BUSINESS MANAGEMENT A+ WORK, ON TIME, NO PLAGARIZING; ON TIME

Shadow profit maximization and a measure of overall inefficiency

Subhash C. Ray

Published online: 14 April 2007

� Springer Science+Business Media, LLC 2007

Abstract Determining the profit maximizing input–output

bundle of a firm requires data on prices. This paper shows

how endogenously determined shadow prices can be used in

place of actual prices to obtain the optimal input–output

bundle where the firm’s shadow profit is maximized. This

approach amounts to an application of the Weak Axiom of

Profit Maximization (WAPM) formulated by Varian [(1984)

The Non-parametric approach to production analysis. Eco-

nometrica 52:3 (May) 579–597] based on shadow prices

rather than actual prices. At these shadow prices, the shadow

profit of a firm is zero. The maximum shadow profit that

could have been attained at some other input–output bundle

is shown to be a measure of the inefficiency of the firm.

Because the benchmark input–output bundle is always an

observed bundle from the data, it can be determined without

having to solve any elaborate programming problem.

Keywords Shadow prices � Weak Axiom of Profit

Maximization � Data envelopment analysis

JEL Classifications C61 � D20

1 Introduction

In data envelopment analysis (DEA), the efficiency of a

firm is measured by comparing its observed input–output

bundle with a reference point on the frontier or the graph

of the technology. Radial measures of technical efficiency

are either input- or output-oriented. In a radial input-ori-

ented model one seeks the maximum equi-proportionate

reduction in all the inputs of a firm that would be possible

without violating the feasibility of its output bundle. In

the output-oriented approach, on the other hand, the

objective is to expand all outputs by the same factor

without using any additional input. When the technology

exhibits non-constant returns to scale, the two approaches

yield different measures of efficiency. In the case of

constant returns to scale, although the efficiency measures

are identical, the reference bundles for comparison are

different, however. In a typical empirical application one

has to choose between an input-oriented and an output-

oriented model.

When input and output prices are available, the refer-

ence bundle is one that maximizes profit and an inefficient

firm attains full efficiency by simultaneously altering its

inputs and outputs as needed. There are, indeed, several

approaches in the DEA literature that allow changes in both

inputs and outputs in order to obtain the efficient projection

of an inefficient input–output bundle even without the

benefit of prices. Färe et al. (Färe, Grosskopf, Lovell—

FGL) (1985) introduced the concept of graph efficiency

and the corresponding hyperbolic distance function mea-

sured by the maximum scalar by which all outputs can be

expanded and all inputs can be contracted at the same time.

Chambers et al. (Chambers, Chung, Färe, CCF) (1996)

introduced the directional distance function and the cor-

responding Nerlove–Luenberger measure of efficiency.

Here one seeks to increase all outputs and reduce all inputs

by the same proportion. In both these approaches, however,

a single parameter determines how the output bundle is

expanded and the input bundle is contracted.

This paper has benefited from very insightful comments from an

anonymous referee on an earlier version of the manuscript. The usual

disclaimer about responsibility for errors applies.

S. C. Ray (&)

University of Connecticut, Storrs, CT 06269-1063, USA

e-mail: [email protected]

123

J Prod Anal (2007) 27:231–236

DOI 10.1007/s11123-007-0036-8

The non-radial Russell efficiency measures defined by

Färe and Lovell (1978) permits outputs or inputs to change

by different proportions. But like the radial measures, they

also are either input- or output-oriented. Pastor et al.

(1999) introduced a generalized Russell measure that al-

lows individual outputs to increase and the individual in-

puts to decrease by different scale factors. In a related

strand in the literature, Briec (1997, 1998), Frei and Harker

(1999), and Briec and Leleu (2003) explore alternative

ways to project an observed input–output bundle on to the

frontier allowing inputs as well as outputs to change

simultaneously.

In none of these approaches, however, does the refer-

ence bundle show an increase in any input or a decrease in

any output compared to observed input–output bundle of

the firm.1 Yet, when the firm maximizes profits the optimal

bundle can show either an increase or a decrease in any

input or output so long as the resulting profit is higher.

