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INFO-GAP VALUE OF INFORMATION IN MODEL UPDATING
Mechanical Systems and Signal Processing (2001) 15(3), 457}474 doi:10.1006/mssp.2000.1377, available online at http://www.idealibrary.com on
INFO-GAP VALUE OF INFORMATION IN MODEL UPDATING YAKOV BEN-HAIM Faculty of Mechanical Engineering, Technion~Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected] (Received 31 October 1999, accepted 30 November 2000) Information is the complement of uncertainty, so model updating*which is a process of acquiring information*entails the reduction of uncertainty. We concentrate on severe uncertainty quanti"ed with information-gap uncertainty models, so updating focusses on the improvement of info-gap models of uncertainty. Our main e!ort is to show how model updating can be evaluated in terms of the performance of the system itself. The central tool in this evaluation is the robustness function, whose value expresses the greatest level of info-gap uncertainty consistent with successful performance. A theorem establishes an irrevocable trade-o! between robustness and reward (with "xed information). The demand value of an increment of information is the increment of improvement in performance (or reward) which the system can achieve without su!ering a loss of robustness, in exchange for applying the information. The demand value of an increment of information is evaluated by comparing the robustness trade-o! curves before and after updating. The second theorem introduces an idea of the &informativeness' of an info-gap model of uncertainty in terms of set inclusion: an uncertainty model is informative to the degree that it more tightly delimits the range of unknown variation. The main implication of the theorem is that information is universally valuable only when it satis"es the set-inclusion criterion. We consider three examples which illustrate the evaluation of the demand value of information. The importance of establishing the comparability of the info-gap models is stressed. It is explained that the demand value can be evaluated either as an a priori estimate of the potential value of information which is not yet in hand, or as an a posteriori assessment of information which has been acquired. 2001 Academic Press
1. INTRODUCTION
Information is a commodity and, like all other commodities, we buy information in many forms, including newspapers, espionage, college degrees and, of course, the updating of mathematical models of technological systems. But information di!ers in fundamental ways from all material commodities. The most peculiar feature of information is that, while it can go stale and thus lose its value, it is not used up when consumed. Information has a permanence and accessibility and, sometimes, an ease of communicability, which belie the tremendous cost which is often paid for information, and which bedevil attempts to quantify its worth. In this paper, we explore an economic (though not necessarily monetary) model of the value of information in technological model updating. The basic question we ask is: what is the worth of information? More speci"cally, what is the value of the information needed to reduce our uncertainty about the model of a system? More directly still, by how much can one con"dently demand better performance from the system as a result of acquiring new information and thereby reducing uncertainty? 0888}3270/01/030457#18 $35.00/0
2001 Academic Press
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This last question carries the seeds of the answer we will develop. Namely, viewing model updating as a process whereby uncertainty is reduced, how do we assess the value of information in the same units by which the system performance is evaluated? In Section 2, we describe information-gap models of uncertainty which quantify uncertainty as a disparity between what is known and what can be known. We formulate the robustness function which assesses the greatest info-gap uncertainty consistent with acceptable performance and discuss the trade-o! between robustness and reward. In Section 3, we formulate a concept of the demand value of information as the increment of improved performance accruing from an increment in information. We show that only when the information is ranked in a set-inclusion sense is the information universally valuable. In Sections 4}6 we discuss three examples.
2. INFO-GAP MODELS AND DECISIONS
2.1. INFO-GAP MODELS OF UNCERTAINTY Our quanti"cation of uncertainty is based on non-probabilistic information-gap models. An info-gap model is a family of nested sets. Each set corresponds to a particular degree of uncertainty, according to its level of nesting. Each element in a set represents a possible realisation of the uncertain event. Info-gap models, and especially convex-set models of uncertainty, have been described elsewhere, both technically [1] and axiomatically [2]. Uncertain quantities are vectors or vector functions. Uncertainty is expressed at two levels by info-gap models. For "xed a, the set U(a, uJ ) represents a degree of uncertain variability of the uncertain quantity u around the centrepoint uJ . The greater the value of a, the greater the range of possible variation, so a is called the uncertainty parameter and expresses the information gap between what is known (uJ and the structure of the sets) and what needs to be known for an ideal solution (the exact value of u). The value of a is usually unknown, which constitutes the second level of uncertainty: the horizon of uncertain variation is unbounded. Let R denote the non-negative real numbers and let S be a Banach space in which the uncertain quantities u are de"ned. An info-gap model U(a, uJ ) is a map from R;S into the power set of S. The basic axiom, which characterises the representation of uncertainty by info-gap models, is that the sets of an info-gap model are nested by the uncertainty parameter a: U(a, uJ )-U(a , uJ )
if a)a .
(1)
In many applications it is found that the relevant info-gap models obey speci"c structural axioms. The most common structural axioms are [2]: Contraction: U(0, 0) is a singleton set containing its centrepoint: U(0, 0)"+0,.
(2)
¹ranslation: U(a, uJ ) is obtained by shifting U(a, 0) from the origin to uJ : U(a, uJ )"U(a, 0)#uJ
(3)
where U#uJ means that uJ is added to each element of U. ¸inear expansion: info-gap models centred at the origin expand linearly: b U(b, 0)" U(a, 0) a
for all a, b'0
where (b/a)U means that b/a multiplies each element of U.
(4)
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In some situations the linear expansion axiom is altered to include non-linear expansion properties [3]. 2.2. INFO-GAP DECISIONS AND THE ROBUSTNESS FUNCTION Let the vector q represent the decision-maker's decisions or actions, which incur a consequence depending upon q and upon an uncertain vector or vector function u belonging to an info-gap model U(a, uJ ). The decision-maker knows a scalar reward function, R(q, u), which could be used, together with uJ , to maximise the nominal reward. The reward function R(q, u) must be construed very broadly, and can represent a performance requirement in vibrational analysis or other engineering design problems [4], or an economic or managerial goal [5]. R need not be a von Neumann}Morgenstern utility function. R is de"ned very generally in [6]. The decision theory developed in this paper is utterly di!erent from von Neumann}Morgenstern game theory [7]. Foremost among the distinctions is that the present theory does not depend on probabilistic information of any sort. Info-gap theory is particularly suited to highly unstructured uncertainty and severe lack of information. Furthermore, info-gap theory is not vulnerable to the di$culties which arise in von Neumann}Morgenstern theory due to its axiom of linear independence. In particular Allais and Ellsberg ¶doxes' [8] are avoided in info-gap theory. The robustness is the greatest value of the uncertainty parameter for which the performance is &acceptable'. In terms of the scalar reward function, &acceptability' is de"ned in one of the two ways: either R is no less than a lowest acceptable value, or R is no greater than a greatest acceptable value. In other words, desirable values of R are either small or large, depending on the application, and &acceptability' means that R is su$ciently small or su$ciently large. Acceptability is used here to refer to a type of satis"cing. The critical reward, r , is a parameter whose value need not be chosen a priori by the decision-maker. A De"ne A(q, r ) as the set of a-values for which the critical reward is guaranteed. When A desirable performance means that the reward function R takes small values, then A(q, r ) is A
(5)
(6)
A(q, r )" a: max R(q, u))r . A A u3U(a, uJ ) On the other hand, if large R is preferred then
A(q, r )" a: min R(q, u)*r . A A u3U(a, uJ )
In either case, the robustness function is the greatest value of the uncertainty parameter for which acceptable performance is assured. That is, the robustness is the least upper bound of the set A(q, r ): A aL (q, r )" sup a. (7) A a3A(q, r ) A
If A(q, r ) is empty then the reward r cannot be achieved at any level of uncertainty and we A A de"ne a( (q, r )"0. When a( (q, r ) is large then the decision-maker is immune to a wide range A A of variation, while if a( (q, r ) is small then even small #uctuations can lead to violation of the A performance requirement. Thus, &bigger is better' for the robustness a( . The robustness function establishes a preference ordering of actions. The decision-maker will prefer action q over action q if the robustness is greater with q than with q at the same critical value: q Y q
if a( (q, r )'a( (q , r ). A A
(8)
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The decision-maker's optimal robust strategy is to choose an action qL (r ) which maximises A the robustness: a( (qL (r ), r )"max a( (q, r ) A A A q3Q
(9)
where Q is the set of available actions. The logical structure of this decision algorithm as a &severe test' is discussed further in [2]. Robustness to uncertainty is important when we concentrate on the pernicious possibilities entailed by unknown variation. However, variations can be propitious and surprises can be bene"cient. The dynamic #exibility which is so important for survival, as Rosenhead has stressed [9], thrives on emergent opportunities. We can study this aspect of decisions with the &opportunity function' which is the logical complement to the robustness function a( and is discussed elsewhere [10]. 2.3. ROBUSTNESS TRADE-OFF The following theorem expresses a fundamental trade-o! between immunity-to-uncertainty and demanded reward. The proof of this theorem [11] depends only on the nesting axiom of info-gap models, equation (1). Theorem 1. ¸et U(a, uJ ) be an info-gap model and let R(q, u) be a uniformly continuous reward function. ¹he resulting robustness function aL (q, r ) is strictly monotonic on the set of critical A rewards r for which aL (q, r ) exists and is positive. aL (q, r ) decreases with improving r -value. A A A A Speci"cally, if small R is preferred, equation (5), then a small value of r is more desirable A and more demanding than a large value. Hence a( (q, r ) decreases with decreasing r : A A a( (q, r )(a( (q, r ) for r (r . (10) A A A A If large R is preferred, equation (6), then a( (q, r ) decreases with increasing r : A A a( (q, r )'a( (q, r ) for r (r . (11) A A A A This is true also for the robustness function evaluated at the optimal action, as stated in the following corollary. Corollary 1. ¸et U(a, uJ ) be an info-gap model, let R(q, u) be a uniformly continuous reward function, and let qL (r ) be a maximally robust strategy as a function of critical reward r . ¹hen A A the optimal robustness, aL (qL (r ), r ), is strictly monotonic in r , decreasing with improving r . A A A A That is, the maximal robustness is strictly monotonic on the set of reachable rewards. Theorem 1 and its corollary show the inexorable trade-o! between reward and immunityto-uncertainty. Robustness is obtained only in exchange for degraded performance. The decision-maker must choose a position on the spectrum of this trade-o!. This theorem also shows that the decision-maker need not make an irrevocable prior choice of the critical reward r . Rather, r is chosen in light of the analysis of the robustness to ambient A A uncertainties. For instance, the decision-maker may "nd that better performance can be demanded without substantially increasing the vulnerability to uncertainty. This (very pleasing) outcome of the analysis may well induce the decision-maker to demand better performance. Alternatively, the analyst may "nd that, by slightly diminishing the demanded performance, the immunity to uncertainty is greatly enhanced. Again, the analysis may lead to a modi"cation of the decision-maker's preferences. In short, the robustness a( (q, r ), A viewed as a function of critical reward, is a decision-support tool with which the decisionmaker assesses the options and their consequences.
