378 European Journal of Operational Research 35 (1988) 378-381 North-Holland

Theory and Methodology

Balancing hydraulic turbine runners: A quadratic assignment problem

Gi lbe r t L A P O R T E a n d H r l ~ n e M E R C U R E l~cole des Hautes F, tudes Commerciales de Montrdal, 5255 avenue Decelles, Montreal H 3 T 1 V6, Canada.

Abstract: Hydraulic turbine runners are used in electricity generation. These consist of a cylinder around which are welded, at regular spacings, a n u m b e r of blades whose weights differ slightly. It is desired to locate the blades around the cylinder in order to minimize the distance between the center of mass of the blades and the geometric center of the cylinder. The problem can be formulated as a quadratic assignment problem. I t is solved by adapting an interchange algorithm devised by Or for the travelling salesman problem. Computat ional results are reported.

1. Introduction

In a recent paper, Mosevich (1986) described the problem of balancing hydraulic turbine runners and presented an algorithm which consisted of selecting the best of a large number of randomly generated solutions. In this article, we reexamine the problem from a different angle and show that it is a quadratic assignment problem (QAP). We then present a solution procedure based on Or's algorithm (1976) which outperforms, in a series of computer tests, that of Mosevich.

2. The problem

We first summarize the problem. A hydraulic turbine runner consists of a cylinder, around which are welded, at regular spacings, a number of blades. In the application considered by Mosevich, there

were between 14 to 18 blades in each turbine runner. These blades weighed around 38000 lbs each, but weight deviations could be as large as +5%. The problem consists of determining the location of the blades around the cylinder so as to minimize the distance between the center of mass of the blades and the geometric center of the cylinder. As in Mosevich, we make three assump- tions in order to model the problem: (i) blade mass is proportional to thickness; (ii) the centers of mass of all blades lie in a common plane, and (iii) around a circle whose center coincides with the axis of the runner. The problem can then be solved in two dimensions: the cylinder is replaced by a circle centered at the origin and the blades by weights to be positioned at regular intervals around the circle.

3. The model

This work was supported by the Canadian National Sciences and Engineering Research Council (grant A4747). Thanks are due to the anonymous referees for their valuable comments.

Received December 1986; revised May 1987

In order to formulate the problem, we intro- duce the following notation:

n : the number of blades; M i : the mass of blade i;

0377-2217/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)

G. Laporte, H. Mercure / Balancing hydraulic turbine runners 379

r : the radius of the circle around which the blades are positioned;

0j= 2 ~ ( j - 1)/n ( j = 1 , . . . , n): the angles at which the blades are positioned;

x;j : a binary variable equal to 1 if and only if blade i is positioned at angle 0j;

X = ( X l l , X12 . . . . , x , , ) : a feasible solution; (X(x) , Y(x)): the coordinates of the center of

mass associated with a feasible solution x. These coordinates are defined by:

X( x ) --- "-~ E M i r cos Ojxij, i = 1 j = l

Y ( x ) = -~ ~ Mir sin Ojxij. t = l j = l

The problem consists of minimizing the following objective:

f(x) = [ X 2 ( x ) + y2 (x ) ]1 /2 . (1)

This is equivalent to minimizing [Mf(x ) / r ] 2. The objective then becomes:

n H n T/

Minimize E E }-". E M, Mk(C°S Oj cos 0; / = 1 j = l k = l ; = 1

+ sin 0j sin O;)xijxk;.

This expression can be simplified and the problem formulated as a QAP (LaMer, 1963):

(P) Minimize E E M, Mk i = 1 j = l k = l 1=1

× cos(O, - ot) x,jxk,,

subject to ?/

E Xi j = 1 ( j = 1 . . . . . n ) , ( 2 )

i = 1

t t

£ xij = 1 (i = 1 . . . . . n) , (3) j = l

x ; j= 0 or 1 (i , j = 1 . . . . . n) . (4)

4. The algorithm

QAP's with general cost functions are, unfor- tunately, very difficult to solve by exact methods (see the recent survey by Finke et al. (1987)). An efficient heuristic procedure is called for. The most common heuristics in use include classical ex-

change algorithms (see Burkard and Stratmann (1978) for a survey) and more recently, simulated annealing (see, for example, Burkard and Rendl (1984)). Most of these methods are inspired from Lin's r-opt algorithms (1985) for the travelling salesman problem (TSP). An initial assignment is gradually improved (or worsened with some prob- ability in the case of simulated annealing) by making permutations of elements taken r at a time until no further improvement is possible. The time complexity of an iteration in an r-opt algorithm is O(nr). Or has proposed an O(n 2) exchange al- gorithm (Or-opt) for the TSP which, according to Golden and Stewart (1985), produces solutions which are just as good as those generated by a 3-opt algorithm. Like r-opt procedures, it is very flexible and can be used for just about any type of assignment problem (i.e. minimize a function of x, subject to (2), (3), and (4)). We now describe the adaptation of Or's algorithm to our problem.

An assignment will be represented by a circular permutation (i 1, i 2 . . . . . i n, il) where ij denotes the element (blade) assigned to position j (angle 0j). In what follows, i 9 should be interpreted as

i j (mod n )"

Step O. Consider a randomly generated permu- tation. Set t = 1.

Step 1. Remove from the permutation the tri- plet of consecutive elements (i,, it+l, it+2) and tentatively insert it between all n - 3 pairs of consecutive elements of the remaining permuta- tion.

- If a tentative insertion decreases the objective function, implement it immediately, thus defining a new initial permutation. Set t = 1 and repeat this step.

