optimal highway safety improvement investments by dynamic

BILLY PAXTON SECRETARY

COMMONWEALTH OF KENTUCKY

DEPARTMENT OF TRANSPORTATION

FRANKFORT, KE~TUCKV 40601

BUREAU OF HIGHWAYS

August 28, 1974

WENDELL H. FORD GOVERNOR

H.3.47

MEMORANDUM TO: J. R. Harbison

SUBJECT:

State Highway Engineer Chairman, Research Committee

Research Report No. 398; "Optimal Highway Safety Improvement Investments by Dynamic Programming;" KYP-73-47; HPR-PL-1(10), Part III.

The impersonal and objective aspects of many decision-making processes consist of orderly quantification of situational descriptors, alternatives, and consequences and a ranking of the consequences. The selection or choice of descriptors remains somewhat subjective or intuitive; unnecessary factors null-out in the quantification process. The principal difficulties arise in quantifying the factors.

The report enclosed herewith demonstrates a specific application; more extensive applications are implied.

JHH:gd Attachment cc's: Research Committee

Respectfully submit!/

O~.Ji~dc•J ~Havens

Director of Research

_ c6~~~ ... ~·~~URN TO: ~I VISION OF RESEA~ ... ~H, 533 ~OUTH LIMESTONE, LEXINGTON, KY. 40508

OI'TIMAL HIGHWAY SAFETY IMPROVEMENT INVESTMENTS

BY DYNAMIC PROGRAMMING

by

J. G. Pigman, K. R. Agent, J. G. Mayes, and C. V. Zegeer

ABSTRACT

The process of determining which projects to implement under a given budget, and which to defer

until later, is central to the planning and management of highway systems. With a limited budget for

construction, maintenance, and safety improvements, investments which will produce the optimal benefits

must be chosen. This is often impossible to accomplish without the aid of a computer because of the

complexity of the problem. Dynamic programming has been tested and verified as an efficient method

for selecting priority projects to derive maximum benefits.

There are several approaches to priority programming as it is related to the capital allocation problem.

Benefit-cost, present worth, and rate-of-return calculations have traditionally been used as an integral

part of the transportation planning process. Construction and maintenance programs continually face

the task of having to assign priorities when insufficient funds are available to complete all projects.

Safety improvement programs, which were initially funded through the Highway Safety Act of 1966

and expanded through the Federal-Aid Highway Act of 1973, have become so large that they are

unmanageable without a clear and concise means of priority allocation.

A dynamic programming procedure was developed in this study which selects the optimal combination

of safety improvement projects for a given budget. Sixty-one projects, each with one or more alternatives,

were evaluated. The input consisted of the designated budget for the safety improvement program, the

improvement cost, and the benefits derived from each improvement. The accuracy and reliability of

dynamic programming is dependent upon the accuracy of benefits and costs used as input.

In a comparison with benefit-cost analyses, it was shown that dynamic programming can yield a

higher return for a given budget. An optimal allocation of funds will always be obtained if the individual

project costs are multiples of the increment used in dynamic programming.

Applicability of dynamic programming to budget allocation in transportation planning is practically

unlimited. In addition to the various highway programs, dynamic programming can be used to optimize

investments for entire transportation departments.

OPTIMAL HIGHWAY SAFETY IMPROVEMENT INVESTMENTS

BY DYNAMIC PROGRAMMING

by

J. G. Pigman, K. R. Agent, J. G. Mayes, and C. V. Zegeer

ABSTRACT


until later, is central to the planning and management of highway systems. With a limited budget for

construction, maintenance, and safety improvements, investments which will produce the optimal benefits

must be chosen. This is often impossible to accomplish without the aid of a computer because of the

complexity of the problem. Dynamic programming has been tested and verified as an efficient method

for selecting priority projects to derive maximum benefits.


Benefit-cost, present worth, and rate-of-return calculations have traditionally been used as an integral

part of the transportation planning process. Construction and maintenance programs continually face

the task of having to assign priorities when insufficient funds are available to complete all projects.

Safety improvement programs, which were initially funded through the Highway Safety Act of 1966

and expanded through the Federal-Aid Highway Act of 1973, have become so large that they are

unmanageable without a clear and concise means of priority allocation.