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3. DEMAND VALUE OF INFORMATION
3.1. ROBUSTNESS PREMIUM AND DEMAND VALUE The basic question we ask is: what is the worth of information? The answer has two parts. We will assess the usefulness of information as a &robustness premium' which it entails. We will determine a value for information as the added reward which the decision-maker can demand without losing robustness when the information is exploited. This will turn out to be a &reward premium' which the information induces and will be recognised as what the economists call a &demand value'. The units of the demand value are the same as those of the reward function R(q, u), which may be dollars, or millimeters, or hertz, etc. Let a( (q, r ) and a( (q, r ) be the robustness functions from two di!erent constellations of A A information, each with its own info-gap model and reward function and underlying system models and performance criteria, as in Fig. 1. In short, these robustness curves are based on two di!erent &decision models'. The information upon which one curve is based may be obtained by updating or improving the information underlying the other curve. Alternatively, one curve may be based on better or simply di!erent analyses of existing data. Or, the curves may re#ect the assessments of uncertainty or choices of models made by di!erent analysts. While the pair of robustness curves in Fig. 1 do not cross, this is not the general case as we will see in subsequent examples. The demand value of the information needed to move from a( (q, r ) to a( (q, r ), *r in A A A Fig. 1, is the increment in reward which can be demanded after acquiring the new information. The horizontal arrows in Fig. 1 show the increment in reward which the decision-maker can con"dently demand as a result of using the di!erent information. This is a &demand value' in the economic sense that the decision-maker would be willing to pay any quantity up to, but not more than, this increment in reward in order to exchange the information. The demand value of information is expressed explicitly in the units of the reward by which the system itself is evaluated. We note that the demand value depends on the level of ambient uncertainty: where on the robustness curve the decision-maker is operating. Information has no absolute value; its value is relative to the environment. In particular, we see from this pair of curves that information is of relatively lower worth at a low level of demanded reward and at a high level of robustness. The robustness premium of information, *a( in Fig. 1, expresses the increment of immunity resulting from the exchange of information, and is shown by the vertical arrows. The
Figure 1. Two robustness curves based on di!erent information.
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robustness premium speaks directly to the issue of immunity, but it lacks the economic overtone of the demand value. The robustness premium, like the demand value, depends on the operating point at which the decision-maker is functioning. We can express the demand value at uncertainty a, for decision model &2' with respect to decision model &1', as the di!erence: *r (a)"r !r (12) A A A where the critical rewards r and r satisfy A A a( (q, r )"a"a( (q, r ). (13) A A If the robustness function decreases with increasing critical reward, equation (11), then *r (a) will be positive or negative if model &2' is more or less valuable than model &1', A respectively. Furthermore, the sign of *r (a) may change with a since the robustness curves A may cross. If the robustness function increases with increasing critical reward, equation (10), then *r (a) is negative for valuable increments and positive otherwise. A The robustness premium at critical reward r , of decision model &2' with respect to model A &1', may be either positive or negative, and is de"ned as *a( (r )"aL (q, r )!a( (q, r ). A A A
(14)
3.2. INFORMATION AND ROBUSTNESS We have indicated that, in general, the sign of the demand value can change with the level of uncertainty, indicating a switch in preference between the decision models. However, an important special case exists in which the sign of the demand value is constant, as implied by the following theorem. The proof of this theorem is found in [6]. The theorem employs a set-inclusion concept for comparing the &informativeness' of two info-gap models of uncertainty, U (a, uJ ) and U (a, uJ ). If U (a, uJ ) is a subset of U (a, uJ ) then the former constrains the ambient uncertainty more strictly than the latter. In this sense, U (a, uJ ) is more informative than U (a, uJ ). Theorem 2. ¸et U (a, uJ ) and U (a, uJ ) be two info-gap models with corresponding robustness functions aL (q, r ) and aL (q, r ), based on the same reward function and performance require A A ment. U (a, uJ ) is more informative than U (a, uJ ), in the set-inclusion sense, if and only if aL (q, r ) A is everywhere no less than aL (q, r ). ¹hat is, A U (a, uJ )-U (a, uJ ) (15) if and only if: a( (q, r )*a( (q, r ). (16) A A Theorem 2 implies that robustness functions will never cross (though they may intersect) if and only if their info-gap models are ranked by informativeness. This implies that the demand value between them will never change sign (though it may become zero). In other words, information-ranked info-gap models generate decision models for which one is always preferable (or equivalent) to the other in terms of their demand value. Theorem 2 is true also if the robustnesses are evaluated at their respective optimal actions. Corollary 2. ¸et U (a, uJ ) and U (a, uJ ) be two info-gap models with corresponding robustness functions aL (q, r ) and aL (q, r ) and optimal actions qL (r ) and qL (r ), based on the same reward A A A A function and performance requirement.
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463
U (a, uJ ) is more informative than U (a, uJ ), in the set-inclusion sense of equation (15), if and only if: a( (qL (r ), r )*aL (qL (r ), r ). A A A A
(17)
4. UNCERTAIN LOADS ON A CANTILEVER
We employ the cantilever, subject to uncertain static loads, to illustrate the demand value of information. We will de"ne a performance criterion, a de#ection model, and an info-gap uncertainty model. We will evaluate the demand value of improvements in the info-gap model of load uncertainty. Let x denote the position along the cantilever, where the clamped end is at x"0 and the free end is at x"¸. Static forces are applied in a single plane and perpendicular to the beam axis, where u(x) is the unknown force density, in units of N/m, at position x. The de#ection pro"le resulting from this load is v(x) which, for small de#ections, obeys the following di!erential equation: EI
dv(x) "u(x) dx
(18)
where E is Young's modulus for the material and I is the area moment of inertia of the cross-section. We assume that EI is constant along the length of the beam. The boundary conditions at the clamped end are v(0)"v (0)"0
(19)
where the prime denotes di!erentiation with respect to x. At the free end the boundary conditions are v (¸)"v (¸)"0.
(20)
The performance criterion is that the de#ection at the free end must not exceed a critical value, v : A "v(¸)")v . (21) A Thus, the performance requirement is of the small-is-better kind [equation (5)]. The force density u(x) is represented as nnx L u(x)"uJ # c sin (22) L ¸ LL "uJ #c2 (x) (23) L L where the nominal load, uJ , is constant, c is the vector of Fourier coe$cients and (x) is L L the vector of corresponding sine functions. The info-gap model for uncertainty in the load-pro"le Fourier-coe$cient vector c, with least and greatest spatial modes n and n , is U (a, uJ )"+u(x)"uJ #c2
(x): c2c)uJ a,, a*0. (24) L L L L The informativeness of the info-gap model U (a, uJ ), in the set-inclusion sense, is L L determined by the bandwidth. A tighter bandwidth implies stricter constraints on the possible load pro"les, and hence greater information about the environment. If n )m and m )n .
(25)
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Then (a, uJ )-U (a, uJ ). (26) U L L K K U (a, uJ ) is a more informative info-gap model than U (a, uJ ). K K L L The solution of equation (18), in response to load pro"le u(x)"uJ #c2 (x), for the L L clamped-free boundary conditions in equations (19) and (20), is
x L c ¸ x ¸x ¸x ! # ! L 24 6 4 2 nn LL L c ¸ x ¸ ¸ nnx # L # x! sin nn 6 nn nn ¸ LL ¸x x L c ¸ # ! L (1!(!1)L). (27) 6 2 nn LL We are particularly interested in the end de#ection, for which equation (27) simpli"es considerably: EIv(x)"uJ
EIv(¸) 1 1 L 1 (!1)L " # c ! L (nn) uJ ¸ 8 uJ 3nn LL GFFHFFI f 1 1 L " # c2f 8 uJ
(28)
(29)
where the elements of the vector f are de"ned in equation (28). The quantity EIv/uJ ¸ on the left-hand side of equation (28) is a dimensionless de#ection. In order to calculate the robustness function we must evaluate the greatest de#ection for any load pro"le in the info-gap model. Using Lagrange optimisation the result is max u3U (a, uJ ) L L
EIv(¸) 1 " #a (f2f . uJ ¸ 8
Denote the dimensionless form of the maximum allowed de#ection v as A EIv A. o" A uJ ¸
(30)
(31)
Equating the maximum normalised de#ection, equation (30), to the normalised performance requirement, o , and solving for a leads to the robustness for this info-gap model: A 8o !1 a( (o )" A . (32) L L A 8(f2f The performance requirement is that the dimensionless end de#ection be no greater than o , A which means that reward increases as o decreases: a small value of o is more desirable or A A demanding than a large value. In equation (32) and in Fig. 2 we see the usual trade-o! (even though the robustness curves have positive rather than negative slopes): the robustness a( (o ) increases as the performance degrades (as o increases), as anticipated by L L A A Theorem 1. Robustness functions versus critical de#ection are shown in Fig. 2 for values of m and G n satisfying relations (25), so the corresponding info-gap models are nested as in equation G (26). The horizontal arrow shows the demand value, *o (a), of the more informative A
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Figure 2. Robustness curves for more and less informative info-gap models.