- If no tentative insertion decreases the objec- tive function, increase t by 1. If t = n + 1, go to the next step. If t < n + 1, repeat this step.

Step 2. Proceed as in Step 1, by considering this time pairs of consecutive elements.

Step 3. Proceed as in Step 1, by now moving only one element at a time.

Step 4. Print the last permutation obtained. Stop.

5. Computational results

Or's algorithm and the random search method proposed by Mosevich were compared on 245

380

Table 1 Basic statistics on the algorithms

G. Laporte, H. Mercure / Balancing hydraulic turbine runners

Number of RLrN1 (Or-opt) blades n F1 ITR1

RUN2 (random) RUN3 (random)

T1 F2 ITR2 T2 F3 ITR3 T3

10 0.525 4860 12 0.015 7478 14 0.010 10815 16 0.005 14034 18 0.006 17816 20 0.004 22886 22 0.002 27610

6.2 0.711 4860 7.8 0.699 3379 6.2 11.2 0.042 7478 14.3 0.043 5236 11.2 18.9 0.031 10815 24.3 0.042 7634 18.9 27.6 0.019 14034 35.9 0.024 9891 27.6 39.3 0.021 17816 51.7 0.018 12557 39.3 55.0 0.009 22886 72.8 0.015 16106 55.0 73.1 0.014 27610 97.8 0.018 19434 73.1

problems generated as follows. Seven values of n were selected (n = 10, 12 . . . . . 22) and for each of these values, 35 problems were generated. In each problem, the values of M i were distr ibuted accord- ing to a normal distr ibution with a mean of 100 and a s tandard deviation of 5 /3 , so that most M / s would fall within + 5% of the mean.

In RtrN1, these problems were solved by Or's algorithm, starting with 10 randomly generated permutations. The total number of function evaluations (ITR1), the CPU time (T1) and the best function value obtained (F1) were then noted. In RUN2, the random search algorithm was executed until ITR1 solutions were produced. The CPU time (T2) and the best function value (F2) were recorded. Final ly in RUN3, random solutions were generated until the CPU time reached the value T1. Again, the number of solutions examined (ITR3) and the best function value (F3) were computed. These results are reported in Tables 1 and 2. All times are CPU seconds on a CYBER 173.

Table 1 provides basic statistics on the algorithms. All entries represent average values over the 35 problems solved for each value of n. The results confirm the superiority of the pro-

posed algorithm. In all cases, F1 is less than both F2 and F3. The value of T1 is always less than that of T2 and ITR3 < ITR1. This last result is due, we believe, to the fact that the random number calls are relatively time consuming.

Table 2 contains comparative statistics. The first (resp. fourth) colunm represents the propor- tion of problems for which F1 was less than F2 (resp. F3). The remaining columns give the ratios of the corresponding values of Table 1. It can be seen that the Or-opt algorithm provides the best solution in most problems. The relative superiority of F1 is measured by the two ratios F 1 / F 2 and F1/F3 . The ratio T 1 / T 2 tends to decrease with increasing values of n whereas for a given running time, the Or-opt algorithm enables approximately 40% more function evaluations to be carried out than the random search approach.

6. C o n c l u s i o n

These observations lead us to conclude that Or's algorithm dominates the random search pro- cedure. Better results may be obtained by devoting more effort to the generation of an initial permu-

Table 2 Comparative statistics on the algorithms

Number of PROOF1 < F2) F1 blades n F2

T1 T2

PROP(F1 < F3) F1 ITR1 F3 ITR3

10 0.63 0.74 12 0.63 0.46 14 0.80 0.31 16 0.77 0.27 18 0.80 0.32 20 0.74 0.52 22 0.86 0.16

0.79 0.71 0.75 1.44 0.78 0.74 0.45 1.43 0.78 0.86 0.23 1.42 0.77 0.77 0.21 1.42 0.76 0.77 0.37 1.42 0.75 0.71 0.30 1.42 0.75 0.91 0.12 1.42

G. Laporte, H. Mercure / Balancing hydraulic turbine runners 381

ta t ion and by using Or 's algori thm as a post-

processor (as did Go lden and Stewart for the

TSP). A similar approach may also yield an effi- cient heuristic for the QAP.

References

Burkard, R.E., and Rendl, F. (1984), "A thermodynamically motivated simulation in procedure for combinatorial opti- mization problems," European Journal of Operational Re- search, 17, 169-174.

Burkard, R.E., and Stratmann, K.-H. (1978), "Numerical in- vestigations on quadratic assignment problems", Naval Re- search Logistics Quarterly 25, 129-148.

Finke, G., Burkard, R.E. and Rendl, F. (1987), "Quadratic assignment problems", in: S. Martello et al. (Eds.), Surveys in Combinatorial Optimization, North-Holland, Amsterdam, 61-82.

Golden, B.L., and Stewart, W. (1985), "Empirical analysis of heuristics", in: E.L. Lawler et al. (Eds.), The Traveling Salesman Problem, Wiley. New York, 207-250.

Lawler, E.L. (1963) "The quadratic assignment problem", Management Science 9, 586-599.

Lin, S. (1965), "Computer solutions of the traveling salesman problem", Bell System Technical Journal 44, 2245-2269.

Mosevich, J. (1986), "Balancing hydraulic turbine runners--A discrete combinatorial optimization problem", European Journal of Operational Research 26, 202-204.

Or, I. (1976), "Traveling salesman-type combinatorial prob- lems and their relation to the logistics of blood banking, Ph.D. Thesis, Northwestern University, Evanston, IL.