multilevel linear programming - Acsu Buffalo

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THE OPERATIONS RESEARCH PROGRAM DEPARTMENT OF INDUSTRIAL ENGINEERING STATE UNIVERSITY OF NEW YORK AT BUFFALO

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O SUNY AT BUFFALO n

MULTILEVEL LINEAR PROGRAMMING

by Wayne F. Bialas Mark H. Karwan

Research Report No. 78-1 May 1978

Department of Industrial Engineering State University of New York at Buffalo Buffalo, New York 14260

MULTILEVEL LINEAR PROGRAMMING

*

Wayne F. Bialas t Mark H. Karwan Technical Report No. 78-1

*Presented at the Joint National Meeting of the Operations Research Society of America and The Institute of Management Sciences, May 1-3, 1978. tAssistant Professor, Department of Industrial Engineering, State University of New York at Buffalo, Amherst, N. Y. 14260. +Assistant Professor, Department of Industrial Engineering, State University of New York at Buffalo, Amherst, N. Y. 14260.

Abstract The general multilevel programming problem is a set of nested optimization problems over a single feasible region. Control over the decision variables is partitioned among the levels, but a decision variable may impact the objective functions of several, if not all, levels. The two-level linear resource control problem is a special case where the level-two planner controls the effective resource space for the levelone planner. This produces a solution structure with the feasible region viewed by level two as a nonconvex subset of the overall feasible region. An algorithm to solve this problem is proposed.

•

Introduction Many planning situations require the analysis of several objectives. Multiobjective optimization techniques have been developed to permit a more faithful analysis of the tradeoffs among competing goals, and assist a planner in reaching an acceptable compromise. Such methods assume that all objectives are those of a single planner, impacting directly on his state of well-being. Multiobjective optimization fails to recognize that many objectives are ordered within an administrative or other hierarchical structure. For example, the energy policies set forth by the Federal government affect the objectives and options, and hence the strategies, of state officials. This process continues within a hierarchy including local governments, planning agencies, and basic economic units such as firms and households. Each unit in the hierarchy wishes to maximize its individual benefits in view of the partial exogenous control exercised at other levels. In the example, actions at the state level affect the benefits sought by the Federal government which can, in turn, constitutionally exercise first, but only partial control over the state.

Multilevel optimization techniques partition control over the decision variables among the levels, and analyze objectives at all levels of the planning process. A planner at one level of the hierarchy may have his objective function determined, in part, by variables controlled at other levels. However, his control instruments may allow him to influence the policies at other levels and thereby improve his own objective function. The problem of multilevel linear programming was offered by Candler and Norton [2].

Unfortunately, their general problem was imprecisely defined

and they did not realize that the effective feasible regions for higher

1

levels were nonconvex sets. This causes the Candler and Norton algorithm to fail since it assumes convexity. Although the general problem formulation and solution technique proposed by them is incorrect, the basic premise of their problem has far-reaching applications. Some particular problems are amenable to this framework:

(i) Control of oil imports: With the desire to limit oil imports, the Federal government can impose import quotas and duties. This action will affect the consumption of oil by economic units which are maximizing profits. The decisions made by these units would then impact on the objectives of the Federal government including a change in consumption patterns and levels of oil imports.

(ii) Floodplain planning: Seeking to reduce flood risk, government can implement floodplain zoning programs and subsidize flood insurance. These policies will influence the land activity patterns based on the budgets and objectives of land users.

(iii)Job incentive

programs: As industry seeks to maximize productivity, it may implement profit-sharing and other job incentive programs. Such programs, in turn, affect the goals of employees whose decisions impact on industrial productivity.

