NOAK 1980

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should be helpful when selecting the appropriate solution procedure to a ..... programming to the optimization of water supply expansion. Their approach .... position will not result in a higher present value of cost, i.e.: (. ) ( )j ...... 100S42/ms2. 3. 68* ... 1 and 2) could be constructed for far less money and provide more capacity ...
Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

HEURISTIC TECHNIQUES IN PROJECT SEQUENCING by Egill B. Hreinsson The National Power Co. Reykjavik, Iceland

Abstract: One of the problems faced by planners when designed an expansion plan for water resources systems is selecting the sequence of projects that satisfies a given demand at minimum cost, some prescribed time into the future. This problem is indeed closely related to what is generally known as the capacity expansion problem although unique local conditions at each project site result in a somewhat different problem formulation. To ensure an optimal solution to the project sequencing problem the usual approach has been to use dynamic programming techniques, although it suffers from the commonly known limits on problem size due to the associated excessive computational requirements. The need of planners in search of a practical screening method useful at all stages in the planning process, involving possibly a large number of projects, has led to the introduction of several heuristic algorithms. While such algorithms do not guarantee an optimal solution, their usefulness is based on the assumption that the solution obtained is close enough to being optimal for some practical purposes. Little has, however, been written about their performance in general. This paper deals with performance evaluation of several such algorithms. The algorithms are tested by creating a number of sequencing problems which are in turn solved by each of them as well as by dynamic programming. By keeping track of each algorithms performance for varying assumptions such as the shape of the demand function and the cost-capacity characteristics for individual projects, a performance profile is established for each algorithm. Such performance measure should be helpful when selecting the appropriate solution procedure to a project-sequencing problem for instance in a practical setting.

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

TABLE OF CONTENTS Abstract: ..................................................................... 1 1. INTRODUCTION............................................................... 3 2. FORMULATION OF THE PROBLEM................................................. 3 3. PREVIOUS RESEARCH ON THE PROJECT SEQUENCING PROBLEM........................ 5 4. SOLUTION METHODOLOGY....................................................... 6 4.1 Dynamic Programming ................................................... 7 4.2 The Heuristic Sequencing Procedure .................................... 7 4.1.1 Ci/xi Sequencing ............................................... 8 4.1.2 TMR sequencing ................................................. 8 4.1.3 ki and Si sequencing........................................... 10 4.1.4 Applying the heuristics to a simple sequences problem ......... 12 4.4 Random Sequencing .................................................... 14 4.5 Max (R, K) and Max (R, S) sequencing ................................. 14 5. PRESENTATION AND EVALUATION OF DATA AND ASSUMPTIONS....................... 15 5.1 Construction cost and capacity data .................................. 15 5.1.1 Randomly selected data. ....................................... 15 5.1.2 Randomly generated data. ...................................... 15 5.2 Demand functions ..................................................... 15 5.3 Other assumptions .................................................... 16 6. DESCRIPTION OF THE COMPUTER RUNS.......................................... 16 7. ANALYSIS OF RESULTS....................................................... 17 8. CONCLUSIONS AND DISCUSSIONS............................................... 20 REFERENCES ................................................................... 21

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

1.

INTRODUCTION

When expanding water resources systems planners are often faced with the task of evaluating and selecting from a large number of feasible projects. The sequence in which these projects are constructed is often extremely critical with respect to the overall cost of the scheme. This problem of finding the optimal sequence, often called the project sequencing problem, is what this paper addresses. In short it involves the determination of the sequence of projects which will satisfy a projected increasing water related demand, over a specified planning horizon, at the minimum discounted cost. Many researchers have approached this problem in the past using primarily dynamic programming techniques (1), (6), (7), (12), (16), to obtain an absolute optimal schedule. However, as pointed out by Tsou et. al. (20) and Morin (13) the problem becomes increasingly difficult to solve when the number of feasible projects exceeds 10 unless specific simplifying conditions are present regarding f. ex. the shape of the demand function. Associated with the dynamic programming solution methods are thus the common problems of excessive storage requirements and computation time, closely related to what Bellman (4) referred to as the “curse of dimensionality”. In an effort to develop sequencing procedures that circumvents the often excessive computational needs of formal mathematical programming techniques several heuristic techniques have been proposed ((6), (7), (14), (20) and (24)) and tested on a limited number of sequencing problems. However, little has been written about these or other algorithms general performance. The objective of this paper is twofold. First, some new intuitively appealing heuristics are proposed to solve the project sequencing problem. These heuristics can serve as complementary to the best available heuristics (the TMR algorithm) and as such they improve its performance. Secondly, performance evaluation is carried out of these and other available algorithms against an optimal solution. While the TMR algorithm has been evaluated for a very limited number of sequencing problems no comprehensive evaluation of the hitherto proposed seems to be available.

2.

FORMULATION OF THE PROBLEM

We will here formulate the single dimensional project sequencing problem (fig. 1). We assume there is given a nondecreasing demand function D(t) of time, to reflect the projected demand for water. Assume that demand equals capacity at the beginning of the planning horizon so that without any loss in generality D(t) may be defined in such a way that D(0) = 0. Further assume that available is a set of N projects each with it’s associated capital cost

Ci , i = 1, 2, L , N and capacity (output)

xi , i = 1, 2, L , N The problem is to find the sequence of N projects

S N = (V(1) , V( 2 ) , V( N ) ) (out of N!

possible permutation sequences) such as to

N  min ∑ C[ i ] exp(− α t[i ] )  i =1  © 1980 Egill B. Hreinsson

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(2.1)

Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

while demand is being met at all times. Here α is the continuous discount rate and t[i ] is the time at which project no i in the sequence starts its operation. We have also followed Conway’s et. al. (5) notation that number j is positioned in the i’th place in the start a project as late as possible due to the of the demand requirement means that project constructed when accumulated capacity of all demand, i.e.

C j = C[i ] if project

sequence. Since it is optimal to discounting effect, the inclusion no i in the sequence should be preceding i − 1 projects equals

 i −1  t[i ] = D −1 ∑ x[ j ]   j =1  where

(2.2)

D −1 (.) is the inverse demand function. (See figure 1)

Thus the problem can be restated as follows: Find a sequence

S N to i −1    −1   min ∑ C[i ] exp − αD  ∑ x[ j ]    i =1  j =1   N

(2.3)

No consideration has to be given to the project benefits since they have been implicitly included by assuming a certain demand function independent of price. Hence the interpretation of the objective function is the total discounted cost of constructing the N projects (but not the net cost). No consideration is given to what happens in the post-planning period (when

 N  t > D −1  ∑ xi  . In this formulation this is unnecessary since investments then  i =1  will be independent of the selected sequence. We have assumed projects to have fixed

capacity

and

they

will

be

fully

utilized

at

a

time

 N  T = D −1  ∑ xi   i =1 

independent of the sequence. A number of simplifying assumptions has been made in the above formulation. 1.

2. 3. 4. 5. 6. 7. 8. 9.