uncertainty model. This demand value is calculated as in equations (12) and (13). The critical rewards o and o are evaluated from A K A L a( (o )"a"a( (o ). (33) K K A K L L A L The demand value of U with respect to U is L L K K *o (a)"o !o (34) A A K A L K L f! f . (35) "a L L LK LL The demand value of the more informative info-gap model, in units of o , is negative A because a small value of o is desirable. We also see that the demand value increases as the A ambient uncertainty, a, rises: an informative model is more useful in the presence of great uncertainty. A word of caution: the more informative info-gap model is more valuable if it is veri,ed. If we know that real variability is described by the tighter, more informative, of the two info-gap models, then we can demand better performance, and in this sense the information needed to update the less informative to the more informative model is valuable. Of course, it would be valuable (in a somewhat di!erent sense) to know that the putatively &more informative' model is wrong, and the &less informative' model in fact describes the ambient uncertainty. This is valuable because it informs us that we must either relax our demands on the system or employ alternative technologies to achieve assurance of better performance. The demand value in equation (35), in units of dimensionless de#ection, is shown in Table 1 for a range of m and n values and with m "n "1 and a"1. Moving from left to right on each row we see that *o gets better (more negative) as the di!erence n !m A increases. This means that the worth of U increases in comparison with U as the latter K L info-gap model becomes less informative than the former. The reverse is true as we move down each column: the worth of U decreases as it becomes more similar to U . These K L increments of value are expressed in units of the performance requirement of the system itself, *o or, equivalently, *v . The decision-maker presumably has a sound qualitative A A understanding of the signi"cance of increments in these parameters, which enables the assessment of the signi"cance of increments of information.
5. CANTILEVER: DISJOINT INFO-GAP MODELS
The set-inclusion criterion for informativeness, equation (15) or (26), captures only one aspect of the full gamut from &simple' to &complex' decision models. Furthermore, in
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TABLE 1 Demand value of an informative info-gap model, equation (35), for m "n "1 and a"1
m "1 2 3 4 5
n "1
2
3
4
5
0
!0.0084 0
!0.013 !0.0045 0
!0.015 !0.0067 !0.0022 0
!0.017 !0.0082 !0.0037 !0.0015 0
connection with equation (12), we explained that a demand value can be calculated for any pair of decision models, regardless of whether or not one is more informative than the other. As implied by Theorem 2, the sign of this demand value will be constant at all levels of uncertainty if and only if the info-gap models are nested. Otherwise, the sign will change indicating a di!erent preference between the models in di!erent circumstances. We will illustrate this by comparing the decision model of Section 4 against a nominally simpler model. The &complex' decision model represents uncertain distributed loads with the Fourier ellipsoid info-gap model of equation (24). We will suppose that spatial models from the 5th to the 10th make up the uncertain load functions, so n "5 and n "10. The fourth-order di!erential equation for beam de#ection, equation (18), is used, leading to the robustness function of equation (32). We will denote this robustness by a( which, for this speci"c A info-gap model, becomes
1 a( (o )"26.46 o ! . A 8 A A
(36)
The &simple' decision model assumes that the loads are concentrated at the mid-point of the beam. Instead of equation (28), the end de#ection resulting from a point force f at the mid-point is f ¸ . v(¸)" 12EI
(37)
The fractional variation of the point load is uncertain, so the info-gap model for variation in f is
U(a, fI )" f :
" f!fI " a ) , a*0 fI (2
(38)
where the nominal point load fI corresponds to the nominal uniform load density uJ in equation (24): fI "uJ ¸.
(39)
The normalisation by (2 in equation (38) is chosen to coordinate the scales of value of the uncertainty parameters in this info-gap model and the Fourier ellipsoid model in equation (24). We want U(a, fI ) and U (a, uJ ) to be comparable in size. In equation (38) the least L L upper bound of the squared fractional variation of the load, up to uncertainty a, is
f!fI a ) . 2 fI
(40)
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Evaluating the analogous quantity for U (a, uJ ) we "nd L L 1 L * nnx 1 * u(x)!uJ dx" c sin df x L uJ uJ ¸ ¸ ¸ LL a ) . 2
(41) (42)
Comparing equations (40) and (42) we see that, though the info-gap models U(a, fI ) and U (a, uJ ) are di!erent, the calibrations of their uncertainty parameters are comparable in L L terms of the horizons of uncertainty of these models. The robustness function for the simple decision model is found by evaluating the maximum de#ection at uncertainty a, equating this to the critical de#ection v , and solving A for a. One "nds a( (o )"12o !1 (43) Q A A where o is the dimensionless critical de#ection in equation (31). A The robustness functions of the &simple' and &complex' decision models, equations (43) and (36), are depicted in Fig. 3. These curves cross at the point (0.16, 0.92) in the o vs aL plane. A The intersection of the robustness curves means that there is no unequivocal demand-value preference for one or the other of these two decision models. The &simple' model is more robust if the normalised critical reward o is less than 0.16. Consequently, the demand value A of the &simple' model over the &complex' model is negative (preferable) in this range. However, for o '0.16 the situation is reversed, and the &complex' model is preferable to the A &simple' model in terms of robustness premium or demand value. In other words, when our performance requirement is o '0.16 we would prefer to verify A and use the &complex' info-gap model. On the other hand, if our system requirement is o (0.16 we would rather verify and employ the &simple' model. Note that the &complex' A uncertainty model is preferred in the case of the less-demanding performance requirement. The predicates &simple' and &complex' assert something about the structure of the uncertainty models, but do not imply anything about the quality of the ambient uncertainty or the performance which can be demanded of the system.
Figure 3. Robustness curves for &simple' and &complex' decision models.
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In comparing the robustnesses of these two decision models we are assuming that the underlying uncertainty parameters of the respective info-gap models are commensurate. As in the present case, this assumption is reasonable when each uncertainty parameter has comparable meaning and calibration as a measure of variation. In the spectral-bound model, equation (24), a is the fractional variation of the force-density spectral amplitudes. In the interval-uncertainty model, equation (38), a is the fractional variation in the magnitude of the point force. In each case the total force nominally applied to the beam is the same, uJ , and the calibrations of the uncertainty parameters have been coordinated as explained earlier.
6. LINEAR SYSTEM WITH UNCERTAIN DYNAMICS
In the previous two examples, Sections 4 and 5, we have evaluated increments in information about uncertain loads on a mechanical system whose properties are completely known. We now consider a system-model updating problem. The concept of modelupdating advocated here is that a model is improved by reducing the uncertainty associated with that model. We start with a nominal model subject to substantial uncertainty (represented by an info-gap model). We contemplate improving the model by reducing the uncertainty in the system model (by improving the info-gap model). The question we ask is: what is the value of the information needed to reduce our uncertainty in the system model? The analysis involves the following elements: *The performance requirements of the system. *The current system model and the current uncertainty model for this system model. *An improved model of the uncertainty in the system model. We do not yet know that this improved uncertainty model is correct. We are evaluating, a priori, the value of obtaining this improved uncertainty model. From this information we evaluate two robustness functions: that based on the initial system and uncertainty models, and that based on the initial system model and the updated uncertainty model. From these functions we estimate the value of the information required to reduce our uncertainty about the system model. This estimate does not explicitly account for the very real possibility that, by improving the uncertainty model, we may wish to revise the nominal system model as well. After updating the uncertainty model and incidentally also revising the nominal system model, we can evaluate the a posteriori value of the information obtained. This would be done by reevaluating the second robustness function using the updated system and uncertainty models. We use a single degree-of-freedom linear system to illustrate the procedure for a priori estimation of the value of reducing the uncertainty in a system-model. The response x (t) to S input u(t) is determined by the impulse response function g(t): R
g(t!q)u(q)dq.
x (t)" S
(44)
The performance requirement is that the response to each of ¸ speci"ed test inputs, u (t), 2 , u (t), be no greater than a critical value: * "x l (t)")r , S A
l"1, 2 , ¸.
(45)
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The actual impulse response function g(t) is unknown, though the current, nominal, model gJ (t) is known. The current info-gap model for uncertainty in the system model is U (a, gJ )"+g(t): "g(t)!g (t)")a,, a*0. (46) As usual, this info-gap model is a family of nested sets, representing two di!erent types of uncertainty in the system model. At "xed a, the actual impulse response function deviates from the nominal model in an unknown way, up to uncertainty a. Furthermore, the horizon of uncertainty is unknown: we do not know the value of a. The uniform-bound info-gap model of equation (46) is extremely &uninformative' and &naive', containing in"nitely many impulse response functions with very high-frequency components. We envision updating the system model by reducing the uncertainty which is inherent in U (a, gJ ). In the present example this is done by excluding all but speci"ed temporal modes. The amplitudes of the remaining modes, with frequencies from n n/¹ up to n n/¹, will remain uncertain. The revised system model will be L nnt (47) g(t)"gJ (t)# c cos L ¹ LL "gJ (t)#c2c(t) (48) where c is the vector of uncertain Fourier coe$cients and c(t) is the vector of the corresponding cosine functions. The updated uncertainty-model for the impulse-response function has the following structure: U (a, gJ )"+g(t)"gJ (t)#c2c(t): c2=c)ha,, a*0 (49) where = is a real, symmetric, positive de"nite matrix. The parameter h is chosen so that U (a, gJ ) is the largest possible subset of U (a, gJ ). That is, h is the largest value satisfying: max max "g(t)!gJ (t)")a (50) 0)t)2¹
g3U (a,gJ )
Note that c2c"g!gJ . Using Lagrange optimisation one "nds max
g3U (a,gJ )
"c2c(t)""ha(c2(t)=\c(t).
