A Mathematical Programming/Economic Equilibrium
Model for the Quantitative Analysis of the
Stability of Japan's Energy System
by
Tatsuo Oyama
MIT EL 86-016WP
August 1986
A Mathematical Programming / Economic Equilibrium Model
for the Quantitative
Analysis
of the Stability
of Japan's Energy System
Tatsuo Oyama
Graduate School of Policy Sciences
Saitama University, Urawa, Saitama
Japan 338
A Mathematical Programming / Economic Equilibrium Model
for the Quantitative
Analysis
of the Stability
of Japan's Energy System
ABSTRACT
Japan's energy supply-demand system is fully dependent on
the import of primary energy resources from foreign countries. So
the availability of primary energy, including crude oil and coal,
is
a
very
important
factor
for
the
stability
of
our
energy
system. In order to measure our energy system's stability under
an uncertain future availability of energy resources, we built a
mathematical programming /
economic equilibrium model based upon
linear programming techniques. In the model analysis uncertain
future availability of primary energy resources is expressed as
random variables with a given probability distribution, and the
economic equilibrium point
is obtained by iterative convergent
computation.
From
our
numerical
results
we
know
an
optimal
energy
supply-demand structure with equilibrium prices of primary energy
resources at the future target year, and obtain supply stability
and instability probabilities of
our energy system. Furthermore,
applicability of decomposition
techniques to our energy model
analysis
and
necessary
and
sufficient
conditions
stability of our energy system are discussed.
for
the
During the years following the Arab
oil
of
embargo
1973,
there have been many energy policy debates throughout the
world,
including Japan. Energy policy debates concern various technical,
environmental, social, economical, political
and
problems. Energy policy modeling efforts have
not only the necessity
of such
also the greater availability
Hoffman [1973]
proposed
supply-demand
energy
even
increased
interdisciplinary
of
high
energy
speed
network
problems,
military
due
research,
but
computers.
systems
various
Since
analyses
systems
for
analysis
approaches have been developed. (See e.g. Charpentier [1974]
Manne,
et
al
[1975, 1977],
[1980], and
[19793
for
energy
models.
Oyama [1980, 1980a, 1983b],
Shapiro
and
White
[1982].)
Japanese electric power system (see Energy
Saito and Oyama
situation
Also
see
Modiano
We
[19803) to see what our energy
and
Shapiro
Group
supply
the
[1979],
and
demand
will be like in the year 2000.
Many economic equilibrium models have
also
been
developed
which use linear and nonlinear programming techniques. (See
Kennedy [1974], Hogan [1975], Griffin [1977],
Hogan
[1980], Daniel and Goldberg [19813. See also Takayama
and
and
[1971] for price and resource allocation models.) Shapiro
discusses
decomposition
techniques
to
show
the
between linear programming and econometric components
planning
and
Shapiro
investigated
Study
to
models.
interpretation
Furthermore,
Shapiro
of the Kuhn-Tucker
[1978]
optimality
economic equilibrium point for certain
1
e.g.
Weyant
Judge
[1978]
relationship
of
energy
presents
conditions
mathematical
the
as
an
programming
models.
However, in most of these modeling
demands and the availability
of
studies,
primary
future
energy
energy
resources
are
demand
and
given exogenously. We believe that our future energy
the availability of primary energy resources should be uncertain.
In our analysis,
we express
the
structure
of
our
energy
supply and demand system as a network as does Hoffman [1973].
then formulate a linear programming economic equilibrium
in which the supply availabilities of primary
energy
We
problem
resources,
such as crude oil and coal, are defined as random variables.
This paper proposes an
iterative
convergent
procedure
model.
compute an economic equilibrium point for our energy
to
The
equilibrium point indicates an energy supply and demand structure
for the
future
target
year.
Our
mathematical
programming
economic equilibrium (MP/EE) model is a mathematical
optimization
model
to find an economic
equilibrium
/
programming
point
as
an
optimal solution of a linear programming problem.
We define
the probability
the stability
that
the
robability of our energy system as
given
linear
programming
feasible under the "randomized" supply availability
model
is
constraints.
In addition to these stability probabilities of our energy system
of prices
and demands
for crude
from the economic
equilibrium
solutions
at the target year, distributions
oil and coal are obtained
of our energy model.
policy in
various
We can also evaluate
demand
sectors
by
analysis with the probabilistic approach.
2
our energy
combining
conservation
shadow
price
In section
1 we explain
the
current
energy
Japan. In Section 2 we outline our MP/EE energy
the
formulation
of
our
energy
model
and
situation
model
the
their
Finally,
theoretical
in Section
analysis
is
5, we summarize
including
computational
procedure to solve it. Numerical results are given in
and
in
described
Section
in
Section
our model and comment
on
3
4.
our
approach.
1.
Introduction
Japan imports almost 90% of
her total primary energy.
imported energy resources are fossil fuels
coal and
liquefied
natural
gas
such
(LNG).
as
These
the South Pacific region,
economically unstable
which
arose during
which
contain
countri es.
many
Considering
primary energy resources. The oil
impact, and created
confusion
engineering system. Therefore,
in
our
whether
enough primary energy resource s to meet
politically
and
difficulties
it
had
a
country's
or
not
future
energy
and
is
an adequate
embargo
oil,
countries
the
the oil emb argo of 1973-'74,
that we will not always be abl e to obtain
crude
primary
resources are mostly imported from Middle Eastern
Major
probable
amount
very
serious
economic
we
can
expected
of
and
obtain
demand
concerns us greatly.
The oil embargo
in
1973 induced
increase from $2.51/bbl in 1972
a
large
to $10.79/bbl
3
crude
in 1974,
oil
price
and it had
serious effects on the world
system.
After
consumption
decreased
ti
his
has
economy
"first oil
stopped
our
crisi
increasing
The "second oil crisis",
which
re
ted
in 197 8, also caused a secon
Our
crucde oil
imports
Iranian
the
increase.
price
in 1978
to
$3
by 10.7% from
1980.
Total prim.ary energy consumption
during
the same p eriod.
conservation has prevailed
energy
Oil
from
large
imports decreas
Through
primary
.8% from 1974 to 1975.
Crude oil prices increased from $13.77/
in 1980.
supply-demand
rapidly.
by 4.4% from 1973 to 1974, an
revolution
energy
and
so
these
2.97/bbl
1979
decreased by
"oil
to
3.4%
energy
crises",
in our indus trial demand sector.
energy system has structurally changed to a system
Our
that does
not
depend heavily on crude oil.