A dynamic programming procedure was developed in this study which selects the optimal combination

of safety improvement projects for a given budget. Sixty-one projects, each with one or more alternatives,

were evaluated. The input consisted of the designated budget for the safety improvement program, the

improvement cost, and the benefits derived from each improvement. The accuracy and reliability of

dynamic programming is dependent upon the accuracy of benefits and costs used as input.

In a comparison with benefit-cost analyses, it was shown that dynamic programming can yield a

higher return for a given budget. An optimal allocation of funds will always be obtained if the individual

project costs are multiples of the increment used in dynamic programming.

Applicability of dynamic programming to budget allocation in transportation planning is practically

unlimited. In addition to the various highway programs, dynamic programming can be used to optimize

investments for entire transportation departments.

Research Report 398

OPTIMAL HIGHWAY SAFETY IMPROVEMENT INVESTMENTS BY DYNAMIC PROGRAMMING

KYP-73-47; HPR-PL-1{10), Part III

by

J. G. Pigman Research Engineer Senior

K. R. Agent Research Engineer

J. G. Mayes Research Engineer

C. V. Zegeer Research Engineer Associate

Division of Research Bureau of Highways

DEPARTMENT OF TRANSPORTATION Commonwealth of Kentucky

The contents of this report reflect the views

of the authors who are responsible for the

facts and the accuracy of the data presented

herein. The contents do not reflect the official

views or policies of the Kentucky Bureau

of Highways. This report does not constitute

a standard, specification, or regulation.

August 1974

INTRODUCTION


until later, is central to the planning and management of the highway system. With a limited budget

for construction, maintenance, and safety improvements, investments which will produce the optimal

benefits must be chosen. This is often impossible to accomplish without the aid of a computer because

of the complexity of the problem. Dynamic programming has been tested and verified by others as

an efficient method for selecting priority projects to derive maximum benefits.

Dynamic programming is an optimization technique which transforms a multistage decision problem

into a series of one-stage decision problems. The decision at each stage depends on the input to that

stage, the feasible set of decisions at that stage, and the conditional set of decisions from the preceding

stages.

There are three main reasons why dynamic programming is needed for transportation planning. First,

dynamic programming is designed to provide the best plan over a period of time inasmuch as the scheduling

of a project is a critical variable. Secondly, dynamic programming makes it possible to obtain the best

combination of projects where some approaches are inaccurate and trial and error methods can become

an impossible task. Thirdly, dynamic programming can determine the optimal investment plan when the

usual benefit-cost, present worth, or maximum rate of return approaches are not practical. When the

amount of money required for a single project is a large portion of the budget, the best set of projects

does not necessarily consist of those which would be chosen by the conventional means of priority

selection. Benefit-cost and rate of return methods may not provide the best overall use of resources

because an efficient implementation of results may not be possible. In addition, the benefit-cost method

of selecting optimal alternatives does not always produce the best results because it focuses narrowly

on immediate benefits and often precludes some future combinations of alternatives which are more

desirable.

Many programs do not require detailed knowledge of the mechanics of dynamic programming. The

input consists only of the costs and benefits anticipated for any project along with the time required

for completion. Dynamic programming, by taking all possible combinations into account, avoids the

possibility of missing an optimal plan which will gnarantee the best economic investment.


Benefit-cost, present worth, and rate of return calculations have traditionally been used as an integral

part of the transportation plarming process. Performance budgeting has been proposed as a means of

highway maintenance management (1). Construction and maintenance programs continually face the task

of having to assign priorities when insufficient funds are available to complete all projects. Safety

improvement programs, which were initially funded through the Highway Safety Act of 1966 and

expanded through the Federal· Aid Highway Act of 1973, have become so large that they are unmanageable

without a clear and concise means of priority allocation. Possibly the most comprehensive and accurate

method of cost allocation for a constrained budget is dynamic programming. The term was coined by

Bellman (2) in an attempt to simplify the phrase definition previously used -- mathematical theory of

multistage decision processes. He has summarized dynamic programming applicability into three types

of projects: single-stage, multistage, and multistage incorporating a time factor.

Single-stage dynamic programming is the evaluation of a single project with several alternatives as

compared to multistage where several projects with several alternatives are evaluated. Multistage with

a time factor involves the allocation of funds by dynamic programming where several projects with several

alternatives are subject to implemention over a period of time.