(51)
Consequently h is chosen as 1 . h" max0)t)2¹ (c2(t)=\c(t)
(52)
Let us emphasise that the updated info-gap model has been carefully constructed so as to be comparable to the initial model. The uncertainty parameters a in both models have the same units. In addition, U (a, gJ ) is explicitly constructed as the largest band-limited subset of U (a, gJ ). That is, these info-gap models are ranked by set inclusion as in equation (15). Furthermore, the information invested in the more informative model is explicitly de"ned: exclusion of all modes outside a speci"ed bandwidth. We know, experimentally, what we must do in order to verify or falsify the updated info-gap model. To evaluate the robustness functions we need the extreme responses*both minimum and maximum*on each of the info-gap models because
max "x (t)""max S g3U G
max x (t) , min x (t) S S g3U g3U G
G
.
(53)
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Consider "rst U (a, gJ ). Let sgn(y) represent the algebraic sign of y. R max x (t)" [gJ (t!q)#sgn(u(q))a] u(q) dq S g3U (a, gJ ) R R gJ (t!q)u(q) dq#a "u(q)" dq " GFFFHFFFI GFHFI
(54)
(55)
xJ (t) uJ (t) S "xJ (t)#auJ (t) (56) S which de"nes the terms xJ (t) (the nominal response) and uJ (t), both of which are known S functions. In similar fashion we "nd min x (t)"xJ (t)!auJ (t). S S
(57)
g3U (a, gJ )
Combining relations (53), (56) and (57) we conclude that max "x (t)"""xJ (t)"#auJ (t). S S
(58)
g3U (a, gJ )
The robustness function for test input u(t) is the greatest value of the uncertainty parameter consistent with not violating the performance requirement. To "nd the robustness we equate the maximum absolute response to the critical value and solve for a: r !"xJ (t)" S . "xJ (t)"#auJ (t)"r NaL (t)" A S A S uJ (t)
(59)
This assumes that the nominal response, xJ (t), satis"es the performance requirement S "xJ (t)")r . (60) S A If this is not true, then the robustness is zero: any deviation entails the possibility of violating the performance requirements. The overall robustness of the initial uncertainty model, considering the ¸ test inputs, is the lowest of the ¸ robustnesses: a( (t)" min aL l (t). S 1)l)¸
(61)
Now consider U (a, gJ ). Combining equations (44) and (48) we can express the response to input u(t) as x (t)"xJ (t)#c2 S S
R
c(t!q)u(t)dq GFFHFFI
(62)
t (t) S
"xJ (t)#c2t (t) S S which de"nes the known vector function t (t). S Using Lagrange optimisation we "nd the extremal values of c2t (t): S max c2t (t)"$ah (t2S (t)=\tS(t) S GFFHFFI o (t) S
g3U (a, gJ )
(63)
(64)
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which de"nes the known function o (t). The maximum absolute response is S max "x (t)"""xJ (t)"#aho (t). S S S g3U (a, gJ )
(65)
As in equation (59), we "nd the robustness for input u(t) by equating the maximum absolute response to the critical response and solving for a: r !"xJ (t)" S a( (t)" A (66) S ho (t) S provided equation (60) holds; if not, the robustness is zero. The overall robustness a( (t) is evaluated as in equation (61). The demand value of the information needed to implement the updated uncertainty model is evaluated as in equations (12) and (13). For a single test input u(t) this becomes *r (a)"a[ho (t)!uJ (t)] A S "a
(67)
(t2(t)=\t (t) R S S ! "u(q)" dq . max0)t)¹ (c2(t)=\c(t)
(68)
The salient features of this result will appear in a simple example. Let the shape matrix = in equation (49) be the identity matrix. Consider the test input
u(t)"
b, 0,
0)t)h t'h.
(69)
The a priori value of updating the info-gap model of uncertainty, for this test input and for t)h, is found from equation (68) to be
L 1 nnt t sin ! (70) n ¹ ¹ LL In Fig. 4 we plot equation (70): the a priori demand value of the information needed to update the system model. The contemplated update reduces the uncertainty about the system model from the uniform-bound info-gap model of equation (46) to the Fourier ellipsoid-bound model of equation (49). Four points are to be noted. First, the demand value is in the units in which the system performance is evaluated, displacement, as speci"ed in equation (45). Thus numerical values of *r can presumably be A interpreted qualitatively (large, small; good, bad) by an engineering analyst who can qualitatively assess levels of performance as expressed by r . The demand value is the A improvement in performance (the increment in r ) which can be assured consequent to A obtaining the new information. A large negative value of *r (a) is desired, since a small value A of r is more stringent than a large value. A Second, *r in equation (70) is an a priori estimate. It is an assessment of the potential A value of the information. We do not know if uncertainty model U (a, gJ ) is veri"able. Equation (70) assesses the demand value accruing from U (a, gJ ) if we are able to establish the veracity of this model, and if the nominal model remains unchanged. We may "nd, in the course of testing the validity of U (a, gJ ), that it is wrong, that the evidence suggests a di!erent frequency band (n , n ), or that the nominal model gJ (t) should be altered in some way. After verifying a new updated info-gap model we would be in a position to evaluate an a posteriori value of the new information. The a priori estimate gives us some early indication of where we might "nd ourselves, informationally, after completing the update. h *r (t)"ab¹ A n
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Figure 4. Demand value vs time, equation (70). n "1, n "5 and ab¹"1. h"0.4714.
Third, the demand value varies in time, depending on the instant or time interval after system initiation at which the performance requirement is imposed. For times in excess of t/¹"0.4 the demand value becomes increasingly attractive (negative). So, if the relevant time frame is beyond t/¹"0.4, then there is a clear and strong incentive to acquire new information which would verify U (a, gJ ). On the other hand, if the relevant times are very short then the new information would be of little value. Finally, it is important to recognise that the demand value depends very intimately upon comparison of the two underlying info-gap models. Care must be taken to assure that the uncertainty parameters are comparable in units and scaling. In the present example this is achieved by de"ning U (a, uJ ) as the largest subset of U (a, uJ ) resulting from the exclusion of a particular class of functions from U (a, uJ ).
7. CONCLUSION
Information is the complement of uncertainty, so model updating*which is a process of acquiring information*entails the reduction of uncertainty. Very often this uncertainty is a stark lack of knowledge, since it is the counterpart or backdrop of existing understanding: uncertainty describes what we do not know; information is what we do know. We have concentrated on this severe type of uncertainty, quantifying it with information-gap models, and we have considered updating to focus on the improvement of info-gap uncertainty models. Our main e!ort has been to show how the result of model updating can be evaluated in terms of the performance requirements of the system itself. That is, the central question has been: by how much can one con"dently demand better performance from the system as a result of acquiring new information and thereby reducing uncertainty? The central tool in this evaluation is the robustness function, a( (q, r ). This is the greatest A level of info-gap uncertainty consistent with successful performance. Theorem 1 establishes the irrevocable trade-o! between robustness-to-uncertainty and reward: enhanced performance of the system is obtained only by sacri"cing immunity to uncertainty (Fig. 1). The same "gure also illustrates the demand value of an increment of information, and shows how new information can be used to circumvent the robustness vs reward trade-o!. The horizontal displacement between an initial and an updated robustness curve assesses the increment in performance which can be demanded of the system, without loss of robustness, as a result of the increment in information. Stated in more classically economic
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terms, the demand value of an increment of information is the increment of improvement in performance (or reward) which the system can achieve without su!ering a loss of robustness, in exchange for applying the information. Theorem 2 introduces the idea of the &informativeness' of an info-gap model of uncertainty in terms of set-inclusion: an uncertainty-model is informative to the degree that it more tightly delimits the range of unknown variation. Theorem 2 implies that the demand value of information will be of constant sign if and only if the info-gap model is updated to a more informative model in the set-inclusion sense. Information is universally valuable only when it satis"es the set-inclusion criterion. We have considered three examples. The cantilever example in Section 4 illustrates the determination of the demand value of an increment of information about uncertain load pro"les, where the updated info-gap model is strictly included in the initial model. In Section 5, the demand value of a &complex' decision model is evaluated with respect to a &simple' model, where neither is set-included in the other. In this case the value of the information changes sign as a function of the level of performance required of the system, indicating that neither model is universally preferred. This means that the increment in information may or may not be worth acquiring, depending on the required level of performance. Interestingly, the &simple' model is preferred under stringent performance requirements, while the &complex' model is preferred in a lax environment. In Section 6, we consider the demand value of information in updating the impulse response function of a linear dynamic system. The updated system-uncertainty model is set-included in the initial model so that information is always valuable. However, it is seen that the actual worth depends on the time frame of interest. In all these examples we have stressed the importance of establishing the comparability of the info-gap models. In particular, the uncertainty parameter must have the same dimensions and calibration both before and after updating. This is a delicate matter and not always unambiguously achieved. Furthermore, the demand value studied here is an a priori assessment of the potential value of information which is not yet in hand. The demand value is a pre-measurement decision-support tool for anticipating the utility of information and for deciding what measurements and analyses to perform. We have indicated how an a posteriori assessment of information can be made.
ACKNOWLEDGEMENT
This research was supported by the Fund for the Promotion of Research at the Technion.
REFERENCES 1. Y. BEN-HAIM 1996 Robust Reliability in the Mechanical Sciences. Berlin: Springer-Verlag. 2. Y. BEN-HAIM 1999 Journal of the Franklin Institute 336, 1093}1117. Set-models of informationgap uncertainty: Axioms and an inference scheme. 3. Y. BEN-HAIM International Journal of Systems Science 31, 1511}1518. Robustness of model-based fault diagnosis: Decisions with set-models of uncertainty. 4. Y. BEN-HAIM 1998 Shock and
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8. A. MAS-COLELL, M. D. WHINSTON and J. R. GREEN 1995 Microeconomic ¹heory, Oxford: Oxford University Press. 9. J. ROSENHEAD (ed.), 1989 Rational Analysis For a Problematic =orld: Problem Structuring Methods For Complexity, ;ncertainty and Con-ict. New York: John Wiley & Sons. 10. Y. BEN-HAIM 1999 Structural Safety 21, 269}289. Design certi"cation with information-gap uncertainty. 11. Y. BEN-HAIM 2000 Journal of the Franklin Institute 337, 171}199. Robust rationality and decisions under severe uncertainty.