Project costs Ci are assumed to be known and deterministic and are incurred as lump sums at the time of construction. Construction period is taken as being small. There exists a single deterministic performance measure on the size or output of a given project. This measure we denote as capacity, xi . Operating costs are either included in the construction cost or proportional to capacity and, therefore, need not be considered. The discount rate is constant with respect to both time and magnitude of investment. Secondary energy (or water) has negligible value. The full capacity xi is available for use instantaneously upon project completion. Projects, once constructed, have infinite lifetime. Projects are independent, that is, capacity or cost of a given project does not depend on other projects, constructed or not constructed. There is an infinite penalty cost incurred when demand is not satisfied.

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

10. 11. 12.

3.

Demand equals capacity at the start of the planning horizon. A project, once constructed, cannot be economically expanded. Inflation is assumed to have a balancing effect on both cost and revenue and hence can be excluded from further consideration.

PREVIOUS RESEARCH ON THE PROJECT SEQUENCING PROBLEM

The problem of optimally sequencing and sizing water resources projects to meet a projected demand is indeed closely related to what is generally termed the capacity expansion problem. However, the distinctive feature associated with expansion of water resources as opposed to e.g. expansion of manufacturing facilities is the lack of possibilities for replication. While factories and machines can be replicated at will to form a series of identical expansion steps, this is generally not possible when expanding water resources. Nature’s diversity ensures that unique geographical conditions prevail at each project site to determine the project's cost and capacity while capacity expansion models often assume a series of identical projects. (10) This is strongly reflected in the research effort to date in this area. Manne (10) presented a simple model where a sequence of identical facilities was selected from a continuum of sizes to minimize the discounted cost of development. The works of Manne (10) have been widely used in water resources planning. Lauria et. al. (9) introduced an extension to Manne’s model involving f. ex. initial deficits. Scarato (17) applied Manne’s model to sizing and timing of pipelines and water treatment. Sengupta and Fox (19) present a general review of capacity expansion models and general applications f. ex. in industry. If due recognition of the unique characteristics of each project is to be included, the problem formulation and methodology takes a different form. A review of some of the research follows although it is far from complete. Although several researches used mathematical programming techniques in the early sixties in the design and operation of specific aspects of water resources system. Butcher et. al. (1) seem to be one of the first to apply dynamic programming to the optimization of water supply expansion. Their approach was based on a n-stage recursive optimization process where n represents the number of independent project with given cost and output (capacity). As a state variable they used a discretation on the cumulative capacity of all projects and their objective was to minimize discounted cost to meet a deterministic demand function some prescribed time into the future. Morin and Esogbue (16) introduced a more efficient dynamic programming algorithm to solve the same problem. Their distinctive feature as a different representation of the state variables based on their definition of the imbedded state space. They recognized that, rather than calculate the minimum cost function for a fixed increment on the projects’ cumulative capacity, accuracy is increased and savings can be made in computation if definition of allowable states is limited to the possible cumulative capacities of the projects corresponding to each stage. Later Morin (15) recognized that the original DP formulation by Butcher et. al. (1) may lead to nonoptimal solutions in certain cases. Several extensions to the basic problem of sequencing fixed projects to meet a deterministic demand, have been introduced. Becker and Yeh (3) introduced reservoir sizing into their problem formulation recognizing at the same time that reservoir storage size is not the critical quantity but rather the firm water output (firm water being defined as the maximum quantity that can be guaranteed to be delivered each year, 100% of the time according to some prescribed monthly or biweekly distribution). In addition project interdependencies were implicitly introduced in their formulation as reservoir operation, by means of heuristic operating rules, was explicitly included in their model. Erlenkotter (8) pointed out that nonoptimal solution could be reached by simultaneously considering these 2 extensions in an imbedded state space forward dynamic programming formulation. Earlier Erlenkotter (6) introduced project interdependencies into dynamic programming formulation (of the imbedded state space type) and noted that "interdependencies in the form of

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

technologically external economies add little to the difficulty of finding an optimal sequence” (6) p. 24)). Becker and Yeh (2) used iterated linear programming to optimize project sizing and system operation while DP formulation was employed to optimize sequencing and timing of 6 projects. Moore (25) developed an extension to Becker and Yeh’s earlier approach (3) by considering demand price elasticity and formulated a forward dynamic programming procedure to maximize net economic benefits. Knudsen and Rosbjerg (31) presented a model based on dynamic programming recognizing the uncertainties in future demand, that is, in their formulation demand was assumed probabilistic rather than deterministic. They included operations and maintenance cost and discretation of time was assumed in their problem formulation which used chance constraints or penalty functions depending on preferences on shortage. Finally, Bogle and O’Sullivan (32) developed an approach where they explicitly incorporated the stochastic nature of streamflow into the system and storage carryover between periods (and years) in a stochastic programming formulation, while in most of the previously mentioned studies these effects were only implicitly included through the determination of project capacity. Whereas these approaches were in whole or in part based on dynamic programming and hence plagued by the curse of dimensionality, several researchers have proposed heuristic algorithms to produce a near-optimal solution to the projectsequencing problem. Erlenkotter (7) proposed a heuristic algorithm base on a simple ranking approach where an index is calculated for each project and the project with the lowest index selected as the next project in the sequence. Tsou Mitten and Russel (20) presented a search procedure almost identical to that of Erlenkotter and tested it on 24 sequencing problems with, they claimed, optimality achieved in all cases. Morin (13) presented a counter-example showing that Tsou et. al. procedure (hereinafter called the TMR algorithm) could produce non-optimal solutions. He, however, developed sufficient conditions for its optimality. Morin developed a similar heuristics designed to sequence projects with multi-dimensional definition of project capacity (33), (34) and presented a branch and bound algorithm that, however, proved inferior to the imbedded state space approach in terms of computational time (34) when applied to solve the multi-dimensional sequencing problem. Although the TMR algorithm is a little more sophisticated, several researches have proposed a heuristic based on a ranking approach with a ratio of cost to capacity as an index (6), (7), (19). However, no evaluation seems to be available on its optimality performance.

4.

SOLUTION METHODOLOGY

A computer program was developed to evaluate the performance of the individual heuristics against a Dynamic Programming solution guaranteeing an optimum. In the program a predetermined number of sequencing problems are created, by specifying both costs and capacities for the projects as well as demand function. Each problem is first solved by DP and then subsequently by each of the heuristics. A comparison among heuristics is made regarding the deviations from a rigorous optimum and for example the average error accumulated for all the sequencing problems. This procedure is repeated for various problem sizes and types of demand functions. We will first discuss the Dynamic Programming algorithm and then each of the heuristics.

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

4.1 Dynamic Programming A dynamic programming algorithm based on the imbedded state space (Morin (16)) along similar lines as presented by Erlenkotter (8) will be formulated. Prior to formulating the problem in this manner, let us introduce the following notation:

(i = 1, 2, L , N )

i

project index

Ci

investment cost (construction cost)

xi

capacity of project no i

α

continuous time, discount rate

X

specified subset of project indices corresponding to a specific partial sequence

ZX

accumulated capacity of projects belonging to the set X i.e.