Johnson, Dare, and Skinner (3) presented dynamic programming as a means of selecting highway

improvement projects to eliminate hazardous locations and therefore maximize the annual cost reduction

benefit. They suggested an optimal solution is assured when several projects are being considered and

construction funds are limited. De Neufville and Mori ( 4) have dealt with a simplified procedure for

determining the optimal construction schedule for additions over time to a highway or similar

transportation network. Only costs and benefits for each project are required as input to determine

the optimum schedule. Funk and Tillman ( 5) used the systems approach to emphasize that the costs

and benefits occurring to all parts of the system must be evaluated to establish the effect upon a specific

route under consideration. Dynamic programming was used to analyze the entire system such that optimal

stages of construction were implemented.

Jorgensen (6) has done extensive work in the identification of high-accident locations and the

development of methods for selecting improvements from among various projects. Benefit-cost, present

worth, or rate of return calculations were recommended by Jorgensen as methods for determining which

project yields the maximum difference between the annual investment cost and the annual expected

safety benefit. Determining priorities with these methods is restrictive because they will not assure the

optimal combination of projects when operating with a limited budget. Lorie and Savage (7) have shown

that, under a constrained budget, the selection of a large initial cost project with a high ratio of present

worth to cost may preclude the selection of several smaller projects which together yield a greater present

worth. Another disadvantage is the inability of previously used methods to evaluate the relative merit

of competing alternatives at varying investment levels.

2

Previous studies have dealt with Kentucky highway budgeting (8, 9). Agent (10) evaluated the

high-accident location spot-improvement program in Kentucky and it was determined that the small

investment in the program had returned significant dividends. It was felt that further study was warranto<!

and Zegeer ( 11) recently completed an investigation of the various methods for selecting high-accident

locations. Favorable results from the studies by Agent and Zegeer, combined with an expansion of the

spot-improvement program as a result of appropriations through the Federal-Aid Highway Act of 1973,

have stimulated for the development of an optimal method for allocating funds within the safety

improvement program. Dynamic programming, as an optimal investment plan with a constrained budget,

is presented here in a rather simplified but effective form for the particular problem.

The State of Alabama Highway Department has done considerable work in the application of dynamic

programming to the optimization of budget allocation for the spot safety improvement program (12).

Significant modifications have been incorporated into the Alabama program to evaluate the data which

were available for the spot-improvement program in Kentucky. The authors wish to acknowledge the

cooperation of the Alabama Highway Department in providing information used to determine the

applicability of dynamic programming to the spot safety improvement program.

PROCEDURE

The problem of optimum utilization of improvement funds can be divided into two distinct steps.

First, the benefits associated with each proposed improvement must be determined. Second, given the

costs and benefits for a set of improvements and given a specific budget, the optimum combination

of improvements to be implemented must be chosen. The computer program presented in APPENDIX

A was used to calculate the costs and benefits in the subroutine COSBEN. These results are printed

out and passed into the subroutine DYNAM along with the budget and output information. DYNAM

then determines and prints out the optimum combination of improvements for the desired budgets. If

no alternative emerges at a particular location, alternative "0" is printed. A range of budgets including

the maximum budget available are considered. In this manner, an optimum budget may be determined.

A list of variables and flow chart for the computer program are presented in APPENDIX B and APPENDIX

C, respectively. Coding instructions are presented in APPENDIX D, and APPENDIX E contains sample

program input and output.

Calculatio11 of Costs and Benefits Using the Present Worth Method

The following equations were used to calculate costs and benefits ( 13 ):

c S + A[(l - i)L- I] /i(l - i)L (I)

3

where

where

where

c = present worth cost of improvement,

s = construction cost,

A = yearly maintenance cost,

= present interest rate = 10 percent, and

L = life of improvement.

B = [[[[(! + t)CL + 1)/(1 + i)] · I]/[[(! + t)/(1 + i)] - I]] - I] il

B = present worth benefit,

t = exponential growth rate factor for traffic volume = 4 percent, and

J 3

il = ( 1; 1; "m Nmn 'Yn)/T m=l n=l

il =

T =

J =

"m =

N = mn

'Yn =

benefit per year associated with the improvement,

time (years) of accident history,

number of accident causes associated with the location,

percent reduction of m-th cause affected by the improvement,

number of accidents associated with m-th cause, and

average cost of an accident:

n = I fatality,

n = 2 nonfatal injury, and

n = 3 property damage only.