Similar work in this general area has been conducted on Stackelberg games [1,3,7].Within the broad definition of such games, a static Stackelberg game with fixed leaders and a continuous decision space could be defined to encompass multilevel optimization problems. However, current methodology does not consider the activity space of one player to be a function of the strategies of other players. Such an extension of Stackelberg games would require the payoff function of one level to have discontinuities dependent on the decisions of other players. This formulation is, at best, unwieldly, and perhaps intractable. The term "multilevel" has been frequently used to describe approaches to similar important planning problems. However, in the context of the multilevel programming problem defined in the next section, these previous approaches are primarily decomposition techniques applied to single level problems [4,5,6]. For example, Haimes, Foley and Yu [5] employ lagrangian duality to decompose and efficiently solve a large model for the control of 2



water quality with a single overall system objective of a central planning agency. The dual variables are then interpreted, as with Dantzig-Wolfe decomposition, to determine prices (taxes) to be charged to each subproblem (polluter) for violating pollution standards. The flexibility of the multilevel approach can answer questions regarding the assignment of control over variables to various levels. For example, in some cases, coalitions of levels can improve the objective functions of all levels.

Furthermore, because of the structure of some problems, a single

level could exercise complete control over all levels even though controlling only a proper subset of the variables. This methodology can assess the value of controlling a particular subset of variables, and with this information, policy makers could determine what control should be relinquished or maintained over certain variables. The next section will provide the formal, general definition of a multilevel programming problem. Then the discussion will focus on a special case of this general problem, the two-level linear resource control problem, and its mathematical structure. This will include a characterization of the solution set and some general algorithmi c approaches to solve the problem. Finally, a specific algorithm for finding local optimal solutions will be presented.

General Definition of Multilevel Programming Problems n Let the decision variable space (Euclidean n-space) R be partitioned among

r

x = (x ,x ,...,x ) n 1 2

levels,

k k, k nk • 4 x k 2 "-- ,xn k=1,...,r , R = (x 1 ' 1 k r where varying by

k=1 only

nk = n. x

k

E

n Denote the maximization of a function f(x) over R by n n k+2 x nk +l r k+1 k+2 ^ R in R R k given fixed x, x

max f(x) k k+1 k+2 ,...x ,x x ix

.

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The general multilevel programming problem can then be defined as max f (x) x

r

r

st: x

max f (x) r-1 r-1 1 r Ix

st: max xr-2

(P

f

ix

r-2 r-1

(x) ,x

r

r-1 ) 4

(P

r-2 ) 4

st: max f (x) 11 2 1 x ix ,x- ,...,x st:xESCR

This establishes a collection of nested mathematical programming problems {P 1 ,.. ,P r }.

The feasible region, S = S l , is defined as the level-one feasible

region. The solutions to P S

2

1

in R

n

1

2

3

for each fixed x ,x ,..., x

r

form a set

1

} called the level-two feasible region over = {x E S :f (x) = max f (x) 1 1 -r xi 1X 2 ,X 3 ,..., x

which f (x) is then maximized by varying x 2

2

3

r

for fixed x ,...,x .

In general, the level-k feasible region is defined recursively

as

^ Sk-1 : S = {t x x E - fk-1 (x)= max fk-1 (x) } x Note that x

k-1

is a function of x lc

k-1 ^k ^r ix ,...,x

r

,...,x

. Furthermore, the problem at

each level can be written as

( Pk )

max f (x) k k k+1 x ix ,...,x x E

which is a function of x

k+1

Sk

r

r

,

and (P ): max f (x) defines the entire x€S r

problem.

4

r

The Two-Level Linear Resource Control Problem The two-level linear resource control problem is the multilevel programming problem, of the form 2

max c x x 1

st: max c x 2 (P ) 4

(P

Here, level 2 controls x one by restricting A

1

x

l

2

x l

A x + A x 2 1 x > 0 .

2

2 (P ) - x + 2x < 1 2 x + x < 2 1

+ x 3 2 = 4

1x

1 2 4

For x = 2, the unique level-one solution is (x 11 x 2, x 3 ) = (2,2,2) with value 4. 2 The corresponding level-two solution value is 3. However, for x 2 = 1, there exists a set of alternate optimal solutions to P 1 X = f(x ,x x ) 1 2' 3 0 < x

1

< 3, x

2

= 1, x

3

= 3} .

