Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of Newfoundlandspkenny@engr.mun.ca

ENGI 5708 Design of Civil Engineering Systems

Lecture 03: Problem Definition and Model Formulation

2 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Assignment #1

Revised Due DateJanuary 28th due in class

3 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Lecture 03 Objective

To understand qualitative and quantitative formulation of the problem definition within systems engineering

4 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-01A city needs a certain amount of water from a river that is 300 feet below and three miles from the city's water treatment plant. The city engineer must decide on the internal diameter of the pipeline to the city and the head capacity of the pump to be installed at the river.

The engineer knows that a smaller pipe is less expensive but results in greater head loss due to friction and therefore requires a greater head capacity for the pump, which increases the cost of the pump. Conversely, a larger pipe is more expensive but permits a less expensive pump to meet the fixed head and flow requirement at the water treatment plant.

The engineer is also aware that the range of pump capacities andpipeline diameters is limited in order to avoid a long waiting period for delivery.

The most economical pump/pipe combination is desired. Describe the optimization problem in qualitative terms.

5 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-01 (cont.)A city needs a certain amount of water from a river that is 300 feet below and three miles from the city's water treatment plant. The city engineer must decide on the internal diameter of the pipeline to the city and the head capacity of the pump to be installed at the river.

The engineer knows that a smaller pipe is less expensive but results in greater head loss due to friction and therefore requires a greater head capacity for the pump, which increases the cost of the pump. Conversely, a larger pipe is more expensive but permits a less expensive pump to meet the fixed head and flow requirement at the water treatment plant.

The engineer is also aware that the range of pump capacities andpipeline diameters is limited in order to avoid a long waiting period for delivery.

The most economical pump/pipe combination is desired. Describe the optimization problem in qualitative terms.

Decision VariablesPump capacity (static head)Pipeline diameter (flow characteristics)

6 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-01 (cont.)A city needs a certain amount of water from a river that is 300 feet below and three miles from the city's water treatment plant. The city engineer must decide on the internal diameter of the pipeline to the city and the head capacity of the pump to be installed at the river.

The engineer knows that a smaller pipe is less expensive but results in greater head loss due to friction and therefore requires a greater head capacity for the pump, which increases the cost of the pump. Conversely, a larger pipe is more expensive but permits a less expensive pump to meet the fixed head and flow requirement at the water treatment plant.

The engineer is also aware that the range of pump capacities andpipeline diameters is limited in order to avoid a long waiting period for delivery.

The most economical pump/pipe combination is desired. Describe the optimization problem in qualitative terms.

Objective StatementMinimize the total cost of the pump and pipeline system

7 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-01 (cont.)A city needs a certain amount of water from a river that is 300 feet below and three miles from the city's water treatment plant. The city engineer must decide on the internal diameter of the pipeline to the city and the head capacity of the pump to be installed at the river.

The engineer knows that a smaller pipe is less expensive but results in greater head loss due to friction and therefore requires a greater head capacity for the pump, which increases the cost of the pump. Conversely, a larger pipe is more expensive but permits a less expensive pump to meet the fixed head and flow requirement at the water treatment plant.

The engineer is also aware that the range of pump capacities and pipeline diameters is limited in order to avoid a long waiting period for delivery.

The most economical pump/pipe combination is desired. Describe the optimization problem in qualitative terms.

ParametersRequired head at treatment plantPump characteristicsRequired flow rate at treatment plantPipeline geometric and material propertiesElevation profile and length of pipelineFriction factors and losses for pipeline

8 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-01 (cont.)A city needs a certain amount of water from a river that is 300 feet below and three miles from the city's water treatment plant. The city engineer must decide on the internal diameter of the pipeline to the city and the head capacity of the pump to be installed at the river.

The engineer knows that a smaller pipe is less expensive but results in greater head loss due to friction and therefore requires a greater head capacity for the pump, which increases the cost of the pump. Conversely, a larger pipe is more expensive but permits a less expensive pump to meet the fixed head and flow requirement at the water treatment plant.

The engineer is also aware that the range of pump capacities andpipeline diameters is limited in order to avoid a long waiting period for delivery.

The most economical pump/pipe combination is desired. Describe the optimization problem in qualitative terms.

ConstraintsPump head capacity ≥ minimum head requirementFlow rate ≥ minimum flow requirement

9 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-02A national space administration is selecting instruments to place aboard an earth-orbiting satellite. Each instrument under consideration has its own specific capacity to gather data with varying quality depending on the feature of interest. The administration has assigned a value to the quality of information each instrument gathers on each earth feature, and it is assumedthat such “values” can be mathematically combined across instruments to give a total value.

A budget has been established for the purchase of instruments (instrument prices are known) and the budgeted amount cannot be exceeded. The instrument total payload weight and volume are also constrained to not exceed certain amounts and the weights and volumes of the various candidate instruments are known.

