Mathematical Modeling for Strategic Freshwater Management

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Summary

Freshwater ecosystems provide essential services for human populations and arehome to the greatest concentration of biodiversity on Earth. As abuse and degra-dation of these ecosystems increase from factors such as pollution and climatechange, countries around the world have strived for a successful water strategyfocusing on methods of storage and movement, de-salinization, and conservation.Using Russia as our subject, our research observed poor water quality that stemsdirectly from the regulation of poor fresh water resource management. In thispaper, we developed a feasible, uniform, and descriptive mathematical model thathas potential in serving as a framework for capturing nearly all the complexitiesof fresh water resource management. Our model is based on an information-flowview, referred to as situation theory, that involves layers upon layers of differentsituations for the country’s fresh water. It allows researchers, government agen-cies, and other interested parties to analyze data of different ’objects’ (i.e. theworkers, de-salinization technology, location of water sources) in a given problemand formally determine an optimal solution.

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Situation Theory and Mathematical

Modeling for Strategic Freshwater

Management

Team # 21629February 4, 2013

Abstract

Freshwater ecosystems provide essential services for human populations andare home to the greatest concentration of biodiversity on Earth. As abuse anddegradation of these ecosystems increase from factors such as pollution and climatechange, countries around the world have strived for a successful water strategyfocusing on methods of storage and movement, de-salinization, and conservation.Using Russia as our subject, our research observed poor water quality that stemsdirectly from the regulation of poor fresh water resource management. In thispaper, we developed a feasible, uniform, and descriptive mathematical model thathas potential in serving as a framework for capturing nearly all the complexitiesof fresh water resource management. Our model is based on an information-flowview, referred to as situation theory, that involves layers upon layers of differentsituations for the country’s fresh water. It allows researchers, government agen-cies, and other interested parties to analyze data of different ’objects’ (i.e. theworkers, de-salinization technology, location of water sources) in a given problemand formally determine an optimal solution.

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Contents

1 Introduction 4

2 Approach 4

3 Understanding Situation Theory 53.1 Infons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Infon Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.5 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Applications of Situation Theory 74.1 Type Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Satisfaction Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Layered Formalism and Zooming . . . . . . . . . . . . . . . . . . . 8

5 Modeling a Strategy 95.1 Identifying a Situation . . . . . . . . . . . . . . . . . . . . . . . . . 105.2 The Hypothetical Results and Analysis . . . . . . . . . . . . . . . . 12

6 Conclusion 146.1 Strengths and Weaknesses . . . . . . . . . . . . . . . . . . . . . . . 146.2 Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156.3 Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

7 Non-Technical Position Paper 16

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1 Introduction

With increasing population growth in this developing world, concern about theshortage of freshwater water also rises. Rich with almost all resources, Russianever had to worry about the their water quantity. Regardless, issues of the equaldistribution of Russia’s freshwater water resources has always been a top priorityup until 2004 as only 8% of the entire country’s water resources are located in themost populated area of the country, west of the Ural Mountains.1 However, inthe past decade, a situation in the utilities sector deteriorated significantly, andnow potable water and international quality standards achievement are extremelyacute issues.1

These issues are the consequences among many by the government. Devel-opment of the state water management policy both at the federal and regionallevel now requires a shift in water resource management paradigm from the equaldistribution of water resources concept, to the concept of proper management andaccountability of the resources.2 To meet the projected fresh water needs in 2025,our research show that the issues Russia currently faces begins with poor waterresource and cycle management, with an ineffective municipal enterprise supply-ing drinking water and cleaning wastewater.2 Such management requires a newapproach and another process. Along these lines, our mathematical model laysout and discusses the integration of a new information support system.1

2 Approach

We believe the development of a feasible, uniform, and descriptive mathematicalmodel has the primary responsibility for serving as a framework for capturingnearly all the complexities of fresh water resource management in Russia. We notethat the primary aspects of interest in our model includes storage and movement,de-salinization, and conservation of the fresh water. The model is projected to becapable of sufficient precision to allow for communication across different domainsboth in the government and the private industry. Our framework is based off aninformation-flow view that will allow for analysis by the government for nearly allaspects of water supply and sanitation needs for optimal management.