The corresponding level-two objective for

this set of solutions ranges continuously from to be returned to level two for fixed x

1

to 3

1 For a unique solution 2'

2 and to induce level one to return

x = 3 for x = 1, a side payment to the level-one objective from level two may 2 1 be employed. For the example, the level-one objective function, max x

+ x 2

3

+ E(x

1

+ 1 -x ) 2

with E > 0 sufficiently small, is a perturbation

which accomplishes this. Given this side payment scheme, x 2 = 1 is the optimal decision for level two.

Nonconvexity In the two-level linear resource control problem, r S 1 = l x > 0 : A x l + A2 x 1

2

< b ,

is a convex set. However, 2 l 1 S = Ix f S : c x = max c 1 xi1 x11i x2

6

2 need not be. Therefore, P , which can be written as 2 max c x x(S 2 involves the optimization of a linear function over a nonconvex region. Consider the geometric example in Figure 1. The set S Figure 1 is a subset of the edges of the boundary of S dimension, S

2

2 1

for the problem in . In problems of higher

is composed of edges and faces of the boundary of S

1

. Consider

the following three dimensional example: max x 2

x st:

2

+ x

3

max x

3 xi ,x3 x 2 x x x x

where S

2

1 1 1 1

+ x - x + x + x

2 2 2 2

- x + x + x

3 3 3

> 2 < 2 < 6

+ x

> 2 3 x < 1 , 3 -

is the hatched region shown in Figure 2.

Relationships to Multiobjective Programming Optimal solutions to the multilevel programming problem may not be Pareto-optimal. While cooperation might improve the objective functions at every level, the order and independence with which decisions are made prevent such cooperation. This rules out any algorithmic approach which seeks only Pareto-optimal solutions and is one of the main distinguishing characteristics between multiobjective and multilevel programming. For an example of this behavior, consider Figure 1. Both levels have higher objective function values at point (a). However, for x

2

fixed at x

2'

2

level one will choose x l = xl (point (b)), thus point (a) is not in S This leads to the best choice of x

2

to be x = 0 with the optimal solution at point (c). 2

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A Sufficient Condition for Complete Control Consider the two-level linear resource control problem. Given any basis B C A for the set of constraints Ax = b, one can write the equivalent set of constraints on x: Bx

B

+ Nx

xB = B

Or

N -1

= b b - B

-1

N xN .

When x N is fixed, x is uniquely determined. Thus to have complete control of B the solution, the level-two planner need only control the complete set of nonbasic variables corresponding to any basis.

Further Characterization of S

2

and P2

The following theorem and its corollaries help to characterize both S2 2

and the optimal solution for P

in the two-level linear resource control

problem. Theorem I Suppose S= {x : Ax = b, x > 0} is bounded. Let S 2 = {x =(x 1 ,x 2 ) E S i : 11 1 1, c x = max c x /. Then the following hold x

l

1X 2

(i) S

2

r be any r points of S i , such that t=1

(ii) Let

{yd

2

x = A y t t t

S

Proof:

s

(i)

1

C S

2

with A

S

1

> 0, t -

X

t

= 1. Then A

t

> 0 implies y

2 by the definition of S .

(ii) (By contradiction) Let y1 , y 1

2

x = (x ,x ) =

2"

r X

t=1

2

A y t t

S

A

= 1.

r > 0 A l > 0, at -

X

t=1

8

t

..' y , and

r

f S

1

with

t

( S2

k optimal

solution

Figure 1

Example of NOnconvexity of S

2

Figure 2 EXample of S 2in Three Dimensions

10

1 2 = (Y1' Y1 ) f S 1 1 1 I with c 17 1 > c y l .

Then there exists y1 1

Suppose y

Using (i), Y l E S .

1 1

CX

1

c

= c 1A 1y1

-1 2 _2 Y 1 Y1 ) (

Therefore,'

=AY + XAyc 1 1 t=2 t t - , 2 Noting that x = x2 and

since S1 is convex.

=

such that

A.

