Energy Resource Allocation Optimization--A Linear Programming ModelSTARS STARS
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BY
RESEARCH REPORT
Submitted in partial fulfillment of the requirements for the degree of Master of Science in Operations Researeh
in the Graduate Studies Program of Florida Technological University
Orlando, Florida 1976
for a Central Energy Plant which minimizes operational
costs for a system involving the generation of chilled
water and high temperature hot water plus the generation
and/or purchase of electric power with equipment using
natural gas or fuel oil energy. The fundamental concepts
developed herein are sufficient for the analysis of any
combination of energy supplies, demands, apd energy con-
version equipment. Utilization of this model is demon-
strated with a case study and computer program results
for high and low temperature environments. The linear
programming model approach estab.lishes a well-defined
framework for the analysis of complex utility systems and
provides valuable results for the economical operation
of a Central Energy Plant.
Director of Research Report
CASE STUDY . . . . . . . . . . . . . . . . .
. . . System Definition.
Model Application. .
5.1 Operating Instructions for the ."LP" Computer
Program. . . . . . . . . . . . . . . . 6.0 LITERATURE CITED . . . . . . . . . . . .
6. Sensitivity Study-- Demand/Capacity Variables . 34
7. Sensitivity Study-- Cost Coefficients • . . . . 36
v
Block Diagram. . . . . . . . . . . . . . . . 2. Case Study Central Energy Plant
Block Diagram. . . . . . . . . . . 3.
Cold Day Energy Rate Demands for
Electricity and Chilled Water Use.
Cold Day Energy Rate Demands for
Hot Water. . . . . . . . . . . .
society to conserve our valuable energy resources. Since
production must continue to increase to meet ever-growing
needs, it is important that energy plants allocate this
limited resource as efficiently as possible. This paper
develops a Central Energy Plant Industrial Model which is
designed to analyze and minimize operational costs for a
system involving the generation of chilled water, high
temperature hot water, plus the generation and/or pur­
chase of electric power with equipment using natural gas
or fuel oil energy. The objective is to minimize opera­
tional costs which must be realized within constraints
peculiar to the system.
demands, energy and , energy flow conservation, maximum
and minimum equipment operating capacities, minimum
equipment operation time, and various other equipment
peculiarities such as maximum number of starts per day.
In the optimization process, the decision- maker must de­
cide which equipment to operate and at what rate. Such
decisions often involve making a choice between increas-
1
2
ing the output rate on one machine or bringing another
on- line. Other options include purchasing energy versus
in- house power generatio~ use of fuel oil versus natural
gas or other -energy sources, and choices between alter­
nate methods of meeting system demands. The intent is
not to duplicate the multitude of efforts in the area of
optimum power dispatch for electrical power systems.
Economic dispatch results are used to determine what com­
bination of electricity generators is economically best
based on the shapes of their cost/hr versus load curves.
The solution to this well documented and classic problem
is for optimum dispatch, individual generators should
operate at equal incremental production costs [1]. The
purpose of this paper is to determine which path of
energy conversions to chose when presented with alter­
nate ones which lead to the same end, and what equipment
and rates of operation are necessary to meet the desired
demands. The fundamental concepts presented herein are
sufficient for the analysis of any combination of energy
supplies, demands, and energy conversion equipment.
A basic linear programming model was used in the
analysis. Each portion of the model is described in de­
tail to provide a clearer understanding of the problem.
The linear programming model approach establishes a well­
defined framework for analysis of complex problems and
provides insight into the relationships of the system
parameters required to analyze and develop an optimal
solution. To demonstrate the usefulness of the derived
model, a case study of a typical Central Energy Plant
operation is presented.
the computer program called "LP" which is discussed in
detail in the appendix. To complete the analysis "LP"
3
is used to perform a sensitivity study which shows the
effect of changes in the different system parameters on
the current optimal solution. This study then determines
the range of a given element for which the solution re­
mains optimal and, therefore, how sensitive the optimum
solution obtained is to the parameter setting.
