Fixed cost minimization over a Leontief substitution system

R

’

,

,

RECHERCHE OPÉRATIONNELLE

. R

G ^ARY J. K ^ŒHLER

A NDREW B. W HINSTON

G ORDON P. W RIGHT

Fixed cost minimization over a Leontief substitution system

Revue française d’automatique, informatique, recherche opé- rationnelle. Recherche opérationnelle, tome 8, n^oV2 (1974), p. 119-124

<http://www.numdam.org/item?id=RO_1974__8_2_119_0>

© AFCET, 1974, tous droits réservés.

L’accès aux archives de la revue « Revue française d’automatique, in- formatique, recherche opérationnelle. Recherche opérationnelle » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/

conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

R.A.I.R.O.

(8 e année, mai 1974, V-2, p. 119 à 124)

FDŒD GOST MINIMIZATION

OVER A LEONTIEF SUBSTITUTION SYSTEM (*)

par Gary J. KŒHLER, Andrew B. WHINSTON and Gordon P. WRIGHT (2)

Résumé. — Using Bender's décomposition method and properties of a Leontief substi- tution systemt a formulation for solving fixed cost minimizations over Leontief substitution Systems is presented.

It has been pointed out in the literature [2] that Bender's décomposition method appears useful as a solution strategy. Using Bender's .method and the properties of a Leontief substitution System, a solution procedure is spe- cified for solving fixed cost minimization problems over a Leontief substitution System. Preliminary results indicate that for some large problems, this pro- cedure is a viable alternative to procedures specified elsewhere [4],

A Leontief substitution System is characterized by a constraint matrix having exactly one positive element per column and non-trivial rows [3].

Substitute activities are identified by columns whose positive éléments fall in the same row. A salient feature of Leontief substitution Systems is that one and only one activity is optimal for each set of substitute activities.

Consider the problem

Min/(;c) subject to

Ax = b x > 0 b ^ 0

where A is m by n and Leontief, x is n by 1, b is m by 1, and ƒ (•) is a quasi- concave function of the fonn :

(1) This research was supported by the Office of Army Research under contract

# DA-31-124-ARO-D-477.

(2) Krannert Graduate School of Industrial Administration Purdue University, West Lafayette.

120 where

G. J. KOEHLER, A. B. WHINSTON, G. P. WRIGHT

+ d^t x^t > 0 d^t ^ 0 0 xf = 0

The above problem is termed a fixed cost minimization over a Leontief substitution System.

Let the Set Jt contain the column identifications for all columns whose positive coefficients fall in row i. Associate with each continuous variable xt

a binary variable yt and reformulate the fixed cost problem as : Min c'x + d'y

subject to

Ax = b x — ly^O

x> 0 b> 0 v€Y where

Y = { y : y is binary and = 1, i = 1, 2,... m } and

for ail possible JC^. The requirement that

is a resuit of the substitution property of Leontief substitution Systems.

The problem given in équation (1) may be reformulated as : Min d'y + Min c'x

X

subject to Ax = b

x^ly x> 0

Revue Française d'Automatique, Informatique et Recherche Opérationnelle

FIXED COST LEONTIEF MINIMIZATION 121

Using Bender's décomposition method [1], an approximation to the outer problem is now given. Let N dénote the number of times the inner problem has been solved. Also, let n[ be the dual multipliers to the constraint set

Ax = b

at the solution of the inner problem at the /th itération. Likewise, let TC| be the dual multiplier to the constraint set

x < ly

at the f1 itération. The approximation to the outer problem is (2) Min z + d'y

subject to

z > b'iz\ + ly'iz\ i = 1,2,.. JV

Given a y € Y9 the inner problem is easily solved. Let y% be the fixed y vector supplied to the inner problem at the beginning of itération L Then the solution to the inner problem is :

(3) <

where

and A'K is the transpose of the submatrix of A consisting of the columns in the set K. Also

(4) ^ = [c-^'7ci] +

where [a]+ is a vector of [at]+ i = 1, 2, ..,> n, and [aj+ = min [ai9 0].

For computational purposes, it is bénéficiai to note that if Ak is recursive [4], as is usually the case, then [^i]"1 is easily computed.

As an example, consider an arborescence-structured production and inven- tory system. Let xf, yf, and rf be, respectively, the production, end-of-period inventory, and known requirements at facility K in period /. For illustrative purposes let the number of facilities be 2 and the number of periods be 3.

The arborescence model is

Minf(xty)

1 2 2 G. J. KOEHLER, A. B. WHINSTON, G. P. WRIGHT

subject to

yUx + *î —y\ - xf = r\ i =1,2,2,

rf_

x?_tf = r? i = 1, 2, 3

where

and

Âx,y)= t t {ffixf) + ftof

J cfxf + df xf>0

ƒ f (xf) =

l o

and ff(yf) is similarly defined. Let

r1» C2— 3

^y — O

V

' 3 ) 2

. 4 J

\ /

C1 — 3 2

1

6 d\ =

\ 7

By the structure of the problem, the set of all feasible basis is

6

where

To start the solution procedure, an arbitrary initial feasible solution is specified. Let the optimal solution to the continuous portion of the problem be the initial starting basis. Then

^ i = { xi» X2> X3> xu X2> yi }

Revue Française d'Automatique, Informatique et Recherche Opérationnelle

FIXED COST LEONTffiF MINIMIZATION 1 2 3

and the outer problem can be written as follows. Let the continuous variable identifications, xf and yf represent their binary counterparts.

Then équation (2) becomes :

subject to

z^ 54

Then

Ki = {xlylylxlylx\}

or

Kz = {xl,ylxlxî

yî

xl}

and the outer problem becomes (using the first K2) : Min z + d£x + d'yy subject to

z > 54

z > 87 — 200x1 — 300x1 — 500x^

where / = 100.

Continuing in like manner and using all alternative optimal points to form constraints, the following was found.

Itération 3 4

5

6

7 STOP

Y1

Xi,

*i,

xl xl xl

x\

xl

1

V1 Y1

~1 vl

x2, x3,

7i. yl

yl xl

x\,y\,

vl vl

x2, x3,

Y1 V1

^ ,

*î,

xl xl

X i ,

Y2

Xi,

Y2

» - M ,

yl xl xl xl xl xl

xl xl xl xl xl xl xl xl

z > 70 — 700x|

z ^ 73 — 100^ — 700^

z>11~ 200x1 - 300x1 — 300^

z ^ 68 — 200x1

z ^ 59 — 100x1 — 300^

z > 56 — 200^

1 2 4 G. J. KOEHLER, A. B. WHINSTON, G. P. WRIGHT

BIBLIOGRAPHY

[1] BENDERS J. F., « Partitioning Procedures for Solving Mixed Variables Program- ming Problems », Numerishe Mathematik, 4, 1962, pp. 238-252.

[2] LASDON L. S., Optimization Theory for Large Systems, MacMillan Co.,|New York, 1970.

[3] VEINOTT Jr., A. F. « Extreme Points of Leontief Substitution Systems », Linear Algebra and its Applications, 1, 1968, pp. 181-194.

[4] VEINOTT Jr. A. F., « Minimum Concave-Cost Solution of Leontief Substitution Models of Muîti Facility Inventory Systems », Opérations Research, 17,|No. 2, 1969, pp. 262-291.

Revue Française d'Automatique, Informatique et Recherche Opérationnelle n° mai 1974, V-2.