Though initially thought of as an unusual approach, our logic in establishingwhat to model is largely due to the definitive role of information and informationflow that comprises a productive strategy. With an element of uncertainty aboutthe future, we acknowledged a strategy is more about a dynamic set of options thana fixed plan. More than a decade of rapid technological advances and exponentialpopulation increases will leave room for a number of errors on one best strategy.Therefore, our mathematical model goes to determine a varying set ofoptions, or multiple strategies, that are considered best for any giventime in the future up until the desired year of 2025 through use of amathematical theory on information called situation theory.

1Russian Federation2Russian Water Sector

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3 Understanding Situation Theory

3.1 Infons

Information cannot be considered an independent entity but it should be takenas information about a situation that is built up from discrete items known asinfons.3 Infons are of the form:

〈〈R, a1, . . . , an, 1〉〉〈〈R, a1, . . . , an, 0〉〉

where R is an n-place relation, the value 0 or 1 represents polarity and the a1, anare appropriate objects for R. An n-place relation is a class of ordered systemsof n elements in the language of set theory and of algebra.3 If the ordered pair(a1, a2) belongs to some relation R, it is said that a1 is related to a2 by R. Thereis a significant relationship between a given situation s and an infon, designatedσ, in the form as:

s |= σ

and we say s supports σ and can be now written in the form as:

s |= 〈〈R, a1, . . . , an, 1〉〉

This means that in the situation s, the objects a1, . . . , an stand in relation to R.Conversely, to state that in a situation s, the objectsa1, . . . , an do not stand inrelation to R, the form is as:

s |= 〈〈R, a1, . . . , an, 0〉〉

Infons are items of information, known more precisely as data. Independently,they are not true or false, but rather will be true or false if applied to a situation.3

3.2 Objects

The objects, called uniformities, in a certain ontology of information include thefollowing3:

Object Denotation Description

indiviudals objects such as chairs, people,hands, etc. that the agent in-dividualates or identifies (essen-tially unitary items)

a, b, c

relations uniformities either individu-alated or identified by the agentthat link together certain otheruniformities

P,Q,R

spatial locations description of where something isphysically located

l, l′, l′′, l0, l1, l2, . . .

temporal locations location in time (usually justcalled time)

t, t′, t′′, t0, t1, t2, . . .

types a constancy that is of higher or-der identified by the agent

S, T, U, V

parameters the limits that identify certaintypes

a, s, t, l

3HHL Situation Theory

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The perception behind this ontology is that in a study of the activity (bothphysical and cognitive) of a particular agent or specids of agent, we notice thatthere are certain regularities or uniformities that the agent either individuates orelse discriminates in its behavior.4 For example, people individuate certain thingsin reality as objects (a cup of water is considered an object in the real-world) butthought of as individuals in situation theory. The cup of water’s behavior canbehave in a methodical way based on situation types (spatial location, temporallocation, etc.). In general we will write:

α : V

to indicate that α is of type V.

3.3 Parameters

Parameters are measurable factors that forms one or more sets that defines asystem or the condition of a situation. They are not individuated by the agentbut theoretical constructs. However, within the theoretical framework, parameterscapture and correspond to important aspects of the agent’s cognitive behavior.2

The notation of parameters is simple to remember as the letters of a type:l- location, t- time (temporal location), a- individual, s- situation etc.

3.4 Infon Logic

Using parameters, it is possible for there to be a conjunction of two infons σ, τusing the form σ ∧ τ . Formally, for any situation s the conjunction is4:

s |= σ ∧ τ iff s |= σ and s |= τ

This conjunction is no longer considered just an infon but a compound infon. Thedisjunction of two infons, σ, τ using the form σ ∨ τ . Formally, for any situation sthe conjunction is4:

s |= σ ∨ τ iff s |= σ or s |= τ (or both)

For any situation, s:if an infon, σ (or compound infon) that involves a parameterz and u is some set, then

s |= (∃z ∈ u)σ

is a compound infon4.For any situation, s: if an infon, σ (or compound infon) that involves the

parameter z and u is some set, then

s |= (∀z ∈ u)σ

is a compound infon4.The bounding set, u, in cases of both universal quantification (for all) and

existential quantification (for some) may be comprised all objects of a certainkind in situation, s.4 Despite sounding unrestricted, the quantification is stillsituated or fixed in this sense. When someone honestly declares

All citizens have equal rights.

the quantification is likely fixed to a country such as the United States but it isnot a globally true claim4.

