... When the Info-Gap Spin Goes Marching in! ...Last modified: Friday, 30-Dec-2011 13:02:16 MST

This page takes up a new short piece of typical Info-Gap Spin. It reads as follows:

" ...Relation between robust-satisficing and min-max.These strategies are interchangeable as tools for describing observed behavior of an agent. However, they can lead to very different choices when used by an agent to select an action, depending on the agent's beliefs. We explain the observational equivalence and behavioral difference between these decision strategies ..."This paragraph is taken from the program description of a workshop:

Workshop on Info-Gap Theory and Its Applications in Design and Strategic Planningorganized by Frank Coolen and Matthias Troffaes.May 17-20, 2010

Department of Mathematical Sciences

Durham University

Durham, UKIn this note I expose, yet again, the info-gap rhetoric in this and similar paragraphs in the Info-Gap literature for what it is: spin, fog and empty rhetoric. This statement, which had been repeated in several publications and presentations, in effect conceals a number of highly embarrassing facts about Info-Gap decision theory.

Before I turn to these facts, it is important first of all to make clear that by using this type of spurious terminology ("observational" and "behvioral") and by raising spurious arguments, Ben-Haim creates a

diversionaway from the simple and straightforward question that he ought to have addressed in the first place, in so many words:

Simple Question:What exactly is the relationship between the following two simple mathematical models?

Info-Gap's Robustness Model (2001) Wald's Maximin Model (circa 1940) max {α ≥ 0: r(d,u) ≤ r* , ∀u∈U(α,û)}

max min f(x,s) x∈X s∈S(x) The simple and straightforward answer to this question has, of course, been available to the public since the end of 2006 and it reads as follows:

Simple answer to a simple question:

Relation between info-gap's robustness model and Wald's famous Maximin model.Contrary to the repeated claims in the info-gap literature that info-gap decision theory is a unique, novel, distinct, revolutionary theory, that is radically different from all current theories for decision under uncertainty, the truth is that, info-gap's robustness model is a simple instance of Wald's Maximin model (1939, 1945, 1950). This instance is known universally as

Radius of Stability modeland has been used extesively in numerical analysis, control theory and optimization theory at least since 1962. Furthermore, this instance of Wald's Maximin model was designed specifically for the treatment of robustness againstsmall perturbationsin a givennominal valueof a parameter, and it is thereforeutterly unsuitablefor the treatment ofsevereuncertainty where the uncertainty space isvastand the estimate of the true value of the parameter of interest is ofpoorquality.However, instead of taking the bull by the horns and dealing directly with this simple question and its straightforward answer, Ben-Haim spins his way around it as follows:

To begin with, take note of the terminology. As a rule, Ben-Haim makes every effort to avoid referring directly and explicitly to the term Maximin. Instead, he systematically uses the term "min-max strategy". I should also add that he equally avoids using (in this context) directly and explicitly the term "info-gap's robustness model". Instead he says "robust-satisficing strategy". Building on this diversionary language he thus dodges the task of dealing directly and unambiguously with the relation between the info-gap's robustness model Wald's famous Maximin model.

He then goes on to present a spurious argument by means of which he suggests (again indirectly, and not in so many words) an alleged difference between info-gap's robustness model and Wald's famous Maximin model by comparing info-gap's robustness model ("robust-satisficing strategy") to some ill-considered min-max model that he created for this purpose.

This leads him to the "conclusion" that whatever the "observational similarities" between these two "strategies" (read models), because of the "behavioral differences" between them the two models are different. In other words, the so called "behavioral" difference between the so-called "min-max strategy" and the so called "robust satisficing strategy" is supposed to imply a "behavioral difference" between Wald's famous Maximin model and info-gap's robustness model.

So, the question is: why is Ben-Haim so unwilling to address the above simple question?

After all, given that Wald's Maximin model is the most famous non-probabilistic model for the treatment of decision problems subject to severe uncertainty, and given that Ben-Haim claims that info-gap decision theory is so radically different from all current theories for decision under uncertainty, it is only natural to expect that he should have asked in so many words: in what way, if any, is info-gap's robustness model similar to and different from Wald's Maximin model?