2.0 THEORY
Programming problems deal with the efficient use or al­
location of limited resources to optimize certain desired
objectives. Often, for each problem there exists a large
number of solutions that satisfy the basic conditions.
The best solution to a problem is selected based upon
some over-all objective that is included in the statement
of the problem. The optimum solution is the one that
satisfies the constraints of the problem and maximizes /
minimizes the given objective [2].
The linear programming problem is a special sub­
class of programming problems. With this type of problem
a mathem~tical model can be stated using linear relation­
ships only which greatly s·implifies problem-solving. The
complete linear programming problem includes a set of
simultaneous linear equations which represent the condi­
tions of the problem and a linear function which expresses
the problem objective [2].
Optimum allocation of energy resources has always
been an important concern in power plant operations.
Much time and money has gone into the development of
techniques for efficient utilization and generation of
electricity in large utility companies. Playing a large
role in this effort are computers which have been used in
areas of system control, simulation, and advancement of
optimization techniques. Power system optimization pro­
blems include economic dispatch, power plant size and lo­
cation, maintenance scheduling and marketing policies [3].
The problem discussed in· this paper deals with a
special type of power system - one with multiple energy
supplies, multiple energy demands, and multiple energy
conversion equipment. This system is called a Central
Energy Plant and is found in large recreational comp~xes,
hotels, education facilities, etc. In most cases deci­
sions in these facilities are made by a control room
operator. These decisions are based on empirical data,
machine efficiency calculations, and trial/error. The
decisions are often economical ones, but as the relation­
ships between various equipment become increasingly com­
plex, the decisions become more difficult to make. Even
for the simpler operations it is difficult to determine
how sensitive a system is to changes in various para-
meters. It is not only the intent of this paper to
allow the operator to make decisions that are optimal,
but also to give him a clear understanding of the com­
plete system -and how changes in system parameters effect
operations.
6
of energy: electricity, natural gas, and fuel oil, which
produce and distribute three demand items: electricity,
chilled water and high temperature hot water. These
three demand items are produced by the following pro­
cesses:
b) produced by a generator fed by natural gas or
fuel oil
tricity
3. High Temperature Hot Water
a) boilers fed by natural gas or fuel oil
b) boilers fed by waste heat from generators run
on gas or fuel oil
This complex operation is shown in Figure 1.
The system inputs are raw electricity, natural gas, and
-- --
·- --
? --
-- --
.
OW ER
8
fuel oil. The raw electricity can be used to meet the
electric power demand directly and the chilled water de-
mand by powering the centrifugal chillers. Natural gas *
and fuel oil can be used to meet all three system demand
outputs. Electric power can be generated by using the
gas or oil to operate electricity generators. High tern-
perature heated water can be produced by waste heat from
the electricity generation process and by boiler overfire
(i.e., by direct burning of gas or oil to heat the water,
but not both). The decision involving the use of natur-
al gas or fuel oil is made inherently in the solution
process. Where there is an alternative, only one fuel
will be selected, the less expensive of the two. Chill-
ed water can be produced by powering the centrifugal
chillers using the generated electricity and by operating
the absorption chillers using high temperature heated
water.
The lines in Figure 1 represent the flow of an
energy form from the supply to demand by way of some
energy conversion equipment. The amount of energy per
unit time required to meet the system demands are re-
presented by the variables x1 through x10 . The energy
costs and equipment efficiencies are coefficients. The
right hand sides of the model equations represent the
maximum/minimum capacities of the various conversion
equipment or the levels of demanded power.
Several assumptions are made in order to main-
tain linearity. Electricity, fuel oil, and natural gas
prices are -assumed to be constant over the range of the
amounts of energy purchased in the problem. The equip-
9
operation.
unique set of values that represent the rates at which
energy sources (oil, gas, and electricity should be pur-
chased and the rates at which the various energy conver­
sion equipment should be operated. The values (rates)
are in the form of energy per unit time (power) and are
expressed in the dimensional units of million British
Thermal Units per hour or m2BTU/hr. Equipment capa­
cities and power demands for electricity (in kilowatts or
megawatts) and refrigeration (in tons/hour) are easily
converted to the m2BTU/hr units. Likewise, fuel oil and
natural gas each have associated heat values with the
units of BTU/gall9n and BTU/cubic foot, respectively,
which also yield a straight forward conversion.