4Stanford Army Report

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3.5 Constraints

Fundamental processes are captured by the mechanism of type procedures whileclassification of the world is captured by a cognitive agent5. Constraints allowsthe agent to make rational inferences and operate in a logical manner by imple-menting a theoretical mechanism for certain situations. In short, the constraintsare linkages between situation types, both of similarity or difference. Examples ofconstraints range from natural laws, to logical rules, to economic rules.

To understand the linkage, the following constraint is conventional:

Heartbeat means life.

In a type of situation, S, where a heartbeat is existant, and a type of situation,S′ where a person is alive, an agent can grasp the information that the person isalive by observing the present of his/her heartbeat. In addition, an agent can beaware of the constraint that links both situation types together which is denotedby5:

S ⇒ S′

In a type of situation, S′′, where someone verbally communicates the existantof a heartbeat to the type of situation after physically feeling the heartbeat’sexistance S′ is denoted as:

Heartbeatmeans heartbeat. −→ S′′ ⇒ S′

Being informed of the rules of linguistics allows an agent to be aware of theconstraint between S′′ and S′.

4 Applications of Situation Theory

4.1 Type Abstraction

Situation theory provides various mechanisms for defining types, ensuring thatan abstract type is distinct from its representative. The two most basic methodsare the type-abstraction procedures for the construction of two kinds of types:situation-types and object-types.5

• Situation-Types: Given a SIT -parameter, s, and a compound infon, σ,there is a corresponding situation-type [s|s |= σ], the type of situation inwhich σ obtains. This process of obtaining a type from a parameter, sis known as (situation-) type abstraction. The parameter s is called theabstraction parameter used in this type abstraction. For example:

[SIT1|SIT1 |= 〈〈running, p, LOC1, T IM1, 1〉〉]

• Object-Types: These include the basic types TIM, LOC, IND, RELn,SIT, INF, TYP, PAR, and POL as well as the more fine-grained uniformitiesdescribed below. Object-types are determined over some initial situation.Let s be a given situation. If x is a parameter and σ is some compoundinfon (in general involving x, then there is a type [x | s |= σ], the type of allthose objects x to which x may be anchored in the situation s, for which


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the conditions imposed by σ obtain.This process of obtaining a type [x | s |= σ] from a parameter, x, a situation,s, and a compound infon, σ, is called (object-) type abstraction.The parameter, x, is known as the abstraction parameter used in this typeabstraction.The situation s is known as the grounding situation for the type. In manyinstances, the grounding situtations, s, is the world or the environment welive in (generally denoted by a w). For example, the type of all people couldbe denoted by:

[IND1 |w |= 〈〈sees, Jon, e, LOC1, T IM1, 1〉〉]

4.2 Satisfaction Diagram

From the previous section, we can conclude that across actual situations, con-straints function by gathering together numerous generalities. The facilitation ofinformation flow stems from the agent’s ability to make a rational inference. Theability of an agent is considered with two requirements:

1. The agent must be able to separate the two types S and S′ at the most basicof cognitive levels.6

2. The agent must be aware of, or behaviorally attuned to, the constraint.

When the agent finds itself in a situation s of type S, the logical inference that asituation s′ of type S′ can be made.

SC

=⇒ S′

s : S ↑ ↑ s′ : S′

s∃−→ s′

Figure 1

So suppose S =⇒ S′ represents the constraint heartbeat means life. An agentA physically feels a heartbeat of someone B. By the constraint, agent A con-cludes that the person B must be alive resulting in a situation s’ of type S’. Theclear separation between two different kinds of entity that are necessary in theformulation and delivery of information is an important feature of this analysismethod.