Determining the profit-maximizing bundle of inputs and

outputs requires data on the prices faced by the firm under

evaluation. This paper shows how endogenously deter-

mined shadow prices of inputs and outputs of a firm can be

used in place of actual prices to obtain the optimal pro-

jection of its observed input–output bundle where its sha-

dow profit is maximized. As shown below, our approach

amounts to an application of the Weak Axiom of Profit

Maximization (WAPM) formulated by Varian (1984). But

our evaluation is based on the shadow prices rather than

actual prices. Moreover, the benchmark input–output

bundle is always an observed bundle from the data and can

be determined without having to solve any elaborate pro-

gramming problem.

The rest of the paper unfolds as follows. In Sect. 2, we

present the basic non-parametric methodology proposed in

this paper. Section 3 describes a simple computational

procedure with a 2-input, 2-output example. Section 4

concludes.

2 The non-parametric methodology

Consider a data set for N firms from an industry. Let yj be

the m-element output vector and xj the corresponding n-

element input vector of firm jðj ¼ 1; 2; . . . ;NÞ. Under the

standard assumptions of convexity of the technology, free

disposability of inputs and outputs, and variable returns to

scale, an inner approximation to the unobserved production

possibility set of this industry is

S ¼ ðx; yÞ : x � XN

1

kjx j; y �

XN

1

kjy j;

(

XN

1

kj ¼ 1; kj � 0; ðj ¼ 1; 2; . . . ;NÞ ) :

ð1Þ

The efficient input-oriented projection of any observed

input–output bundle (x0,y0) is (h0x0,y0), where

h0 ¼ min h : ðhx0; y0Þ 2 S: ð2Þ

The corresponding input-oriented measure of technical

efficiency is

s0 x ¼ h0: ð2aÞ

Similarly, the output-oriented efficient projection is

ðx0;u0y0Þ, where

u0 ¼ max u : ðx0;uy0Þ 2 S: ð3Þ

The output-oriented measure of technical efficiency is

s0 y ¼

1

u0 : ð3aÞ

Note that employing either (2) or (3) involves a prior

judgment on whether expanding outputs or contracting

inputs is more important in a given context.

For FGL’s graph efficiency measure, the efficient pro-

jection of ðx0; y0Þ is ð 1 d0 x0; d0y0Þ obtained from the hyper-

bolic distance function

d0 ¼ max d : 1

d x0; dy0

� � 2 S: ð4Þ

As can be seen for the 1-input, 1-output case, the actual

and the projected input–output bundles lie on a rectangular

hyperbola. For an efficient projection, d0 must be greater

than or equal to unity. Note that here inputs are reduced

and outputs are increased simultaneously.

Another measure of graph efficiency based on simulta-

neous changes in inputs and outputs is the Nerlove–Lu-

enberger efficiency measure that is derived from CCF’s

directional distance function

b0 ¼ max b : ð1� bÞx0; ð1þ bÞy0 � �

2 S: ð5Þ

In both (4) and (5), however, a single parameter deter-

mines how both inputs and outputs change.

Note further that because ðx0; y0Þ 2 S; d equal to unity is

always a feasible solution in (4). Hence at the optimal

solution d0 � 1. Similarly, b0 � 0 in (5). That is, in both of

these models outputs may only increase and inputs may

only decrease.

1 Restricting inputs to decrease and/or outputs to increase avoids any

trade off between input conservation and output expansion. This may

be a valid approach when no behavioral assumption is made.

232 J Prod Anal (2007) 27:231–236

123

Now suppose that we had information on the output and

input prices for the firm under review. Specifically, assume

that p0 and w0 were the output and input price vectors,

respectively. In that case, the optimal projection of the

observed input output bundle would be ðx0 �; y

0 �Þ satisfying

the inequality

p00y0 � � w00x0

� � p00y� w00x 8ðx; yÞ 2 S: ð6Þ

Define p0 � � p00y0

� � w00x0 � and p0 � p00y0 � w00x0. A

measure of the unrealized profit of the firm is

D0 ¼ p0 � � p0. It may be noted that in order to get to the

profit-efficient projection, the firm does not increase all of

its outputs or decrease all of its inputs by the same pro-

portion. In fact, it may increase or reduce individual inputs

or outputs appropriately so long as the resulting bundle

maximizes profit.