10
function consist of the product of a cost coefficient and
a variable. As stated earlier, the variable has the
units, m2BTU/hr. The cost coefficients have the units of
dollars per m2BTU. Therefore, each term and the total
objective function have the units, $/hr. This means that
the model minimizes the rate at which money is being
spent during operation. The product of .optimal objective
function value and the number of hours during which the
particular solution is used will yield the total cost of
operation during the given time period. The length of
the time periods should be chosen judiciously to repre-
sent intervals of generally uniform demand while not vio-
lating equipment constraints that involve either a mini-
mum operation time or maximum number of starts per time
period.
minimize:
where
fuel oil ($/m2BTU),
x1 is the rate at which electricity is purchased to
meet the electric power demand (m2BTU/hr)~
x2 is the rate at which electricity is purchased to
operate the centrifugal chillers (m2BTU/hr)~
x3 is - ~he rate at which natural gas is purchased to
operate the generators (m2BTU/hr)~
x4 is the rate at which natural gas is purchased to
operate the boilers (m2BTU/hr)~
x5 is the rate at which fuel oil is purchased to 2 operate the generators (m BTU/hr)~
x6 is the rate at which fuel oil is purchased to
operate the boilers (m2BTU/hr).
equipment operating rates,. the equipment energy conver-
sion efficiencies, the flow rate of energy demanded, and
the relationships of the various system components. The
coefficients of the constraint equations~ often called
technological coefficients, represent the various equip-
ment efficiencies. The right hand sides of the con-
straints are either the demanded energy rat e or the
equipment capacity. The constraint equations represent­
ing equipment capacity (i.e., the maximum energy rate
output) are:
a 4
a13 is the efficiency of the natural gas to electri­
city conversion~
a15 is the efficiency of the fuel oil to electri­
city conversion,
(m2BTU/hr),
water conversion (via generator waste heat),
a 25 is the efficiency of the fuel oil to hot water
·conversion (via generator waste heat),
a 34 is the efficiency of the natural gas to hot
water conversion (via boiler overfiring)~
a 36
is the efficiency of the fuel oil to hot water
conversion (via boiler overfiring),
waste heat m2BTU/hr)~
overfiring, m2BTU/hr),
a 42 is the efficiency of the electricity to chilled
water conversion,
for -the centrifugal chillers (m2BTU/hr),
b4 is the centrifugal chiller maximum output rate
(m2BTU/hr),
water conversion.
x 9
is the rate at which hot water is produced for
the absorption chillers, and
overfiring).
where
(m2BTU/hr),
tor waste heat, m BTU/hr),
b8 is the boiler minimum output rate (via boiler
overfiring, m2BTU/hr),
(m2BTU/hr), and
overfiring).
rate are:
~0 > bl3 -
bll is the electric power demand,
bl2 is the chilled water demand, .
xlO is the rate at which hot water is produced to
meet the hot water demand, and
b13 is the high temperature hot water demand.
14
mediate energy flow rates are defined in terms of the
energy supply variables x3 , x4, x5, x6 as follows:
Lr + xg = ~3x3 + ~5x5 and
x_ 9
+ x 10
. 23 3 34 4 25 5 36
15
These are equivalent to
a13X3 + a15x5 - ~ - x8 = o, and a23x3 + a34x4 + a25x5 + a36x6 -x - x = 0.
9 10 Together, the objective function, the above con-
straints, plus the constraint that all x1 through x10 be
nonnegative, combine to yield the total system linear
programming model. The system equations which represent
the Central Energy Plant in Figure 1 are the following.