ZX =

∑x i∈X

i

X −i

set X with index i deleted

D(t )

a nondecreasing demand function of time

D −1 ( Z )

time at which demand exhausts total capacity Z

P( X )

total discounted cost of projects in the set X

Then we can formulate a forward dynamic programming recursive relationship:

P( X ) = min {Ci exp(−αD −1 (Z X −i )) + P( X − i )} i∈ X

(4.1)

In visualizing the function of this DP algorithm, figure 2, may be helpful and some comments regarding that figure seem appropriate. The figure illustrates the tree that the DP algorithm has to search through when seeking the optimal solution in the simple case of 3 or 4 projects. Referring for instance to figure 2B, which shows a search tree for sequencing four projects, (N = 4), this tree reflects a DP formulation with 5 stages and in each stage the no of states is

N   where n is the stage no. Each node in the tree corresponds to a set X, as n 

previously defined with index no for the projects belonging to X shown adjacent to the node. The number of branches emanating from a node in stage n is equal to N − n while the number of branches merging into node no n is equal to n. Since the stage number is equal to the number of projects already selected to form a partial sequence (X) the emanating branches correspond to the choices left (N − n ) when n projects have been selected. (See figure 2)

4.2 The Heuristic Sequencing Procedure Common to all the heuristic algorithms employed in this study is the procedure below where the difference between individual algorithms appears in step 2. The procedure is based on a ranking approach where projects are ranked according to some index that should reflect the desirability or economic attractiveness of a project (pseudo-utility). The projects are sequenced according to ascending order of magnitude of the index and it is the definition of the index that distinguishes the heuristics from each other. The procedure is as follows: Step 1. Set current accumulated capacity

© 1980 Egill B. Hreinsson

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Z = 0.

Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Step 2. For each of the remaining projects to be selected into a nearoptimal sequence calculate a specific index. The index is some measure of the attractiveness or feasibility with the project having the highest attractiveness assigned the lowest index. Step 3. Identify the project with the lowest index and select that project as the next (first) project in sequence. Ties may be resolved arbitrarily. Step 4. Increase accumulated capacity by capacity of the project selected in step 3 and go to step 2 unless all projects have been selected. Then stop. Each of the different indices will now be described and the intuitive reasons for their introduction and their economic interpretation be explained.

4.1.1 Ci/xi Sequencing The simplest approach is to use the ratio of construction cost to capacity (or unit cost) for each project as an index, i.e.:

k fi =

Ci xi

(4.2)

This index has been suggested by several researchers (6), (7), (19) and it implies that it is generally economical to start by constructing projects with the lowest unit cost of energy, where unit cost is here defined according to eq. (4.2).

4.1.2 TMR sequencing The index used in the TMR algorithm has the following form suggested by Tsou, Mitten and Russel (20) and Erlenkotter (7).

Ri =

where

∆ ti

as

Ci 1 − exp(− α ∆ ti )

originally

(4.3)

is the time period that elapses from the construction date of project

no i until it is fully utilized (the period of excess capacity) i.e.:

∆ ti = ti +1 − ti = D −1 (xi + Z ) − D −1 (Z ) where

(4.4)

ti = D −1 (Z ) is the time at which project no i is started up and Z is

accumulated capacity of all projects proceeding project no i:

Z =

i −1

∑x j =1

j

The choice of this index is based on considerations regarding necessary conditions for sequencing 2 adjacent projects as follows. In order that a project (no i) should precede another project (no j) given that the two projects are adjacent in a sequence of N projects, it is necessary that an exchange of position will not result in a higher present value of cost, i.e.:

Ci + C j exp(− α∆ ti ) ≤ C j + Ci exp(− α ∆ j ) Rearranging we get:

Ci (1 − exp(− α∆ t j )) ≤ C j (1 − exp(− α∆ ti ))

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Cj Ci ≤ 1 − exp(− α∆ ti ) 1 − exp(− α∆ t j ) or by referring to equation (4.3)

Ri ≤ R j The sequence produced by the TMR algorithm has the property that it is at least locally optimal, i.e. an exchange of any 2 adjacent projects in the sequence will result in a higher (or equal) present value of cost. It may not, however, be globally optimal unless certain conditions are present. These sufficient optimality conditions were pointed out by Morin (13) and are loosely stated in the following: If the sequence produced by the TMR algorithm is invariant for all possible accumulated capacities of 1 and N projects, then the TMR procedure will lead to an optimal sequence. However, note that these conditions are only sufficient conditions; even though they would be violated the TMR-sequence could still be optimal. One of the results obtained by Erlenkotter (7) and Morin (13) is that given a linear increase in demand, (that is D (t ) = q ⋅ t ) the TMR algorithm guarantees an optimal

solution.

The

Ri

indices

will

then

be

independent

of

accumulated

capacity, Z and an optimal sequence is readily found by ordering the projects in a sequence with increasing (or nondecreasing) value of Ri . As pointed out interpretation:

by

Morin

Ri =

(13),

the

Ci = 1 − exp(− α∆ ti )

Ri

index

has

an

interesting

economic



∑ C exp(− jα ∆ t ) j =0

i

i

To quote Morin (13) (with a change in notation) " Ri can thus be interpreted as the discounted cost of an infinite series of capital investments of $ Ci each at times

0, ∆ ti , 2∆ ti , L "

If the demand was such that the time interval D (xi + Z ) − D ( Z ) = ∆ ti remained constant for all values of Z (as in the case for linear demand projections), then Ri could be interpreted as the (minimum) discounted cost of satisfying all future demands with an infinite number of projects, each of which is identical to project i“. However, the Ri index has another interesting economic −1

−1

D −1 ( Z ) the index Ri is proportional to the cost to benefit ratio k i of

interpretation. For a demand function that increases linearly between

and D (Z + xi ) the project, the benefit being the present value of the firm energy or water delivered by the project during its (infinite) lifetime. Alternatively the Ri −1

index is proportional to the unit price

k i of energy or water that the user has

to pay in order to ensure a rate of return α on the project.

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Let

∆ ti = D −1 (Z + xi )− D −1 (Z ) = ti +1 − ti , which is the time interval during which

project i becomes fully utilized. (See fig. 3). Then in the interval

D(t ) = Z −

xi −1 x D (Z ) + i t ∆ ti ∆ ti

∆ ti i.e. D(t ) increases linearly for ti < t < ti + 1 .

