optimal procedures for some constrained selection problems

Optimal Procedures for Some Constrained Selection ProblemsAuthor(s): David A. HarvilleSource: Journal of the American Statistical Association, Vol. 69, No. 346 (Jun., 1974), pp. 446-452Published by: American Statistical AssociationStable URL: http://www.jstor.org/stable/2285676 .

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Optimal Procedures for Some Constrained Selection Problems

DAVID A. HARVILLE*

Suppose that Yo,- *, YQ are q + 1 criterion variates and that X is a p-dimensional column vector of predictor variates. A principal result of this article is the analytical solution of the problem of determining a p X 1 vector b to maximize the correlation p(Yo, b'X) subject to the constraints p(Yi, b'X) = wi, i = 1,- * *, q, where wl, - -, wq are specified constants. Some frequently encountered selection problems can be solved by applying this result.

1. INTRODUCTION

A problem frequently encountered by educational institutions, by the military services and by animal and plant breeders is that of selecting people, animals or other objects from a group of such items. Naturally, they wish to make their selections in such a way that the selected items are of the highest possible "quality." Generally, quality cannot be assessed directly, at least not at the time the selections must be made; however, several measurements that are thought to be related to quality may be available on each item. A (uestion of considerable interest is that of how to best use these measurements in making the selections.

Problems of this type are especially common in the geneticist's domain. There the objective may be to select from some group of animals or plants a smaller group to be used for breeding purposes. An individual's quality would be interpreted as its "breeding value" for a trait or as a weighted average of its breeding values for several different traits. The observed or phenotypic values of these traits, both on the individual itself and possibly on its relatives, would comprise the information on which the selections are based.

The Air Force Academy is faced with a problem whose characteristics are much the same [7]. Each year, those to be enrolled in the freshman class at the Academy must be chosen from a group of candidates. A primary measure of a candidate's quality is taken to be his academic performance during his freshman year at the Academy, as reflected in a variable referred to as academic score. Here, the selections are based on a battery of "selection tests" that is administered to all the candidates.

In one mathematical formulation for this selection problem, the candidates for selection are conceived to be

* David A. Harville is mathematical statistician, Applied Mathematics Research Laboratory, Aerospace Research Laboratories, Wright-Patterson AFB, Ohio 45433. The technical report [4] served as a basis for the present article. That report also contains some supplementary material. The author is indebted to a referee for some helpful suggestions and to Dr. R.E. Miller of the Air Force Human Resources Laboratory, who provided the information in Table 1.

a random sainple from some population of items. It is supposed that the quality of an item is indicated by an unobservable variate Yo, and that the value of a vector X = (X1, *, X,)' of p "predictor" variates is known for each item. The joint probability distribution of (Y0, X) is assumed to be known. When X is a vector of continuous variates, it is supposed that a selection rule is wanted which has the form that an item is selected if X E S and rejected otherwise, where S is some subset of the sample space of X. Furthermore, we require that the selection rule have some specified intensity a (0 < a < 1); i.e., that S satisfy

Pr (X & S) = a. (1.1)

The best selection rule is commonly defined to be the one which maximizes the expected value of Y0 for a selected item. The problem of determining the best rule is then mathematically that of choosing S to maximize the conditional expectation E (Yo X C S) subject to the constraint (1.1). As shown by Cochran [1], it follows from the Neyman-Pearson lemma that the best rule consists of a truncation from below on the regression function E (Yo X), where the truncation point is deter- mined so as to satisfy (1.1).

It is the above mathematical formulation that serves as a starting point for this article. This forinulation has led to selection procedures that have, at least in some instances, been applied with apparent success. It is beyonid the scope of this article to review the many other formulations that have been proposed. The interested reader is referred to such articles as those by Lehmann [6], Cochran [1], and Williams [13]. Lehmann covers what he refers to as Model I selection problems, in which the candidates are regarded as "fixed." Cochran and Williams deal with Model II problems, where the items are, as in this article, assumed to be a random sample from a population.