(2)

(3)

Dynamic Programming Algorithnm

STEP 1 Divide budget into N equal intervals.

STEP 2 (STAGE I) Determine the best alternative at Location I to maximize the return using j

increments, j = 1, 2, ... , N; i.e.,

=

where

=

(4)

total optimum return after STAGE I for an investment of j increments,

return from Location I for an investment of j increments, and

4

o1 G) = chosen alternative for Location I for an investment of j increments.

STEP 3 (STAGE 2 through STAGE M) Repeat STEP 2 for each STAGE.

where

OiG) = Max [(~(k) + oi . 10 · k)], (5)

j = 1, 2, ... , N and

k ;;::: 1, 2, ... , j

M = number of locations considered,

OiG) = total optimum return after STAGE for an investment of j

increments,

~(k) = return from Location i for an investment of k increments (k .;;

j),

Oi . 10 · k) = total optimum return after STAGE (i · I) for an investment of

0 · k) increments, and

DiG) = chosen alternative at Location i for an investment of j increments.

STEP 4 The optimum alternative at each location can now be obtained by determining the best

alternative for Location M at STAGE M with N increments. The remaining incremen.ts can

now be used at STAGE (M · 1), etc. Therefore,

= ~(N), leaving NM increments,

AM . 1 = DM .J(NM), leaving NM . 1 increments,

AM . 2 = ~ . 2(NM . 1 ), leaving NM . 2 increments, etc .

• •

=

where A;_ = alternative chosen at the i-th location.

Development of Benefit and Cost Values

Some of the major inputs into the dynamic progranuning model are the benefits assigned to each

improvement at a location. For example, upgrading a traffic signal at an intersection will affect accident

patterns differently than will installing channelization. To quantify the effect of various improvements

~n accidents, approximately 300 spot locations improved in Kentucky since 1968 were studied to

determine the accident reduction (or increase) associated with each at various location types.

Various improvements on curves, intersections, and other (general) locations are given in Table I

along with corresponding number of projects included, total accident reduction, service life of

s

TABLE I

SUMMARY OF IMPROVEMENT COSTS AND BENEFITS

NUMBER TOTAL ANNUAL TYPE OF OF ACCIDENT SERVICE MAINTENANCE

IMPROVEMENT PROJECTS REDUCTION LIFE COST (PERCENT) (YEARS) ($)

Signs and Markings 9 36 3 25 Warning Signs 23 35 5 25 Regulatory Signs 16 22 5 25 Guidance Signs 10 14 5 25 Sign Combinations 16 20 5 25 Markihgs 8 16 2 0 Si$ht .Distance. Imp. 9 28 2 50 Post Delineators 3 25 5 20 Comb.- Deline_ators, Markings, 11 22 5 25 'iii · Signs,- MWnteiiance .~

·:! ShOUlder IIDprovements 7 23 10 100 r3 Coni!:>. Resurface, Patching, 22 16 10 100 Drainage, Di::slick; Culvert ' R\llllb1e Stdps 8 29 5 0

Remove J4edian Crossovers 2 29 20 0 Lighting 1 -58 10 500 Li$htlng & Rumble Strips 1 17 7 300 Rumble Stdps & Beacon 2 32 7 50 Side Road Sign Only 31 19 5 25 !'repare for Sudden Stop Sign Only 19 25 5 25 Side Road Sign & Warning Sign 15 27 5 25

Signing 34 30 5 25 Post Delineators 4 32 5 25 Signs & Delineators 16 28 5 25

~ Signs & Mainteriance 6 47 3 25 ~ Comb. Delineators, Markings, 16 24 5 25 8 Signs, Maintenance

Resurfacing, Patch, Dr~age, 22 33 10 100 Deslick,_ Super, Culvert, Guardrail

Re-alignment (Relocate) 3 32 20 100

Signs & Markings 21 24 3 25 Warning Signs 11 27 5 25 Regulatory. Signs 5 48 5 25 Regulatory & Warning Signs 20 16 5 25