1. Abstract types and the constraints that link them.

2. Actual situations in the world that the agent encounters or whose existenceis inferred.

4.3 Layered Formalism and Zooming

Every object, from individual to situation, can be divided into smaller constituents;a possibility of depicting and evaluating a domain from any scope of smallerentities. This implies an agent has the capability of moving up and down thegranularity scale, zoomın on a level, and review any issues of the analysis. This


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technique was developed by Devlin K. and Rosenburg D called Layered Formal-ism and Zooming (LFZ). 7 The initial analysis of data is largely non-mathematicalbased on ethnomethodology, but uses the mathematical formalisms of situationtheory.

A process of increased formalism and stepwise refinement begins with the ini-tial analysis until a problem is encountered in the specified domain. Mathematicalprecision is increased for zooming to understand in detail what the problem is.Once resolved, an agent zooms out and begins with a new initial analysis of adifferent area. At each step of the refinement process, you adopt the minimalpossible level of formalism and the minimal possible level of precision, therebyminimizing the likelihood of any inadvertent alteration to the data under consid-eration.5 Therefore, a check is conducted between the analysis and the data aftereach cycle of analysis to prevent errors.

5 Modeling a Strategy

In using situation theory to capture all aspects of managing fresh water resources,we needed to represent several different ontologies in situation-to-theoretic terms:

• Methods of Desalinization: geothermal, multi-stage flash distillation, ion ex-change, reverse osmosis, nanofiltration, solar, low-temperature thermal, etc.(each method can be broken down into smaller ontologies such as technol-ogy)

• People: actions, capabilities, expertise, dependability, position of power, etc.

• Buildings/Constructs: operation, size, time to build, amount of workersneeded, energy required to run, materials, surroundings, etc.

• Technology/Equipment: operation, capabilities, effects, reliability, etc.

• Water Resources: river, groundwater reservoir, lakes, sea, etc.

• City/Town: population, number of businesses, terrain, location, etc. (manydifferent cities each with their own ontology)

• Regulations: many different rules pertaining to separate local governments

The interaction of so many different ontologies, some within each other, lead tocommunication difficulties, but we now advocate the adoption of a single uniformframework.

7Soft Mathematics

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Figure 2 : A dynamic-network conception of management offreshwater resources in a city.

The framework we are using can be understood easily by viewingthe Russian local government water resource management as a dy-namic network in which various different kinds of entities flow andinteract. Using satisfaction diagrams, we achieve a uniform descrip-tion of each action by each entity in a managerial view.8

5.1 Identifying a Situation

We begin by identifying situation types and the ontologies involved:

1. We let S be the type of situation that contains de-salinizationmethods for producing fresh water as its object.

2. We let S ′ be the type of situation that contains people(i.e work-ers) as its object.

3. We let S ′′ be the resulting situation from the situations S andS ′. That means S ′′ will have two situations where the S is fixedand the S ′ is flexible. The flexibility of S ′ comes from changing

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its objects. Each time an object in S ′ changes, S ′′ may or maynot be a different result.

We determine that the following sets will be used as our objects inthis example of situation S, S ′, and S ′′:

• People = {laborers, managers, engineers, inspectors, drivers}

• De-salinization Methods = {geothermal, solar, ion exchange}

• Water Sources = {lake, sea, river, groundwater reservoirs} *Note:Later, in our satisfaction diagram, we will give proper names forRussian water sources*

• City/Town = {Moscow, St. Petersburg, Ufa, Chelyabinsk}We now can write a case as:

• S = [s|s |= 〈〈geothermal, t, 1〉〉]

• S = [s|s |= 〈〈laborers, t, 1〉〉]

• S = [s|s |= 〈〈calculating, a, t, 1〉〉︸︷︷︸stable situation

∧ 〈〈 budget︸︷︷︸stable object

, a,Lake Baikal︸︷︷︸flexible object

, t, 1〉〉

︸︷︷︸flexible situation

]

Thus, we constract the constraint figure as follows:

C = [S =⇒ S ′]

SC

=⇒ S ′

s : S ↑ ↑ s′ : S ′

s∃−→ s′

Figure 3

From Figure 3, S =⇒ S ′ represents the constraint people needto take over the technology and the Russian government eventuallyneeds to consider calculating that the cost of technology, people, andwater resource (in this case Lake Baikal) for building the necessarywater supply plant will not exceed the budget.