The discounted value of the firm energy from the project as of the project completion time will then be:

PEi =

D −1 (Z + xi )



D −1 ( Z )

 x x  − D −1 ( Z ) i + i ∆ ti ∆ ti 

 t  exp(− α (t − D −1 ( Z ) 

))dt +



∫ x exp(− α (t − D (Z )))dt −1

i

D −1 ( Z + xi )

(4.6) which, after evaluation of the integrals, becomes:

PEi =

xi (1 − exp(− α∆ ti )) ∆ tiα 2

(4.7)

The ratio Ci to PEi is the cost benefit ratio or alternatively the unit cost from project no. i.

ki = Ci PEi

ki

(4.8)

By substituting equation (4.7) into equation (4.8):

ki =

∆ t iα 2 C i xi (1 − exp(− α ∆ ti )

(4.9)

From equations (4.9) and (4.3) we conclude:

ki =

∆ ti Ri ⋅ α 2 xi

(4.10)

k i is the same quantity as Ri except for the factor α 2 ∆ ti xi which, however, is constant for all projects when we have linearly increasing demand. An intuitively appealing interpretation of the TMR algorithm is now obtained as follows (assuming a linear increase in demand): The project with the lowest unit cost of energy k i should be constructed first, continuing by constructing projects with increasing energy cost moving from relatively favorable to more economically unfavorable project sites. The above procedure is illustrated in figure 3 and it suggests the definition of two new indices.

4.1.3 ki and Si sequencing As outlined above, it would seem an intuitively appealing approach to sequence projects according to a rising unit cost of the output (energy). This has in fact been suggested in the definition of the index Ci xi as described above (in section 4.2.1) However, no consideration has in that definition been given to the fact that during the period of excess capacity, ∆ ti , the project is not fully utilized. The projects actual energy (or water) production is in this

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

period below capacity and this fact should somehow be reflected in the actual unit cost of energy from the project. In order to incorporate this limited utilization let us, instead of defining unit cost as Ci xi , define a project’s unit cost as follows:

Pc discounted value of cost (4.11) = PE discounted value of output In the above definition the discounted value of cost will be the construction cost, and by using the construction (start up) date as a reference for discounting, the nominator (Pc ) in equation (4.11) will simply be Ci . The Unit cos t =

denominator, PE , will however not be quite so simple and will correspond to the “discounted hatched areas” in figure 3. That is:

PE =

∆ ti

∫ (D(t + t )− Z ) e i

−α t

dt +

Si =

∫xe i

−α t

dt

∆ ti

0

We can now define a new index



Si : Ci

∆ ti

∫ (D(t + t ) − Z )e

−α t

i

dt +

0

(4.12)



∫x

i

e

−α t

dt

∆ ti

This expression representing the actual unit cost can be simplified if D ( t ) is assumed to increase linearly between

ti and ti +1 as previously outlined. This is

a plausible assumption if the projects are large relative to the irregularities of the demand function. (See figure 3) Define an approximating demand function

De (t )

t i xi x + i t ti < t < ti + 1 (4.13) ∆ ti ∆ ti which increases linearly between ti and ti + 1 . By substituting De (t ) for D ( t ) in De (t ) = Z −

equation (4.12) we get (in similar manner as equations 4.6, 4.7 and 4.8) the index k i .

ki =

∆ t iα 2 C i xi (1 − exp( − α∆ ti ))

(4.14)

also reflecting the unit cost of energy, if the demand function is reasonably smooth but now taking into account the limited utilization of the project’s capacity during the period ∆ ti . The definitions of unit cost k fi (eq. (4.2),

k i (eq. (4.14) and Si (eq. 4.12) will

be equivalent if the period of excess capacity ∆ ti → 0 , i.e. full utilization of the project of excess capacity is assumed instantaneously upon project completion. Consider the limit

∆ tiα 2 Ci ∆ t → 0 x (1 − exp( − α∆ t )) i i lim

© 1980 Egill B. Hreinsson

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(4.15)

Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Both the nominator and the denominator applying L’Hopitals rule we get:

will

be

0

as

∆t → 0.

However,

α 2 Ci α Ci lim = ∆ t → 0 x α exp( − α∆ t ) xi i

by

(4.16)

This is the unit cost (of energy) assuming instantaneous full utilization and is the same expression as equation (4.2) except for the factor α which, since it is constant, can be omitted from the definition of the index k fi = Ci xi . Thus the

k i and Si are both defined for the case, ∆ ti = 0 where they both become identical to Ci xi , as opposed to the index Ri which is not defined when ∆ ti = 0 . This is clearly advantage of the k i and Si algorithms. indices

Both indices can easily be incorporated into a heuristic algorithm. Index

Si

(eq. (4.12) can be calculated by numerically integrating the function D ( t ) from

ti to ti +1 . The simplifying assumption is made that demand can be represented as a piecewise linear function. This will simplify considerably the numerical integration since exact representation of integrating each linear segment is incorporated, this assumption should not either limit in any way the possible shapes of demand functions to be represented. Index

k i is as previously pointed out a simple extension of index Ri as seen

from eq. (4.10). However, since the factor α in eq. (4.10) is a constant it has been omitted from the index, which then will be defined as follows: 2

ki =

∆ t i Ci xi (1 − exp(− α∆ ti ))

with the only difference between factor

∆ t i xi

(4.17)

k i (eq. (4.17) and Ri (eq. (4.3) being the

generally varying from project to project.

For a linear demand function D ( t ) = q ⋅ t (q being constant) both indices

k i and Ri

Si now being identical to k i ) will, however, be equivalent since the above factor ∆ ti xi will be constant ( = 1 / q ) . Therefore, these indices k i , Ri and Si will in such a case all guarantee an optimal solution.

(as well as

4.1.4 Applying the heuristics to a simple sequences problem

To illustrate the above heuristic algorithms (Ci xi , TMR , k i and S i ) they will be applied to a simple sequencing problems involving only 3 projects. The problem is identical to a counterexample presented by Morin (13) where he demonstrated the nonoptimality of the TMR algorithm. We have a nonlinear demand function as shown in figure 4. The planning horizon is 25 years and during that period, the demand rises to 45 units. Furthermore, to simplify the calculations we assume that the demand function is piecewise linear between the following points:

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

TABLE 1 − DEMAND FUNCTION IN MORIN’S COUNTEREXAMPLE Point no. 1 2 3 4 5

Time (units) 0 2 5 20 25

Demand (units) 0 10 15 20 45

There are three projects to be sequenced to satisfy this demand, their costs and capacities being shown below. TABLE 2 − COSTS AND CAPACITIES IN MORIN’S COUNTEREXAMPLE

Project Project Project (units) Project (units)

i

number name cost

Ci xi

capacity

1 A 10

2 B 40

3 C 15

10

20

15

Applying the heuristics to this problem results summarized in table 3. The heuristics applied are:

in

calculations

which

k i algorithm B. Ci xi algorithm A.

C. TMR algorithm D. S i algorithm TABLE 3 − SUMMARY OF CALCULATIONS REQUIRED TO APPLY THE HEURISTICS TO A SIMPLE SEQUENCING PROBLEM Project name (and number) Cost

Ci

Capacity

(x i )

A. k i seq. k i (eq. (4.17) ) Select project A and calculate

A(1) 10

B(2) 40

C(3) 15

10

20

15

21.5

64.2

23.1

64.2

24.8

1

2

1

107.5

64.2

69.3

ki

when demand is = 10

ki k i sequence is ACB B. Ci xi seq. Ci xi

Select project C i.e.