It is interesting to note that the maximum profit can be

easily obtained as

p0 � ¼ max p00yj � w00xj

� � ðj ¼ 1; 2; . . . ;NÞ: ð7Þ

Suppose that for a given data set

p00yk � w00xk � p00yj � w00xj for j ¼ 1; 2; . . .;N: ð7aÞ

Then, for any set of non-negative kjs adding up to unity,

p00yk � w00xk � p00 XN

1

kjy j

! � w00

XN

1

kjx j

! :

But, by assumption, for any ðx; yÞ 2 S; there exist some

non-negative kjs adding up to unity such that

y � XN

1

kjy j and x �

XN

1

kjx j:

Hence,

p00yk � w00xk � p00yj � w00xj for all ðx; yÞ 2 S: ð7bÞ

Varian’s Weak Axiom of Profit Maximization (WAPM)

argues that if the input–output bundle of a particular firm

evaluated at the prices it faces yields a lower profit than what

could be earned if it had chosen the observed input–output

bundle of some other firm in the sample, then the firm under

consideration could not be maximizing profit. An implica-

tion of (7a–b) above is that if the firm k does satisfy WAPM

then it actually is maximizing profit over the production

possibility set S. Further, (7) can also be expressed as

p0 � ¼ min p

s:t: p � p00yj � w00xj ðj ¼ 1; 2; . . . ;NÞ: ð8Þ

Lacking the necessary price information, we cannot take

this approach. We may, however, use endogenously deter-

mined shadow prices to look for the input–output bundle

that maximizes profit over the entire production possibility

set S at these prices. Consider some output price vector u0

and input price vector v0 such that at these shadow prices

the input–output bundle ðx0; y0Þ yields zero profit. That is

u00y0 � v00x0 ¼ 0: ð9Þ

We now look for the optimal bundle ðx�; y�Þ such that

P� � u00y� � v00x� � u00y� v00x 8ðx; yÞ 2 S: ð10Þ

The maximum profit P* provides a measure of the

overall inefficiency of the firm producing y0 from x0. One

problem that remains, however, is that one can change the

shadow prices of inputs and output by any given proportion

and P* also changes by the same proportion without vio-

lating the requirement of zero profit at the observed input–

output bundle. As a result, the maximum shadow profit P*

would be unbounded. One way to overcome this problem is

to normalize the shadow prices separately so that

u00y0 ¼ v00x0 ¼ 1: ð11Þ

Following (8), the shadow profit maximization for the

firm under review can be specified as

min P s:t: P � u00yj � v00xj; ðj ¼ 1; 2; ::;NÞ

u00y0 ¼ 1; v00x0 ¼ 1;

u0 � 0; v0 � 0; P unrestricted

: ð12Þ

The dual of this linear programming problem is

max u� h

s:t: PN

1

kjy j � uy0;

PN

1

kjx j � hx0;

PN

1

kj ¼ 1;

kJ � 0; /; h unrestricted

ð13Þ

Note that (13) combines the features of both the output-

and the input-oriented radial models for a variable returns to

scale technology. In fact, setting h equal to unity, we get the

measure of the firm’s output-oriented inefficiency, (/0 � 1).

Similarly, when / is preset at unity, the model yields the

firm’s input-oriented inefficiency, (1� h0). Clearly, the

optimal value of the objective function will be at least as

large as both (/0 � 1) and (1� h0). Thus, the optimal value

J Prod Anal (2007) 27:231–236 233

123

of the objective function in (13) can be interpreted as a

generalized measure of the inefficiency of a firm which is no

lower than the average of its output- and input-oriented

technical inefficiencies. Note that no matter what the input

and output prices actually are, the optimal value of

ðu� � 1Þ in (13) shows the proportionate increase (de-

crease) in the revenue without changing the output mix.

Similarly, ð1� h�Þ shows the proportionate decrease (in-

crease) in the cost with the input mix unchanged. When

revenue increases (u�[1) and cost falls ðh�\1Þ both

contribute to an increase in profit. But even when cost in-

creases, so long as revenue increases even more ðu�[h�Þ, profit would increase. The same will be true when u�\1

and revenue falls but h�\u� so that cost falls even further.

One problem with the normalization of the shadow

prices of output and inputs specified in (11) above is that

when multiple outputs or multiple inputs are involved,

individual shadow prices may be zero at the optimal

solution of (12) leading to positive output or input slacks at

the optimal projection of ðx0; y0Þ obtained from (13). One

possible way to resolve this problem is to replace the

restrictions in (11) by the multiplier bounds

u0 r y0

r � ar for each output r ðr ¼ 1; 2; . . . ;mÞ ð14aÞ

and

v0 i x0

i � ci for each input i ði ¼ 1; 2; . . . ; nÞ ð14bÞ

where ar ðr ¼ 1; 2; . . . ;mÞ and ci ði ¼ 1; 2; . . . ; nÞ are

pre-specified positive quantities.