Often, while one facet of quality may be of primary interest, others may be of secondary concern. Thus, we may want the selected items to rate highly on the primary trait, but at the same time want them to conform to, or to exceed, certain standards on the auxiliary traits. To incorporate these ideas into our mathematical formulation, we suppose that the primary criterion of quality

? Journal of the American Statistical Association June 1974, Volume 69, Number 346

Theory and Methods Section

446



Optimal Procedures for Constrained Selection 447

is reflected in the unobservable variate Yo and that the secondary criteria depend on q other unobservable variates Yj, , Yq The joint distribution of Yo, , Yq with X is assunmed known. Here, we suppose that we want to choose the set S, on which the selection rule is to be based, to maximize E (Yo X C S) subj ect both to the constraint (1.1) and the q equality or inequality constraints

E(Yi I X S)t =, > }E(Yi) + ci, i = 1, *.. q, (1.2)

where cl, , c, are specified constants. Kempthorne and Nordskog [5] describe the following

situation. In selecting hens for breeding purposes, a procedure was desired for which the selected hens would, on the average, be at some intermediate level with respect to breeding value for egg size but, subject to that constraint, rate as highly as possible on a weighted average of breeding values for several other traits including egg production. They obtained the solution to the constrained selection problem when all of the constraints (1.2) are equality constraints and ci = 0 for all i. Rao considered the problem when all of the constraints are inequality constraints. In [8], he gave the solution for the special case where all of the ci's are zero. In [10], he mentioned the more general situation where the ci's are possibly nonzero as a problem, but did not carry through to a solution.

By making use of the Neyman-Pearson lemma, it can be shown that a given selection rule is optimum for the constrained selection problem if and only if the set S on which it is based satisfies (1.1) and (1.2) and there also exist constants t, k1, . - ., kI such that S differs by only a set of zero probability from the set

{X: E(Yo I X) -4- kiE(Y1 I X) + * * + k,E(YQ I X) > t}.

That is, the best rule essentially takes the form of a truncation on a linear combination of the regressions of the Yi's on the predictor variates. Thus, if the criterion and predictor variates are normally distributed, we need only consider selection rules consisting of a truncation on an index that is linear in X1, . - ., X,. Moreover, for any real number t and for any linear combination of the predictor variates, say b'X, we have, in the case of normality,

E(Yilb'X > t)= E(Yi) + (O,/c)aip(Yi, b'X),

where c = Pr (b'X > t), Xc is the derivative of the standardized normal distribution function taken at the point where that function assumes the value c, ai is the standard deviation of Yi, and p(Yi, b'X) denotes the correlation between Yi and the linear combination. [When b = 0, we put p(Yi, b'X) = 0.] Consequently, in the normal case, the problem of maximizing E ( Yo X E S) with respect to S subject to the restrictions (1.1) and (1.2) is equivalent to the problem of maximizing p (Yo, b'X) with respect to b subj ect to the constraints

p( Yi,b'X) {=, > }(aci)/(+aai), i = 1, * ,q (1.3)

The equivalence is in the sense that if b* is a solution to the latter problem, then S = {X: b*'X > t* , where t* is defined by Pr (b*'X > t*) = a, is a solution to the first problem.

We now turn our attention to the followinig problem.

Problemi 1: M\Iaximize p(Yo, b'X) with respect to b subject to the constraints p(Yi, b'X) = wi, i = 1, *--, q, where w = (w1, *, iv)' is any q X 1 vector of constants.

In Section 2, we relate Problem 1 to Problem 2 and, in Section 3, present some results on Problem 2. In Section 4, we use these results to determine when a feasible solution to Problem 1 exists, to obtain explicit expressions both for the solution to Problem 1 and for the maximum of the objective function p(Yo, b'X) in that problem, and to examine the dependency of this maximum on the vector w. (We shall refer to any vector which satisfies the constraints of a maximization problem as a feasible solution to that problem. A vector at which a maximum is attained will be called a solution.)

These results on Problem 1 are of interest for at least three reasons:

1. When all of the constraints in the constrained selection problem are equality constraints, that problem can, in the case of normality, be resolved immediately by direct application of the Problem 1 results with wi = (aci)/( Iai).

2. Even when the constrained selection problem contains one or more inequality constraints, the task of computing an optimum selection rule can, again in the case of normality, be reduced to solving problems having the form of Problem 1. The mechanics of such a computing procedure will be indicated in Section 5.