It
is probable
that
wle
will
have
another
induced by a disruption in oil supplies due
events
in oil exporting
quantitatively
various
countries.
evaluate
levels
of
our
primary
Therefore,
energy
energy
"stability" of our energy system
to
is
to be "stable"
crisis"
unexpected
it is important
supply
under
constraints.
The
Our
or "unstable",
upon
energy
to meet our
"supply stability
future
energy
probability"
of
demand.
our
By
energy
as
probability that our mathematical programming energy model has
feasible solution, the "supply stability" of
4
our
energy
to
energy
defining
system
the
system
corresponding
whether or not we can have a sufficient supply of primary
resources
to
stability
dependent
possibility of importing crude oil and coal.
can be determined
some
system's
fully
"oil
the
the
a
system
can be quantitatively
investigated.
2. MP/EE Energy Model
2.1 Energy System and Linear Programming Model
Our energy system, which involves energy flow
supply regions
to final
sectors,
demand
is
from
and
Four
stockpile-transfer.
hydro-nuclear,
crude
oil,
regions,
kinds
coal,
of
and
as
illustrated
network system in Figure 1. The supply sector consists
divisions, including five supply
various
of
a
seven
domestic
production,
primary
energy
LNG-natural
gas
of
are
transformed into petroleum products, coal products, and secondary
energy of electricity and
consists of four
city
categories:
gas.
The
industry,
final
demand
sector
residential-commercial,
transportation, and stockpile-transfer.
A feasible
future
energy
energy
demand
flow in the network
under
various
supply
physical and engineering constraints. The
arcs of the network
correspond
to unknown
and network constraints are linear
has
to
satisfy
constraints,
energy
flows
variables
equalities
the
and
on
the
of the model,
and
inequalities
using those variables. Thus the problem of finding an equilibrium
energy
flow can be formulated
The goal is to
corresponding
to
an
obtain
as a linear programming
a
desirable
economic
feasible
equilibrium
problem.
energy
point
in
flow
our
mathematical programming energy model. A desirable energy flow in
5
the network
energy system
of Figure
1 can be determined
as a flow
attaining a maximum economic surplus criterion; i.e., minimizing
supply cost less demand cost. One can obtain
flow by solving linear programming
problems
an
optimal
energy
iteratively,
until
satisfying a convergence criterion.
2.2 Structure
of the MP/EE Energy Model
In the MP/EE
variables ({xi, yj,
correspond
Zk,
to energy
the remaining
demands
energy model there are five kinds of endogenous
wi, dij). Four kinds of them (xi,yj,zk,w})
flows, as in the network
endogenous
for imported primary
xi, iM
energies
= (1,...,17) : primary
indicate
{dij)
variables
of Figure
1,
while
"flexible"
of crude oil and coal.
energy
transported
the
from
supply regions and the stockpile-transfer
node
yj, jN
= {1,...,14} : primary energy transformed into petroleum
products, coal products, or secondary
energy; primary energy directly consumed
in demand sector
Zk,
: petroleum
ksK = (1,...,28}
and coal products
into secondary
transformed
energy or consumed
in
demand sector
wl, le
= {1,...,5}
:
secondary
energy
consumed
in final demand
sector
d'j, iI
= {R,L}, jJ
= { ±1, ±2,...,
±6)
variables indicating the perturbation of
6
primary energy resources' (R:crude oil
L:coal) demand from their standard demands
Using
the
programming
above
variables,
model are expressed
constraints
in
the
linear
as follows:
(1) Availability Constraints of Primary Energy Resources
In the energy network of Figure 1, primary energy
enter the system through supply
nodes.
The
amount
resources
of
primary
energy at each supply node has an upper bound determined
by
the
physical, economical or, sometimes, political situations
in
the
supply regions. The physical availability of each primary
energy
resource from each supply region is given as follows:
xi
b,
isM.
(1)
(2) Flow Conservation Constraints
The set of nodes
in the network
is divided
into three groups
supply nodes corresponding to supply regions, demand nodes
indicating
demand
sectors
nodes.
intermediate
At
in-flow has to be equal
=
Z
x
Z
yj = Z
ieMp
jsNp
jeNp
keKp
from
each
p=13
to
p=16,
intermediate
to out-flow.
node
and
remaining
p(1,...,12),
Therefore,
yj
p=1,2,3
(2)
Zk
p=4,...,10
(3)
p=11,12
(4)
Z yj + Z Zk = Z wl
jsNp
k6Kp
l6Lp
7
(3) Upper and Lower Bounding Capacity Constraints
Upper
and
lower
bounds
are
given
for
the
variables
indicating production of petroleum and coal products (Zk,
and consumption
constraints
of electricity
are written
and
city
gas
eL).
These
as follows:
LBZk S Zk
UBZk,
kEK'CK
(5)
LBWi
UBWI,
E1L
(6)
wi
where LBZk, UBZk, LBWI and UBW l are lower
{Zk}
({wi,
kK'CK),
and {wi), respectively.
constraints
(5) are given
and
K' is a proper
upper
subset
for some variables
bounds
of
K,
of
hence
of (k}.
(4) Yield Constraints of Petroleum Products
Refinery systems have their
own
restrictions regarding the yields
physical
of
petroleum
and
engineering
products.
Each
petroleum product has both lower and upper production bounds.
YLjC
yj
YUjC
C = X
+ X2 + X4 + X7 + Xl
3<j:6
(7)
+ X12 + X15 - Y2
where YLj and yuj are the lower and upper bounds
petroleum
product
indicated
of the yield
by yj, and C is the total
crude
of
oil
entering the refineries.
(5) Demand Requirement Constraints for
Imported
Primary
Energy
Resources
Imports of primary energy resources are
restricted
by
following constraint:
Z
Xi
iEMk
Dk
+
Z
jEJ+
dkj -
8
Z
jEJ-
dk
ke {R,L)
(8)
the
where J+ and J- indicate
J=(±l,
±2,...,
sets of positive
±6), respectively.
The
inequality expresses the flow of
and negative
left
each
side
primary
indices of
of
the
energy
above
resource
from each supply region to Japan, while the right side
expresses
the perturbed demand of each primary
from
standard demand. The
deleted,
is illustrated
corresponds
dkj
in Figure
0
dkj
(dkj},
in Figure
to the standard
Each variable
interval
variable
has
an
energy
demand
upper
with
2, where
Dko
the
the
the
supe:rscript
the demand
in
bound
resource
k
q uantity Do
constr aint
correspondin g
(8).
to
the
2.