Ci xi seq. Ci xi

C. TMR seq. Ri

(eq. (4.3))

Select project B and calculate

Ri

when

demand is 20 Select project A, i.e. TMR sequence is BAC

D. S i seq. S i

21.5

Select project A and calculate

Si

when demand is 2 Select project C i.e. © 1980 Egill B. Hreinsson

107.5

Si

sequence is ACB - 13-

110.1 49.6

22.4

106.3

41.7

are

Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Table 4 summarizes the results of an exhaustive enumeration of the simple sequencing problem, with results from table 3. Since the problem involves 3 projects there are 3! = 6 possible sequences. Table 4 − RESULTS FROM AN EXHAUSTIVE ENUMERATION OF THE SIMPLE SEQUENCING Sequence

Discounted cost

ABC ACB

51.41 37.96

BAC BCA CAB

48.90 48.91 37.19

CBA

49.60

Comment

„worst“ solution k i , S i and Ci xi algorithm solution TMR solution Optimal and

In table 4 it is readily observed that the

Ci xi solution

k i -algorithm, the Si algorithm and

the Ci xi algorithm all give a solution fairly close to the optimal one, while the TMR algorithm, however, gives a solution much closer to the worst than to the best. Further, if the tie in applying the Ci xi -algorithm is resolved in such a way to select the sequence BAC, the optimal solution is obtained using the Ci xi algorithm. This is, as we shall see in chapter 7, (Analysis of results) fairly typical for problems with an ill-behaved demand function (i.e. an oddly shaped nonlinear demand function) with cost-capacity data that cause the TMR-algorithm to provide a non-optimal solution. (See figure 4)

4.4 Random Sequencing To provide a reference point against which a failure of some of the heuristics to provide an optimal solution, can be measured, a procedure involving random sequencing of projects was included. The procedure utilizes a pseudo-random number generator to randomly select each project from the set of available projects. The evaluation of a heuristics performance inevitably involves calculating the discounted cost of a sequence and possibly the associated error, in the case of non-optimality of the heuristics. It is useful in such a case to compare the error to the expected error when projects are selected randomly since the magnitude of the error in such a case will in general be problem-specific. This is the main reason for including the random sequencing procedure.

4.5 Max (R, K) and Max (R, S) sequencing The function of the two algorithms max (R,K) and max (R,S) is to select the better of two heuristics. The max (R,K) utilizes the results of both TMR sequencing and k i sequencing and selects for each sequencing problem the sequence (with the associated discounted cost and possible sequencing error) which gives results closer to being optimal. The function of the max (R,S) algorithm is analogous except it selects the better of the TMR and S i algorithm. The objective of defining and testing these algorithms is as follows. While the TMR-algorithm shows the best overall performance of the heuristic algorithms defined in the previous sections, it fails for certain sequencing problems. An example of such a problem was shown in section 4.2.4 above. In these cases the k i and Si algorithms often give considerably better results. Thus we have a procedure involving sequencing both according to the © 1980 Egill B. Hreinsson

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Ri index and either of the

Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

two indices k i and S i , and then choosing the better. Although such a procedure increases the computational effort needed to apply the heuristics it is a definite improvement over the TMR-algorithm.

5. PRESENTATION AND EVALUATION OF DATA AND ASSUMPTIONS 5.1 Construction cost and capacity data The cost and capacity data used in the study were of two types. A. Randomly selected data B. Randomly generated data

5.1.1 Randomly selected data. This approach is based on the utilization of cost and capacity data from actual hydroelectric project plans in Iceland. Studies and cost estimates have been carried out to a differential level of detail on Iceland’s hydroelectric power potential which so far has only been harnessed to a limited extent or about 10% (29). With the idea that these estimates are to some degree representative of cost and capacities in other areas of the world it was decided to use these data in the study. The reason for using them was also lack of any comprehensive data from elsewhere. The data can be summarized in a diagram such as figure 5. They consist of costs and capacities of 44 projects and each project can be represented as a dot in the diagram with it’s associated cost and capacity. The data are based on various studies performed in recent years. Some of the studies have not been published while others have, but a comprehensive survey can be found in (21). The plans and cost estimates are continuously being reevaluated and some of the fugues of the 44 projects have been adjusted accordingly. But in the light of the present application of the data in this study, as representing the overall cost capacity relationship for hydroelectric projects, they should be quite satisfactory. (See figure 5) In order to create a sequencing problem with a practical number of projects (< 15) solvable by dynamic programming and the heuristics, projects were randomly selected from the above list of 44 projects. Thus it was possible to create many different sequencing problems each with f. ex. 10 projects by randomly selecting for each problem, 10 projects out of the list.

5.1.2 Randomly generated data. In order to increase the flexibility of the study, the cost capacity data were not restricted to the 44 projects in Iceland as previously described. Cost and capacity data were also randomly generated from a probability distribution. Figure 5 shows the zone in the cost-capacity diagram within which random generation cost capacity pairs was allowed. More specifically the cost capacity pairs for each project were generated from a linear distribution with respect to both Ci and xi limited as shown in figure 5. This zone was selected more or less arbitrarily but it’s nevertheless somewhat representative of the distribution of the 44 projects according to the figure. In addition to the random selection and random generation of data as outlined above it was possible, with the current design of the program, to include a prespecified number of fixed sets of projects. Thus it was possible to define completely a given project sequencing problem, without any random procedures, and solve it with all sequencing procedure.

5.2 Demand functions The criterion, when selecting demand functions, was to include as many different shapes of demand functions as possible both to analyze the influence of different shapes on the performance of the heuristics as well as to evaluate the © 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

overall performance of the heuristics. Figure 6 illustrates the 10 different demand functions used in the study. Many of them are quite oddly shaped (such as no. 1, 3, 4 and 10) while others closely resemble the “normal exponential increase” one could expect to represent actual growth of demand. In fact, demand function no. 2 represents an electrical energy forecast for Iceland during the next 25 years as presented in (22). (See figure 6) All of the demand functions have been represented as consisting of a number of linear segments. While the figure only shows the functions up to the point marked P, they have been assumed to continue beyond that point in a linear fashion, as a continuation of the last linear segment. The point P was selected having a demand of 3834 units at t = 25 due to characteristics of demand function number 2, since the energy forecast for Iceland assumes an increase of 3834 GWh during the next 25 years. However, the demand functions can be scaled arbitrarily and the length of the planning horizon is not necessarily restricted to 25 year as will be explained in chapter 6.

5.3 Other assumptions In addition to specifying cost-capacity data and demand functions it is necessary to define some additional assumptions to specify a project sequencing problem (or a class of such problems). The discount rate ( α ) f. ex. is an important variable and greatly affects the selection of the optimal sequence in a given project sequencing problem. However, it was decided to use the single discount rate of 5% per year throughout this study. The reasons for this are twofold. First a great controversy remains among economists, planners, engineers and even the public policy-makers what is the appropriate discount rate in such studies, even when a given problem is to be analyzed such as a specific project sequencing problem. Secondly since the variations of discount rate is equivalent to changing the scale factor on the time variable (the discount rate always appears as a product, α t or α ∆ t , in the heuristics and dynamic programming formulation) it should not basically alter the performance of the heuristics. Other assumptions will be defined in chapter 6 as individual computer runs are outlined.