Indeed, had Ben-Haim asked himself this very question before he launched info-gap decision theory, this would have saved him the major embarrassment caused by the following simple, obvious fact:

Maximin Theorem:Info-gap's robustness model is a simple instance of Wald's famous Maximin model (circa 1939). Specifically,

Info-Gap's Robustness model A corresponding instance of Wald's Maximin model max {α ≥ 0: r(d,u) ≤ r* , ∀u∈U(α,û)} ≡

max min f(α,u) α ≥ 0 u∈U(α,û) where f(α,u) = α if r(d,u) ≤ r*; and f(α,u) = -∞, otherwise. Note that d is treated as a given entity.

Maximin Corollary:Info-gap's decision model is a simple instance of Wald's famous Maximin model (circa 1939). Specifically,

Info-Gap's Decision model A corresponding instance of Wald's Maximin model

max max {α ≥ 0: r(d,u) ≤ r* , ∀u∈U(α,û)} d∈D ≡

max min g(d,α,u) d∈D,α ≥ 0 u∈U(α,û) where g(d,α,u)=α if r(d,u) &le r*; and g(d,α,u)=-∞ otherwise.

Proof of the Maximin Theorem:

Instance of Wald's Maximin Model Equivalent Math Programming formulation

max min f(α,u) α ≥ 0 u∈U(α,û) ≡

max { v: v ≤ f(α,u), ∀u∈U(α,û) } α ≥ 0 v ∈ ℜ

≡

max { α: α ≤ f(α,u), ∀u∈U(α,û) } α ≥ 0

≡

max { α: α ≤ f(α,u), ∀u∈U(α,û) }

≡

max { α: r(d,u) ≤ r*, ∀u∈U(α,û) } Info-Gap's Robustness Model where ℜ denotes the real line.

In other words, info-gap's robustness model is an instance of Wald's generic Maximin model that is determined by the correspondence

and is specified explicitly and unambiguously as follows:

- x ≡ α

- s = u

- X = [0,∞)

- S(x) = U(x,û)

- f(x,s) = x if r(d,s) ≤ r*, else it is equal to -∞.
Similarly, info-gap's decision model is an instance of Wald's generic Maximin model that is determined by the correspondence

and is specified explicitly and unambiguously as follows:

- x ≡ (d,α)

- s = u

- X = D×[0,∞)

- S(x) = U(α,û) , x=(d,α)

- g(x,s) = α if r(d,s) ≤ r*, else it is equal to -∞, where x=(d,α).
In due course I shall explain the details.

For now I want to draw attention to the following simple facts.

- Wald's generic Maximin model is incomparably more general and poweful than info-gap's robustness model and info-gap's decision model. Which means of course that there can be no talk whatsoever of any type of equivalence or "interchangeability" between them.

- That instance of Wald's Maximin model which is deployed in the Maximin Theorem is equivalent to info-gap's robustness model. Meaning of course that info-gap's robustness model is an instance of Wald's Maximin model. Both generate the same results.

- Similarly, that instance of Wald's Maximin model which is deployed in the Maximin Corrolary is equivalent to info-gap's decision model. Meaning of course that info-gap's decision model is an instance of Wald's Maximin model. Both generate the same results.

- Ben-Haim's cryptic statements do not address the Maximin Theorem.

- Ben-Haim's cryptic statements do not address the Maximin Corollary.

- Ben-Haim's cryptic statements address (by means of an ill-conceived comparison) the relationship between info-gap's decision model and the following Minimax model which, of course, is another instance of Wald's famous Maximin model:

Ben-Haim's ill-concidered Minimax model

min max r(d,u) d∈D u∈U(α,û) where D denotes the set of feasible decisions and α is a given non-negative number.

Comments

- In case you have not encountered Wald's Maximin model before, note that this stalwart of classical decision theory is not only one of the central models used in classical decision theory for the treatment of severe uncertainty, it has been used widely, (for decades) in many disciplines: economics, optimization, control theory, statistics and so on.

- The proof demonstrating that the Info-gap's robustness model is a simple instance of Wald's Maximin model is short and straightforward. It has been on my website since the beginning of 2006. It has been on WIKIPEDIA for a long time. It is included in many articles --- some published in refereed journals.

- The same applies to info-gap's decision model.

- Ben-Haim is fully aware of this fact.

- Indeed, on at least two occasions Ben-Haim stated in public that the proof of the Maximin Theorem is mathematically correct.