INFO-GAP VALUE OF INFORMATION IN MODEL UPDATING YAKOV BEN-HAIM Faculty of Mechanical Engineering, Technion~Israel Institute of Technology, Haifa 32000, Israel. E-mail: [email protected] (Received 31 October 1999, accepted 30 November 2000) Information is the complement of uncertainty, so model updating*which is a process of acquiring information*entails the reduction of uncertainty. We concentrate on severe uncertainty quanti"ed with information-gap uncertainty models, so updating focusses on the improvement of info-gap models of uncertainty. Our main e!ort is to show how model updating can be evaluated in terms of the performance of the system itself. The central tool in this evaluation is the robustness function, whose value expresses the greatest level of info-gap uncertainty consistent with successful performance. A theorem establishes an irrevocable trade-o! between robustness and reward (with "xed information). The demand value of an increment of information is the increment of improvement in performance (or reward) which the system can achieve without su!ering a loss of robustness, in exchange for applying the information. The demand value of an increment of information is evaluated by comparing the robustness trade-o! curves before and after updating. The second theorem introduces an idea of the &informativeness' of an info-gap model of uncertainty in terms of set inclusion: an uncertainty model is informative to the degree that it more tightly delimits the range of unknown variation. The main implication of the theorem is that information is universally valuable only when it satis"es the set-inclusion criterion. We consider three examples which illustrate the evaluation of the demand value of information. The importance of establishing the comparability of the info-gap models is stressed. It is explained that the demand value can be evaluated either as an a priori estimate of the potential value of information which is not yet in hand, or as an a posteriori assessment of information which has been acquired. 2001 Academic Press
1. INTRODUCTION
Information is a commodity and, like all other commodities, we buy information in many forms, including newspapers, espionage, college degrees and, of course, the updating of mathematical models of technological systems. But information di!ers in fundamental ways from all material commodities. The most peculiar feature of information is that, while it can go stale and thus lose its value, it is not used up when consumed. Information has a permanence and accessibility and, sometimes, an ease of communicability, which belie the tremendous cost which is often paid for information, and which bedevil attempts to quantify its worth. In this paper, we explore an economic (though not necessarily monetary) model of the value of information in technological model updating. The basic question we ask is: what is the worth of information? More speci"cally, what is the value of the information needed to reduce our uncertainty about the model of a system? More directly still, by how much can one con"dently demand better performance from the system as a result of acquiring new information and thereby reducing uncertainty? 0888}3270/01/030457#18 $35.00/0
2001 Academic Press
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This last question carries the seeds of the answer we will develop. Namely, viewing model updating as a process whereby uncertainty is reduced, how do we assess the value of information in the same units by which the system performance is evaluated? In Section 2, we describe information-gap models of uncertainty which quantify uncertainty as a disparity between what is known and what can be known. We formulate the robustness function which assesses the greatest info-gap uncertainty consistent with acceptable performance and discuss the trade-o! between robustness and reward. In Section 3, we formulate a concept of the demand value of information as the increment of improved performance accruing from an increment in information. We show that only when the information is ranked in a set-inclusion sense is the information universally valuable. In Sections 4}6 we discuss three examples.
2. INFO-GAP MODELS AND DECISIONS
2.1. INFO-GAP MODELS OF UNCERTAINTY Our quanti"cation of uncertainty is based on non-probabilistic information-gap models. An info-gap model is a family of nested sets. Each set corresponds to a particular degree of uncertainty, according to its level of nesting. Each element in a set represents a possible realisation of the uncertain event. Info-gap models, and especially convex-set models of uncertainty, have been described elsewhere, both technically [1] and axiomatically [2]. Uncertain quantities are vectors or vector functions. Uncertainty is expressed at two levels by info-gap models. For "xed a, the set U(a, uJ ) represents a degree of uncertain variability of the uncertain quantity u around the centrepoint uJ . The greater the value of a, the greater the range of possible variation, so a is called the uncertainty parameter and expresses the information gap between what is known (uJ and the structure of the sets) and what needs to be known for an ideal solution (the exact value of u). The value of a is usually unknown, which constitutes the second level of uncertainty: the horizon of uncertain variation is unbounded. Let R denote the non-negative real numbers and let S be a Banach space in which the uncertain quantities u are de"ned. An info-gap model U(a, uJ ) is a map from R;S into the power set of S. The basic axiom, which characterises the representation of uncertainty by info-gap models, is that the sets of an info-gap model are nested by the uncertainty parameter a: U(a, uJ )-U(a , uJ )
if a)a .
(1)
In many applications it is found that the relevant info-gap models obey speci"c structural axioms. The most common structural axioms are [2]: Contraction: U(0, 0) is a singleton set containing its centrepoint: U(0, 0)"+0,.
(2)
¹ranslation: U(a, uJ ) is obtained by shifting U(a, 0) from the origin to uJ : U(a, uJ )"U(a, 0)#uJ
(3)
where U#uJ means that uJ is added to each element of U. ¸inear expansion: info-gap models centred at the origin expand linearly: b U(b, 0)" U(a, 0) a
for all a, b'0
where (b/a)U means that b/a multiplies each element of U.
(4)
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In some situations the linear expansion axiom is altered to include non-linear expansion properties [3]. 2.2. INFO-GAP DECISIONS AND THE ROBUSTNESS FUNCTION Let the vector q represent the decision-maker's decisions or actions, which incur a consequence depending upon q and upon an uncertain vector or vector function u belonging to an info-gap model U(a, uJ ). The decision-maker knows a scalar reward function, R(q, u), which could be used, together with uJ , to maximise the nominal reward. The reward function R(q, u) must be construed very broadly, and can represent a performance requirement in vibrational analysis or other engineering design problems [4], or an economic or managerial goal [5]. R need not be a von Neumann}Morgenstern utility function. R is de"ned very generally in [6]. The decision theory developed in this paper is utterly di!erent from von Neumann}Morgenstern game theory [7]. Foremost among the distinctions is that the present theory does not depend on probabilistic information of any sort. Info-gap theory is particularly suited to highly unstructured uncertainty and severe lack of information. Furthermore, info-gap theory is not vulnerable to the di$culties which arise in von Neumann}Morgenstern theory due to its axiom of linear independence. In particular Allais and Ellsberg ¶doxes' [8] are avoided in info-gap theory. The robustness is the greatest value of the uncertainty parameter for which the performance is &acceptable'. In terms of the scalar reward function, &acceptability' is de"ned in one of the two ways: either R is no less than a lowest acceptable value, or R is no greater than a greatest acceptable value. In other words, desirable values of R are either small or large, depending on the application, and &acceptability' means that R is su$ciently small or su$ciently large. Acceptability is used here to refer to a type of satis"cing. The critical reward, r , is a parameter whose value need not be chosen a priori by the decision-maker. A De"ne A(q, r ) as the set of a-values for which the critical reward is guaranteed. When A desirable performance means that the reward function R takes small values, then A(q, r ) is A
(5)
(6)
A(q, r )" a: max R(q, u))r . A A u3U(a, uJ ) On the other hand, if large R is preferred then
A(q, r )" a: min R(q, u)*r . A A u3U(a, uJ )
In either case, the robustness function is the greatest value of the uncertainty parameter for which acceptable performance is assured. That is, the robustness is the least upper bound of the set A(q, r ): A aL (q, r )" sup a. (7) A a3A(q, r ) A
If A(q, r ) is empty then the reward r cannot be achieved at any level of uncertainty and we A A de"ne a( (q, r )"0. When a( (q, r ) is large then the decision-maker is immune to a wide range A A of variation, while if a( (q, r ) is small then even small #uctuations can lead to violation of the A performance requirement. Thus, &bigger is better' for the robustness a( . The robustness function establishes a preference ordering of actions. The decision-maker will prefer action q over action q if the robustness is greater with q than with q at the same critical value: q Y q
if a( (q, r )'a( (q , r ). A A
(8)
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The decision-maker's optimal robust strategy is to choose an action qL (r ) which maximises A the robustness: a( (qL (r ), r )"max a( (q, r ) A A A q3Q
(9)
where Q is the set of available actions. The logical structure of this decision algorithm as a &severe test' is discussed further in [2]. Robustness to uncertainty is important when we concentrate on the pernicious possibilities entailed by unknown variation. However, variations can be propitious and surprises can be bene"cient. The dynamic #exibility which is so important for survival, as Rosenhead has stressed [9], thrives on emergent opportunities. We can study this aspect of decisions with the &opportunity function' which is the logical complement to the robustness function a( and is discussed elsewhere [10]. 2.3. ROBUSTNESS TRADE-OFF The following theorem expresses a fundamental trade-o! between immunity-to-uncertainty and demanded reward. The proof of this theorem [11] depends only on the nesting axiom of info-gap models, equation (1). Theorem 1. ¸et U(a, uJ ) be an info-gap model and let R(q, u) be a uniformly continuous reward function. ¹he resulting robustness function aL (q, r ) is strictly monotonic on the set of critical A rewards r for which aL (q, r ) exists and is positive. aL (q, r ) decreases with improving r -value. A A A A Speci"cally, if small R is preferred, equation (5), then a small value of r is more desirable A and more demanding than a large value. Hence a( (q, r ) decreases with decreasing r : A A a( (q, r )(a( (q, r ) for r (r . (10) A A A A If large R is preferred, equation (6), then a( (q, r ) decreases with increasing r : A A a( (q, r )'a( (q, r ) for r (r . (11) A A A A This is true also for the robustness function evaluated at the optimal action, as stated in the following corollary. Corollary 1. ¸et U(a, uJ ) be an info-gap model, let R(q, u) be a uniformly continuous reward function, and let qL (r ) be a maximally robust strategy as a function of critical reward r . ¹hen A A the optimal robustness, aL (qL (r ), r ), is strictly monotonic in r , decreasing with improving r . A A A A That is, the maximal robustness is strictly monotonic on the set of reachable rewards. Theorem 1 and its corollary show the inexorable trade-o! between reward and immunityto-uncertainty. Robustness is obtained only in exchange for degraded performance. The decision-maker must choose a position on the spectrum of this trade-o!. This theorem also shows that the decision-maker need not make an irrevocable prior choice of the critical reward r . Rather, r is chosen in light of the analysis of the robustness to ambient A A uncertainties. For instance, the decision-maker may "nd that better performance can be demanded without substantially increasing the vulnerability to uncertainty. This (very pleasing) outcome of the analysis may well induce the decision-maker to demand better performance. Alternatively, the analyst may "nd that, by slightly diminishing the demanded performance, the immunity to uncertainty is greatly enhanced. Again, the analysis may lead to a modi"cation of the decision-maker's preferences. In short, the robustness a( (q, r ), A viewed as a function of critical reward, is a decision-support tool with which the decisionmaker assesses the options and their consequences.