6.

DESCRIPTION OF THE COMPUTER RUNS

Computer runs were performed as described in the following table (table 5). Table 5 − CHARACTERISTICS OF THE COMPUTER RUNS Run

Problem size* (N)

No of demand functions

No of randomly selected data sets

No of randomly generated data sets

Total no of data sets

Total no of sequencing problems

Scale factor for project capacity

3 5 8 10

10 10 10 10

500 250 30 5

500 250 30 5

1000 500 60 10

10000 5000 600 100

1 0.32 0.2 0.16

1 2 3 4

*Problem size indicates the number of projects in each sequencing problem. Basically the work was performed in 4 runs where in each run a specific number of sequencing problems was solved, each with the same number of projects. In each run the same data sets (where a data set is defined as the M pairs of number: Ci , xi , i = 1, 2, L , M was applied to each of the 10 different demand functions according to figure 6. The data sets were both randomly selected as well as randomly generated with half of the data sets in each category. The scale factors indicated were selected to fit the data sets to the demand functions. © 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Since the average capacity of the 44 projects is 1078 GWh/year and the expected capacity of the randomly generated data is 1950 GWh/year the expected capacity of a project with 50% randomly selected data and 50% randomly generated data will be 1514 GWh. This directed the choice of the scale factors. As previously outlined, it is not necessary to define any specific planning horizon but rather let the planning horizon be controlled by the total capacity in each data set. The time when that demand equals capacity will define the corresponding planning horizon, even though the demand at this time is into the last linear segment of the demand function. The program was implemented on a Digital Equipment Co. PSP11/34 Minicomputer owned and operated by the National Power Co., Reykjavik. It was written in a FORTRAN IV language run under the RSX-11/M multi-user operating system.

7.

ANALYSIS OF RESULTS

In order to evaluate the algorithms' performance the following statistics were calculated for each algorithm (j) and each demand function. 1. No of sequencing failures: m sj

msj =

m

∑x k =1

kj

where and

where

Pkj

is

the

xkj = 1 if Pkj > Pk*

xkj = 0 if Pkj = Pk*

total

discounted

cost

P

* k

sequence produced by algorithm j and

in

sequencing

problem

no

k

of

the

is discounted cost of the optimal

sequence. m is the number of sequencing problems1 created with the same demand function ( = number of data sets, as previously defined). 2. Failure rate r j = m sj

m

3. Average relative failure (%) erj

erj =

100 msj

m

∑d k =1

kj

where

(

d kj = Pkj − Pk*

)P

* k

4. Average absolute failure (%), eaj :

eaj =

100 m

m

∑d k =1

kj

5. Maximum failure (%), emj :

emj = 100{max k =1, 2,L, m (d kj )}

6. No of sequencing problems when algorithm is best, mbj

mbj = where

1

m

∑y k =1

kj

(j goes for all algorithms except DP)

 1 if Pkj ≤ min j {Pkj } y kj =  0 otherwise

m is equal to the number of data sets as given in table 5.

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

7. Superiority matrix,

sij

indicating for how many sequencing problems

algorithm no i is superior to algorithm no j, i.e.:

sij =

m

∑Z k =1

where

ijk

 1 if d ki ≤ d kj Z ijk =  0 otherwise

The following algorithm indices were used: 1. Ci xi − algorithm 2. TMR − algorithm 3. ki − algorithm 4. S i − algorithm 5. Random sequencing 6. Max (K,R) − algorithm 7. Max (S,R) − algorithm Table 6 − SUMMARY OF RESULTS Average absolute failure in discounted cost

Heuristics Max(R,S)

Max(R,K)

TMR

Si

ki

Ci/xi

Rand.seq.

Proble size 3 5 8 10 3 5 8 10 3 5 8 10 3 5 8 10 3 5 8 10 3 5 8 10 3 5 8 10

1*

2

3

4*

5

0,110 0,316 0,277 0,282 0,299 0,409 0,312 0,282 0,835 0,639 0,623 0,288 2,720 2,80 3,00 3,38 4,430 3,96 4,50 4,23 4,22 2,00 1,59 1,70 10,3 8,02 9,08 8,26

0,002 0,003 0,004 0,005 0, ,0003 0,005 0,005 0,001 0,001 0,008 0,075 0,060 0,060 0,085 0,043 0,119 0,072 0,095 0,043 2,29 0,38 0,14 0,203 7,70 4,78 5,32 5,25

0,048 0,041 0,059 0,029 0,076 0,045 0,059 0,029 0,114 0,057 0,066 0,030 1,83 0,961 1,13 0,671 2,73 1,24 1,18 0,684 6,00 3,22 2,46 2,03 10,4 5,90 5,95 6,00

0,072 0,144 0,072 0,107 0,116 0,162 0,072 0,107 0,197 0,255 0,109 0,131 1,36 1,46 1,60 2,05 2,00 1,85 1,87 2,25 3,08 1,03 0,74 0,849 8,93 6,49 7,31 6,81

0,003 0,003 0,006 0,011 0,003 0,003 0,037 0,012 0,010 0,012 0,027 0,024 0,390 0,321 0,222 0,073 0,568 0,361 0,398 0,400 3,60 1,47 1,06 1,07 8,60 5,13 5,43 5,64

Demand function number 6 7* 0,001 0,001 0,002 0,010 0,001 0,001 0,002 0,010 0,003 0,001 0,002 0,010 0,725 0,251 0,272 0,148 1,67 0,450 0,280 0,148 5,72 1,79 1,04 1,07 10,1 6,41 6,78 7,13

0,009 0,007 0,007 0,010 0,011 0,007 0,007 0,010 0,098 0,007 0,017 0,010 1,57 0,266 0,063 0,115 2,45 0,410 0,067 0,170 0,882 0,057 0,036 0,064 7,26 2,89 2,97 2,73

8

9

10*

,0002 0,005 0,002 0,018 ,0002 0,005 0,002 0,018 ,0003 0,005 0,002 0,018 0,355 0,124 0,026 0,040 0,531 0,167 0,028 0,066 0,779 0,046 0,048 0,058 6,75 3,64 4,03 3,81

,0005 0,001 0,004 0,014 ,0005 0,001 0,004 0,014 ,0001 0,001 0,004 0,028 1,52 0,643 0,833 0,357 3,53 1,14 1,18 0,897 8,72 3,79 2,84 2,40 12,3 7,74 7,98 8,33

0,189 0,465 0,466 0,344 0,224 0,628 0,601 0,655 0,972 0,906 0,983 0,655 4,26 3,63 3,45 1,21 5,55 4,54 5,97 4,58 3,68 2,98 2,44 2,24 11,1 8,26 9,37 8,32

Notes: * The performance of the algorithms is the same as the order in the table except for demand functions marked (*) where the Ci/xi algorithm is superior to Si and ki algorithms.