- The question is then: given the well-documented proofs showing that info-gap's robustness model and info-gap's decision model are simple instances of Wald's Maximin model, how is it that this fact is not stated in so many words in the workshop's program? Worse, how is it that the relation between info-gap's decision model and Wald's Maximin model -- which is clearly defined by the Maximin Corollary -- is unacceptably misrepresented by the claim that the two models are "observationally equivalent" but "behaviorally different"?

- I therefore urge you (especially participants in the Durham workshop) to read about this issue in the discussion on spin, fog and rhetoric and in the references therein. At present I shall only reiterate the following.

In brief:in his books, Ben-Haim launched info-gap with great fanfare as a distinct, novel, revolutionary theory that is radically different from all current theories for decision under uncertainty. However, as it became public knowledge that Info-Gap's robustness model is in fact a simple instance of Wald's Maximin model, Ben-Haim began to spin his way around this simple fact. The two code terms that he has been using to "explain" the alleged difference between Wald's Maximin model and info-gap's decision model are "observational equivalence" versus "behavioral difference". That this is nothing short of mathematical revisionism ought to be clear even to high-schooll students. For how can a simple instance of a prototype be different from the prototype?As I have been arguing all along, the upshot of all this is that not only is info-gap's robustness model not new, it is a reinvention of a well-established well-oiled wheel that ends being a square wheel! This is so because that particular instance of Wald's Maximin model, which is info-gap's robustness model, is implemented in the theory to seek out

localrobustness. This renders the model thoroughly unsuitable for the treatment of severe uncertainty, as it is described in the info-gap literature. This is the reason for my labeling info-gap decision theory a voodoo decision theory par excellence.

Now that we have a clear picture of the situation, let us go back to the paragraph in question in order to examine Ben-Haim's reasoning in his attempted (indirect) "analysis" of the relationship between info-gap' decision model and Wald's Maximin model.

" ...Relation between robust-satisficing and min-max.These strategies are interchangeable as tools for describing observed behavior of an agent. However, they can lead to very different choices when used by an agent to select an action, depending on the agent's beliefs. We explain the observational equivalence and behavioral difference between these decision strategies ..."The first thing to observe is that inexplicably Ben-Haim compares his info-gap's decision model to a

min-maxmodel, not a Maximin model. So the question that you would no doubt ask is: How come?After all, info-gap's decision model

maximizesthe size (α) of the "safe" region around the estimate û, whileNature(i.e. uncertainty) selects from U(α,û) the worst value of u (with respect to the performance constraint r(d,u) &le r*). So, it only stands to reason that from the decision maker's point of view, the modeling framework at work here is that of a Maximin game rather than a Minimax game. In short, an experienced decision theory specialist will immediately conclude that the "game" described by info-gap's decision model is a Maximin game, not a Minimax game.Hence, it is immediately clear that, as an instance of the Maximin model, info-gap's decision model should be compared to:

max min g(d,α, u) d∈D,α ≥ 0 u∈U(α,û) where the objective function g stipulates the outcome (α) of the game insofar as the constraint r(d,u) &le r* is concerned. Thus, we define g so that g(d,α,u) is equal to α if r(d,u) &le r* and is equal to -∞ otherwise.

It seems that part of the problem in this muddled handling of the info-gap - Maximin relation is due to Ben-Haim's apparent lack of familiarity with this type of modeling techniques which, I need hardly point out, are used routinely to incorporate

constraintsin a Maxim style worst-case analysis. That this is so is clear from the fact that he seems to be laboring under the mistaken, indeed groundless, notion that Wald's Maximin paradigm can deal only with a worst-case approach tooutcomes, namelyrewardsbut that it is unable to handle a worst-case approach toconstraints.He is apparently unaware that in the area of robust optimization, constraints are routinely incorporated in Maximin models.As a matter of fact, the

classic formulationof the generic Wald Maximin model has an equivalent formulation namely, theMath programming formulation. The latter appears of course in the first line of the proof of the Maximin Theorem. Here are these two formats:

Classic format Math Programming format

max min f(x,s) x∈X s∈S(x) ≡

max { v: v ≤ f(x,s), ∀s∈S(x)} x∈X v∈ℜ Undergraduate students are instructed in this simple modeling technique when, among other similar modeling issues, they are taught the transformation of a 2-person zero sum game (Maximin or Minimax) formulation into an equivalent standard linear programming formulation. In the robust optimization literature these formats are used interchangeably.