3. The Section 1 results, when applied to the constrained selection problem in the case of normality with all equality constraints, enable us to examine the maximum in that problem as a function of either wi = (aci) / (,,gi) or ci itself. 'Displaying' the 'penalty' that is extracted by the constraints as a function of the 'severity' of the constraints can give invaluable information in deciding what values of the ci's to impose in a particular application. Such a display was attempted by Cunningham, Moen, and Gjedrem [2] for a sheep-breeding problem. Their results could have been obtained much more efficiently if they had had access to the results to be presented here.

Subsequently, the covariance matrix of X will be denoted by V and will be assumed to have full rank. Also, we let ai represent the p X 1 vector of the co- variances of the standardized variate Yi/of with Xi, *, Xp, respectively, i = 0, *, q, and put A = (a1, *.., aJ). We do not require that A have rank q. Using well-known results, we find p(Yi, b'X) = (a'b)/ (b'Vb)iy i = 0, , q, unless b = 0 in which case these correlations are taken to be zero by definition. Thus, the information about the distribution of Yo, *, Yq, X that is relevant in resolving Problem 1 is all contained in the parameters ao, *-*, aq, and V. No assumption that these vrariates are normal, or even continuous, is needed or will be used in conjunction with that problem. Normality need only be invoked in applying the Problem 1 results to the selection problem.



448 Journal of the American Statistical Association, June 1974

2. A SECOND PROBLEM AND ITS RELATIONSHIP TO PROBLEM 1

Consider the following problem.

Problem 2: Maximize a'n with respect to c subject to the constraints

= 1 (2.1) and

A's= w. (2.2)

Lemma 1: The entire class of nonnull solutions to Problem 1 can be generated from b = -y by allowing y to range over the positive real numbers and r to range over the solutions to Problem 2. The class of nonnull feasible solutions to Problem 1 can be generated from the feasible solutions for Problem 2 in similar fashion.

The proof of Lemma 1 is straightforward and will be omitted. The significance of this lemma is that it allows us to work with Problem 2 rather than Problem 1. Both of these problems involve nonlinearities; however, Problem 2 appears more amenable to solution by analytical techniques.

In conjunction with Lemma 1, we note that b 0 is a solution to Problem 1 only when w = 0 and when in addition rank (ao, ..., aq) = rank (A). Even then, there are also nonnull solutions unless rank (A) = p, in which case b = 0 is the unique feasible solution in Problem 1.

3. SOLUTIONS FOR PROBLEM 2 3.1 Existence of Solutions and Feasible Solutions

It is not hard to show that the set consisting of those vectors t for which (2.1) is satisfied is bounded, so that, in Problem 2, we are maximizing a continuous function over a closed and bounded set. It follows that, if Problem 2 has a feasible solution, i.e., if there exists a vector c that simultaneously satisfies (2.1) and (2.2), then necessarily Problem 2 has a solution, i.e., then there is a vector at which a maximum is attained.

We next investigate the existence of a feasible solution in Problem 2. Obviously, the consistency of the linear system (2.2) is a necessary condition for the existence of such a vector; however, in itself, it is not sufficient. The additional condition that is needed is supplied by Lemma 2.

Here and in what follows, we denote by T- a generalized inverse of an arbitrary matrix T; i.e., T- is any matrix such that TT-T- T. Also, we use q* inter- changeably with rank (A).

Lemma 2: Suppose that the vector w is such that the linear system (2.2) is consistent. Then,

(i) if q* < p, a necessary and sufficient condition for the existence of a feasible solution to Problem 2 is that

w' (A'V-'A)-w < 1; (3.1)

(ii) if q* = p, a feasible solution exists if and only if w'(A'V-'A)-w = 1; and

(iii) in either case, if w'(A'V'-A)-w = 1, Problem 2 has the unique feasible solution

- = V-'A(A'V-'A)-w. (3.2)

Proof: It follows from result (ii) of [9, p. 49] that

minimum ''V-c = w'(A'V-1A)-w, (3.3) T : A'r= w I

which establishes the "necessity" part of (i). Moreover, the minimum value (3.3) is uniquely attained at the point (3.2) which implies (iii) and the "if" part of (ii). The "only if" part of (ii) follows from the fact that, when q* = p, the equation A's = w has a unique solution; namely, the vector (3.2). The "sufficiency" part of (i) is demonstrated in Section 3.2 where a vector v is constructed which, when the linear system (2.2) is consistent and the condition (3.1) is met, is not only a feasible solution for Problem 2 but a solution as well.