Akj,
k{(R,L},
jeJ.
(9)
(6) Final Energy Demand Requirement Constraints
In
the
four
demand
residential-commercial,
nodes
corresponding
transportation
and
to
industry,
stockpile -transfer,
the following final energy demand requirement constrain ts have
to
be satisfied:
Z yj
jgNp
+
Z
kEKp
Zk
+
Z
lcLp
(10)
p=13,...,16
Dp
Wl
(7) Objective Function
The total supply
defined
cost of the energy
as the sum of fuel costs
this model, we define
and
system
in
Figure
transformation
the fuel cost to be the cost
of
1
is
costs.
In
obtaining
the resource in each supply region and transporting it to
i.e., the CIF cost. The transformation
cost is defined
Japan,
to be
the
cost for transforming primary energy resources into petroleum and
9
coal products, electricity, and city gas, and consists mainly
the capital
cost necessary
for
energy
fuel cost per thermal unit (010
transformation
be d,
jN.
transformation.
kcal) be ci, iM,
cost per unit kcal of petroleum
Let
the
let
the
and
and coal
products
The transformation cost per kcal of secondary
is given by e,
lL.
Then
the total energy
supply
of
cost
energy
can
be
written as follows:
Z cxi
+
i6M
However,
djyj +
leL
the demand
total cost for meeting
(11)
eiwi.
jEN
cost of the energy
a forecast
system,
energy demand,
approximation to the area under the nonincreasing
By using a step function
=
g(q)dq
E
in Figure
2, the cost
defined
is given
demand
as
as
a
an
curve.
is
Pjdj
(12)
dj,
(13)
jSJQ
0
where
Q = Qe +
.
jasJQ
and JQ is a set of indices
contained
in the
corresponding
range
from
to the demand
from (12) the constant
{j}
corresponding
0
to Q,
in
demand
the
and
Pj
j-th
is a
the
intervals
commodity
interval.
cost corresponding
10
to
price
Subtracting
to the
integral
of the demand
following
from 0 to Q
curve
in
f(q)dq =
where sgn(j)
we
obtain
the
sgn(j)Pjdj
Z
(14)
j6Ja
indicates
sgn(j)
=
the sign of index j
1
if
(j+O), i.e.,
j>O
if j<O.
= -1
Thus adding (12) for each iI=(R,L),
Minimize
2,
sum:
Qe0
be given
Figure
our objective
function
can
as follows:
Z cixi +
ieM
Z djyj
jEN
+
eiwi lEL
Z Z sgn(j)Pijdij.
ieIjeJ
The negative of the above objective function can
as maximizing
the demand
be
(15)
interpreted
cost less supply cost, while meeting
the
future energy demand requirements. Hence the optimization problem
corresponds to finding the economic equilibrium point
maximizing
economic surplus.
The
probabilistic
aspects
of
our
energy
model
are
as
follows: many energy supplying countries are somewhat politically
and
economically
unstable.
Hence,
we
assume
availabilities of the primary energy resources
11
that
which
supply
correspond
bi in
to the right hand side values
Suppose
that the upper
(1)
are
bound of the total
random
variables.
of
availability
some
primary energy resources from the overseas supply region to Japan
follow beta
denoted by b
and
q,
whose
distributions
bm,
and
respectively.
and
whose
Then
the
upper
and
parameters
random
lower
bounds
are
p
integers
are
b
variable
has
the
following probability density function:
f(b) =
(b-bm) P- (bM-b) q-
,
K
where K is a constant.
When parameters
bm
b
b
(16)
p and q are integers,
K is
given by
K =
r(p)r(q)
r(p+q)
(p-l)
! (q-l) !
=
(p+q-l)!
(bM-bm)P+q-
where r(-) is a Gamma function
r(p)
=
e-xxp-dx.
.0
12
1
(17)
Suppose parame ters p and q satisfy p>1 and q>l, then
variable
b has the following mean ,
variance
2,
the
and mode m.
bmq+ bp
(18a)
=
p
random
+ q
- bm )
pq(bM
2
(18b)
(p+q)2(p+q-1)2
bm(q-1)+ b(p-1)
m
(18c)
=
p+q-2
2.3 Computational Method
Our energy model described in the previous
formulated
in a vector-matrix
Minimize
cx - pd
subject to
Aix
variables
form as follows:
(19)
A2x = b2
(20b)
s
a
b3 + Kd
(20c)
d ~ b4
(20d)
d
(20e)
0
variable
{xi, yj, Zk,
w)
vectors
and
x
and
d
consist
(dij), respectively.
of
(20c)
are
in (14), respectively.
the
resource
Constraints
availability
13
the
Here, p and d
are price and commodity vectors, whose elements are given by
and sgn(j)dij
be
(20a)
x,
the unknown
can
bi
A3x
where
section
(20a),
constraints,
Pj
(20b) and
balancing
equations and demand requirement constraints, respectively. (20d)
indicates
the
bounding
constraints
in
(9)
for
the
demand
variables (dkj).
We define our resource supply cost minimization submodel
as
follows:
c s xs
Minimize
subject
to
ASixs
AS3Xs 2 bs3
(2:2c)
by (19)-(20 ').
imply
2 0
i=1,2 ,3,
for
subvectors
that
The
of
cs
and
the
and
are,
corresponding
subvectors
supply
resource
xs
respecti vely,
Ai,
b;
for
inal linear programming problem given
submatrices
our
2b)
(2:2d)
c and x in the orig
i=1,2,3,
(22a)
(2
bSi
submatrices and
bSi
AS2Xs = bS2
xs
where Ai,
(21)
submodel
consists
components related to crude oil and coal only,
above
described
rather
of
the
than
the
whole suppl y model.
Let
the
thLe shadow
demand
resource
requirement
is I be
i .e ., the
price,
n'i.
optimal dual
co ns traint
Then
th.e
equi
1
(22c)
of
ibrating
solution,
primary
condition
for
energy
for
our
MP/EE energgy model can be wr.i t ten as follows:
t'i = gi(Di
where g
g(q),
indicates
corresponding
+
di'j -
jE +
the i-th
j Ej -
component
di'j)
of
(23)
the
demand
to the primary energy resource iI.