Minimize the objective function:
subject to
a x +a X < b 23 3 25 5 - 2
a34x4 + a36x6 < b3
a42 + a42Xs < b4
~4x4 + ~6x6 < b8
a42x2 + a42x8 < bg
al3x3 + al5x5 -X -xs = 0 7
a23x3 + a24x4 + a25x5 + a36x6 -x ~xlO = 0 9
xi, i=l 2 ••• 10 > 0 , , , -
In this form it should be noted that minimum capa­
city constraints will force all of the equipment into the
solution to at least the minimum capacity rate. Since
this is certainly undesirable (and most likely not optim­
al), it is necessary to solve the linear programming pro-
blem with the minimum capacities equivalent to zero in-
itially. If the results yield a solution that would vio-
late a minimum capacity constraint, the problem is solved
again with the appropriate minimum capacity rate present.
3.0 CASE STUDY
3.1 System Definition
which produce three main energy items: electricity,
chilled water, and high temperature hot water.
Electric power can be either purchased from a
power corporation or generated within the Central Energy
Plant by two fuel-powered turbine engin~s. This equip­
ment can provide a maximum of approximately 11 MW of e~c­
tric __ power using either natural gas or number two diesel.
On the average test data has shown that 29% of the energy
used to operate the turbines is converted into electricit~
43% is waste heat, and 28% of the input energy is lost.
When using natural gas, the overall efficiency is approx­
imately 7% higher than these nominal values.
Chilled water for air conditioning is also gene­
rated ip the Central Energy Plant. Six absorption chil­
lers using high temperature hot water as an energy source
can produce 8,000 tons/hr of refrigeration. Electrical
centrifugal chillers can generate 11,000 tons/hr. This
yields a total of 19,000 tons/hr installed . . The absorp-
17
chillers 75% in energy utilization.
18
electricity· generation via the hot gases from the turbine
drive. Waste heat from these generators is used to heat
water in boilers to 400°F, 500 psi. at a maximum rate of
41 m2BTU/hr. The boilers can also be fired by burning
natural gas or fuel oil at a maximum of 139 m2BTU/hr,
giving the total high temperature hot wat·er capacity of
180 m2BTU/hr. The energy recovery for the waste heat
system is 70% efficient. Using natural gas increases
this efficiency by approximately 5%.
It can be assumed that the minimum operating
rate of all equipment is 10% of the maximum. The tur­
bines and boilers must be operated a minimum of four
hours at a time. The centrifugal chillers are limited
to a maximum of three starts a day.
A complete description of the utility system
complex is shown in Figure 2. Each different type of
energy conversion equipment is represented by a single
block because for this model there is no difference in
the efficiencies of equipment of the same type. It is
not in the scope of this paper to determine the best
combination of generator or centrifugal chiller loading.
- - - - - - - - - ,.
- - - ,
20
sions to take and at what ene~gy rates should these paths
be pursued. The paths are labelled by variables which
will represent the optimum flow rate of energy necessary
to meet the -system demands.
The case study analysis will consist of two part~
The first will involve the application of the linear pro­
gramming model to determine the optimum energy rates for
both hot and cold temperature days. The second part con­
tains a sensitivity analysis of the cost coefficients and
the right hand side values for two extreme operating con­
ditions -- maximum energy flow for a hot day and the same
for a cold day. The data used for this study is found in
Figures 3, 4, and 5.
All data included in this study will be converted
to common units, m2BTU's. The relationships for making
conversions are contained in Table 1 which define equip­
ment capacities. Energy cost calculations are listed in
Table 2.
ENERGY RATE
--HISIDRICAL DATA
0 ,----L_ __ _
21
100 HIS'IDRICAL DATA
8
and Chilled Water Use
-- HIS'IDRICAL DATA
TIME OF DAY (hrs)
Figure 5 . Cold Day Energy Rate Demand for Hot Water
23
Natural gas price = $2.14/1000rt3
Fuel oil price = $0.35/gal
26
chilled water, and heated water demands while minimizing
hourly costs can be found by applying the linear pro­
gramming model. Assignment of the variable coefficients
can now be made and are listed below.