However, as Ben-Haim seems to be unfamiliar with modeling the Maximin in this manner, his approach to "elucidating" the relationship between info-gap's robustness model and a Wald-Type robustness, so as to demonstrate the purported differences and similarities between them is based on the following argument:

Regard α as a

parameterrather than adecision variable, and conisder some fixed given value of α. To determine whether α is admissible with respect to the constraint r(d,u) ≤ r*, minimize the value of r(d,u) over u∈U(α,û) and compare it to the value of r*. If the constraint is satisfied, increase the value of α. If the constraint is not satisfied, decrease the value of α. Thus, the Minimax model model he invokes is the following:

min max r(d,u) d∈D u∈U(α',û) where D is the decision space and α' is some given positive number.

Now, let α(d) denote the largest value of α' such that the optimal value of r(d,u) in this optimization problem does not exceed the value of r*. Then. clearly the most robust decision is a decision whose α(d) value is the largest.

Ben-Haim's conclusion is then that as info-gap's decision model performs similarly to this Minimax model, it follows that they are "observationally similar". However, because in the case of the Minimax model the value of α must be fixed in advance, it follows that the two models are "behaviorally different".

What Ben-Haim’s “argument” actually obscures is that he is in fact comparing his info-gap's decision model to the "wrong" instance of Wald's maximin model. Meaning that what he identifies is at most a difference between info-gap's decision model and an ill-conceived instance of the Maximin model, but not a difference between info-gap's decision model and a proper instance of Wald's Maximin model, such as the one specified by the Maximin Corollary.

Worse, because Ben-Haim makes every effort to dodge the Maximin Theorem and its Corollary, he misrepresents the relationship between info-gap's decision model and Wald-Type decision models. For, the Maximin Corollary indicates in no uncertaint terms that the relationship between info-gap's decision model and a Wald-Type robustness model is far more direct and intimate than the convoluted relation depicted by Ben-Haim's analysis. Indeed, the direct implication of info-gap's decision model being a simple instance of Wald's Maximin model means that Info-gap's decision model and the simple instance of Wald's Maximin model that is equivalent to it always generate the same solutions.

Still, as the above quoted statement indicates, Ben-Haim prefers to continue to spin around this issue.

But this cannot go on much longer because two info-gap scholars have already conceded that info-gap's robustness model is a Maximin model:

This theory has recently been shown by Sniedovich (2008) to be formally equivalent to Wald's maximin model in classical decision theory (French, 1988).Bryan Beresford-Smith and Colin J. Thompson

An info-gap approach to managing portfolios of assets with uncertain returns

Journal of Risk Finance

10(3), 277-287, 2009.See my review of this article for more details.

## Recent Articles, Working Papers, Notes

Also, see my complete list of articles

Moshe's new book! - Sniedovich, M. (2012) Fooled by local robustness,
Risk Analysis, in press.

- Sniedovich, M. (2012) Black swans, new Nostradamuses, voodoo decision theories and the science of decision-making in the face of severe uncertainty,
International Transactions in Operational Research, in press.

- Sniedovich, M. (2011) A classic decision theoretic perspective on worst-case analysis,
Applications of Mathematics,56(5), 499-509.

- Sniedovich, M. (2011) Dynamic programming: introductory concepts, in
Wiley Encyclopedia of Operations Research and Management Science(EORMS), Wiley.

- Caserta, M., Voss, S., Sniedovich, M. (2011) Applying the corridor method to a blocks relocation problem,
OR Spectrum, 33(4), 815-929, 2011.

- Sniedovich, M. (2011)
Dynamic Programming: Foundations and Principles, Second Edition,Taylor & Francis.

- Sniedovich, M. (2010) A bird's view of Info-Gap decision theory,
Journal of Risk Finance, 11(3), 268-283.

- Sniedovich M. (2009) Modeling of robustness against severe uncertainty, pp. 33- 42, Proceedings of the 10th International Symposium on Operational Research, SOR'09, Nova Gorica, Slovenia, September 23-25, 2009.

- Sniedovich M. (2009) A Critique of Info-Gap Robustness Model. In: Martorell et al. (eds),
.Safety, Reliability and Risk Analysis: Theory, Methods and Applications,pp. 2071-2079, Taylor and Francis Group, London.