The instrument package that gives the greatest detection effectiveness is desired. Describe the optimization problem in qualitative terms.

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Example 3-02 (cont.)A national space administration is selecting instruments to place aboard an earth-orbiting satellite. Each instrument under consideration has its own specific capacity to gather data with varying quality depending on the feature of interest. The administration has assigned a value to the quality of information each instrument gathers on each earth feature, and it is assumed that such “values” can be mathematically combined across instruments to give a total value.

A budget has been established for the purchase of instruments (instrument prices are known) and the budgeted amount cannot be exceeded. The instrument total payload weight and volume are also constrained to not exceed certain amounts and the weights and volumes of the various candidate instruments are known.

The instrument package that gives the greatest detection effectiveness is desired. Describe the optimization problem in qualitative terms.

Decision VariablesInstrument type to be placed on satellite

11 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-02 (cont.)A national space administration is selecting instruments to place aboard an earth-orbiting satellite. Each instrument under consideration has its own specific capacity to gather data with varying quality depending on the feature of interest. The administration has assigned a value to the quality of information each instrument gathers on each earth feature, and it is assumed that such “values” can be mathematically combined across instruments to give a total value.

A budget has been established for the purchase of instruments (instrument prices are known) and the budgeted amount cannot be exceeded. The instrument total payload weight and volume are also constrained to not exceed certain amounts and the weights and volumes of the various candidate instruments are known.

The instrument package that gives the greatest detection effectiveness is desired. Describe the optimization problem in qualitative terms.

Objective StatementMaximize the total value (cost, quality) of information gathered

12 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-02 (cont.)A national space administration is selecting instruments to place aboard an earth-orbiting satellite. Each instrument under consideration has its own specific capacity to gather data with varying quality depending on the feature of interest. The administration has assigned a value to the quality of information each instrument gathers on each earth feature, and it is assumed that such “values” can be mathematically combined across instruments to give a total value.

A budget has been established for the purchase of instruments (instrument prices are known) and the budgeted amount cannot be exceeded. The instrument total payload weight and volume are also constrained to not exceed certain amounts and the weights and volumes of the various candidate instruments are known.

The instrument package that gives the greatest detection effectiveness is desired. Describe the optimization problem in qualitative terms.

ParametersInstrument cost, weight and volumeQuality or value of instrument data Total budgetMaximum total weight and volume

13 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-02 (cont.)A national space administration is selecting instruments to place aboard an earth-orbiting satellite. Each instrument under consideration has its own specific capacity to gather data with varying quality depending on the feature of interest. The administration has assigned a value to the quality of information each instrument gathers on each earth feature, and it is assumed that such “values” can be mathematically combined across instruments to give a total value.

A budget has been established for the purchase of instruments (instrument prices are known) and the budgeted amount cannot be exceeded. The instrument total payload weight and volume are also constrained to not exceed certain amounts and the weights and volumes of the various candidate instruments are known.

The instrument package that gives the greatest detection effectiveness is desired. Describe the optimization problem in qualitative terms.

ConstraintsInstrument package cost ≤ total budgetInstrument package weight ≤ maximum total weightInstrument package volume ≤ maximum total volume

14 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-03The North Atlantic oil refinery purchases light crude and heavy crude. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of light crude is 0.40, 0.20 and 0.35. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of heavy crude is 0.32, 0.30 and 0.25. The process cost per barrel is $10/barrel of light crude and $22/barrel of heavy crude. The annual production targets are 1.5MMBBL/yr of gasoline, 0.6MMBBL/yr kerosene and 0.05MMBBL/yr jet A.

Formulate (set-up) the problem in mathematical terms.

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Example 3-03 (cont.)The North Atlantic oil refinery purchases light crude and heavy crude. The barrel fraction (including production losses) of gasoline,kerosene, and jet A produced per barrel of light crude is 0.40, 0.20 and 0.35. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of heavy crude is 0.32, 0.30 and 0.25. The process cost per barrel is $10/barrel of light crude and $22/barrel of heavy crude. The annual production targets are 1.5MMBBL/yr of gasoline, 0.6MMBBL/yr kerosene and 0.05MMBBL/yr jet A.

Decision VariablesxL = # BBL/yr light crude purchasedxH = # BBL/yr heavy crude purchased

16 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-03 (cont.)The North Atlantic oil refinery purchases light crude and heavy crude. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of light crude is 0.40, 0.20 and 0.35. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of heavy crude is 0.32, 0.30 and 0.25. The process cost per barrel is $10/barrel of light crude and $22/barrel of heavy crude. The annual production targets are 1.5MMBBL/yr of gasoline, 0.6MMBBL/yr kerosene and 0.05MMBBL/yr jet A.

