Geometric Programming Applications

1
Multiple Stage Distillation (1)

• The most common way to extract fresh water from
  the ocean is distillation, i. e., vaporize fresh
  water from seawater. This technique is very
  expensive.

• To reduce the cost of distillation, excessive
  heat in the vaporized water is often exploited to
  vaporize more fresh water from seawater. This
  procedure is called multiple stage distillation
  (Zener, 1971).

2
Multiple Stage Distillation (2)
Seawater
Seawater
Seawater
Input Heat
Vapor
Vapor
Vapor
Vapor
Fresh Water
Fresh Water
Fresh Water
Seawater
Seawater
Seawater
Multiple Distillation (totally
stages)
3
Multiple Stage Distillation (3)
Parameters in the problem
Temperature drop on every distillation stage.
Temperature difference between the first and the
last stage.
The heat for vaporizing 1 unit water.
The heat transfer coefficient at every
distillation stage.
The plate area at every stage.
The velocity of fresh water vaporized at each
stage.
The cost of the plate per unit area.
The cost per unit heat.
4
Multiple Stage Distillation (4)
The objective is to minimize the cost for one
distillation stage
where
And CRF is capital-recovery factor, which is
used to calculate the rate of amortization
5
Ship Operation (1)

• There are several possible applications of
  geometric programming in shipbuilding, for
  example, global design of ships, main
  construction of a ship and ship operations.

• The problem of interest here concerns long-time
  charters, where a contract calls for
  transportation of a fixed quantity per year of
  some material between two ports (Beightler and
  Phillips, 1976). The objective is to operate
  ships at the minimum cost.

6
Ship Operation (2)
Variables and Parameters of the Model
A Discounted building costs of ship, exclusive
of machinery, per ton light weight C Capitalized
fuel cost, per ton D Ships deadweight capacity,
tons E Discounted building costs of machinery,
per SHP F Fuel consumption, tons per SHP per
hour K Admiralty coefficient M Loading rate,
deadweight tonnage handled per hour N Number of
ships
P Power of propulsion machinery, SHP Q Total
quantity to be transported in one direction R
Range, distance in one direction, nautical
miles U Utilization factor, ratio of hours at
sea to total number of operating hours V
Service speed, knots W Total number of operating
hours per year Y Number of years for discounting
and capitalizing, charter term Z Ships loading
displacement, tons
7
Ship Operation (3)

• Objective function
• Minimize discounted shipbuilding cost
  discounted machinery cost fuel cost

Discounted shipbuilding cost NA(Z -
D) Discounted machinery cost NEP Fuel cost
NCFPUWY
8
Ship Operation (4)

• Constraints
• 1. There must be enough ships and voyages to
  meet the required tonnage per year
• N(DV/R-FP)UW gt 2Q
• 2. The utilization factor should be bounded
  by the maximum possible value
• U lt R/V/(R/V2(D/M)
• 3. An additional constraint is that the
  optimum number of ships must be integer, but in
  practice a simple round-off solution is valid
  (Beightler and Phillips, 1976).

9
Ship Operation (5)

• The formulation of the optimization problem
  after substituting the parameters by proper
  values
• Min go 200ND 0.5ND2/3V3
  0.8NUD2/3V3
• s.t. 2.75106N-1D-1V-1U-1
  210-3V2D1/3 lt 1
• 6.6710-7DVU U lt 1
  

• This is a typical constrained geometric
  programming problem. The optimal solution is
• N 15.21, D 17220, V 12.10, U 0.88,
  g0 73.96106