- Sniedovich M. (2009) A Classical Decision Theoretic Perspective on Worst-Case Analysis, Working Paper No. MS-03-09, Department of Mathematics and Statistics, The University of Melbourne.(PDF File)

- Caserta, M., Voss, S., Sniedovich, M. (2008) The corridor method - A general solution concept with application to the blocks relocation problem. In: A. Bruzzone, F. Longo, Y. Merkuriev, G. Mirabelli and M.A. Piera (eds.),
11th International Workshop on Harbour, Maritime and Multimodal Logistics Modeling and Simulation,DIPTEM, Genova, 89-94.

- Sniedovich, M. (2008) FAQS about Info-Gap Decision Theory, Working Paper No. MS-12-08, Department of Mathematics and Statistics, The University of Melbourne, (PDF File)

- Sniedovich, M. (2008) A Call for the Reassessment of the Use and Promotion of Info-Gap Decision Theory in Australia (PDF File)

- Sniedovich, M. (2008) Info-Gap decision theory and the small applied world of environmental decision-making, Working Paper No. MS-11-08

This is a response to comments made by Mark Burgman on my criticism of Info-Gap (PDF file )

- Sniedovich, M. (2008) A call for the reassessment of Info-Gap decision theory,
Decision Point,24, 10.

- Sniedovich, M. (2008) From Shakespeare to Wald: modeling wors-case analysis in the face of severe uncertainty,
Decision Point,22, 8-9.

- Sniedovich, M. (2008) Wald's Maximin model: a treasure in disguise!,
Journal of Risk Finance,9(3), 287-291.

- Sniedovich, M. (2008) Anatomy of a Misguided Maximin formulation of Info-Gap's Robustness Model (PDF File)

In this paper I explain, again, the misconceptions that Info-Gap proponents seem to have regarding the relationship between Info-Gap's robustness model and Wald's Maximin model.

- Sniedovich. M. (2008) The Mighty Maximin! (PDF File)

This paper is dedicated to the modeling aspects of Maximin and robust optimization.

- Sniedovich, M. (2007) The art and science of modeling decision-making under severe uncertainty,
Decision Making in Manufacturing and Services,1-2, 111-136. (PDF File) .

- Sniedovich, M. (2007) Crystal-Clear Answers to Two FAQs about Info-Gap (PDF File)

In this paper I examine the two fundamental flaws in Info-Gap decision theory, and the flawed attempts to shrug off my criticism of Info-Gap decision theory.

- My reply (PDF File) to Ben-Haim's response to one of my papers. (April 22, 2007)
This is an exciting development!

- Ben-Haim's response confirms my assessment of Info-Gap. It is clear that Info-Gap is fundamentally flawed and therefore unsuitable for decision-making under severe uncertainty.

- Ben-Haim is not familiar with the fundamental concept
point estimate. He does not realize that a function can be a point estimate of another function.So when you read my papers make sure that you do not misinterpret the notion

point estimate. The phrase "A is a point estimate of B" simply means that A is an element of the same topological space that B belongs to. Thus, if B is say a probability density function and A is a point estimate of B, then A is a probability density function belonging to the same (assumed) set (family) of probability density functions.Ben-Haim mistakenly assumes that a point estimate is a point in a Euclidean space and therefore a point estimate cannot be say a function. This is incredible!

- A formal proof that
Info-GapisWald's Maximin Principlein disguise. (December 31, 2006)

This is a very short article entitled Eureka! Info-Gap is Worst Case (maximin) in Disguise! (PDF File)

It shows thatInfo-Gapis not a new theory but rather a simple instance ofWald's famous Maximin Principledating back to 1945, which in turn goes back to von Neumann's work on Maximin problems in the context ofGame Theory(1928).

- A proof that
Info-Gap's uncertainty model is fundamentally flawed. (December 31, 2006)

This is a very short article entitled The Fundamental Flaw in Info-Gap's Uncertainty Model (PDF File) .

It shows that becauseInfo-Gapdeploys a single point estimate under severe uncertainty, there is no reason to believe that the solutions it generates are likely to be robust.

- A math-free explanation of the flaw in Info-Gap. ( December 31, 2006)

This is a very short article entitled The GAP in Info-Gap (PDF File) .

It is a math-free version of the paper above. Read it if you are allergic to math.

- A long essay entitled
What's Wrong with Info-Gap? An Operations Research Perspective(PDF File) (December 31, 2006).