Objective StatementMinimize total cost to meet production targets

Objective FunctionMinimize annual cost

10 22L Hz x x= +

17 ENGI 5708 Civil Engineering Systems – Lecture 03© 2007 S. Kenny, Ph.D., P.Eng.

Example 3-03 (cont.)The North Atlantic oil refinery purchases light crude and heavy crude. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of light crude is 0.40, 0.20 and 0.35. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of heavy crude is 0.32, 0.30 and 0.25. The process cost per barrel is $10/barrel of light crude and $22/barrel of heavy crude. The annual production targets are 1.5MMBBL/yr of gasoline, 0.6MMBBL/yr kerosene and 0.05MMBBL/yr jet A.

ParametersProduction fractions for gasoline, kerosene and jet A

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Example 3-03 (cont.)

Constraints

, 0L Hx x ≥

0.40 0.32 1,500,000L HBBLx x yr+ ≥

0.20 0.30 600,000L HBBLx x yr+ ≥

0.35 0.25 50,000L HBBLx x yr+ ≥

Gasoline production

Kerosene production

Jet A production

Non-negativity constraint

The North Atlantic oil refinery purchases light crude and heavy crude. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of light crude is 0.40, 0.20 and 0.35. The barrel fraction (including production losses) of gasoline, kerosene, and jet A produced per barrel of heavy crude is 0.32, 0.30 and 0.25. The process cost per barrel is $10/barrel of light crude and $22/barrel of heavy crude. The annual production targets are 1.5MMBBL/yr of gasoline, 0.6MMBBL/yr kerosene and 0.05MMBBL/yr jet A.

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Example 3-03 (cont.)

Common Characteristics?Objective function

Constraints

, 0L Hx x ≥

0.40 0.32 1,500,000L HBBLx x yr+ ≥

0.20 0.30 600,000L HBBLx x yr+ ≥

0.35 0.25 50,000L HBBLx x yr+ ≥

10 22L Hz x x= +

Linear Functions

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Example 3-04A bank makes four kinds of loans to its personal customers and these loans yield the following annual interest rates to the bank:

1. First mortgage 14%2. Second mortgage 20%3. Home improvement 20%4. Personal overdraft 10%

The bank has a maximum foreseeable annual lending capability of $250 million and is further constrained by the policies:

1. first mortgages must be at least 55% of all mortgages issued and at least 25% of all loans issued (in $ terms)

2. second mortgages cannot exceed 25% of all loans issued (in $ terms)3. to avoid public displeasure and the introduction of a new windfall tax the

average interest rate on all loans must not exceed 15%.

Formulate the bank loan problem as a Linear Programming (LP) problem so as to maximize interest income whilst satisfying the policy limitations. Note here that these policy conditions, whilst potentially limiting the profit that the bank can make, also limit its exposure to risk in a particular area. It is a fundamental principle of risk reduction that risk is reduced by spreading money (appropriately) across different areas.

Ref: Beasley (2007)

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Example 3-04 (cont.)

Decision Variablesxi = amount loaned in sector i (M$/yr)•

x1

= 1st

mortgage•

x2

= 2nd

mortgage•

x3

= Home improvement•

x4

= Personal overdraft

Objective StatementTo maximize annual interest income

Objective FunctionMaximize

1 2 3 40.14 0.20 0.20 0.10z x x x x= + + + Ref: Beasley (2007)

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Example 3-04 (cont.)

ConstraintsLimit on total annual lending capability

Policy 1: first mortgages ≥ 0.55 total mortgage lending

Policy 1: first mortgages ≥ 0.25 total loans

1 2 3 4 250x x x x+ + + ≤

( )1 1 20.55x x x≥ +

( )≥ + + +1 1 2 3 40.25x x x x x

Ref: Beasley (2007)

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Example 3-04 (cont.)

ConstraintsPolicy 2: second mortgages ≤ 0.25 total loans

Policy 3: average interest rate on all loans must not exceed 15%

( )2 1 2 3 40.25x x x x x≤ + + +

Ref: Beasley (2007)

( )1 2 3 4 1 2 3 40.14 0.20 0.20 0.10 0.15x x x x x x x x+ + + ≤ + + +

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ReVelle et al., (2004)

Relevant ChaptersCh. 1 Explaining Systems Analysis•

Section 1A through 1E

Ch. 2 Models in Civil and Environmental Engineering•

Section 2A through 2C

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References

Beasley (2007). http://people.brunel.ac.uk/~mastjjb/jeb/or/lp.htmlReVelle, C.S., E.E. Whitlatch, Jr. and J.R. Wright (2004). Civil and Environmental Systems Engineering 2nd Edition, Pearson Prentice Hall ISBN 0-13-047822-9