3.2 Derivation of Solutions

Problem 2 can now be solved with the aid of Lagrange- multiplier techniques. We shall assume that the vector w is such that the linear system (2.2) is consistent and that in addition w satisfies the condition of Part (i) or (ii) of Lemma 2. Then necessarily a feasible solution and a solution for Problem 2 exist.

Subsequently, we take A* and P to be as defined by

A* = [A, ao] and

P = [I -V-1A (A'V-'A)-A']V-1.

The following lemma will be needed.

Lemma 3:

a6Pao > 0, if rank (A*) = q* + 1, = 0, if rank (A*) = q-.

Proof: Partition A*'V-lA* as

FA'V-'A A'V-'ao] La'V-'A a'V-'ao' o o

and observe that rank (A'V-'A) = q* and rank (A*'V-lA*) = rank (A*). The first part of the lemma now follows directly from a lemma of Rhode [11]. A proof for the second part can easily be constructed from Rhode's proof.

The Lagrangian function for Problem 2 is

F(r, 2) = Xoa's + X1(1 - c'V) + A'(w - A's), (3.4) where I = (X0, X1, A')' is the vector of Lagrange multipliers.

The derivative of (3.4) with respect to the vector 's is

dF - X0a0- 2X1V-s- A.




The objective function a6i assumes its maximum, for c satisfying (2.1) and (2.2), at the point co only if there is a solution for I to the system

Xoao - 2X1Vro - AA =01 ItV=C o 1, (3.5) A',ro w,

(Xo, X1) ? 0. J

We distinguish three cases where feasible solutions to Problem 2 exist:

Case A: w'(A'V-'A)w - 1. Case B: w'(A'V-'A)-w < 1, rank (A*) = q* + 1. Case C: w'(A'V-1A)-w < 1, rank (A*) = q* < p.

The solution of Problem 2 for Case A is made trivial by Lemma 2(iii). There is a unique feasible solution, given by (3.2).

In Case B, after considerable matrix manipulations requiring Lemma 3, as well as some knowledge of existing results on generalized inverses (see, e.g., [12, Ch. 1]) and linear equations, we find that the Lagrangian system (3.5) is equivalent to the system

Xo/(2X1) = i [1 - w'(A'V-'A)-w]1(a6Pao)-i,

(A'V-'A)A XoA'V-1ao - 2X1w,

t E[Xo/(2X1)]Pao + V-'A(A'V-'A)-w

(equivalent in the sense that they have identical solutions for vo and 2).

Thus,

a/V-1A(A'V-'A)-w + [1 - w'(A'V-'A)-w]i(a'Pao)1 is the maximum value of a&s for v satisfying (2.1) and (2.2). The maximum is achieved at the unique point

= V-A(A'V-'A)-w + 1- w'(A'V-lA)-w]i(a'Pao)-TiPao. (3.6)

A corresponding solution for the Lagrange multipliers is Xo = 1,

i = (1/2)(a'Pao)I[1 -w'(A'V-'A)-w]-i,

A - (A'V-'A)-EA'V-'ao - (aPao)f X { 1 - w'(A'V-lA)-w I-iwv].

In Case C, there exists a q X 1 vector r for which

ao = Ar. (3.7)

In particular, rO= [(A'V-'A)-]'A'V-lao

satisfies (3.7), so that for any vector t satisfying (2.2),

aol = r;A'" = row = aXV-lA(A'V-'A)-w. (3.8)

Thus, in Case C, every feasible solution to Problem 2 is a maximizing vector and the maximum value is given by the right side of (3.8). Note that an explicit feasible solution can easily be obtained by replacing a0 in (3.6) with any p X 1 vector a* such that rank [A,a*] = q*? 1.

By again making use of Lemma 3, it can be demonstrated that, in Case C, the Lagrangian system (3.5) has no solution for co, I for which Xi 54 0. Thus, in Case C, the Lagrangian system is equivalent to the system

Xoao - AA = 0, = 1,

A'? o ?, w =O xo FZo,0 Xi 0.