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3. DEMAND VALUE OF INFORMATION
3.1. ROBUSTNESS PREMIUM AND DEMAND VALUE The basic question we ask is: what is the worth of information? The answer has two parts. We will assess the usefulness of information as a &robustness premium' which it entails. We will determine a value for information as the added reward which the decision-maker can demand without losing robustness when the information is exploited. This will turn out to be a &reward premium' which the information induces and will be recognised as what the economists call a &demand value'. The units of the demand value are the same as those of the reward function R(q, u), which may be dollars, or millimeters, or hertz, etc. Let a( (q, r ) and a( (q, r ) be the robustness functions from two di!erent constellations of A A information, each with its own info-gap model and reward function and underlying system models and performance criteria, as in Fig. 1. In short, these robustness curves are based on two di!erent &decision models'. The information upon which one curve is based may be obtained by updating or improving the information underlying the other curve. Alternatively, one curve may be based on better or simply di!erent analyses of existing data. Or, the curves may re#ect the assessments of uncertainty or choices of models made by di!erent analysts. While the pair of robustness curves in Fig. 1 do not cross, this is not the general case as we will see in subsequent examples. The demand value of the information needed to move from a( (q, r ) to a( (q, r ), *r in A A A Fig. 1, is the increment in reward which can be demanded after acquiring the new information. The horizontal arrows in Fig. 1 show the increment in reward which the decision-maker can con"dently demand as a result of using the di!erent information. This is a &demand value' in the economic sense that the decision-maker would be willing to pay any quantity up to, but not more than, this increment in reward in order to exchange the information. The demand value of information is expressed explicitly in the units of the reward by which the system itself is evaluated. We note that the demand value depends on the level of ambient uncertainty: where on the robustness curve the decision-maker is operating. Information has no absolute value; its value is relative to the environment. In particular, we see from this pair of curves that information is of relatively lower worth at a low level of demanded reward and at a high level of robustness. The robustness premium of information, *a( in Fig. 1, expresses the increment of immunity resulting from the exchange of information, and is shown by the vertical arrows. The
Figure 1. Two robustness curves based on di!erent information.
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robustness premium speaks directly to the issue of immunity, but it lacks the economic overtone of the demand value. The robustness premium, like the demand value, depends on the operating point at which the decision-maker is functioning. We can express the demand value at uncertainty a, for decision model &2' with respect to decision model &1', as the di!erence: *r (a)"r !r (12) A A A where the critical rewards r and r satisfy A A a( (q, r )"a"a( (q, r ). (13) A A If the robustness function decreases with increasing critical reward, equation (11), then *r (a) will be positive or negative if model &2' is more or less valuable than model &1', A respectively. Furthermore, the sign of *r (a) may change with a since the robustness curves A may cross. If the robustness function increases with increasing critical reward, equation (10), then *r (a) is negative for valuable increments and positive otherwise. A The robustness premium at critical reward r , of decision model &2' with respect to model A &1', may be either positive or negative, and is de"ned as *a( (r )"aL (q, r )!a( (q, r ). A A A
(14)
3.2. INFORMATION AND ROBUSTNESS We have indicated that, in general, the sign of the demand value can change with the level of uncertainty, indicating a switch in preference between the decision models. However, an important special case exists in which the sign of the demand value is constant, as implied by the following theorem. The proof of this theorem is found in [6]. The theorem employs a set-inclusion concept for comparing the &informativeness' of two info-gap models of uncertainty, U (a, uJ ) and U (a, uJ ). If U (a, uJ ) is a subset of U (a, uJ ) then the former constrains the ambient uncertainty more strictly than the latter. In this sense, U (a, uJ ) is more informative than U (a, uJ ). Theorem 2. ¸et U (a, uJ ) and U (a, uJ ) be two info-gap models with corresponding robustness functions aL (q, r ) and aL (q, r ), based on the same reward function and performance require A A ment. U (a, uJ ) is more informative than U (a, uJ ), in the set-inclusion sense, if and only if aL (q, r ) A is everywhere no less than aL (q, r ). ¹hat is, A U (a, uJ )-U (a, uJ ) (15) if and only if: a( (q, r )*a( (q, r ). (16) A A Theorem 2 implies that robustness functions will never cross (though they may intersect) if and only if their info-gap models are ranked by informativeness. This implies that the demand value between them will never change sign (though it may become zero). In other words, information-ranked info-gap models generate decision models for which one is always preferable (or equivalent) to the other in terms of their demand value. Theorem 2 is true also if the robustnesses are evaluated at their respective optimal actions. Corollary 2. ¸et U (a, uJ ) and U (a, uJ ) be two info-gap models with corresponding robustness functions aL (q, r ) and aL (q, r ) and optimal actions qL (r ) and qL (r ), based on the same reward A A A A function and performance requirement.
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463
U (a, uJ ) is more informative than U (a, uJ ), in the set-inclusion sense of equation (15), if and only if: a( (qL (r ), r )*aL (qL (r ), r ). A A A A
(17)
4. UNCERTAIN LOADS ON A CANTILEVER
We employ the cantilever, subject to uncertain static loads, to illustrate the demand value of information. We will de"ne a performance criterion, a de#ection model, and an info-gap uncertainty model. We will evaluate the demand value of improvements in the info-gap model of load uncertainty. Let x denote the position along the cantilever, where the clamped end is at x"0 and the free end is at x"¸. Static forces are applied in a single plane and perpendicular to the beam axis, where u(x) is the unknown force density, in units of N/m, at position x. The de#ection pro"le resulting from this load is v(x) which, for small de#ections, obeys the following di!erential equation: EI
dv(x) "u(x) dx
(18)
where E is Young's modulus for the material and I is the area moment of inertia of the cross-section. We assume that EI is constant along the length of the beam. The boundary conditions at the clamped end are v(0)"v (0)"0
(19)
where the prime denotes di!erentiation with respect to x. At the free end the boundary conditions are v (¸)"v (¸)"0.
(20)
The performance criterion is that the de#ection at the free end must not exceed a critical value, v : A "v(¸)")v . (21) A Thus, the performance requirement is of the small-is-better kind [equation (5)]. The force density u(x) is represented as nnx L u(x)"uJ # c sin (22) L ¸ LL "uJ #c2 (x) (23) L L where the nominal load, uJ , is constant, c is the vector of Fourier coe$cients and (x) is L L the vector of corresponding sine functions. The info-gap model for uncertainty in the load-pro"le Fourier-coe$cient vector c, with least and greatest spatial modes n and n , is U (a, uJ )"+u(x)"uJ #c2
(x): c2c)uJ a,, a*0. (24) L L L L The informativeness of the info-gap model U (a, uJ ), in the set-inclusion sense, is L L determined by the bandwidth. A tighter bandwidth implies stricter constraints on the possible load pro"les, and hence greater information about the environment. If n )m and m )n .
(25)
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Then (a, uJ )-U (a, uJ ). (26) U L L K K U (a, uJ ) is a more informative info-gap model than U (a, uJ ). K K L L The solution of equation (18), in response to load pro"le u(x)"uJ #c2 (x), for the L L clamped-free boundary conditions in equations (19) and (20), is
x L c ¸ x ¸x ¸x ! # ! L 24 6 4 2 nn LL L c ¸ x ¸ ¸ nnx # L # x! sin nn 6 nn nn ¸ LL ¸x x L c ¸ # ! L (1!(!1)L). (27) 6 2 nn LL We are particularly interested in the end de#ection, for which equation (27) simpli"es considerably: EIv(x)"uJ
EIv(¸) 1 1 L 1 (!1)L " # c ! L (nn) uJ ¸ 8 uJ 3nn LL GFFHFFI f 1 1 L " # c2f 8 uJ
(28)
(29)
where the elements of the vector f are de"ned in equation (28). The quantity EIv/uJ ¸ on the left-hand side of equation (28) is a dimensionless de#ection. In order to calculate the robustness function we must evaluate the greatest de#ection for any load pro"le in the info-gap model. Using Lagrange optimisation the result is max u3U (a, uJ ) L L
EIv(¸) 1 " #a (f2f . uJ ¸ 8
Denote the dimensionless form of the maximum allowed de#ection v as A EIv A. o" A uJ ¸
(30)
(31)
Equating the maximum normalised de#ection, equation (30), to the normalised performance requirement, o , and solving for a leads to the robustness for this info-gap model: A 8o !1 a( (o )" A . (32) L L A 8(f2f The performance requirement is that the dimensionless end de#ection be no greater than o , A which means that reward increases as o decreases: a small value of o is more desirable or A A demanding than a large value. In equation (32) and in Fig. 2 we see the usual trade-o! (even though the robustness curves have positive rather than negative slopes): the robustness a( (o ) increases as the performance degrades (as o increases), as anticipated by L L A A Theorem 1. Robustness functions versus critical de#ection are shown in Fig. 2 for values of m and G n satisfying relations (25), so the corresponding info-gap models are nested as in equation G (26). The horizontal arrow shows the demand value, *o (a), of the more informative A
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Figure 2. Robustness curves for more and less informative info-gap models.
