote by Z, the set of the indices i for which 3pZ= 1. M)

.

Lemma 1

The components NizP, p=k, kf 1, . . . , m, i=Z,, corresponding to vertices of any branch passing through the given vertex, do not decrease asp increases from k to m Two k-th level vertices, with values 5, h= (1, ri) for which all and G,,=(Qz), the previous k - 1 values f,j) j-1, 2, . . . , k-l, are the same, will be called adjacent. Branches through two adjacent vertices, and having, for p > k, the same values of tip, will be called similar. Consider two adjacent k-th level vertices with values

tik El (1, r) and cik=(l,

ri), r,(r.

Lemma 2

N&, p=k, k+l, . . . , m, The components corresponding to vertices of the same level of two similar branches passing through two given adjacent vertices, do no increase as r1 decreases from r to 1, Lemma 2 is proved by similar arguments to those used for proving Lemma 1. 7. Now consider a k-th level vertex with value crr( = (1, r) . This vertex belongs to a branch joining it with the zero level. We define for this branch the set Z1. If, for at least one i, i E II, the value ofNfzk is greater than the right-hand boundary N, of the interval [N,+ N,] , then this excess is retained, by Lemma 1, for all branches through the given vertex, including, for vertices of the m-th level, the branches corresponding to completely integer-valued solutions. But none of these vertices can contain an optimal solution of the discrete programming problem, since, in view of the fact that the expenses function is monotonically increasing and concave, the ~th approximating section for the i-th system gives, to the right of the right-hand end of interval a greater level of expenses than does an approximating section with order number [N,-1, Nrl greater than r. In short, we have proved: l%eorem 1 If, for some k-th level vertex, with cik = ( 1, r), it turns out that, for at least one i, i E N1 zk is greater than the right-hand boundary of the rth interval [N,_,, N, 1, then no branch passing through the given vertex can contain the optimal solution. On the basis of Theorem 1, given the property mentioned, we can disregard (reject) all the branches through the given vertex, regardless of the value of the lower bound at the vertex, whose value, from the point of view of the branch and bound algorithm, may require further branching.

Solution of standardization and unification problems

107

Now assume that all branches through a given k-th level vertex, with cik = (1, T), are considered. At all the vertices lying at levels p > k, let the values of Npk x belong to intervals to the so that the maximum value Of Npikx belongs to some interval left of the interval [N,+, N,], Then, in the set of similar branches, passing through the adjacent vertex [NV,-i, NJ, ricf. l>rl, 3;.kP(19r- l), the values of Np, , willalso,byLemma2,betotheleftofN,1.1frthen no branch through this adjacent vertex can contain the optimal solution, since, in view of the fact that the expenses total function is monotonically increasing and concave, the (r - 1)th approximating section gives in this case higher expense than does a section with number less than r - 1. In this case, therefore, it is pointless to investigate the branches passing through adjacent and after rejecting these vertices with their vertices (1, r-l), (1, r-2), . . . , (1, r,+l), continuations, we sensibly start with the vertex l&k= (1, r,) . We have thus proved: Theorem 2 If, for all the branches though a k-th level vertex with {ik = (1, r), the values of Npikx at the vertices at levels p > k, do not exceed the right-hand endNrr of some rr-th interval [N,,_,, N,,], ri
I. B. Vapnyarskii

108

The results of Theorems 14 may easily be taken into account in the strategy of one-way circuit of the tree of versions (see [4,5]); hence we can greatly reduce the number of vertices at which 1.~. problems have to be solved, which in turn leads to a reduction in the computing time and volume of computations when finding the stable solution of a standardization and unification problem.

Tmnslated by D. E. Brown

REFERENCES 1. CHUEV, Yu. V., Method for selecting optimal series of technicaI systems, Sfandarly i kachesfvo, No. 7, 52-54,1969.

2. BERESNEV, V. L, GIMADI, E. Kh., and DEMENT’EV,V. T., Exfremal problems ofstandardisafion (Ekstrermll r@e zadachi standartkatsii), Nauka, Novosibirsk, 1978. 3. VAPNYARSKII, L B., On mnnerlcaI methods of solving problems of the mathematiad theory of standardization, Zh. v~chfsl.Mat. mar. Fir, 18, No. 2,484-487,1978. 4. FINKEL’SHTEIN, Yu. Yu., Approximate methods and applied problems of discrete programming (Priblizhennye metyody i prikkdnye zadachi diskretnogo programmirovaniya), Nauka, Moscow, 1977. 5. ROMANOVSKII, I. V., and SOROKINA, M. G., Orwway circuit of the tree in the Land and Duig method, Zh. vj%hM. Mat. mat. Hz., 13, No. 1,221-227,1973.