If we keep changing the object that stands in the situation S, S ′, S ′′

(only as the flexible object), we will still be able to get the same con-straint construction of Figure 3.

However, there are a lot of cases that can occur by substitutingthe objects of S, S ′, S ′′ with any element from the sets of people, de-salinization methods, and city/town.

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5.2 The Hypothetical Results and Analysis

The following are three examples of cases, that only looks at the wa-ter supply network (supply, purification stations) in which objects aresubstituted into different situation sets:

Hypothetical CASE 1 Lake Baikal has started becoming a popular tourist destinationthe last several years. In order to keep the business of tourismup, conservation of the lake is closely scrutinized. In one area ofthe lake, however, conservation efforts of the lake is considered tobe very poor. The following case sets up a satisfaction diagramfor the government to analyze the problem at what they believeare the primary factors.

S = [s|s |= 〈〈Agricultural Farms, t, 1〉〉]S ′ = [s|s |= 〈〈managers, t, 1〉〉]S = [u|u |= 〈〈calculating, a, t, 1〉〉∧〈〈budget, a,Lake Baikal, t, 1〉〉]

Figure 4 : Satisfaction Diagram for Hypothetical Case 2

Possible Conclusion: The management in charge of conservingthe Lake came from very experienced backgrounds with superiorratings from their workers. Even with the ability to conservetheir region of Lake Baikal, statistics show that the row of agri-cultural farms that lie near the lake have had significant amountsof runoff due to record rainfall in the region over the past month.The government knows where the problem lies, and can continuezooming in on issues within the farm to solve the initial crisis.

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Hypothetical CASE 2 In a town near the Amur River, a water purification plant hasbeen failing quality inspections after switching to new de-salinizationmethods. The government believes the objects of primary inter-est, which lead on to failing quality inspections, revolve aroundthe method switch, the Amur River, and poor management atthe plant. To determine a resulting situation S ′′ while noting thebudget cap, the government inputs information on their objects,and uses the satisfaction diagram to zoom in on the issue.

S = [s|s |= 〈〈reverse osmosis, t, 1〉〉]S ′ = [s|s |= 〈〈managers, t, 1〉〉]S = [u|u |= 〈〈calculating, a, t, 1〉〉∧〈〈budget, a,Amur River, t, 1〉〉]


Possible Conclusion: Despite having a more efficient methodof de-salinization than ion exchange at the plant,9 the situationS ′′ depicts more signs of management issues than the other ob-jects. The reason of failed satisfaction is easily observed and thegovernment now has the information and data to take the nec-essary steps of improving the plant management at the AmurRiver by narrowing their satisfaction diagram into smaller on-tologies within the managers.

Hypothetical CASE 3 S = [s|s |= 〈〈multi-stage flash distillation, t, 1〉〉]S ′ = [s|s |= 〈〈inspectors, t, 1〉〉]S = [u|u |= 〈〈calculating, a, t, 1〉〉∧〈〈budget, a, St. Petersburg, t, 1〉〉]

9DOW Water Solutions

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Possible Conclusion: Like CASE 2, this CASE has an objectchange of the people ontology and a different de-salinization method.


As seen with the hypothetical cases, there is a large abundance ofpossible objects that can be used from the sets listed on page 8.

As a mathematical structure the satisfaction diagram is extremelysimple, but its simplicity hides a lot of power and depth. The same istrue to the commutative diagram of category theory, on which it wasoriginally based. Keith Devlin makes an analogy of satisfaction dia-grams to Lego blocks because of how both fit together into a complexnetwork. Although the basic lego blocks are all identical, by fittingthem together appropriately, objects of considerable complexity canbe created.10. Use of our framework for government analysists in theRussian Federation Water Sector.