uncertainty model. This demand value is calculated as in equations (12) and (13). The critical rewards o and o are evaluated from A K A L a( (o )"a"a( (o ). (33) K K A K L L A L The demand value of U with respect to U is L L K K *o (a)"o !o (34) A A K A L K L f! f . (35) "a L L LK LL The demand value of the more informative info-gap model, in units of o , is negative A because a small value of o is desirable. We also see that the demand value increases as the A ambient uncertainty, a, rises: an informative model is more useful in the presence of great uncertainty. A word of caution: the more informative info-gap model is more valuable if it is veri,ed. If we know that real variability is described by the tighter, more informative, of the two info-gap models, then we can demand better performance, and in this sense the information needed to update the less informative to the more informative model is valuable. Of course, it would be valuable (in a somewhat di!erent sense) to know that the putatively &more informative' model is wrong, and the &less informative' model in fact describes the ambient uncertainty. This is valuable because it informs us that we must either relax our demands on the system or employ alternative technologies to achieve assurance of better performance. The demand value in equation (35), in units of dimensionless de#ection, is shown in Table 1 for a range of m and n values and with m "n "1 and a"1. Moving from left to right on each row we see that *o gets better (more negative) as the di!erence n !m A increases. This means that the worth of U increases in comparison with U as the latter K L info-gap model becomes less informative than the former. The reverse is true as we move down each column: the worth of U decreases as it becomes more similar to U . These K L increments of value are expressed in units of the performance requirement of the system itself, *o or, equivalently, *v . The decision-maker presumably has a sound qualitative A A understanding of the signi"cance of increments in these parameters, which enables the assessment of the signi"cance of increments of information.
5. CANTILEVER: DISJOINT INFO-GAP MODELS
The set-inclusion criterion for informativeness, equation (15) or (26), captures only one aspect of the full gamut from &simple' to &complex' decision models. Furthermore, in
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TABLE 1 Demand value of an informative info-gap model, equation (35), for m "n "1 and a"1
m "1 2 3 4 5
n "1
2
3
4
5
0
!0.0084 0
!0.013 !0.0045 0
!0.015 !0.0067 !0.0022 0
!0.017 !0.0082 !0.0037 !0.0015 0
connection with equation (12), we explained that a demand value can be calculated for any pair of decision models, regardless of whether or not one is more informative than the other. As implied by Theorem 2, the sign of this demand value will be constant at all levels of uncertainty if and only if the info-gap models are nested. Otherwise, the sign will change indicating a di!erent preference between the models in di!erent circumstances. We will illustrate this by comparing the decision model of Section 4 against a nominally simpler model. The &complex' decision model represents uncertain distributed loads with the Fourier ellipsoid info-gap model of equation (24). We will suppose that spatial models from the 5th to the 10th make up the uncertain load functions, so n "5 and n "10. The fourth-order di!erential equation for beam de#ection, equation (18), is used, leading to the robustness function of equation (32). We will denote this robustness by a( which, for this speci"c A info-gap model, becomes
1 a( (o )"26.46 o ! . A 8 A A
(36)
The &simple' decision model assumes that the loads are concentrated at the mid-point of the beam. Instead of equation (28), the end de#ection resulting from a point force f at the mid-point is f ¸ . v(¸)" 12EI
(37)
The fractional variation of the point load is uncertain, so the info-gap model for variation in f is
U(a, fI )" f :
" f!fI " a ) , a*0 fI (2
(38)
where the nominal point load fI corresponds to the nominal uniform load density uJ in equation (24): fI "uJ ¸.
(39)
The normalisation by (2 in equation (38) is chosen to coordinate the scales of value of the uncertainty parameters in this info-gap model and the Fourier ellipsoid model in equation (24). We want U(a, fI ) and U (a, uJ ) to be comparable in size. In equation (38) the least L L upper bound of the squared fractional variation of the load, up to uncertainty a, is
f!fI a ) . 2 fI
(40)
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INFO-GAP VALUE
Evaluating the analogous quantity for U (a, uJ ) we "nd L L 1 L * nnx 1 * u(x)!uJ dx" c sin df x L uJ uJ ¸ ¸ ¸ LL a ) . 2
(41) (42)
Comparing equations (40) and (42) we see that, though the info-gap models U(a, fI ) and U (a, uJ ) are di!erent, the calibrations of their uncertainty parameters are comparable in L L terms of the horizons of uncertainty of these models. The robustness function for the simple decision model is found by evaluating the maximum de#ection at uncertainty a, equating this to the critical de#ection v , and solving A for a. One "nds a( (o )"12o !1 (43) Q A A where o is the dimensionless critical de#ection in equation (31). A The robustness functions of the &simple' and &complex' decision models, equations (43) and (36), are depicted in Fig. 3. These curves cross at the point (0.16, 0.92) in the o vs aL plane. A The intersection of the robustness curves means that there is no unequivocal demand-value preference for one or the other of these two decision models. The &simple' model is more robust if the normalised critical reward o is less than 0.16. Consequently, the demand value A of the &simple' model over the &complex' model is negative (preferable) in this range. However, for o '0.16 the situation is reversed, and the &complex' model is preferable to the A &simple' model in terms of robustness premium or demand value. In other words, when our performance requirement is o '0.16 we would prefer to verify A and use the &complex' info-gap model. On the other hand, if our system requirement is o (0.16 we would rather verify and employ the &simple' model. Note that the &complex' A uncertainty model is preferred in the case of the less-demanding performance requirement. The predicates &simple' and &complex' assert something about the structure of the uncertainty models, but do not imply anything about the quality of the ambient uncertainty or the performance which can be demanded of the system.
Figure 3. Robustness curves for &simple' and &complex' decision models.
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Y. BEN-HAIM
In comparing the robustnesses of these two decision models we are assuming that the underlying uncertainty parameters of the respective info-gap models are commensurate. As in the present case, this assumption is reasonable when each uncertainty parameter has comparable meaning and calibration as a measure of variation. In the spectral-bound model, equation (24), a is the fractional variation of the force-density spectral amplitudes. In the interval-uncertainty model, equation (38), a is the fractional variation in the magnitude of the point force. In each case the total force nominally applied to the beam is the same, uJ , and the calibrations of the uncertainty parameters have been coordinated as explained earlier.
6. LINEAR SYSTEM WITH UNCERTAIN DYNAMICS
In the previous two examples, Sections 4 and 5, we have evaluated increments in information about uncertain loads on a mechanical system whose properties are completely known. We now consider a system-model updating problem. The concept of modelupdating advocated here is that a model is improved by reducing the uncertainty associated with that model. We start with a nominal model subject to substantial uncertainty (represented by an info-gap model). We contemplate improving the model by reducing the uncertainty in the system model (by improving the info-gap model). The question we ask is: what is the value of the information needed to reduce our uncertainty in the system model? The analysis involves the following elements: *The performance requirements of the system. *The current system model and the current uncertainty model for this system model. *An improved model of the uncertainty in the system model. We do not yet know that this improved uncertainty model is correct. We are evaluating, a priori, the value of obtaining this improved uncertainty model. From this information we evaluate two robustness functions: that based on the initial system and uncertainty models, and that based on the initial system model and the updated uncertainty model. From these functions we estimate the value of the information required to reduce our uncertainty about the system model. This estimate does not explicitly account for the very real possibility that, by improving the uncertainty model, we may wish to revise the nominal system model as well. After updating the uncertainty model and incidentally also revising the nominal system model, we can evaluate the a posteriori value of the information obtained. This would be done by reevaluating the second robustness function using the updated system and uncertainty models. We use a single degree-of-freedom linear system to illustrate the procedure for a priori estimation of the value of reducing the uncertainty in a system-model. The response x (t) to S input u(t) is determined by the impulse response function g(t): R
g(t!q)u(q)dq.
x (t)" S
(44)
The performance requirement is that the response to each of ¸ speci"ed test inputs, u (t), 2 , u (t), be no greater than a critical value: * "x l (t)")r , S A
l"1, 2 , ¸.
(45)
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INFO-GAP VALUE
The actual impulse response function g(t) is unknown, though the current, nominal, model gJ (t) is known. The current info-gap model for uncertainty in the system model is U (a, gJ )"+g(t): "g(t)!g (t)")a,, a*0. (46) As usual, this info-gap model is a family of nested sets, representing two di!erent types of uncertainty in the system model. At "xed a, the actual impulse response function deviates from the nominal model in an unknown way, up to uncertainty a. Furthermore, the horizon of uncertainty is unknown: we do not know the value of a. The uniform-bound info-gap model of equation (46) is extremely &uninformative' and &naive', containing in"nitely many impulse response functions with very high-frequency components. We envision updating the system model by reducing the uncertainty which is inherent in U (a, gJ ). In the present example this is done by excluding all but speci"ed temporal modes. The amplitudes of the remaining modes, with frequencies from n n/¹ up to n n/¹, will remain uncertain. The revised system model will be L nnt (47) g(t)"gJ (t)# c cos L ¹ LL "gJ (t)#c2c(t) (48) where c is the vector of uncertain Fourier coe$cients and c(t) is the vector of the corresponding cosine functions. The updated uncertainty-model for the impulse-response function has the following structure: U (a, gJ )"+g(t)"gJ (t)#c2c(t): c2=c)ha,, a*0 (49) where = is a real, symmetric, positive de"nite matrix. The parameter h is chosen so that U (a, gJ ) is the largest possible subset of U (a, gJ ). That is, h is the largest value satisfying: max max "g(t)!gJ (t)")a (50) 0)t)2¹
g3U (a,gJ )
Note that c2c"g!gJ . Using Lagrange optimisation one "nds max
g3U (a,gJ )
"c2c(t)""ha(c2(t)=\c(t).