This system has as one solution for the Lagrange multipliers Xo = 1, Xi = 0,

A = (A'V-1A)-A'V-lao.

4. RESULTS FOR PROBLEM 1 4.1 Solution

In light of the results and discussion of Sections 2 and 3, the resolution of Problem 1 is now a simple matter.

Problem 1 has a feasible solution if and only if the linear system A'b w is consistent and in addition either (i) w'(A'V-'A)-lw= 1, (ii) q* < p and w'(A'V-'A)-lw < 1, or (iii) q* = p and w = 0.

If there exists a feasible solution to Problem 1, then the maximum value attained by the objective function p(YO, b'X) for vectors b satisfying the constraints is

a/V-'A(A'V-'A)-w + [1 - w'(A'V-'A)-w]1 X (a6Pao)i; (4.1)

and a maximizing vector is

b = V-1A(A'V-'A)-w + [1 -w'(A'V-1A)-w] X (a*'Pa*)-iPa*,

where a* = ao, if rank (A*) = q* + 1, and otherwise a* is any p X 1 vector for which rank [A, a*] = q* + 1. (In the trivial case where q* p and w = 0, the maximizing vector is b = 0.)

4.2 Dependence of the Maximum on the Constraints

Take p*(w) to be the maximum value of p(Yo, b'X) for b satisfying the constraints in Problem 1, i.e., p*(w) is given by (4.1). As pointed out in Section 1, it may be instructive, in an application to a selection problem, to examine p*(w) for different values of w. Accordingly, we give some results on p* (w), considered as a function of w. Here, we assume that q* = q, an assumption that proves convenient, yet results in no substantive loss of generality.

When q < p and w'(A'V-iA)-'w < 1, the partial derivative of p* with respect to the components of w are given by

c1p*/dw = (A'V-'A)-'[A'V-lao - (aoPao)i X {1 - w'(A'V-'A)-w}-iw].

This formula was derived fromn the results of Section 3.2, where expressions were obtained for the vector A of Lagrange multipliers. This result can of course also be obtained by direct differentiation of (4.1).




If rank (A*) = q + 1, then, for w such that

w'(A'V-'A)-'w < 1,

p* is the sum of a linear function and a strictly concave function and consequently is a strictly concave function itself. Otherwise, p* is linear in w.

In studying the dependence of p* on w in an application, we may wish to fix all of the components of w except one, say the ith, and deterImine wi to satisfy

p*(w) = p, (4.2)

where p is a constant, i.e., we may want to plot contours of the function p*. It will be assumed that rank (A*) = q + 1. Any vector w that satisfies (4.2) necessarily (Lemma 2) satisfies

(w', p) (A*'V--'A*)-l (w', p) '=1. (4.3) By making use of well-known results on inverses of partitioned matrices and on the solution of quadratic equations, it can be shown that solutions to (4.3) for wi exist if and only if the fixed components of w satisfy

?'(B'V1B)' ?< 1, where

El = (W1l . , Wi-i, Wi+, , W. q P)

and B = (a, a * , a i_1, .,. a, ao);

in which case the solutions are

-, afV-1B(B'V-1B)-1; 4 [l -'(B'V-1B)-1E]i

X Ea'I - V-1B(B'V-1B)-'B'}V-1a]. (4.4)

Thus, our task is reduced to checking which (if either) of the values (4.4) satisfy (4.2).

5. INEQUALITY CONSTRAINTS

When all of the constraints in the constrained selection problem are equality constraints, that problem can be resolved by direct application of the results given in Section 4 for Problem 1. We now show that, even when one or more of those constraints are inequality constraints, an optimum selection procedure can still be computed from the Problem 1 results. It will be assumed that rank (ao, *, a,) = 1 + rank (a,, aq) < p, as will often be the case in practice.

It will be supposed that the constraints (1.3) have been labeled so that the first q' of them are equality constraints and the remainder are inequality constraints. Take Q to be the set consisting of the first q integers. If b* maximizes p(Yo, b'X) with respect to b subject to the constraints (1.3), then, for some subset K* of Q,

p(Yi, b'X) = (acJ)/1( aai), for i E K*, (5.1) and

p(Yi, b'X) > (ac )/(4h-rT), for i Q- K*.