The revised form of the problem (12) is

min P

P � Pm

1

u0 r yj

r � Pn

i v0

i xj i; ðj ¼ 1; 2; ::;NÞ

s:t: u0 r y0

r � ar; ðr ¼ 1; 2; . . . ;mÞ;

v0 i x0

i � ci; ði ¼ 1; 2; . . . ; nÞ; u0

r � 0; v0 i � 0;

ðr ¼ 1; 2; . . . ;m; i ¼ 1; 2; . . . ; nÞ P unrestricted:

ð15Þ

The dual of this problem is

max Pm

1

arur � Pn

1

cihi

s.t. PN

1 kjy j r � ury

0 r ; ðr ¼ 1; 2; . . . ;mÞ;

PN 1 kjx

j i � hix

0 i ; ði ¼ 1; 2; . . . ; nÞ;PN

1 kj ¼ 1; kj � 0; /r; hi

ðr ¼ 1; 2; . . . ;m; i ¼ 1; 2; . . . ; nÞ unrestricted.

ð16Þ

Clearly (16) combines the features of both output- and

input-oriented non-radial models of measuring Russell

efficiency.2 Several points need to be noted about the

optimization problem (15). First, unless the input and

output prices are appropriately normalized, the objective

function would remain unbounded. One may set each

u0 r ¼ ar

y0 r

and increase the shadow prices of inputs v0 i indef-

initely reducing the objective function value without a

lower bound. A possible way out would be to select ars and

cis that satisfy P

r ar ¼ P

i ci to ensure that P� is not less

than 0. It is clear from (16) that one could rescale the

objective function weights by P

r ar ¼ P

i ci

� � .

This would not affect the optimal solution except the

objective function value. The objective function weights in

(16) may then be selected such that P

r ar ¼ P

i ci ¼ 1: In

that case, however, each constraint u0 r y0

r � ar and

v0 i x0

i � ci in (15) must hold as strict equality. That would

amount to setting a priori the shadow prices

u0 r ¼ ar

y0 r ðr ¼ 1; 2; . . . ; mÞ and v0

i ¼ ci

x0 i

ði ¼ 1; 2; . . . ; nÞ. That, unfortunately, would mean that the shadow prices

are not choice variables in (15).3 This is even less desir-

able than allowing individual shadow prices to become

zero at the optimal solution. We therefore, prefer the

(primal-dual) models in (12–13).

A remarkable feature of the optimization problem in

(12) is its computational simplicity. The optimal solution

can be obtained through a simple enumeration process

without any need to solve an elaborate programming

problem. The optimal value of the objective function, P*,

in (12) is attained at some observed input–output bundle

ðxk; ykÞ in the sample.4 Consider a sub-problem

maxPk¼ Pm

r¼1

uryrk� Pn

i¼1

vixik

s:t: Pm

r¼1

uryr0¼1;

Pn

i¼1

vixi0¼1;

ur�0; vi�0; ðr¼1;2;...;m; i¼1;2;...;nÞ:

ð12aÞ

The Kuhn–Tucker conditions for maximizing the

Lagrangian

2 Pastor et al. (1997) introduced an extended Russell measure of

efficiency defined as C ¼ min 1 n

Pn

1 hi

1 m

Pm

1 ur

. Ray (1998) and Ray and Jeon

(2003) suggest using a linear approximation to the objective function

C at all /r and hi set equal to unity. This amounts to minimizing 1 n

Pn 1 hi � 1

m

Pm 1 ur . With this approximation, minimizing C would

amount to maximizing the objective function in (16) with ar ¼ 1 m for

each output r and ci ¼ 1 n for each input i.