~

6 Markings 17 16 2 0

"' Marking, Maintenance, & Signing 9 35 5 25 g ~ -Channelization ~ Storage Lane 13 15 10 100 ;; Charmelization & Signs 2 37 7 75 .,.

lnstall Beacons 13 2 10 100 Upgrade Beacons 10 5 10 100 Install . Signals 10 23 10 300 Upgrade Signals 2 18 10 250

Total Improvements 447 24

6

improvement, and annual maintenance cost. Using the total accident reduction value (in percent reduction)

at each location under consideration, an approximate benefit was calculated. Accidents unrelated to the

location such as brake failures. drunk driving, or tire blowouts were disregarded in the calculation of

expected benefits after improvement.

The subroutine COS BEN was used to compute monetary benefits from expected accident reductions.

Accident costs used were recent National Safety Council values ( 14):

Fatality $45,000,

Injury

Property Damage Only (PDQ)

2,700, and

400.

The accident occurrence at each location is multiplied by the expected percent reduction for the

improvement alternative. The cost of accidents are then multiplied by the expected accident reduction

to give annual benefits. These annual benefits are then multiplied by an exponential growth, present-worth

factor (Equation 2) to obtain the benefits for the entire service life of the improvement.

The costs used in the calculations are the sum of the improvement cost for each project and the

maintenance cost. A present-worth factor (Equation 1) was used to adjust the maintenance cost from

L future date to the present.

It should be understood that the process of accurately estimating benefits and costs can be very

difficult. Even with a large sample of before-and-after data for locations improved by various alternatives,

accident reduction estimates may be inaccurate. This is partially attributable to the varying characteristics

of specific highway locations. Spuriousness in accident occurrence makes it impossible to accurately predict

future accidents. Predictions of expected accidents after a particular improvement should be based on

large samples combined with careful engineering judgement. Dynamic programming can give near-perfect

results if all input is exactly correct. However, if benefit and cost input is carelessly or incorrectly

estimated, results of dynamic programming will be equally in error.

RESULTS

A group of 61 11high-accident 11 locations previously improved under the Kentucky spot-improvement

program were selected as test data for the dynamic programming model. Accident reports at each location

were reviewed, and improvement alternatives were actual improvements made at the locations. Input

into the computer program for each alternative at each location consisted of accident data, expected

accident reduction, project costs, service life of improvement, maintenance costs, and interest rate.

The dynamic programming model computed benefits for each alternative. Then, as the available

budget was varied from $10,000 to $80,000, an optimal scheme of alternatives was generated for each

7

budget.

A similar calculation of return and benefit·cost ratio was made using a benefit-cost analysis. There

was very little difference in the benefit-cost analysis and the dynamic programming analysis for the test

locations. This is shown in Figure I where expected return versus available budget is plotted for both

dynamic programming and benefit-cost analyses. Details of the data used to plot Figure I are presented

in Table 2.

Comparison of Dynamic Programming and Benefit-Cost Ratio

Theoreticaliy, dynamic programming computer techniques will produce a scheme for aliocating funds

under a fixed budget such that the optimal return is obtained. After testing the computer model, it

was found that this is true as long as each project cost is an exact multiple of the budget increment.

For example, if computer storage constraints permit an increment of $250 with a budget of $100,000,

then the cost of each improvement should be a multiple of $250 in order to obtain an optimal

improvement scheme. An increment was defmed as some fraction of the budget used in the computer

analysis for weighing benefits against costs. In general, the smalier the increment, the better the solution

obtained. The number of increments into which the maximum budget may be divided, however, is largely

governed by the computer storage capacity as well as computer time required. Practicaliy, then, the

increment cannot be made as small as desired. If the majority of costs are at least twice the increment,

the results seem to be reasonably good.

A simplified example (Table 3) was developed to demonstrate how the monetary return using dynamic

programming techniques will exceed the return from a benefit-cost analysis if project costs are multiples

of the increment. As shown in Figure 2, the dynamic prograrmning return is the best at nearly every

budget level from $5000 to $34,000. Although the two are fairly close at some points, the return from

the benefit-cost curve is inferior to the dynamic prograrmning curve by about $50,000 at a budget of

$20,000 and by $40,000 at a budget of $30,000. The two curves are equal at budgets of $25,000

and $34,000. In this example, the $34,000-budget was divided into 34 increments of $1,000 each. Each

project cost is a multiple of $1,000.