6 Conclusion

6.1 Strengths and Weaknesses

Strengths:

• The implementation of the model can be widely used for many differentmunicipalities within Russia.

• Additional objects can be inputted into the model depending on the needsof each municipality.

• Captures all complexities of Russian freshwater management.

10Army Report

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• Capable of sufficient precision to allow for cross-domain communication forall parties, both in the government and private sector.

• Sufficiently natural for intuitive understanding by all parties.

• Flexible in clarification and disambiguation.

Weaknesses:

• Conceptulization of model is complicated and maybe not as easy to followas it should to start.

• It does not give exact methods for solving the fresh water managementconcerns of Russia but instead lets the government have the ability, usinginformation at the time, to solve certain problems and concerns.

• Our data is not exact and does not show how much certain municipalitiescan really save until implementation by the Russian government.

6.2 Improvements

Our model is based on concepts and information flow with a mathematical for-mat underlying the entire system. Calculations with numbers, however, were notmade and in our results; only situations in regards to supply station. Our modelwould be easier to understand if we had the ability to input enormous amounts ofaccurate data into a computer program which would be able to crunch numbersand provide output data. This would make our model the best water strategychoice because of its ability to be applied to many different situations or differentmunicipalities.

Also, our research led us to find many new technologies that are more efficientand advanced. If the Russian government had the means to purchase these tech-nologies, then Russia could be a world leader in terms of fresh water management.

6.3 Considerations

Some research we found interesting is the amount of fresh water resources inRussia. They have access to many resources such as, Lake Baikal, which is thelargest, by volume, fresh water lake in the world.11 This is just one of manyfresh water sources the Russian government should think about pumping to otherparts of the world. There are numerous countries in Asia and Africa who wouldbenefit from the potentially great amounts of fresh water. By doing this, theRussian government could make more money on the exportation of fresh waterthan they currently do from fossil fuels. With this extra money coming into thecountry, Russia could then spend that money on the new technologies availablefor fresh water management such as geothermal de-salinization. They could alsouse the money to fund research and development projects related to fresh watermanagement.

11Wikipedia, Lake Baikal

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7 Non-Technical Position Paper

Russian Federation General Council

Presented by: Team #21629

Non-Technical Position Paper:

Situation Theory and Mathematical Modeling for StrategicFreshwater Management

The issue before the General Council is: the creation of a fresh water strategyand management program that will assist local governments in improving freshwater supply, quality, and overall management.

Situation Theory Application

Our goal for creating the best fresh water strategy starts with defining the basicprinciples with regards to fresh water management. These principles include: thestorage, movement, de-salinization, and conservation of fresh water. We ensureeach principle of fresh water management follows our goal of limiting the economic,physical, and environmental implications associated with creating an effective,feasible, and cost efficient strategy.

Our strategy uses a mathematical model based on information flow that viewsaspects of water supply and sanitation needs for optimal fresh water management.Though an unusual approach, it is largely due to the definitive role of informationflow in the strategical scheme. Due to an element of uncertainty regarding thefuture, this strategy is more a set of options than a fixed plan. The sets of optionsare calculated and analyzed through a mathematical theory of information calledsituation theory.

Some of the variables, objects, and constraints we use in our theory are de-salinization, conservation, economics, storage and movement, and consumption.After implementing each of these factors into our situation theory model, we findthat any group in charge of fresh water management could use our model. Eachgroup, or local municipality, will take our model and input their own objects intothe model. This is what makes our model useful for all potential municipalities.

As an example, we reference a hypothetical town along the Amur River. Thistown has just updated its de-salinization plant but is still failing fresh water qualityinspections. We took this situation and input it into our model and find that thereis a managerial problem.

Conclusion

We believe this strategy can be implemented in many municipalities where thereare problems with fresh water management. It will allow government leaders toapply their situations to the model and determined what needs to be solved. Ifthey have fresh water quality problems, it will allow them to delve deeper into theproblem to find the underlying cause. Therefore, local municipalities will be ableto save money and resources.

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Mathematical Modeling for Strategic Freshwater Management

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