3 This point was also noted by Zhu (2003). 4 This may, of course be true for multiple observations.

234 J Prod Anal (2007) 27:231–236

123

L ¼ Xm

r¼1

uryrk � Xn

i¼1

vixik

þ uk 1� Xm

r¼1

uryr0

! þ hk

Xn

i¼1

vixi0 � 1

!

are

@L

@ur ¼ yrk � ukyr0 � 0 ðr ¼ 1; 2; . . . ;mÞ;

@L

@vi ¼ �xik þ hkxi0 � 0 ði ¼ 1; 2; . . . ; nÞ;

ur @L

@ur ¼ uryrk � urukyr0 ¼ 0 ðr ¼ 1; 2; . . . ;mÞ;

vi @L

@vi ¼ �vixik þ vihkxi0 ¼ 0 ði ¼ 1; 2; . . . ; nÞ:

ð12bÞ

Clearly, u�k ¼ minfyrk

yr0 ; r ¼ 1; 2; . . . ;mg and h�k ¼

maxfxik

xi0 ; i ¼ 1; 2; . . . ; ng. Also, by virtue of the comple-

mentary slackness conditions, ur equals 0 if yrk[u�yr0.

Similarly, vi * equals 0 if xik\h�kxi0. Thus the optimal value

of the objective function in (12a) is

P�k ¼ u�k � h�k : ð17Þ

Hence, in (12), which is the substantive problem, the

optimal value is

P� ¼ max u�k � h�k � �

; k ¼ 1; 2; . . . ;N � �

: ð18Þ

Clearly, all we need to do is to evaluate the expression

on the right hand side of (18) at the observed data points to

obtain the optimal value of the objective functions in (15)

and (16). The corresponding input–output bundle repre-

sents the relevant efficient projection of the bundle ðx0; y0Þ on to the graph of the technology.

3 A simple computation procedure

As we noted earlier, the most attractive feature of Varian’s

WAPM is its computational simplicity. One can determine

the optimal solution of the profit maximization problem

shown in (6) through simple enumeration. We now show

that an equally simple solution procedure exists for the dual

problems in (12–13) as well. It is clear that the optimal

value of the objective function in (12) is attained at some

actually observed input–output bundle in the sample. Of

course, by standard duality results, the optimal values of

the objective functions in (12) and (13) will be identical.

Hence, we may just as well solve (13) instead of (12). Let

xj ¼ ðx1j; x2j; . . . ; xnjÞ be the input bundle and

yj ¼ ðy1j; y2j; . . . ; ymjÞ the corresponding output bundle of

firm j (j ¼ 1; 2; . . . ;N) in the sample. Now suppose that we

wish to evaluate the efficiency of firm s using model (13).

This can be achieved in the following steps:

1. For each firm j 6¼ s, compute

ur j ¼

Yrj

Yrs and u�j ¼ minfu1

j ;u 2 j ; . . . ;um

j g:

2. For each firm j 6¼ s, compute

hi j ¼

xij

xis and h�j ¼ min h1

j ; h 2 j ; . . . ; hn

j

n o :

3. Compute P�j ¼ u�j � h�j : 4. Now get P� ¼ max P�1;P

� 2; . . . ;P�N

� � .

Note that Step 1 ensures that yrj � u�j yrs for each output

r while Step 2 ensures that xij � h�j xis for each input i.

Hence, fu ¼ u�j ; h ¼ h�j ; kj ¼ 1; kk ¼ 0 ðk 6¼ jÞg is a fea-

sible solution for the optimization problem in (13) and the

corresponding value of the objective function is

P�j ¼ u�j � h�j . Because the optimal solution is attained at

one of the observed bundles, Step 4 leads to the optimal

solution.

The following numerical example illustrates the com-

putational procedure described in Steps 1–4.

Table 1 shows the input–output quantities of six hypo-

thetical firms for the 2-output 2-input case.

Suppose that we seek to solve the DEA problem (13) for

firm D. The relevant calculations are shown in Table 2

below.

We find that the optimal shadow profit-oriented pro-

jection of the firm D is firm F. At the optimal projection,

both of the outputs increase while one input (x2) decreases

but the other input (x1) increases. In this process, the sha-

dow profit increases from 0 to 2 15 :

Following Ray (2004), we may define the lost or unre-

alized return on outlay as

D ¼ P� � P0

v0x0

where P* and P0 are, respectively, the maximum and actual

shadow profits of a firm and v0x0 is the shadow value of its

input bundle. In this particular context, P0 equals 0 and v¢x0

equals 1 by definition. Therefore, P* itself is a measure of

the rate of unrealized (shadow) return on outlay.

Finally, unless the CRS technical efficiency of a firm is

unity, it cannot be shadow-profit efficient. As shown by

Banker (1984), unless the CRS efficiency of a firm with an

observed input–output bundle ðx0; y0Þ equals 1, there exists

a feasible input–output bundle ðbx0; ay0Þ within the VRS

production possibility set such that a[b. Obviously, for

this firm, the value of P* will be at least as large as a� b.