Benefits from benefit-cost techniques may sometimes equal benefits obtained from dynamic

programming techniques and, in some cases, will produce undesirable results. Dynamic programming,

however, will always produce the optimal scheme if project costs are expressed as multiples of the

increment.

Use of Dynamic Programming

Application of dynamic programming techniques to the highway safety improvement program in

Kentucky involves several steps. First, a list of potentialiy hazardous locations, based on accident data,

8

U) 0:: <l ...1 ..J 0 0

lL 0

U) 0 z <l U) ;::)

2 1-~

z 0::

~ "' 0::

1000

900

BOO

700

0- DYNAMIC PROGRAMMING

A- B/C ANALYSIS

600

500

4oo.L-----~~----~------...1-------~----~------~------L-----~ o ID ~ ro ~ ~ ro ro oo

Fisure l.

AVAILABLE BUDGET (THOUSANDS OF DOLLARS)

Expected Return versus Available Budget for Dyuamic Programming .., Benefit-Cost Analyses

9

T~'

INfU'I DATA FOR DYNAMIC _PROGRAMMING BENEFIT-COST ANALYSIS COMPAIUSON

WC"ATION LOCATION ALTERNATIVE "' NO. ,~, NO. =' RETIJRN AATIO

' N-&S~lli.9 150(). 7b20o 5.06

' :ov-t.t.-1~.1 42~0. 1!\6707· 37o34

' 30-b0-15.2 2000, ll.08~- S,b'l

' ~4-41A-U .0 sco. 9792. ·~·'~'~ , 73-45-6.~ ~oo, 973· 1.95

' ?J-•s-r.z soo, n. o.o ' 73...,~-7-~ soo, ,_ o,o ' 73-45-7.2 750, t>7~6· eo\17

' 4P-42l-l4o4 ~oo. ,, o.o

' 48-4~1-J ,.~ SOli, 5PSo 1.1e

' .. ~-'<n-'"·" 750. o. '·" ' <'0-'>l-lol 75C, -"'~67· 3.9a

' 102-25-9 .... 1000. 3365- 3,37 w 102-25-5'< 500. 2~6- 0.45 H J0-60-15" ?oo. lbl4So 32,29 u iiO-M-1~.~ 4750. :.~24. o. 7 .. u 30-M.I-L5oi' sooo, :!JBb~. t..31

" 72-b41-fio5 750 .• 40568• 54.12

" 30-t>~-4.1 1250. lll':to 0,65

" 3~-171-63.·1 -4500, 21.()29o ~- 78

" n-~4t- 1 8.q ' 7?0o <.''>48o ... 73 ,, !>4-~lA-12.4 • ;5('0 • 14~98. L9o20 ,, 54-4lA-l2.4 , sco. 344~- t..~a ,, 54-'<H-12.4 7«(1. te~·40o 24.05 n .:~-Lb-9 ,J 500, e14. lob3 n Z4-l,h-~, 1 '!>00. 916- 1.~3 n ~-6~-9~1 1000, 1729, 1.13

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" %-rt~KH-1~0.1 1500. 11111· 7.Bl n <>3-(5-10.'1 7,5G, 746~ o.s9 " 1.3-~'o-!.0,4 ~~~o. ,, o.o " 4l-!7~-1~5.L 4000, HZ094· ~IJ.Ql

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" 70-l/Sb!.o-'l\oJ ' 4~~0. J<;~47o J-40

" .. 1-nw-;:~."' ' 2000, 41<'14- ;,:,45

" ,3-2;-u,.;. • ~oo. ~735· &.H

" ~>3-.!~-u,.;. ' sao. ~31l0- 104:b

" t.3-25-lo ... ' lVOOo 1Ul15. Ii'l.n

" 6~-~5-H .• 4 ' 1500. HB9. h2t

" ~i>-25-to.~ 1750. 10£00. h;s.~

" b3-~S-!0.4 200>:>. ll334o '5jlb7

" b~-;;~-10.~ <'250. 1964-!io Sll'l'l'l

" 10?-75.,-ll.~ ~c~. ~'134o 7~&7

" ~~-31J.i-L,2 ~~oo. ~1~0. 2.11

" ~1-'<1-~o.~ 500. l079;. 21.~~

'" u~-nw-1.a ~oo. 1~21· ;3.e4 ~'! 30-t>0-16., 500. 1731. ~-"~ " :;u-,o0-1-:..4 !>00. ~33Ho 6ob8