14
function
Die
and
d'ij indicate the standard energy demand for the resource iI
an optimal solution for the demand variable dj,
respectively.
The optimization problem given by (19)-(20) is
surplus
maximization
problem.
A
general
problem cannot always be transformed
into
maximization
the
problem.
In
order
for
e.g. Hurwicz [1971).
demand
function
has
Therefore,
to
be
the
an
economic
equilibrium
economic
surplus
transformation
be
i.e.
to
integrable
Jacobian
symmetric,
an
market
possible, the demand function g(q) needs to
and
matrix
the
be
(see
of
cross
the
price
elasticities between two different commodities must be symmetric.
In our energy model, cross price elasticities
commodities
were assumed
between
different
to be zero, and thus our Jacobian
matrix
is symmetric.
Let us look at a computational
economic equilibrium
point
in
procedure
our
energy
for
obtaining
model.
Firstly,
sequence of random numbers with a beta distribution (beta
each
pair
of
these
a
random
numbers) are generated. Two sequences of beta random numbers
generated simultaneously, and
an
numbers
are
is
assigned to the corresponding right side in the constraints given
by (1). Then the linear programming economic equilibrium model is
solved to obtain an optimal energy
flow
meeting
demand. The computational method in our MP/EE model
future
energy
analysis
is
presented in the flow chart of Figure 3. Solving our MP/EE energy
model
iteratively
is
a
principal
part
of
our
analysis.
details of the solution algorithm are given in Figure 4.
15
The
The correcting process at the t-th iteration, given
energy prices pt
primary
and supply costs cs, is written as follows:
1
Pt+l =-(7r't
2
+ pt)
(24)
cSt+i = C + cst
(25)
where
Acst
=
A(x't - c),
0
A
1.
(26)
The convergence of this iterative computation
is
attained
when
the shadow price of each primary energy resource of crude oil and
coal equals
that
obtained
from
the
approximate
demand
curve
corresponding to optimal commodity demand.
3. Numerical Results
3.1 Assumptions
We define
and Input Data
the year 1983 as the base year, and then
the year 1990 as our
future
target.
Firstly,
average annual growth rates of final energy
we
demand
look
assume
industry, residential-commercial, transportation
transfer,
respectively.
Final energy demands
given in Table I. Supplies of primary
base year and the target year are
energy
shown
in
and
2.0%
the
for
stockpile-
in 1983 and 1990 are
resources
Table
II.
Table, Other Middle East Region denotes the oil-exporting
16
that
between
base year and the target year are 2.0%, 3.0%, 2.0% and
at
in
the
In
the
Middle
East countries
excluding
Saudi Arabia,
i.e., Iran, Iraq, Bahrein,
Kuwait, Neutral Region, Qatar, Oman and the United Arab Emirates.
The South Region consists mainly of
Southern
such as Indonesia, Brunei and Australia.
crude
oil
includes
African
Algeria and Nigeria. The
Pacific
The
Other
oil-exporting
coal-exporting
countries
Region
countries
Other
for
such
Region
as
includes
South Africa, China, and Soviet Union.
Upper bound availabilities of primary
1990 are estimated as follows. The
import from the Middle East
increase
1990. In
of 4.0% between
estimating
is
upper
based
energy
bound
upon
resources
for
an
average
the base year 1983 and the
upper
bounds
for
crude
coal
oil
annual
target
import
in
year
from
the
Southern Region, North-South America and Other Region, an average
annual increase 4.0% is assumed. Estimates for
supplies
from the Other Middle East Region, crude oil
and
LNG
North-South America Region, crude
the
Other
oil
from
of
LNG
from
the
Region,
domestic crude oil, coal, natural gas and stockpile-transfer
all based on the average annual increase rates
2.0 - 4.0%
are
from
1983 to 1990. The upper bound availability of hydro-nuclear power
is estimated according to an average annual increase of 4.0 - 5.0
% from the base year.
CIF prices of primary energy resources from
various
regions in 1983 and their estimates for the year 1990
supply
are
in Table III. Crude oil prices in 1983 indicate 'average'
given
prices
in oil-exporting countries in the region. For example, the
oil price
in
Saudi Arabia
is that of Arabian
17
light, and
the
crude
oil
price in Other Middle East Region is based
Emirates Murban.
Prices
in
the
on
Southern
Crude
oil price estimates
except
Sumatra
respectively.
for the target year 1990
that the increase
Arab
North-South
Indonesian
Blend,
from 1983 data by assuming an average annual
4.0%,
United
Region,
America Region and Other Region are those of
Light, Mexican Isthmus and Algerian Sahara
the
are
price
obtained
increase
rate is 5.0% for Other Middle
as
East
Region.
Coal prices
in 1983 are the weighted
mean of s team coal
material coal from each supply region. Coal prices in
Region and North- South America
Australian
Region's
Chinese
and
the
coal pri ce
United
is
Region
the
Natural
on an average
mean
weighted
annual
gas and LNG prices
on
The
coal
incr ease of 5.0%
in 1983 are those
and
North-Sou th
South
those
Sout]h
of
for fut ure
LNG for the Middl e East Region, Brunei
South Region, and Alaskan for
based
r espectively.
States,
and Russi an coals. Estimates
1990 are based
are
the
and
of
Other
African,
prices
in
f:rom 1983.
Abu
of
Indonesia n
America
Dhabi
for
the
R,egion.
LNG
price estimates for the target year 1990 are obtained
from
1983
data by assuming an average annual incre ase of 4.0%.
Transformati on costs for petroleum
and secondary
Table
energy
IV. Costs
(electricity and
for fuel oil,
products, coal
products,
city
gas)
are
given
kerosi ne-gas
oil,
and
gasoline-
naphtha are weighted mea ns minus fuel costs .
in
The other petroleum
products' transformation cost is basi cal ly the LPG price, and the
coke transformation
cost is the coke
18
price
minus
the
material
coal price. Electricity transformation costs
for
both
industry
and transportation are the capital costs in the electricity
for industry,
and those for residential-commercial
are
rate
also
the
capital costs in the rate for residences. City gas transformation
costs are the capital costs in the
industrial
and
residential*
commercial city gas rates.
Upper and lower bounds for
respect
to variables
constraints
(5)
and wi, are presented
Zk
and
(6)
in Tables
V and VI.