1

>

1 1 1 cAy y < cA Y + ttt 1 1 1 t t t t=2

t=2 2 and A l > 0, we have established an x with the S

1 2

Given, x = (x ,x ) following properties: (a) x= - 2 (b) x =

sl 2

1 1 < c1-1 x .

(c) c x

This contradicts the definition of S 2 for the fixed value of x .

2

since x

t

1 1 2 S should have maximized c x

Therefore Al > 0 implies y

1

2 S . Since the

choice of y l 'among the y's was arbitrary, we have proven A t > 0 implies yt c S Any point which positively contributes in any convex combination forming a point in S

2

, must also be in S2. Since this is true of any point,

including y which are extreme points of S 1 , Corollary 1.

the following corollary results:

If x is an extreme point of S 2 , then x is an 1 extreme point of S .

Proof: (By contradiction) Let x be an extreme point of S te .

not an extreme point of S1 . A

1

> 0,

>

Suppose x is

Then there exist extreme points y ,...,y r c S1, and Such that

t=1

x

y t=1

From Theorem 1, this implies point of S

2

.,Yr t S and hence x cannot be an extreme

a contradiction.

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Recalling that P may be formulated as

2 max c x and noting the 2 x(S

correspondence of extreme points in S2 S and S 1 , the following result is derived.

Corollary 2.

An optimal solution to the two-level linear resource

control problem (if one exists) occurs at an extreme point of the constraint set of all variables (S ).

Proof: The two-level linear resource control problem can be written as

2 max c x 2 x(S 2 Since c x is linear,if a solution exists, one must occur at an extreme 2 point of'S (alternative optimal solutions at nonextreme points may exist), By Corollary 1, this must be an extreme point of S 1 . This result justifies extreme point search procedures as a basis for algorithmic approaches to solving the two-level linear resource control problem.

Algorithmic Approaches It has been shown that an optimal solution to the two-level linear resource control problem occurs at an extreme point of the level-twO feasible , 2 region, S 2 . Let LS ] denote the convex hull of S 2 . Since the sets of extreme points for S

2

and [S 2 ] are identical, the problem (P2 )

max st:

2 c x

x ( [S 2 ]

is an equivalent formulation for the two-level linear resource control problem. This suggests a search for cutting plane procedures to approximate the convex 2 hull of S as a direction for future research..

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Any desirable algorithm for the two level linear resource control problem should exhibit some particular properties. ^1 ^2 Consider the solution, x = (x ,x ) to the following problem: 2

max c x 6; ) st: Ax=b, x>0 In (P), the level two planner is given full control over all variables. Now fix x x f

2

^2

= x

and solve the following problem with solution x to determine if

S2 :

1 max c x st:

2 1 = (b-A x ) A1 x 1 x > 0

If x = x, then x

t

S

2 is an optimal solution to the overall problem. For

example, note that in Figure 1, the vector c 2 could be changed to produce a solution to (P) at any extreme point of S 2 ..

9

The set $ does not vary with

changes in the second level objective, and, hence quite different choices of c 2 can produce an optimal solution after solving (P). For the example shown in 2 Figure 1, two particular choices of c which lead to such a condition for the 1 2 level-one objective shown are both c2= c and c = -c 1 . Thus both highly complementary and highly conflicting objectives (as well as many inbetween) may lead to solutions after solving the two linear programming problems (P) and (P). Any reasonable algorithm should have the ability to easily solve any problems for which x e S 2 . An Algorithm to Find Local Optimal Solutions

Consider the following portion of abounded simplex tableau to be employed in the proposed algorithm:

13

2 xl

2

2

RHS

.._

x

B

1

x B . 2 xB

m

2 1

2 k

r 22

r

Y 11

Y1

Y lk

Y21 .. Yml

Y22 — - — '''''' " Y2k

r

r

2

-b2 1-; . _

Y m2

Ymk

b

m

2 k represent the nonbasic level-two variables which are at nonzero values, and r i2 ,...,rk2 represent the reduced costs of these variables The variables, x 2

x

with respect to the level-two objective function. In terms of the present basisBCA,b=B_ b- y yx 2 where (y ,Y j lj 2j j= 1 and xB x denotes the ith basic variable.