10
Alkylation Process Optimization (1)
Simplified Alkylation Process Flow Diagram
11
Alkylation Process Optimization (2)
Alkylation Process Variables and Their bounds
Lower bound
Upper bound
Variable
2000 19200 120 5000 2000 93 95 12 4 162
X1 Olefin feed (barrels per day) X2
Isobutane recycle (barrels per day) X3 Acid
addtion rate (thousands of pounds per day) X4
Alkylate yield (barrels per day) X5 Isobutane
makeup (barrels per day) X6 Acid strength
(weight percent) X7 Motor octane number X8
External isobutane-to-olefin ratio X9 Acid
dilution factor X10 F-4 performance number
1 1 1 1 1 85 90 3 1.2 145
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Alkylation Process Optimization (3)
Constraints Regression Relationships
We will express regressing relationships in
the form of two inequality constraints, which
specify the range for which these relationships
are valid. For regression equation Y
f(x), it would be expressed as
dlY lt f(x) lt duY , which is
f(x) lt duY
-f(x) lt - dlY
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Alkylation Process Optimization (4)
There are four regression equations in the
process model
x4 x1(1.120.13167x8-0.00667x82)
x7 86.351.098x8-0.038x820.325(x6-89)
x9 35.82-0.222x10
x10 -1333x7
In this problem, we could set
dl 99/100
du 100/99
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Alkylation Process Optimization (5)
The reformulated regression constraints
0.005955x820.882929x1-1x4-0.117563x8lt1 1.1088x1x
4-10.130353x1x4-1-0.006603x1x4-1x82lt1 0.000662x8
20.017240x7-0.005660x6-0.019121x8lt1 56.7597x7-1
1.08702x7-1x80.321750x6x7-1-0.037620x7-1x82lt1 0.
006198x102462.31x3x4-1x6-25.1256x3x4-1lt1 161.190
x10-15000x3x4-1x10-1-489510.0x3x4-1x6-1x10-1lt1 4
.4333x7-10.330000x7-1x10lt1 0.022556x7-0.007595x1
0lt1
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Alkylation Process Optimization (6)
Constraints Mass Balances
Assume we could express a mass balance
equation as Yf(x), which indicates we want to
eliminate Y by this equation. However, we have to
keep the upper and lower bounds of Y in the
problem, so we need the following constraints in
addition YLB lt f(x)
lt YUB , which is
f(x) lt YUB -f(x) lt
-YLB
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Alkylation Process Optimization (7)
There are three mass balance equations in the
process model
Volumetric reactor balance x5
1.22x4 - x1
Acid mass balance x9
98000x3/(x4x6) - 1000x3/x4
Isobutane recycle balance x2
x1x8 1.22x4 x1
Note x5, x9, x2 are to be eliminated in the
problem.
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Alkylation Process Optimization (8)
The reformulated mass balance constraints
0.000610x4-0.0005x1lt1 0.819672x1x4-10.819672x4-1
lt1 24500.0x3x4-1x6-1-250.0x3x4-1lt1 0.010204x60.
000012x3-1x4x6lt1 0.0000625x1x80.0000625x1-0.0000
7625x4lt1 1.22x1-1x4x1-1-x8lt1
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Alkylation Process Optimization (9)

• Profit function

Alkylate product value 0.063 per
octane barrel Olefin feed cost 5.04
per barrel Isobutane recycle cost
0.035 per barrel Acid addition cost
10.00 per thousand pounds Isobutane
makeup cost 3.36 per barrel So the total
profit per day, to be maximized, is then
Profit 0.063x4x7-5.04x1-0.035x2-10x3-3.36x5
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Alkylation Process Optimization (10)
To make the profit function fit for geometric
programming, we should 1. Minimize the negative
profit instead of maximizing profit. 2. Eliminate
x5 and x2, which have been eliminated in
constraints. 3. Add a large positive constant
(e.g. 3000) to the profit function to make the
optimal value positive. Thus the objective
function of the problem is Min y
1.715x10.035x1x84.0565x410.0x3-0.063x4x73000
20
Alkylation Process Optimization (11)

• The objective function, the regression and mass
  balance constraints, as well as the upper and
  lower bounds of the variables construct the
  alkylation process optimization problem.
• This problem is not in a posynomial form
  because some terms in the objective function and
  the constraints have negative coefficients.
  Generally, this kind of problem can be written in
  the following form

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Summary

• The mathematical formulations of many
  engineering problems can be transformed into
  geometric programming form and solved
  efficiently.
• Examples from various engineering areas have
  been presented to show the details of the
  application of the geometric programming.
• Signomial problem is worth attention for
  application in chemical engineering.

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References

• S.Boyd, L.Vandenberghe, Convex Optimization,
  Cambridge University Press (2004)
  http//www.stanford.edu/boyd/cvxbook.html
• M.Avriel, M.Rijckaert, D.Wilde, Optimization and
  Design, Prentice-Hall, New Jersey (1973)
• C.Beightler, D. Phillips, Applied Geometric
  Programming, Wiley, New York (1976)
• D.Wilde, C.Beightler, Foundations of
  Optimization, Prentice-Hall (1967)
• C.Zener, Engineering Design by Geometric
  Programming, Wiley, New York (1971)
• R.Duffin, E.Peterson, C.Zener, Geometric
  Programming Theory and Application, Wiley, New
  York (1967)
• P.Nijkamp, Planning of industrial complexes by
  means of geometric programming, Rotterdam
  University Press (1972)