Lower bounds for petroleum and coal products are either
those whose consumption
is relatively
small,
either
the consumption
of the base year or a 50%
for
0.0
or the amount
base year's consumption when they are large. The upper
with
of the
bound
increase
added
when they are relatively small. The average annual increase
is assumed from 1983 to
large.
Lower
bounds
1990
for
for
those
electricity
whose
and
4.0%
consumption
city
gas
is
is
are
the
consumption in the base year. Upper bounds are obtained from
the
base year's consumption by assuming an average annual increase of
5.0%.
The upper and lower bounds for
given
petroleum
in Table VII are based on the assumptions
products'
that
yields
demand
light petroleum products such as kerosine, gas oil, gasoline
naphtha will increase in
the
future,
while
demand
for
for
and
heavy
petroleum products such as heavy fuel oil will decrease.
The main sources of Japanese energy data used in
our
model
analysis are Energy Statistics [1985], Handbook of Electric Power
Industry
[1985],
Industrial
Statistics
19
Table
[1984],
and
Petroleum Statistics [1984].
bm and b
Parameters
minimum
and
estimates
maximum,
the
beta
p and q are determined
to the expected
future
distribution
respectively,
for the future availability
Parameters
equal
the
for
are
indicating
of
crude
oil
so that mean values
availability
of
extreme
and
are
these
the
coal.
nearly
resources.
These parameters are presented in Table VIII.
Beta random numbers are generated by
transformation method to uniformly
applying
distributed
the
random
inverse
numbers.
Random numbers following uniform distribution between 0 and 1 are
generated by using the square method. (For
more
information
on
random number generation, see e.g. Fishman [1973], Bratley, et al
[1983].)
Approximate demand curves for imported crude
are based upon their own and cross
price elasticity
decrease ()
ij,
price
coal
data.
The
represents
the
of commodity i's demand corresponding to a unit % of
to
analysis in Oyama [1983c], own and cross
primary energy resources in 1980 are
7 4
LL=-0.
and
elasticity
i,je(R:crude oil, L:coal),
commodity j's price increase. According
0.69,
oil
.
the
price
RR=-0.07,
translog
model
elasticities
ERL=0.04,
Hence from the above data on eij's
we
of
LR=
can
say
that in Japan crude oil is rather price insensitive compared with
coal, and these resources are substitutes
positivity
of
RL
and
LR.
20
each
other
from
the
3.2 Supply Stability Probability and Equilibrium Prices
We wrote a FORTRAN computer program to
The program consists of nearly 3800
our
analyze
most
statements,
model.
of
which
(around 80%) comprise the product form simplex method for solving
the linear programming problem. Others relate
generation,
figures,
iterative
histgrams
procedures,
and
slack
17 xi's,
variables)
output
random
formatting
5 wi's,
14 yj's, 28 Zk',
and
51
contains
constraints
24
121
a second
requiring
about
of CPU
time on
140 pivots
the
if
simplex technique. In order
d'j's
and
(excluding
we
obtained
IBM
3033
computer
system,
start
from
phase
of
obtain
to
an
1
necessary to solve 4 - 6 linear
programming problems
Since
is rathe r simple
about
of
the
CPU
total
equilibrium problems are solved
in
12 CPU mi nutes by the IBM 3033 system.
As shown
cases.
problems.
250 pairs of beta random numbers. So, a
of 250 cases of these economic
is
it can be so ived quickly.
Hence, solving the MP/EE energy model takes up most
time. We generated
it
alternating
el
e nergy model and the supply submod,
the latter mo del
the
equilibrium
economic
availability constra ints,
the MP/EE
33
bounding
point for each pair of resource
between
for
variables
constraints). An optimal solution for each iteration is
within
number
and so on.
The linear programming MP/EE model
(including
to
i n Figure
3,
we
For some combination s
be infeasible
insufficient
s olved
of
N(=250)
beta random numbers the model may
since either crude oil
to meet our future
our MP/EE model for
or
coal
supplies
final energy demand.
21
may
be
Figure 5 shows model feasibility results for
beta
random
numbers.
In
Figure
5,
the
each
vertical
pair
of
coordinate
indicates the availability of imported coal, while the horizontal
coordinate indicates the availability of imported crude oil.
each combination
of these energy resources'
I indicates whether
Let the number
Ni. Then we define
the model
is feasible
of infeasible
availability,
an F or
or infeasible.
cases among
the "supply stability
For
total
N
probability"
energy system by the ratio of the feasible
cases
cases
be
(Ps) of our
to
the
total
number of cases.
Ps =
N - NI
(27)
N
Our energy system can be
probability
understood
Ps and "unstable"
Ps and 1-Ps as stability
and
to
be
"stable"
with the probability
instability
with
1-Ps. We
probabilities
the
call
of
our
stability
and
energy system, respectively.
Our
numerical
experiments
show
that
the
instability probabilities of Japan's energy system in the
year 1990 can be presented
205
Ps
250
= 0.82,
target
as follows:
45
1-Ps = --250
0.18.
We should consider the instability probability as an
22
(28)
implication
that
an
extremely
probability
di fficult
of 0.18 unl ess
si
tuation
e nergy
our
changed. An infeasibili ty result
may
occur
system
is
m ay be changed
with
structurally
into a "feasible"
case by adding more infrastructure to our energy
system,
transforming our energy system into a more flexibl e one
it
can
meet
variable
substitution among primiatry
Figure
Figure
6 is obtain
ed
and
by
infeasible
by
t hat
prom ot ing
secondary energy resources.
the feasibi lity resul ts in
5 with the equil i brium prices' results of
feasible and
or
so
by
demands
c ombining
oil and coal. The figur e first
divided
energy
final
the
divides
area.s.
the
Then
into nine parts depend ing
on
equilibrium prices PR and PL. Note
the
the
that
imported
whol e
this
into
region
regio n
feasi ble
crude
crude
oil
and
di vision
is
coal
of
the
feasible region is not very accuratee , since the eq uilibrium price
of each primary
availability
energy resource
can vary depending on the
of the ot her. However,
know approximately how much each
this partition
energy
ing
resource' s
price will be changed by the degrees of supply
s upply
helps us to
equilibrium
avai lability.
4. Theoretical Analysis
4.1 Application of Decomposition Techniques
Decomposition techniques, which were originally proposed for
solving multi-stage block diagonal linear
programming
have recently been applied to energy policy analysis
revised
forms.
Shapiro
[1978]
has
23
given
an
idea
problems,
in
for
various
using
decomposition
submodels
techniques
with
a
to
linear
creating a complete energy
combine
econometric
programming
planning
forecasting
optimization
model.