c1 = purchased electric power cost ($/m2BTU) = 7.68
= natural gas cost ($/m2BTU) = 2.00
= fuel oil costs ( $/m2BTU) = 2.57
= natural gas to electricity efficiency
= fuel oil to electricity efficiency
= natural gas to hot water efficiency
' (via turbine waste heat)
(via boiler overfiring)
(via turbine waste heat)
(via boiler overfiring)
efficiency
= 0.30
= 0.29
= 0.33
= 0.73
= 0.30
= 0.70
= 0.75
= turbine maximum rate (m2BTU/hr) = 37.6
= boiler maximum rate (waste heat
m2BTU/hr) = 41.0
b 3
b4 = centrifugal chiller maximum rate
b 9
(m BTU/hr)
2 = electric power demand (m BTU/hr)
= 1-39.
= ~32.
= 96.
= 3.76
= 5.6
= 6.2
= 24+1.44
= 1.~0
It should be noted in t.he above equations and
from Figure 2 that it has been assumed that the effi-
ciency of the conversion from waste heat to hot water is
the same as the fuel input conversion (73% for natural
gas and 70% for fuel oil). If additional data was found
to show these efficiencies to be different, the variabl$,
a 23 , a 25 , a 34, and. a 35 could be altered to describe the
28
pressions describing this problem can now be written and
are shown in Figure 6.
3.3 Results
puter program called "LP" was used. This program solved
the linear program problem and gave the ranges over which
the cost coefficients, c~, and the right-hand side vari- ~
ables, b1 , can vary without changing the optimality or
feasibility of the problem. The values used for the
sensitivity analysis are represented as data points in
Figures 3, 4, and 5. For the twenty-four hour examples,
each day was divided into three periods: midnight to
8:00 AM, 8:00 AM to 4:00 PM, and 4:00 PM to midnight.
This choice insures that the turbine and boiler minimum
run time and the centrifugal chiller maximum number of
starts constraints are not violated.. The demands are
also generally constant during these time periods. For
a more accurate representation of the data provided in
Figures 3, 4, and 5, shorter periods could be used with­
out violating the minimum run time and maximum number of
starts constraints. It is recommended that four hour
periods be considered in future analyses since they
appear to better accommodate the changes in operation
demand. A summary of the results of the case study are
Minimize:
5 +2.57x
*EPD = Electric Power Demand ¥
CWD - Chilled Water Demand
HWD = Heated Water Demand
29
30
shown in Tables 3 through 7.
Tables 3 and 4 contain the results of the hot day
and cold day cases described in Figures 3, 4, and 5. For
example, from- 12 midnight to 8 AM for the hot day case
the demand for electricity is 60 m2BTU/hr, for hot water,
20 m2BTU/hr and cold water, 90 m2BTU/hr (see Figure 3).
From Table 3 one can see that the most economical mode of
operation will be the purchase of electricity at 60
m2BTU/hr to satisfy the electric power demand, the opera­
tion of turbines at peak capacity of 37.55 m2BTU/hr (using
natural gas) to provide power for the centrifugal chillers,
and operation of the boilers at 120 m2BTU/hr (100 for the
absorption chillers and 20 for the hot water demand).
The sum of the centrifugal and absorption chillers output
(28 m2BTU/hr and 62 in m2BTU/hr, respectively) meet the
total chilled water demand of 90 m2BTU/hr.
Table 5 lists the cost of operation for each time
period. The value of the objective function from 12 mid­
night to 8 AM on the hot day is $927/hr. This yields a
total of $7,416 for the eight hour period. This can be
accomplished for each interval and summed to compute the
total cost of operation for the hot day case, $30,040.
The sensitivity analysis for the demand/capacity
variables is shown in Table 6 for the hot and cold day
maximum usage times (see Figures 3, 4, and 5). The first
31
- -
'0 >., Time .p Q) C) rll ....-.. eM H Interval rll eM ....-..
~ ....-.. rll 0
()~ wi 1! H H N co H r-i ~ itl .p ~ >< >< Q) >< C) C) ~
~ ~ r-i ~ .p ::::
~ eM 0'\ ~~ s~ r-i P-t >< 0 >< II:! '-" '-" o:l '-" r-i II:!