This is a paper that I presented at theASOR Recent Advances in Operations Research(PDF File) mini-conference (December 1, 2006, Melbourne, Australia).## Recent Lectures, Seminars, Presentations

If your organization is promoting Info-Gap, I suggest that you invite me for a seminar at your place. I promise to deliver a lively, informative, entertaining and convincing presentation explaining why it is not a good idea to use — let alone promote — Info-Gap as a decision-making tool.

Here is a list of relevant lectures/seminars on this topic that I gave in the last two years.

- ASOR Recent Advances, 2011, Melbourne, Australia, November 16 2011. Presentation: The Power of the (peer-reviewed) Word. (PDF file).

- Alex Rubinov Memorial Lecture The Art, Science, and Joy of (mathematical) Decision-Making, November 7, 2011, The University of Ballarat. (PDF file).

- Black Swans, Modern Nostradamuses, Voodoo Decision Theories, and the Science of Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Invited tutorial, ALIO/INFORMS Conference, Buenos Aires, Argentina, July 6-9, 2010).

- A Critique of Info-Gap Decision theory: From Voodoo Decision-Making to Voodoo Economics(PDF File) .

(Recent Advances in OR, RMIT, Melbourne, Australia, November 25, 2009)

- Robust decision-making in the face of severe uncertainty(PDF File) .

(GRIPS, Tokyo, Japan, October 16, 2009)

- Decision-making in the face of severe uncertainty(PDF File) .

(KORDS'09 Conference, Vilnius, Lithuania, September 30 -- OCtober 3, 2009)

- Modeling robustness against severe uncertainty (PDF File) .

(SOR'09 Conference, Nova Gorica, Slovenia, September 23-25, 2009)

- How do you recognize a Voodoo decision theory?(PDF File) .

(School of Mathematical and Geospatial Sciences, RMIT, June 26, 2009).

- Black Swans, Modern Nostradamuses, Voodoo Decision Theories, Info-Gaps, and the Science of Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Department of Econometrics and Business Statistics, Monash University, May 8, 2009).

- The Rise and Rise of Voodoo Decision Theory.

ASOR Recent Advances, Deakin University, November 26, 2008. This presentation was based on the pages on my website (voodoo.moshe-online.com).

- Responsible Decision-Making in the face of Severe Uncertainty (PDF File) .

(Singapore Management University, Singapore, September 29, 2008)

- A Critique of Info-Gap's Robustness Model (PDF File) .

(ESREL/SRA 2008 Conference, Valencia, Spain, September 22-25, 2008)

- Robust Decision-Making in the Face of Severe Uncertainty (PDF File) .

(Technion, Haifa, Israel, September 15, 2008)

- The Art and Science of Robust Decision-Making (PDF File) .

(AIRO 2008 Conference, Ischia, Italy, September 8-11, 2008 )

- The Fundamental Flaws in Info-Gap Decision Theory (PDF File) .

(CSIRO, Canberra, July 9, 2008 )

- Responsible Decision-Making in the Face of Severe Uncertainty (PDF File) .

(OR Conference, ADFA, Canberra, July 7-8, 2008 )

- Responsible Decision-Making in the Face of Severe Uncertainty (PDF File) .

(University of Sydney Seminar, May 16, 2008 )

- Decision-Making Under Severe Uncertainty: An Australian, Operational Research Perspective (PDF File) .

(ASOR National Conference, Melbourne, December 3-5, 2007 )

- A Critique of Info-Gap (PDF File) .

(SRA 2007 Conference, Hobart, August 20, 2007)

- What exactly is wrong with Info-Gap? A Decision Theoretic Perspective (PDF File) .

(MS Colloquium, University of Melbourne, August 1, 2007)

- A Formal Look at Info-Gap Theory (PDF File) .

(ORSUM Seminar , University of Melbourne, May 21, 2007)

The Art and Science of Decision-Making Under Severe Uncertainty(PDF File) .

(ACERA seminar, University of Melbourne, May 4, 2007)

- What exactly is Info-Gap? An OR perspective. (PDF File)

ASOR Recent Advances in Operations Researchmini-conference (December 1, 2006, Melbourne, Australia).

Disclaimer:This page, its contents and style, are the responsibility of the author (Moshe Sniedovich) and do not represent the views, policies or opinions of the organizations he is associated/affiliated with.