Shapiro
[1982] have applied decomposition techniques to the
integration and optimization of coal supply
submodel,
and
and
White
constructive
demand
models.
The decomposition technique is not an efficient way to solve
the
energy model,
are
but its interpretation
and economic
implication
significant in many model analyses. In our MP/EE model
we show the possibility of applying the decomposition
analysis,
techniques
to solve a market equilibrium problem.
We can rewrite our MP/EE problem (19)-(20) as follows:
Minimize
cx - pd
subject to
Ax - Kd
Bx
(29)
qi
(30a)
2 q2
(30b)
-Id a -q3
0
x, d
We define polyhedra
X = ( x
D =
Let x,
points
{ d
2
-qa,
d
and
d,
of the above polyhedra
of extreme
bounded)
as follows:
0 )
q2, x
iI=(1,2,...,nx)
nd are numbers
(30d)
X and D (assumed
Bx
| -Id
(30c)
(31)
0 )
jJ=(1,2,...,nd)
X and D, respectively,
points
(32)
be
and
in X and D, respectively.
the price directive master problem for (29)-(30) can
be
extreme
nx
and
Then,
written
as follows:
Minimize
Z cAix i iEI
Z pjd
jEJ
24
j
(33)
subject
Z AAix
to
-
isI
Z
j
Kjd
Ai
(34a)
q1
jEJ
= 1
(34b)
=
(34c)
isI
Z aj
jeJ
Ai,
Denoting
the
constraints
problem
dual
variables
by
7,
7ql
+ nx +
respectively,
dual
nx
nd,
and
for
d
(35)
i
nd -
Kd j
cx
i
I
(36a)
jsJ
(36b)
i
-pdi
O.
(36c)
in
III=IJ=O
from
above
prices)
Ax
2
If we start
the
(shadow
nx +
to
(34d)
can be wri tten as:
to the above problem
subject
i I , jeJ.
0
gaj
(34a)-(34c)
Maximize
1
iteration we add new extreme points
xi
( ini
using the optimal dual solution
(35 )-(36),
dj
each
at
then
in the following
tial ly
way:
solve
7r=nx=7rd=O), we
the following problems:
zx = min.(c -
A)x
(37a)
subject to xX.
K)d
(38a)
subject to dD.
(38b)
Zd = min.(-p
Suppose
(37b)
+
zxznx and Zd7rd;
in the dual problem
add a new extreme
then the s olution
(35)-(36).
(n,
x,
7d)
is
ptimal
If n ot, i.e., zx<rx or Zd <d ,
point solution
to generate a new
form
25
column
in
then
the
CX r
AX r
-pdr
Kd r
or
1
0
1
xr and d r are extreme
where
respectively.
Then,
we
point solutions
return
(33)-(34). Thus, an optimal
to
the
solution
to
to
problem
the
(29)-(30) can be obtained by using an optimal
'j
(37)
a nd
of
(38),
the
problem
original
solution
form
A'i
and
for (33)-(34) as follows:
=
d'
Now,
=
isI A' ixi
(39)
j
(40)
Z ujd
j6J
the
key
to
solving
our
MP/EE
problem
using
decomposition technique is that the problem (38a)-(38b)
is
the
very
easy to solve since the polyhedron D is just a hypercube.
Hence,
one can solve the problem analytically, without applying
simplex
methods. This problem is still easy to solve even when the matrix
K contains both nonzero own and
cross
also when demand variable vector d
is
price
elasticities,
given
many
elements
and
by
defining more endogenous demand variables. Incidentally, applying
the
resource
directive
master
decomposition technique does
not
problem
help
much,
approach
since
in
we
the
cannot
obtain a simple, analytically solvable linear programming problem
as
in
(38a)-(38b).
26
4.2 Analysis on Instability Probability
In our MP/EE model analysis, the instability probability
our energy system was defined
to
be
the
probability
of
that
the
Figure
6,
the dividing line between feasible and infeasible regions may
be
energy model results are infeasible. As we can see in
expressed by a straight line. From
equation
of the dividing
our
line in Figure
numerical
results,
the
6 can be approximated
as
follows:
u + v = 3.16x10 5
where
u and v indicate
the
(41)
supply
crude oil and coal, respectively.
availabilities
We can expect
(u,v) is feasible if u+vZ3.16x1O5 . It
is
that
line since
to zero. This is because
of various
an
can
lower bounding
point
otherwise.
be
neither of these two variables
imported
the
infeasible
However, the above equation given by (41) cannot
dividing
of
'exact'
be
close
constraints
on
some of the decision variables in our energy model.
Let X and Y be random variables following beta distributions
with aisX:bi and a25Y:b2, respectively. Then we can illustrate an
"approximate" infeasible region as in a shaded area in Figure
Suppose the equation of the boundary line
between
feasible
7.
and
infeasible regions in Figure 7 is expressed by
pX + qY = r ,
Then the infeasible
region
p, q, r: constants.
of the
shaded
area
(42)
is
the
set
of
points (X,Y) satisfying
pX + qY s r
p>O, q>O, ai~Xsb,
(43a)
a2SYsb2.
27
(43b)
The infeasible region given above can
be
transformed
into
the
case of the standard beta distribution, with random numbers x and
y
distributed
0
between
and
1.
Therefore
the
instability
probability for our energy system is approximated as follows:
ax+byc
TI
1
B(p,q)B(r,s)
xp- (l-x)q-lyr-1(1-y)5ldxdy
0&x I
0 yl1
I (1-x) q - I
XP
x)
b
yr-l(1-y)s-ldydx
0
0
11
XP
-1(1-x),Q-'F(x)dx
(44)
0
The last expression in the
above
formula
can
be
numerical integral since we have a detailed table
given
of
by
values
a
of
beta functions.
Now let us look at our instability
The necessary and sufficient condition
prob abi 1 i ty more closely.
for
the
problem (19)-(20) to be feasible are given a s
Farkas' lemma (see
e.g.
Shapiro
[1979],
fo
p28)
linear
llows
in
program
by
a
using
slightly
modified form.
zy
(45)
O
0
where
zy = max.( - ybi
subject
to
+ Y2b2
+ y3b3
-
y4b4
)
- yiAi + y2A2 + y3A3
yi,
y3,
28
(46a)
(46b)
- y3K - y4I
(46c)
0
(46d)
y4
If zySO for all the random variable
components
of
bi,
then
the
model given by (19)-(20) can always be feasible, i.e., Ps=l.0.