This will be true even when the firm is technically efficient

under the VRS assumption.

J Prod Anal (2007) 27:231–236 235

123

4 Conclusion

Even when actual prices are not available, one may

endogenously obtain shadow prices of inputs and outputs

compute the maximum shadow profit that a firm could earn

at those prices. Because the shadow cost of the actual input

bundle and the shadow value of the actual output bundle

are both normalized to unity, the shadow profit from the

observed input–output combination is zero. Thus, the

maximum shadow profit can be interpreted as the rate of lost

return on outlay. It is also shown to be no lower than the

average of the radial output- and input-oriented technical

inefficiencies of the firm. Thus, it can be interpreted as a

measure of overall inefficiency of a firm.

References

Banker RD (1984) Estimating the most productive scale size using

data envelopment analysis. Euro J Operat Res 17:1(July) 35–44

Briec W (1997) A graph type extension of farrell technical efficiency

measure. J Product Anal 8:95–110

Briec W (1998) Hölder distance function and measurement of

technical efficiency. J Product Anal 11:111–131

Briec W, Leleu H (2003) Dual representation of non-parametric

technologies and measurement of technical efficiency. J Product

Anal 20:71–96

Chambers RG, Chung Y, Färe R (1996) Benefit and distance

functions. J Econ Theory, 70:407–419

Färe R, Grosskopf S, Lovell CAK (1985) The measurement of

efficiency of production. Kluwer-Nijhoff, Boston

Färe R, Lovell CAK (1978) Measuring the technical efficiency of

production. J Econ Theory 19:1(October) 150–162

Frei FX, Harker PT (1999) Projections onto efficient frontiers:

theoretical and computational extensions to DEA. J Product Anal

11:275–300

Pastor JT, Ruiz JL, Sirvent I (1999) An enhanced DEA Russell Graph

efficiency measure. Euro J Operat Res 115:596–607

Ray SC (1998) Pareto-Koopmans measure of efficiency in manage-

ment education: how well managed are America’s Top-40

business schools? Presented at the 2000 North American

Productivity Workshop

Ray SC (2004) Data envelopment analysis: theory and techniques for

economics and operations research. Cambridge University Press

Ray SC, Jeon Y (2003) Reputation and efficiency: a nonparametric

assessment of America’s top-rated MBA programs. Working

Paper 2003–13, Department of Economics, University of

Connecticut

Varian HR (1984) The nonparametric approach to production

analysis. Econometrica 52(3):(May) 579–597

Zhu J (2003) Quantitative models for performance evaluation and

benchmarking: data envelopment analysis with spread sheets.

Kluwer Academic Press, Boston

Table 1 Input–output quantities of hypothetical firms

Firm Output (y1) Output (y2) Input 1 (x1) Input 2 (x2)

A 4 2 2 3

B 9 4 7 5

C 6 3 6 7

D 8 6 5 8

E 7 5 8 4

F 11 8 6 3

Table 2 Calculations for firm D

ðu�j ; h�i Þ ðu�j � h�i Þ

u�A ¼ min 4 8 ; 2

6

� � ¼ 1

3 ; h�A ¼ max 2

5 ; 3

8

� � ¼ 2

5 P�A ¼ u�A � h�A ¼ � 1

15

u�B ¼ min 9 8 ; 4

6

� � ¼ 2

3 ; h�B ¼ max 7

5 ; 5

8

� � ¼ 7

5 P�B ¼ u�B � h�B ¼ � 11

15

u�C ¼ min 6 8 ; 3

6

� � ¼ 1

2 ; h�C ¼ max 6

5 ; 7

8

� � ¼ 6

5 P�C ¼ u�C � h�C ¼ � 7

10

u�E ¼ min 7 8 ; 5

6

� � ¼ 5

6 ; h�E ¼ max 8

5 ; 4

8

� � ¼ 8

5 P�E ¼ u�E � h�E ¼ � 23

30

u�F ¼ min 11 8

; 8 6

� � ¼ 8

6 ; h�F ¼ max 6

5 ; 3

8

� � ¼ 6

5 P�F ¼ u�F � h�F ¼ 2

15

P� ¼ P�F ¼ 2 15

236 J Prod Anal (2007) 27:231–236

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