,t mj

= Y, = B l j

-1

(a ,a 2j

Assume that S is boUnded with no degeneracy and no alternative optimal solutions exist for P 1 for any feasible x 2 . The following algorithm guarantees a local optimal solUtion. ^1 ^2

Step 1. Solve the following problem with optimal solution x = (x ,x and

optimal tableau T via the simplex method:

Step 2. Set x2= x

2

and solve the following problem via bounded simplex

(k=u=x 2 ) beginning with tableau T: 1 max c x st: Ax=b 2 ^2 x =x x >0 Let the optimal solution be x. If x = x, stop; x is a global optimal solution. Otherwise, go to step 3a with current tableau T and relax the constraints x

2

^2 = x .

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Step 3a. If all nonbasic variables are equal to zero, go to step 4 with current tableau T. Otherwise go to step 3b. Step 3b. If b. > 0 for all i, go to step 3c. Otherwise, without loss of such that 1 < j < k and yQj into the basis via a degenerate pivot. Go to

generality, consider b t = 0. Choose Bring

0. 3 step 3a.

y

x2

3

Step 3c. Consider any nonbasic variable which is at a strictly positive value, 2 2 2 If r. < 0, increase xj until it enters the basis. If r, %P. 0, 3 decrease x 2 until either it reaches zero or it must enter the basis. 2 say x..

Go to step 3a. Step 4. Beginning with tableau T solve the following problem via a modified simplex procedure: max c

st:

2

X

Ax=b x>0

The modification is as follows. Given a candidate to enter the basis (one for which c

2 x will increase) only allow it to enter if the

resulting basic solution,

x,

will be contained in S

2.

This is

determined by obtaining the solution x to the following problem: max st:

1 c x A

2 i < (b-A x ) 2 2 2 1 x > 0, x = x

1x

via dual simplex on repeated applications of step 4. If x = x

1

then enter the candidate into the basis. Repeat step 4 until no more candidates exist which satisfy the above mqdification, then stop,

15

Validation and Convergence The algorithm begins by finding the maximum of the second level objective over the entire feasible region, S 1 . A check in step 2 is then made to determine if the resulting solution is in S 2 . If so, the algorithm terminates with a global optimal solution and has solved what was previously termed an easy problem. If termination does not occur in step 2, the resulting solution from step 2 is by definition contained in S 2 . Since the bounded simplex algorithm was employed, a number of nonbasic level-two variables may be at nonzero values 2 corresponding to appropriate components of x. introduced by fixing the components of x

2

Degeneracy may also have been

from step 1.

2 is indeed contained in 5 . Since bounded simplex was employed, a number of nonbasic level-two variables may be at nonzero values corresponding to appropriate components of x 2 . Degeneracy may also have been introduced by fixing the components of x 2 from step 1. The purpose of step 3 is to relax the constraint x = x 2 and to move to an extreme point x° which satisfies x o

S

2

and cx

o

> cx. If a right hand side,

, from the current tableau is equal to zero then step 3b is entered to perform a degenerate pivot. Some nonbasic variable, x 2 , j=1,2,...,k, is then brought into the basis at its current positive level and x B becomes nonbasic at its current value of zero. Thus the number of basic variables at level zero is reduced by one. This is repeated until no degeneracy is present. Note that such a pivot is always possible, that is, yk

0 for some j=1,...,k. Suppose

that y 2, = 0 for all j=1,...,k. Then repeated applications of step 3c would result in a degenerate extreme point of the original feasible region, S 1 , since xB will remain zero no matter how x 2l ,...,x are varied. original nondegeneracy assumption.