It follows that p (Yo, b 'X) assumes a relative maximum at b* for those b satisfyinlg (5.1). (Note that b* is necessarily nonnull.)

Now consider the following procedure for resolving the constrained selection problem:

1. For each possible subset K of Q that includes the first q' integers, determine all vectors at which p(Yo, b'X) attains a relative maximum for b satisfying the constraints

p(Yi, b'X) = (aci)1(40ii), i E K. (5.2)

2. From the totality of the vectors computed in Step 1, delete those that violate one or more of the constraints (1.3).

3. If no vector remains, then there exists no vector that satisfies all of the constraints (1.3). If one or more vectors are left, choose one for which p(YO, b'X) is the largest. The chosen vector necessarily maximizes p(Yo, b'X) for b satisfying (1.3).

In conjunction with the preceding procedure, we note that every vector at which p (Yo, b'X) attains a relative maximum for b satisfyingf (5.2) also yields a global maximum for those b and that all of these vectors are equivalent in the sense that the value of p(Yi, b'X) is, for any i, the same for all of them. Thus, in Step 1, for each K we need only compute a single vector, which can be obtained from the Section 4 results. The necessary computations can be further reduced by making use of the technique described in Hadley [3, p. 71-2].

6. AN ILLUSTRATION In the Air Force Academy example introduced in

Section 1, academic score during the freshman year was identified as being the primary criterion of quality. In addition, there are other aspects of a cadet's performance during that year that are of secondary interest. The variates labeled "military rating" and "early motivational elimination" mneasure two of these. The first variate is self-descriptive. Early motivational elimination is a zero-one variate: 0 is assigned if the cadet leaves the Academy within one year of entry for any reason judged to have a significant motivational component; otherwise 1 is scored. The 16 tests identified in the table are among those that have been considered for inclusion in the battery of selection tests.

We now attempt to phrase this example in terms of our mathematical formulation, thus permitting utili- zation of our selection procedure. In doing so several difficulties, common to many of the potential applications, are encountered. These necessitate some mnodifi- cations of the selection procedure.

In the Academy setting, p = 16, q = 2, X1, ,X16 correspond to the test scores, and YO, Y1 and Y2 refer to academic score, military rating and early motivational elimination, respectively.

The variance-covariance matrix for the variates YO, Y1, Y2, X1, ..., X16 is not known but an estimate is available. Estimates of correlations between these variates are provided in the table. These estimates were based on data from 746 individuals, except those involving early motivational elimination which were based on data from 895 individuals. Here we adopt the customary practice of proceeding as though the estimates are the true parameter values. The sensitivity of the results for the




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constrained selection problem to errors of estimation is an area that remains to be investigated, though some work has been done on the sensitivity of the solution for the unconstrained problem (see, e.g., [13] .

In the Air Force Academy setting, it is to be expected that there will be an upper limit on the number of candidates that can be accepted for admission. Also, the Academy may want the numnber of incoming freshmen to exceed some minimum number. Note that, while our selection rule would select a specified fraction a of the candidates on the average, in a given application, it could specify the selection of a fraction that is un- acceptably larger or smaller than a. Here, a reasonable way to proceed might be to start with the selection procedure that is optimum for a value of a internmediate to the upper and lower limits on the selected fraction, and to then add or delete candidates according to their rank on the associated selection index b'X so as to bring the number to be enrolled within tolerance

The optimality of basing our selection on the index b*IX, where b* maximizes p(Yo, b'X) subject to the. constraints of (1.3), was established under the assumption of normality. The question arises whether it is reasonable to base our selections on this index when one or more of the variates are nonnormal, as in the Academy setting where the variate early motivational elimiation is not only nonnormal but is in fact an indicator random variable. Obviously no general answer is possible - however, the fact that this index was chosen so as to correlate as highly as possible with Yo subject to constraints on its correlation with Y1, , * -, Y would seem to have a certain intuitive appeal even in the absence of normality.