(51)
Consequently h is chosen as 1 . h" max0)t)2¹ (c2(t)=\c(t)
(52)
Let us emphasise that the updated info-gap model has been carefully constructed so as to be comparable to the initial model. The uncertainty parameters a in both models have the same units. In addition, U (a, gJ ) is explicitly constructed as the largest band-limited subset of U (a, gJ ). That is, these info-gap models are ranked by set inclusion as in equation (15). Furthermore, the information invested in the more informative model is explicitly de"ned: exclusion of all modes outside a speci"ed bandwidth. We know, experimentally, what we must do in order to verify or falsify the updated info-gap model. To evaluate the robustness functions we need the extreme responses*both minimum and maximum*on each of the info-gap models because
max "x (t)""max S g3U G
max x (t) , min x (t) S S g3U g3U G
G
.
(53)
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Y. BEN-HAIM
Consider "rst U (a, gJ ). Let sgn(y) represent the algebraic sign of y. R max x (t)" [gJ (t!q)#sgn(u(q))a] u(q) dq S g3U (a, gJ ) R R gJ (t!q)u(q) dq#a "u(q)" dq " GFFFHFFFI GFHFI
(54)
(55)
xJ (t) uJ (t) S "xJ (t)#auJ (t) (56) S which de"nes the terms xJ (t) (the nominal response) and uJ (t), both of which are known S functions. In similar fashion we "nd min x (t)"xJ (t)!auJ (t). S S
(57)
g3U (a, gJ )
Combining relations (53), (56) and (57) we conclude that max "x (t)"""xJ (t)"#auJ (t). S S
(58)
g3U (a, gJ )
The robustness function for test input u(t) is the greatest value of the uncertainty parameter consistent with not violating the performance requirement. To "nd the robustness we equate the maximum absolute response to the critical value and solve for a: r !"xJ (t)" S . "xJ (t)"#auJ (t)"r NaL (t)" A S A S uJ (t)
(59)
This assumes that the nominal response, xJ (t), satis"es the performance requirement S "xJ (t)")r . (60) S A If this is not true, then the robustness is zero: any deviation entails the possibility of violating the performance requirements. The overall robustness of the initial uncertainty model, considering the ¸ test inputs, is the lowest of the ¸ robustnesses: a( (t)" min aL l (t). S 1)l)¸
(61)
Now consider U (a, gJ ). Combining equations (44) and (48) we can express the response to input u(t) as x (t)"xJ (t)#c2 S S
R
c(t!q)u(t)dq GFFHFFI
(62)
t (t) S
"xJ (t)#c2t (t) S S which de"nes the known vector function t (t). S Using Lagrange optimisation we "nd the extremal values of c2t (t): S max c2t (t)"$ah (t2S (t)=\tS(t) S GFFHFFI o (t) S
g3U (a, gJ )
(63)
(64)
471
INFO-GAP VALUE
which de"nes the known function o (t). The maximum absolute response is S max "x (t)"""xJ (t)"#aho (t). S S S g3U (a, gJ )
(65)
As in equation (59), we "nd the robustness for input u(t) by equating the maximum absolute response to the critical response and solving for a: r !"xJ (t)" S a( (t)" A (66) S ho (t) S provided equation (60) holds; if not, the robustness is zero. The overall robustness a( (t) is evaluated as in equation (61). The demand value of the information needed to implement the updated uncertainty model is evaluated as in equations (12) and (13). For a single test input u(t) this becomes *r (a)"a[ho (t)!uJ (t)] A S "a
(67)
(t2(t)=\t (t) R S S ! "u(q)" dq . max0)t)¹ (c2(t)=\c(t)
(68)
The salient features of this result will appear in a simple example. Let the shape matrix = in equation (49) be the identity matrix. Consider the test input
u(t)"
b, 0,
0)t)h t'h.
(69)
The a priori value of updating the info-gap model of uncertainty, for this test input and for t)h, is found from equation (68) to be
L 1 nnt t sin ! (70) n ¹ ¹ LL In Fig. 4 we plot equation (70): the a priori demand value of the information needed to update the system model. The contemplated update reduces the uncertainty about the system model from the uniform-bound info-gap model of equation (46) to the Fourier ellipsoid-bound model of equation (49). Four points are to be noted. First, the demand value is in the units in which the system performance is evaluated, displacement, as speci"ed in equation (45). Thus numerical values of *r can presumably be A interpreted qualitatively (large, small; good, bad) by an engineering analyst who can qualitatively assess levels of performance as expressed by r . The demand value is the A improvement in performance (the increment in r ) which can be assured consequent to A obtaining the new information. A large negative value of *r (a) is desired, since a small value A of r is more stringent than a large value. A Second, *r in equation (70) is an a priori estimate. It is an assessment of the potential A value of the information. We do not know if uncertainty model U (a, gJ ) is veri"able. Equation (70) assesses the demand value accruing from U (a, gJ ) if we are able to establish the veracity of this model, and if the nominal model remains unchanged. We may "nd, in the course of testing the validity of U (a, gJ ), that it is wrong, that the evidence suggests a di!erent frequency band (n , n ), or that the nominal model gJ (t) should be altered in some way. After verifying a new updated info-gap model we would be in a position to evaluate an a posteriori value of the new information. The a priori estimate gives us some early indication of where we might "nd ourselves, informationally, after completing the update. h *r (t)"ab¹ A n
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Y. BEN-HAIM
Figure 4. Demand value vs time, equation (70). n "1, n "5 and ab¹"1. h"0.4714.
Third, the demand value varies in time, depending on the instant or time interval after system initiation at which the performance requirement is imposed. For times in excess of t/¹"0.4 the demand value becomes increasingly attractive (negative). So, if the relevant time frame is beyond t/¹"0.4, then there is a clear and strong incentive to acquire new information which would verify U (a, gJ ). On the other hand, if the relevant times are very short then the new information would be of little value. Finally, it is important to recognise that the demand value depends very intimately upon comparison of the two underlying info-gap models. Care must be taken to assure that the uncertainty parameters are comparable in units and scaling. In the present example this is achieved by de"ning U (a, uJ ) as the largest subset of U (a, uJ ) resulting from the exclusion of a particular class of functions from U (a, uJ ).
7. CONCLUSION
Information is the complement of uncertainty, so model updating*which is a process of acquiring information*entails the reduction of uncertainty. Very often this uncertainty is a stark lack of knowledge, since it is the counterpart or backdrop of existing understanding: uncertainty describes what we do not know; information is what we do know. We have concentrated on this severe type of uncertainty, quantifying it with information-gap models, and we have considered updating to focus on the improvement of info-gap uncertainty models. Our main e!ort has been to show how the result of model updating can be evaluated in terms of the performance requirements of the system itself. That is, the central question has been: by how much can one con"dently demand better performance from the system as a result of acquiring new information and thereby reducing uncertainty? The central tool in this evaluation is the robustness function, a( (q, r ). This is the greatest A level of info-gap uncertainty consistent with successful performance. Theorem 1 establishes the irrevocable trade-o! between robustness-to-uncertainty and reward: enhanced performance of the system is obtained only by sacri"cing immunity to uncertainty (Fig. 1). The same "gure also illustrates the demand value of an increment of information, and shows how new information can be used to circumvent the robustness vs reward trade-o!. The horizontal displacement between an initial and an updated robustness curve assesses the increment in performance which can be demanded of the system, without loss of robustness, as a result of the increment in information. Stated in more classically economic
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473
terms, the demand value of an increment of information is the increment of improvement in performance (or reward) which the system can achieve without su!ering a loss of robustness, in exchange for applying the information. Theorem 2 introduces the idea of the &informativeness' of an info-gap model of uncertainty in terms of set-inclusion: an uncertainty-model is informative to the degree that it more tightly delimits the range of unknown variation. Theorem 2 implies that the demand value of information will be of constant sign if and only if the info-gap model is updated to a more informative model in the set-inclusion sense. Information is universally valuable only when it satis"es the set-inclusion criterion. We have considered three examples. The cantilever example in Section 4 illustrates the determination of the demand value of an increment of information about uncertain load pro"les, where the updated info-gap model is strictly included in the initial model. In Section 5, the demand value of a &complex' decision model is evaluated with respect to a &simple' model, where neither is set-included in the other. In this case the value of the information changes sign as a function of the level of performance required of the system, indicating that neither model is universally preferred. This means that the increment in information may or may not be worth acquiring, depending on the required level of performance. Interestingly, the &simple' model is preferred under stringent performance requirements, while the &complex' model is preferred in a lax environment. In Section 6, we consider the demand value of information in updating the impulse response function of a linear dynamic system. The updated system-uncertainty model is set-included in the initial model so that information is always valuable. However, it is seen that the actual worth depends on the time frame of interest. In all these examples we have stressed the importance of establishing the comparability of the info-gap models. In particular, the uncertainty parameter must have the same dimensions and calibration both before and after updating. This is a delicate matter and not always unambiguously achieved. Furthermore, the demand value studied here is an a priori assessment of the potential value of information which is not yet in hand. The demand value is a pre-measurement decision-support tool for anticipating the utility of information and for deciding what measurements and analyses to perform. We have indicated how an a posteriori assessment of information can be made.
ACKNOWLEDGEMENT
This research was supported by the Fund for the Promotion of Research at the Technion.
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8. A. MAS-COLELL, M. D. WHINSTON and J. R. GREEN 1995 Microeconomic ¹heory, Oxford: Oxford University Press. 9. J. ROSENHEAD (ed.), 1989 Rational Analysis For a Problematic =orld: Problem Structuring Methods For Complexity, ;ncertainty and Con-ict. New York: John Wiley & Sons. 10. Y. BEN-HAIM 1999 Structural Safety 21, 269}289. Design certi"cation with information-gap uncertainty. 11. Y. BEN-HAIM 2000 Journal of the Franklin Institute 337, 171}199. Robust rationality and decisions under severe uncertainty.