12 MIDNIGHT 60 - - 37.5 100 20 60 20 to 8 AM
8 AM to 47.5 70 37.5 - 141 24 85 24 4 PM
4 PM to 42.5 55 . 37-5 - 143 22 80 22 12 MIDNIGHT
r-i r-i
rll .Pr-i r-i Q)
~~ r-i itl .p
~ 0
12 MIDNIGHT 125 108 - - 28 62 90 to 8 AM
8 ArJI to 125 170 - - 52.5 87.5 140 4 PM
4 PM to 125 170 - - 41 89 130 12 MIDNIGHT
t- J
T:ime Number Objective Function Cost for
of Value Given Time
Interval Hours ($/hr) Period ($)
8 AM to 8 1,491 11,928
4 PM
12 IVITDNIGHT
12 MIDNIGHT 8 456 3,648
to 8 AM
4 PM
12 MIDNIGHI'
- - Feasibility Range (m2BTU/hr)
!
Boiler Waste Heat Max. b5 31 56 00 0
Boiler Overf~e Max .. b6 8 124 143.5 -3.5
Centrifugal Max. b7 60 132 00 0
Absorption Max. b8 84 96 00 0
* V is the amount (in dollars per hour) by which the 0
objective function is increased per unit increase of the corre-
sponding righthand side value (in m2BTU/hr) .
35
TABLE 6 - Continued
- -- Feasibility Range (m2BYU/hr)
Right Hand Side Minimum Nominal Maximum vo Variable Value Value Value
Maximum Cold Day Case
Boiler Waste Heat Max. b5 31 56 00 0
Boiler Overfire Max. b6 39 124 135 -3.5
Centrifugal Max. b7 7 132 00 0
Absorption Max. bs 53 96 00 0
36
Raw Electricity cl.,c2 6.67 7.68 00
Natural Gas c3,c4 - 00 2.00 2.30 Fuel Oil cL)_,c6 2.18 2.57 00
37
three columns of numbers describe the range over which
the demands or capacity may vary and still insure that a
feasible solution exists. The column labelled V0 is a
particularly important result for planning purposes. For
example, this column shows that for each single unit in­
crease in the turbine capacity, the objective function is
decreased by $8 per hour.
Finally, Table 7 lists the range of values for
which the costs of the various energy sources may vary
without changing the optimality of the solution. These
values represent the extremes over which the costs may
vary before another energy source should be used. For
example, when the price of fuel oil decreases below
$2.18/ m2BTU, this fuel will enter the solution as a sourre
of energy.
for a Central Energy Plant which minimizes operational
costs for a system involving the generation of chilled
water and high temperature hot water~ plus the generation
and/or purchase of electric power with equipment using
natural gas or fuel oil input. From this general model
other complex utility systems can be analyzed for maxi­
mum utilization of energy resources.
To demonstrate the usefulness of the linear pro­
gramming model a case study was conducted on a complex
utility system. The problem set up~ which consisted of
a conversion to common computational units and applica­
tion of the block diagram~ yielded a clear representation
of the Central Energy Plant. From this achievement im­
portant results were learned and documented in Tables 3
through 7. Tables 3 and 4 contain the optimum rates at
which the various equipment should be operated. The
daily costs for operating at these rates are listed in
Table 5. Finally~ Tables 6 and 7 contain the results of
a sensitiyity analysis that reveal how the system is af-
fected by changes in various parameters an excellent
38
39
It is also wo.rth noting that the total cost for
the twenty-four computer runs made was approximately $12,
or 50¢ per run~ ~ - This cost could probably be cut in half
by eliminating computations for output (such as sensiti­
vity analysis) which is ·not needed for each run. A seriffi
of runs could thus be economically made to provide opera­
ting data for a handbook with nomographs which could be
used by the operator to determine the most economical
operational mode for a given set of demand values.
The model presented in this paper lends an under­
standing to the complex utility system problem and pre­
sents a straight forward method for its solution. The
system of equations can be expanded or reduced to accommo­
date any variety of system combinations. Other costs
such as for equipment maintenance could be added by sim­
ply introducing another coefficient in the objective
function. In the same manner additional constraints,
energy flows, equipment, demands, etc. can be handled
with no disturbance to the basic model structure.