Those w, w2 values for which there exist feasible solutions to Problem 1 (those for which w'(A'V"'A)-"w < 1) are identified in the figure for the Academy example. (Here, the correlations between the variates are being taken equal to their estimates.) Contours for p*(w), the maximum value of p (Yo, b'X) in Probleni 1, are also

Contours for p*

[w2

W!(A}VHA) I W'- I

1 2~~~~~~~~~~~~~. / X < X /XX~~~~0

-A -2 24 W




displayed in the figure, which was constructed by using formula (4.4).

By making simple changes of scale, the figure can be used to display, for any given a, the maximum value of (,a/a))aop(Yo, b'X) } subject to the constraints p(Yi, b'X)

= (aci),' (4,co-), i = 1, 2, as a function of cl and c2. This function is of obvious interest in conjunction with the constrained selection problem, even when the constraints are inequality constraints as we would undoubtedly want them to be in the Academy setting. It indicates how the expected level of academic score among selected individuals is affected by imposing constraints of various degrees of severity on the expected levels of Y1 and Y2, information that would be valuable in settling on values for ci and c2. Conceivably one might also want to plot or tabulate the coefficients bi, *, b16 of the optimum selection index b'X as a function of w, and w2 (or of cl and C2); however, this information would seem to be of lesser interest and is not given here. The Academy example serves to illustrate that there is nothing magical about the particular choice cl =c = 0. There, we would undoubtedly want to use values of cl and c2 greater than zero, leading to expected levels for military rating and early motivational elimination that are higher after selection than before.

A striking feature of the figure is the ever-increasing steepness of the p*-surface as we approach the boundary, w'(A'V-'A)-lw = 1, along lines parallel to either the w1-axis or w2-axis. That this characteristic is a general phenomenon, and not just an anomaly associated with the particular example that we have chosen, can be seen by examining the analytical expression for ap*/aw derived in Section 4.2. That expression indicates that the partial derivative of p* with respect to wi becomes arbitrarily close to either + cc or - when wi is chosen so that w'(A'V-'A)-lw is sufficiently close to 1. In terms of the constrained selection problem, the significance of

this phenomenon is that the average value of the primary trait among selected items might be altered considerably by relatively small changes in the severity of the constraints if the constraints were already stringent.

[Received July 1972. Revised October 1973.]

REFERENCES [1] Cochran, W.G., "Improvement by Means of Selection,"

Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, J, Neyman, ed., Berkeley: University of California Press, 1951, 449-70.

[2] Cunningham, E.P., Moen, R.A., and Gjedrem, T., "Restric- tion of Selection Indexes," Biometrics, 26 (March 1970), 67-74.

[3] Hadley, G., Nonlinear and Dynamic Programming, Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1964.

[4] Harville, D.A. and Reeves, T.E., "Optimal Linear Indexes for Some Selection Problems," Technical Report No. 72-0123, Aerospace Research Laboratories, Wright-Patterson AFB, Ohio, 1972.

[5] Kempthorne, 0. and Nordskog, A.W., "Restricted Selection Indexes," Biometrics, 15 (March 1959), 10-19.

[6] Lehmann, E.L., "Some Model I Problems of Selection," Annals of Mathematical Statistics, 32 (1961), 990-1012.

[7] Miller, R.E., "Predicting First Year Achievement of Air Force Academy Cadets, Class of 1968," Technical Report No. 68-103, Personnel Division, Air Force Human Resources Laboratory, Lackland AFB, Texas, 1968.

[8] Rao, C.R., "Problems of Selection with Restrictions," Journal of the Royal Statistical Society, Ser. B, 24, No. 2 (1962), 401-5.

[9] , Linear Statistical Inference and Its Applications, New York: John Wiley and Sons, Inc., 1965.

[10] , "Problems of Selection Involving Programming Tech- niques," Proceedings of the IBM Scientific Computing Sym- posium on Statistics, White Plains, New York: IBM Data Processing Division, 1965, 29-51.

[11] Rhode, C.A., "Generalized Inverses of Partitioned Matrices," Journal of the Society for Industrial and Applied Mathematics, 13 (December 1965), 1033-35.

[12] Searle, S.R., Linear Models, New York: John Wiley and Sons, Inc., 1971.

[13] Williams, J.S., "The Evaluation of a Selection Index," Bio- metrics, 18 (September 1962), 375-93.



optimal procedures for some constrained selection problems

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