We can give a sufficient
(20)
to
be
feasible.
condition
Since
the
problem's objective function is
for our MP/EE model
general
convex
and
respect to the right hand side values,
corresponding
problem
the
to the inf.(infimum)
programming
nondecreasing
the
should have the greatest lower bound of
values,
linear
objective
function
of the vector
bi. Let
(47)
for all random variable
and no bi' such that b'5b
variable
components
subject
1
bi'~bi
be defined
+ y2b2 + y3b3 -
MP/EE
variable vector bi. Note
that
for
vector
all
bi
random
(48a)
5 0
(48b)
- y3K - y4I ~ 0
(48c)
y4
is
feasible
the
a
y4b4 }
y3,
model
of
by
- yiAi + y2A2 + y3A3
to
yi,
Then, if zySO0, our
components
satisfies
of bi. Let z
zOy = max { - yib0
with
(46a)-(46d)
bB1 = inf. bi
i.e., bibi
(19)-
above
0
(48d)
for
sufficient
any
random
condition
guarantees the feasibility of the MP/EE model by solving a single
linear programming problem, while the
necessary
and
sufficient
condition given by (45)-(46) for the model's feasibility requires
that (45) holds
for all possible
bi.
5. Summary
In this
analysis,
we
have
29
investigated
the
effects
of
primary energy resources' supply constraints
on
system.
a
It
is
generally
true
that
when
Japan's
energy
primary
resource's availability is high, its commodity
price
and vice versa. However, since we have considered
energy
goes
two
down
kinds
of
primary energy resources simultaneously,
and
furthermore
price elasticities are
the
price
effects
of
more
complex
to
resources
supply
analyse.
Through
very
different,
constraints
our
MP/EE
are
a
model
little
we
obtain
an
their
economic
equilibrium point for each energy resource as shown in Figure
However, the
regions
splitting
based
on
of
the
commodity
feasible
prices,
as
region
in
into
the
complicated because the equilibrium points may
be
6.
smaller
figure,
is
dependent
on
the computational method and its convergence criteria.
Our computational technique is fundamentally similar to
PIES model
Ahn
(see e.g. Hogan
[1975], and Hogan,
[1979], and Ahn and Hogan
for special
cases)
in
its
[1982]
main
for
et al [19 78);
also
arguments
convergence
framework,
the
except
that
the
iterative procedures and some assumptions about supply and demand
functions are different. In the PIES model, the
was assumed
to be continuously
differentiable,
demand
and the
supply mapping was a point-to-set mapping, while we
function
(inverse)
assumed
for
our model that both supply and demand functions were po int-to-set
mappings.
The assumptions of the PIES model guarantee
existence and the uniqueness
of an equilibrium
MP/EE
the
model
assumes
only
existence
solution.
30
point,
of
an
both
the
while
our
equilibrium
Our iterative computational method worked very well, and
could obtain an equilibrium point after
all feasible
cases.
Although
the
several
convergence
iterations
for
proof
for
our
believe
the
computational method is not given in this paper, we
convergence
is guaranteed
by showing
the
fact
that
the
shadow
price of the demand requirement constraint (25c) is expressed
the
approximate
demand
function
value
we
corresponding
to
by
the
optimal resource demand. We are presently working on this proof.
Approximating a demand function is another problem. In
paper we assumed the existence of nonzero own price
for
primary
energy
resources
only,
neglecting
this
elasticities
cross
price
elasticities between two distinct primary energy resources.
Both
own and cross price elasticities can be simultaneously considered
in our model
matrix
price
K
analysis
of
by incorporating
(20c).
elasticities
The
does
this information
consideration
not
make
of
solving
into
nonzero
the
the
cross
problem
more
difficult, but rather changes the problem formulation slightly by
adding more nonzero elements in the coefficient matrix.
Applying
decomposition techniques should also be very effective in solving
our MP/EE model
in this case.
The stability probability was defined to be the
that the MP/EE model was feasible.
random
numbers
obtained the
as
our
stability
import
and
We tried a single
supply
instability
probability
sequence
availability, and
probabilities
of
of
then
our
energy system. We know that if the substitutability between crude
oil and coal increases,
then the stability
31
probability
will
also
increase, since there are more ways to meet the
forecast
energy
demand.
We can conclude that the Japanese energy system needs to
more flexible,
so that it can structurally
primary energy supply availability.
For
adjust
example,
(from 1979 to 1980).
of
variations
the
Japanese
cement industry changed almost totally from fuel oil to
one year
be
In another good example,
coal
our
in
power
industry is introducing mixed fuel thermal power plants consuming
fuel oil, coal and LNG.
If our energy system were well
coal,
it will greatly heighten
organized
the stability
primary energy supply, and also lower
cost. We would
the
to
consume
probability
total
not have to depend so heavily
more
of
energy
the
system
on crude oil,
which
has higher supply uncertainty and instability. We must note
coal transportation and storage infrastructure and
countermeasures for SOx and NOx emissions and
very important in the case where we consume
environmental
burned
a
that
great
are
ashes
of
amount
coal.
Thus, the substitutability among primary energy resources is
a very important factor in
our
energy
system's
stability.
In
order to further elucidate the relationship between the stability
probability and the energy resources
substitutability,
we
need
more numerical experiments, trying different values for lower and
upper bounds of certain energy flows, and varying the yields
and
efficiencies of petroleum and coal products.
The model described in Section 2 is a single
32
period
static
a part of the right hand side
model. Letting
optimization
linear programming model be a random
we
variable,
in
could
the
apply
probabilistic and stochastic analyses, obtaining supply stability
and instability probabilities. We can
and
increasing the number of periods
probability
in
the
more
distant
dynamic
add
estimating
future.
In
analysis
the
this
stability
case,
following difficulties occur: uncertainty with respect to
primary
energy
variations
of
prices
and
final energy
supply
availability,
forecast
and
to obtain an
economic
equilibrium
availability
another difficulty. Therefore we believe
representing
the next 10 -
techniques
of
two
or
will
three
15 years will be the largest
and
the
necessary
efficiently
point
future
solutions,
reliability of data. Obtaining the large scale structure
linear programming model and computational
the
subsequent
optimal
justification of probability distribution, and
by
be
stages,
time span
we can deal with reasonably.