16

This contradicts the

2 If all b.> 0 but there are still nonbasic level-two variables, x2 . .. ,x k' 2 at nonzero values,. then step 3c is entered. Any variable x j , j=1,...,k, is chosen to be increased or decreased depending on its reduced level-two cost, 2

2

r.. Since there are no explicit upper bounds on x 3

increase is limited

by a current basic variable reaching zero. The original problem is bounded,so this must occur. If x

2 is decreased, again a current basic variable may reach

2 itself will become zero. In either case, the number of nonzero J nonbasic variables is decreased by one. zero or else x

The points generated in step 3c can be shown to be contained in S

2

which

is assumed when step 4 is entered. Recall that 1;1 0 for all i as a result of step 3b. Thus there exists two scalars, 0 1 > 0 and 0 2 > 0 such that any increase or decrease in x, by an amount less than or equal to 01 and 02 respectively results in a feasible solution (i.e., a point in S 1 ). This implies that the current solution, which is in S 2 , is a convex combination of two feasible points resulting from a strict increase and a strict decrease in x 23 . By Theorem I, such points must also be in S 2 . Thus each point resulting from step 3c must be contained in S 2 . Step 4 is entered when an extreme point of S has been obtained. A modified 2 simplex method is used to take steps in S along which the level two objective increases. This is accomplished by using the normal simplex rules with objective c 2x along with a check that no basis change results in leaving S 2 . The algorithm terminates with an extreme point solution in S 2 which has the property that all adjacent extreme points either lead to a decrease in c 2 x or are not in S 2 . Thus a local optimal solution is obtained. Convergence of the algorithm is established by noting the following facts:

17

(i) The feasible region defined by S



Ax = b, x > 0t is bounded

and each basis is nondegenerate. (ii) Steps 1, 2 and 4 are finite since the simplex, bounded simplex and dual simplex procedures are finite under fact (i). (iii) Each application of step 3b strictly decreases the number of basic variables at level zero and also the number of nonzero nonbasic variables. (iv) Each application of step 3c reduces the number of nonzero nonbasic variables by one. Conclusions Multilevel mathematical prograMming problems, if carefully defined, can serve as useful tools in modelling structured economic units. Such models can predict the inefficiencies of non-Pareto-optimal decisions and identify the seats of true control within hierarchical organizations. This paper has proposed a general mathematical structure for such problems, and specifically characterized the two-level linear resource control problem. For this problem, Theorem I illustrates a key property of the nonconvex feasible region viewed by level two. As a foundation, it justifies extreme point solution techniques and obviates the need for methods to establish the convex hull of the level two feasible region. Towards this goal, this paper has offered an adjacent extreme point method which can find local, and sometimes global, optimal solutions to the two-level linear resource control problem. Acknowledgement The authors wish to expreSs their sincere gratitude to Professbr Daniel P. Loucks for introducing them to this problem

18

Figure Captions FigUre 1. Example of Nonconvexity of S

2

2 Figure 2. Example of S in Three Dimensions

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References 1. Basar, T., "On the Relative Leadership Property of Stackelberg Strategies," Journal of Optimization Theory and Applications. Volume II, No.6, 1973, pp.655-661. Candler, W. and R. Norton; Multi-Level Programming, Unpublished Research Memorandum, DRC, World Bank, Washington, D.C. August 1976. 3. Cruz, J.B., "Stackelberg Strategies for Multilevel Systems," in Directions in Decentralized Control, Many Person Optimization and Large-Scale Systems, Y.C. Ho and S.K. Mitter, Eds., New York, Plenum Press, 1976, pp.139-147. 4. Goreaux, L.M. and A.S. Manne, Multi-Level Planning: Case Studies in Mexico, North-Holland, Amsterdam, 1973. 5. Haimes, Y.Y., J. Foley and W. Yu, "Computational Results for Water Pollution Taxation Using Multilevel Approach," Water-Resources Bulletin, Volume 8, No.4, August 1972, pp.761-771. 6. Haimes, Y.Y., W.A. Hall and H.T. Freedman, Multiobjective Optimization in Water Resources Systems, Elsevier, Amsterdam, 1975. 7. Simaan, M. and J.B. Craig, "On the Stackelberg Strategy in Nonzero-Sum Games," Journal of Optimization Theory and Applications, Volume 11, No,5, 1973, pp.533-555.

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