The performance measure by which the individual algorithms were compared is the average absolute failure as defined above and it is illustrated in table 6. Using this performance measure, the results can in simple terms be stated that the algorithms can be ranked generally in the same order as listed in table 6. This is generally not depending on the demand function except for the case of the Ci xi algorithm. For demand functions that behave fairly normally (no. 2, 3, © 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

5, 6, 9) the ranking is as indicated in the table. However, for ill-behaved demand functions (no. 1, 4, 10) and demand functions that rise steeply at the beginning (no. 7, 8) Ci xi sequencing is to be preferred to k i and S i sequencing. While the TMR-algorithm is found on the average to be superior to the k i and S i algorithms it may fail badly. An example of such a sequencing problem was given above in section 4.2.4. In such cases k i and especially the S i −algorithms often give better results, i.e. they can act as complementary to the TMR-algorithm. Even for ill-behaved demand functions (no. 1, 4, 7, 8, 10) this does not seem to apply to the Ci xi algorithm. Results for all the statistics as previously described are given in table 7 for problem size (N) of 3 and demand function no. 1. Of the 1000 sequencing problems generated the TMR algorithms failed in 63 cases, out of which the S i algorithm performed better in 43 cases the k i algorithm performed better in 26 cases while the Ci xi algorithm performed better in only 15 cases. Even though the results have not been enumerated here, they are fairly consistent for other problem sizes and ill-behaved demand functions. Table 7 − RESULTS FOR DEMAND FUNCTION NO. 1, PROBLEM SIZE 3 Number of sequencing problems: 1000 Algorithm Algorithm number No of seq fail. Failure rate Average relative failure (%) Average absol. failure (%) Maximum failure (%) Number of sequencing problems when best (except DP) Ci/xi Better (1) TMR Better (2) ki Better (3) Si Better (4) Random seq. Better (5) Max(K,R) Better (6) Max(S,R) Better (7)

Ci xi

TMR

1

2

407 0.407

ki

Si

RAND SEQ

MAX(K,R)

MAX(S,R)

3

4

5

6

7

63 0.063

531 0.531

441 0.441

802 0.802

49 0.049

31 0.031

10.4

13.3

8.33

6.16

12.8

6.11

3.55

4.22

0.835

4.43

2.72

10.3

0.299

0.110

73.3

51.0

60.5

32.5

71.3

34.7

12.7

605

949

480

570

204

972

990

653 784 586 630 0

2 0 0 18 13

1 0 0 0 9

793 795

0 18

0 0

0 372 109 173 158

15 0 26 43 20

377 386

26 43

Superiority matrix: 267 212 515 426 0 27 141 0 227 185 515 516

426 426

In the following table (table 8) we compare the two algorithms Max (S,R) and TMR. The table first indicates the estimated reduction in average absolute error for problem size N = 3 and N = 5. The factor (100 ⋅ ea 7 ea 2 ) thus shows what the Max(S,R) error

(ea 7 )

is compared to the TMR-error

(ea 2 ).

indicates the fraction of the cases where TMR failed

The factor

(m2 )

but

(100 S 42

ms 2 )

Si gave better

results (S 42 ) . Thus for the 1000 sequencing problems with 3 projects and demand function no. 1, TMR failed to find an optimal solution in 63 cases out of which Si performed better in 43 cases (68%). The average absolute error was reduced from 0,835% to 0,11% and thus was with Max(S,R) .11/.835 = 13% of what it was using TMR. (Refer to table 7).

© 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

Table 8 − PERFORMANCE COMPARISON OF THE TMRAND THE MAX(S,R) ALGORITHMS Demand Function No. Basis for comparison

100 ea7/ea2

Problem size (N) 3

100S42/ms2 100 a7/ea2

5

100S42/ms2

Average 1 13*

2 0

3 42

4 36

5 31

6 45

7 9

8 52

9 45

10 19

29

68*

50

45

49

56

86

74

50

33

63

57

49

27

71

56

25

84

100

100

77

51

64

33

66

29

32

76

33

0

7

29

31

34

(*see explanations in the text above the table)

The table shows a significant reduction in average absolute error for almost all demand functions. When the TMR algorithm fails the S i algorithm is better on the average about 30-60% percent of the time resulting in an average absolute error 30-60% of what it was using only TMR. Thus these results indicate that considerable reduction error may be obtained by calculating both the TMR sequence and the S i -Sequence and selecting the better (i.e. use the Max(S,R)algorithm). Since the Max(S,R) algorithm is by definition better or at least equal to the TMR algorithm, no rigorous statistical tests were carried out to verify the ranking of the two algorithms. However, it is worth noting that the increased performance of the Max(R,S) method above the TMR-method is obtained at the cost of increased computational effort. It may take twice as long to calculate the algorithms and then select the better, but the computational effort will for the Max(S,R) algorithm be heavily dependent upon the demand function.

8.

CONCLUSIONS AND DISCUSSIONS

The main conclusions of the study seem to be the following: 1. The TMR-algorithm is on the average superior to all other heuristic algorithms tested in the study. For certain sequencing problems, however, it fails to produce an optimal solution (e.g. when the demand function is “ill-behaved”). 2. In the cases when the TMR-algorithm fails the k i algorithm and especially the S i algorithm often perform better. Thus they can act in a complementary manner with the TMR-algorithm. 3. Even though the Ci xi algorithm may perform as well or even better than the S i or k i algorithms, it does not complement the TMR algorithm in the same manner as the 2 others. One may speculate what characteristics of a sequencing problem cause the TMR algorithm to fail while the k i or S i algorithms perform better. One such problem has been described in section 4.2.4. but another striking example of the possible fallacies of using the TMR algorithm on ill-behaved demand functions is shown in figure 7. The two problems are quite similar but the latter has in many respects a simpler structure. The problem involves one very expensive project (per unit of capacity) and two inexpensive but similar projects. Consider first the TMR algorithm. When any 2 of the 3 projects are compared against each other for the first place in sequence the inexpensive project (no. 3) wins both of the others, and hence gets place, in spite of the fact that both these projects (no. 1 and 2) could be constructed for far less money and provide more capacity © 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

together and still overcome the demand barrier (D ( t ) = 10 ) . Thus the fact that only 2 projects are sequenced at a time (although optimally) prevents the TMR algorithm from checking project 1 and 2 together, in spite of their lower cost. This is when the unit cost criteria of the algorithms S i , k i and even Ci xi perform better, since they are not confined to a partial sequence of 2 projects. (See figure 7)