We believe
that the approach
introduced
in this paper can be
useful to quantitatively analyse the energy system
countries like Japan which depend
heavily
on
stability
imported
primary
energy resources. We are considering further modification of
energy systems approach by incorporating dynamic terms
modeling of national economic structures.
33
of
and
our
more
Acknowledgements
This research was initiated when the author was
Researcher at the Economic Research
Institute
of
a
Visiting
the
Economic
Planning Agency in Japan in 1983, then was finished while he
was
a
the
Postdoctoral
Fellow
at
the
Energy
Laboratory
Massachusetts Institute of Technology in the U.S.A.
1985 to July,
1986.
Shapiro
Massachusetts
from
of
the
Institute
Wood of the Energy Laboratory
Operations
of
Research
Technology
for
Center
their
very
Institute
Research
of
grateful
Tokyo
Institute
and
in
to
Doctors
M.
Japan
S.
Ishikawa
for
their
computational aspects of this research.
34
Mori
of
of
at
and
the
invaluable
suggestions, comments and their sincere cooperation.
is also
June,
The author would like to express his sincere
thanks to Professors David O.
Jeremy F.
in
The
the
Science
Electronics
assistance
author
Device
with
the
TABLE I. Final Energy Demand
(1010 kcal )
Sectors
1983
1990
211858
243358
Residential-Commercial
Transportation
99465
122329
56869
65324
Stockpile-Transfer
45844
52660
Industry
TABLE III. Energy Prices by Supply Region
(106 yen/100l
Energy
Crude
Oil
Coal
Natural
Gas
LNG
Supply
Region
1983 Data
kcal)
1990 Price
Saudi Arabia
45.996
60.488
Other Middle East
47.437
66.749
South Region
North-South America
48.964
47.398
64.433
62.373
Other Region
50.729
66.756
South Region
22.712
31.958
North-South America
19.184
26.994
Other Region
17.761
24.992
Middle East
48.072
63.259
South Region
North-South America
45.219
59.505
46.042
60.588
1
TABLE II. Primary Energy Resources by Supply Region
(1010 kcal)
Energy
Crude
Oil
Supply Region
1983 Data
Saudi Arabia
59904
74100
Other Middle East
91649
113368
South Region
38882
48096
9196
11375
13213
16344
447
588
Stockpile-Transfer
South Region
28531
37545
26890
35385
North-South America
21312
28045
Other Region
Domestic
8834
11173
11625
Stockpile-Transfer
7197
8851
Middle East
2411
2965
23655
29093
1389
1708
North-South
America
Other Region
Domestic
Coal
Natural
1990 Supply
Availability
South Region
America
12834
Gas
North-South
LNG
Domestic
2154
2474
Stockpile-Transfer
1335
1642
2
TABLE
IV.
Transformation
Costs of Secondary Energy
Resources
(1010 yen/
Energy
Resources
1 0 10
kcal)
Transformation costs
Fuel Oil
2.01
Kerosine-Gas Oil
Naphtha-Gasoline
32.96
Other Petroleum Products
49.85
24.85
Coke
19.29
Coke Gas-Blast Furnace Gas
Electricity (Industry, Transport)
31.39
Electricity (Residential)
215.52
City Gas (Industry)
284.42
71.95
City Gas (Residential)
106.57
TABLE VI. Upper and Lower Bounds for Secondary Energy
(1010 kcal)
Secondary Energy
Energy
Lower Bound
Use
Industry
Electricity
Upper Bound
101,500
120,500
60,000
70,000
Transportation
4,000
5,500
Industry
2,400
Residential
9,500
3,500
13,500
Residential
City Gas
3
TABLE V. Upper and Lower Bounds for Petroleum and Coal Products
(10'1
Products
Energy
Crude Oil
Lower Bound
Use
Electricity
Stockpile
18,600
41,000
32,000
37,000
41,200
49,400
Residential
Transportation
Industry
0.0
0.0
0.0
12,600
Residential
Transportation
25,000
7,000
12,000
33,000
13,500
17,600
Industry
22,000
29,000
0.0
1,400
35,000
41,500
2,600
3,500
12,500
0.0
16,500
9,500
2,000
3,500
2,000
2,900
32,000
42,500
0.0
50
Stockpile
1,300
1,800
City Gas
2,000
2,900
4,200
10,300
5,600
14,500
Fuel Oil
Gas Oil
Naphtha
Gasoline
City Gas
Transportation
Electricity
Petroleum
Products
Industry
Residential
Transportation
City Gas
Industry
Coke
Coke Gas
Upper Bound
0.0
32,000
Electricity
Industry
Kerosine
kcal)
Residential
Electricity
Industry
4
TABLE VII. Upper and Lower Bounds
for Petroleum Products Yields
Lower Bounds
Petroleum Products
Upper Bounds
Fuel Oil
0.40
0.55
Kerosine-Gas Oil
Naphtha-Gasoline
0.10
0.30
0.20
0.30
Other Petroleum Products
0.0
0.15
TABLE VIII. Parameters for Beta Distribution
Parameters
Energy Resource
b
p
q
200,000
300,000
4.0
2.0
45,000
135,000
2.0
4.0
b m
Crude
Coal
Oil
5
co
II
CL
1-1
L)
-Z
cq
II
v
,,
Qa)
-
L
0a)C
oC
a)
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C
c=
Qa
C.
C/)
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Olo
L.
a)
c
Lz.3
w
L.
w
LL-
II
P-2
P0
Pl
P2
P3
d-3
Figure
2.
d-2
d-i
Do di
d2
Approximate demand function and supply functions
|START
I
Generate two sequences of beta random numbers
Bl(i),
B2(i),
i=l,...,N.
For each pair of Bl(i), B2(i),
i=l,...,N,
solve the MP/EE model and obtain an economic
equilibrium solution.
I
Obtain the probability of supply stability
and distributions of total energy system
cost, primary energy prices and their quantities.
I
END
Figure 3. Computational
procedure for the model analysis
I
START
|
I
I
II
--
Initial p,
pe=Q(q0). t=O.
q,
Solve the MP/EE model and supply submodel.
Optimal solution
t.
X+t, yt,
-
3L
I
-
t-f t E?
Yes
.1
"Converged".
Equilibrium
point obtained.
l-
i
--
No
t+l -
t.
Cs t=Cs+Acs t
Figure 4.
pt=(r't+pt-1)/2.
1
I
END
Algorithm
for solving
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