5.0 APPENDIX
5 .1 o·perating Instructions for the "LP" Computer Program
"LP" is an interactive version of "LPGOGO". It
can handle up to 60 variables (including slack and sur­
plus) and 25 constraints. These parameters could be in-
creased without much difficulty. "LP" uses the two phase,
full tableau form of the simplex method and requires all
right hand sides to be nonnegative.
Since all constraints must be converted to equali-
ties, the user must have a working knowledge of linear
programming to be able to insert slack and surplus vari­
ables which in most cases require a sound understanding.
Multiple inputs are read in free format and
should be separated by spaces or commas. All zero en-
tries must be entered. An example of "LP" run is shown
on the following pages.
irf4x 1p
LP WILL SOLVE ANY MAXIMIZATION PROBLEM OF UP rro 25 CONSTRAINTS AND 60 VARIABLES(INCIDDING SLACK AND SURPLUS). 00 YOU DESIRE INSTRUCTIONS? yes
41
LP USES THE TvVO PHASE, FULL TABLEAU FDRM OF THE SIMPIEX METHOD. IT REQUIRES .ALL RIGHT HAND SIDES 'ID BE NON-NEGATIVE AND STARrS FROM A FULLY ARTIFICIAL BASIS. IT ASSUMES THAT .ALL CONSTRADlTS HAVE BEEN CONVERIED 'ID EQUALITIES BY THE ADDITION OF SLACK AND SURPLUS VARI­ ABLES.
00 YOU DESIRE AN EXAMPIE? yes
MAX: 2Xl + X2 SUBJECT '!0 :
Xl<=4 X2<=3
Xl+X2<=8
PROBLEM FORMULATION: MAX: 2X01 + 1X02 + OX04 + OX05 SUB. 'ID : lX01 + OX02 + + 1X03 + OX04 + OX05 = 4
OX01 + IX02 + OX03 + 1X04 + OX05 = 3 1X01 + 1X02 + OX03 + OX04 + 1X05 = 8
INPUT 'ID PROGRAM: X01 X02 X03 X04 X05 RHS
CONST: 1 0 1 0 0 4 0 1 0 1 0 3 1 1 0 0 1 8
OBJ: 2 1 0 0 0
ENTER TITLE OF PROJECT this is a test ENTER NUMBER OF CONSTRADJTS, AND # OF VARIABLES ? 3 5 CONSTRADJTS?? X01 X02 X03 X04 XOJRHS ? 1 0 1 0 0 4
? 0 1 0 1 0 4
? 1 1 0 0 1 8
OBJECTIVE FUNCTION?? ? 1 1 0 0 0
THIS IS A TEST
CONSTRADJTS = 3 VARIABLES= 5 SOLUTION O~DlAL AFrER 3 ITERATIONS
MAXIMAL OBJECTIVE = 0.800000E+Ol
VARIABLE STATUS VALUE · DELTAJ XOl BASIC 0.400000E+Ol 0.0 X02 BASIC 0.400000E+Ol 0.0 X03 0.0 -O.lOOOOOE+Ol X04 0.0 -0.100000E+01 X05 BASIC 0.0 0.0
CONSTRADJT STATUS VALUE DECREASE INCREASE 1 BINDING 0.100000E+01 0.400000E+Ol 0.0 2 BINDING 0.100000E+Ol 0.400000E+01 0.0 3 SLACK 0.0 0.0 OPEN
00 YOU DESIRE 'ID RUN ANOTHER PROBLEM? no
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1. Elgerd, 0. I. 1971. Electric energy systems th~ory:
an introduct.ion, pp. 274-284. New York: McGraw-H111.
2. Gass, S. I. 1975. Linear programming, methods and
applications, pp. 3-6. New York: McGraw-Hill.
3. Tinney, W. F. and Enns, M.K. 1974. Controlling and
optimizing power systems. IEEE Spectrum, 11: 56-60.
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