REFERENCES (1) Butcher, W.W., Y.Y. Haines and W.A. Hall: Dynamic Programming for the Optimal Sequencing of Water Supply Projects, Water Resources Research 5 (6), 1196-1204, 1969. (2) Becker, L., Yeh, W. W-G. Timing and Sizing of Complex Water Resources Systems, Journal of the Hydraulics division, ASCE, October 1974, pp. 1457-1469. (3) Becker, L., Yeh, W. W-G. Optimal Timing, Sequencing and Sizing of Multiple Reservoir Surface Water Facilities. Water Resources Research, February 1974, PP 57-62. (4) Bellman, R., Dreyfus, S. Applied Dynamic Programming, Princeton University Press, Princeton, N.Y. 1962. (5) Conway, R.H., Maxwell, W.L. and Miller, L.W., Theory of Scheduling. Addison Wesley Publishing Co., Inc., Reading, Mass., 1967. (6) Erlenkotter, D., Sequencing of Interdependent Hydroelectric Projects, Water Resources Research, February 1973, pp. 21-27. (7) Erlenkotter, D., Sequencing Expansion Projects, Operations Research 21.2, 1973, pp. 542-553. (8) Erlenkotter, D., Comment on Optimal Timing, Sequencing and Sizing of Multiple Reservoir Surface Water Supply Facilities by L. Becker and W. W-G Yeh, Water Resources Research, Apr. 1975, pp. 380-381. (9) Lauria, D.T. Schlenger, D.L., Wentworth, R.W., Models for Capacity Planning of Water Systems, Journal of the Environmental Engineering Division, ASCE, April 1977, pp 273-291. (10) Manne, A.S., Capacity Expansion and Probabilistic Growth, Econometrica, Vol. 29, No. 4, October 1961, pp 632-649. (11) Manne, A.S. ed., Investments for Capacity Expansion: Size, Location and Time-phasing, MIT Press, Cambridge, Max., 1967. (12) Morin, T.L.: Optimal Sequencing of Capacity Expansion Projects, Journal of the Hydraulics division, ASCE, Sept. 1973, pp 1605-1622. (13) Morin, T.L.: Optimality of a Heuristics Sequencing Technique, Journal of the Hydraulics division, ASCE, August 1974, pp 1195-1202. (14) Morin, T.L., Esogbue, A.M.O.: A Useful Theroem in the Dynamic Progamming Solution of Sequencing and Scheduling Problems Occuring in Capital Expenditure Planning, Water Resources Research, Febr. 1974, pp 49-50. (15) Morin, T.L., Pathology of a Dynamic Programming Sequencing Algorithm, Water Resources Research, October 1973, pp 1178-1185. (16) Morin, T.L., Esogubue, A.M.O. Some Efficient Dynamic Programming Algorithm for the Optimal Sequencing and Scheduling of Water Supply Projects, Water Resources Research, June 1971, pp 479-484. (17) Scarato, R.F., Time Capacity Expansion of Urban Water Systems, Water Resources Research, Vol. 5, no. 5, October 1969, pp 929-936. (18) Scherer, C.R., Chiulli, R., Optimal Capacity Expansion with Off-Design Costs. Journal of the Water Resources Planning and Management Division, ASCE, November 1977, pp 213-226. (19) Sengupta, J.K., Fox, K.A., Optimization techniques in Quantitative Economic Models, North-Holland Publishing Company, Amsterdam 1969. © 1980 Egill B. Hreinsson

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Presented at NOAK 1980 The 9th Nordic Congress on Operations Research September 18-19, 1980 REYKJAVIK, ICELAND

(20) Tsou, C.A., Mitten, L.G., Russel, S.O., Search Technique for Project Sequencing, May 1973, Journal of the Hydraulics Division, ASCE, May 1973, pp 833-838. (21) Vatnsafl Íslands. Yfirlit yfir áætlanir gerðar á tímabilinu frá maí 1975 til maí 1976. Samanburður á orkuverði við stöðvarvegg. Endurskoðun kostnaðaráætlana miðuð við verðlag í maí 1976. Verkfræðistofa Sigurðar Thoroddsen, Reykjavík, júní 1976. (Hydroelectric power potential in Iceland. Review of studies made during the period May 1975 to May 1976. Revision of cost estimates, based on May 1976 price levels. Comparison of the cost of energy excluding transmission cost. Thoroddsen and Partners, Consulting Engineers, Reykjavik, 1976). In Icelandic. (22) Raforkuspá 1976-2000. Yfirlit eftir landshlutum. Orkuspárnefnd, Reykjavík, 1977. (Electrical energy forecase 1976-2000. A Regional Survey. Energy forecasting Committee, Reykjavik 1977). In Icelandic. (23) Sultartangi Hydroelectric Project, Project Planning Report. Prepared by Harza Engineering Company International and Thoroddsen and Partners, Reykjavik, December 1975. (24) Morin, T.L., Multidimensional Sequencing Rule, Operations Research (U.S.A.), Vol, 23, No 3 May-June 1975, pp 576-580. (25) Moore, N.Y., Optimal Solution to the Timing Sequencing and Sizing of Multiple Reservoir Surface Water Supply Facilities When Deman Depends on Price. A Ph.D. dissertation, University of California, Los Angeles 1977. (26) Linsley, R.K., Kohler, M.A., Paulhus, J.L.H. Hydrology for Engineers. McGraw-Hill Book Co., New York, 1958. (27) Erlenkotter, D., Scherer, C.R. An Economic Analysis of Optimal Investment Scheduling for Salinity Control in the Colorado River. University of California, Los Angeles. Water Resources Centre, 1977. (28) Armstrong, E.L., Hydraulic Resources, in: Renewable Energy Resources; The Full Report to the Conservation Commission of the World Energy Conference, IPC Science & Technology Press, New York 1978. (29) Hreinsson, E.B. Economical Expansion of Water Power. Tímarit Verkfræðingafélags Íslands (Journal of the Associations of Engineers in Iceland). No. 1, 1980. (30) Erlenkotter, D., Coordinating Scale and Sequencing Decisions for Water Resources Projects; from: Economic Modeling for Water Policy Evaluation North-Holland/American Elsevier New York, 1976. (31) Knudsen, J. and Rosberg, D. Optimal Scheduling of Water Supply Projects; Nordic Hydrology, 8, 1977, pp 171-192. (32) Bogle, M.G.V. and O’Sullivan, M.J. Stochastic Optimization of Water Supply Expansion, Water Resources Research, October 1979, pp 1229-1237. (33) Morin, T.L. Multidimensional Sequencing Rule, Operations Research (U.S.A.) Vol 23, no 3, May-June 1975, pp 576-580. (34) Morin, T.L., Solution of Some Combinatorial Optimization Problems in Water Resources Development, Engineering Optimization (U.K.), 1975, Vol 1, pp 155-167. (35) Bellman, R.E., Dynamic Programming, Princeton University Press, Princeton, New Jersey, 1957. (36) M. Held and R.M. Karp: A Dynamic Programming Approach to Sequencing Problems, J. Soc. Indust. and Appl. Math., 10, 196-210 (1962). (37) Bellman, R., Mathematical Aspects of Scheduling Theory, J. Soc. Ind. App. Math., 4, 168-205 (1956). (38) Electric Power Potential in Iceland, Landsvirkjun (The National Power Co.) Reykjavik, Iceland, June 1979.

© 1980 Egill B. Hreinsson

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Figure Dynamic in

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