"" i';2-.,C:-Zb,2 750. 2<?1. 0.29

" o3-~5-IU.O 50~. 2e1. 0,57

"' 63-~~~to.n 500. 326· 0.65 d i>~-25-H,,(, ' looo. f>l.:!o Q.6)

" 63-25-10,0 ' 15M. ll4• o.oa "' b3-25-l0o\i 11~0. bl8o o.~5

" 0'>-2!>-lV.U 2000. 68"1. Q.34

" o3-ZS-lu.o ~250. 1191· 0.53

10

TABLE 3

INPUT DATA FOR DYNAMIC PROGRAMMING BENEFIT-COST ANALYSIS COMPARISON

(EXAMPLE PROBLEM)

LOCATION ALTERNATE BENEFIT-COST NUMBER NUMBER COSTS BENEFITS RATIO

I I $1000 $20000 20 2 1 1000 15000 15 3 1 1000 12000 12 4 I 3000 30000 !0 5 I 5000 45000 9 6 1 10000 80000 8 7 I 1000 7000 7 8 1 9000 54000 6 9 I 2000 6000 3

10 I 1000 2000 2

$34000 $271000

l I

!f) 0

;) I, 0

0 - DYNAMIC PROGRAMMING

A - B/C ANALYSIS

5 10 15 20 25 30

AVAILABLE BUDGET {THOUSANDS OF DOLLARS)

2, Expected Return versus Available .Budget for Dynamic Programming - Benefit Cost Analyses (Example Problem)

12

must he identified. A recommended location-identification procedure for Kentucky identifies hazardous

0.3-mile (0.48-km) spots and 3-mile ( 4.8·km) sections based on fatal accidents, total number of accidents,

accident severity rating (the "equivalent-property-damage-only" number), and accident rate (applying

~mJity control techniques). Locations should be identified based on duall-year and 2-year time intervals.

,.. o, locations identified by citizens, engineering personnel, and state police should be c0nsidered. Ali

locations identified as possibly hazardous should then be reviewed. Locations considered worthy of a

field inspection should be investigated for possible corrective measures.

The proposed program requires that all warranted minor improvements such as signs, paint striping,

flashing beacons, and delineators be implemented without dynamic programming considerations. Major

improvements such as resurfacing, bridge widening, realignment, and intersection channelization should

be selected by dynamic progranuning techniques.

Project costs, expected benefits, maintenance costs, and expected service life of the improvement

should be determined for each alternative at every location to be considered under dynamic progranuning.

After the warranted minor improvements are considered, the remaining money should be budgeted for

use in other projects where the dynamic programming may apply. An optimal set of improvement

alternatives would then be generated.

SUMMARY AND CONCLUSIONS

The objective of this study was to develop or adopt appropriate dynamic programming methods

that would assist in establishing optimal budgeting procedures for various highway programs. Dynamic

programming is a multistage operation which involves the evaluation of several projects with several

alternatives. A dynamic programming procedure was developed to select the optimal combination of

safety improvement projects for a given budget. The following major fmdings may be cited:

1. Use of dynamic programming is relatively simple. Input consists of the budget, costs, and benefits.

Estimating the benefits derived from a particular improvement presents the most difficulty.

2. Table I, which lists accident reduction by type of improvement for past safety improvements, was

developed from past accident experience for use in estimating savings.

3. The accuracy and reliability of dynamic programming is dependent upon the accuracy of benefits

and costs used as input.

4. A prerequisite in the use of dynamic programming for the safety improvement program is an efficient

method of systematically identifying locations based on accident data. In-depth field investigations

are also needed so that only necessary improvements are recommended as input for the dynamic

programming model.

13

5. It was shown that improvements selected by dynamic programming can yield a higher return for

a given budget than those chosen entirely on the basis of benefit-cost ratios (Figure 2).

6. If individual project costs are multiples of the increment used in the dynamic progranuuing, the

optimum allocation of funds will always be obtained. In general, the smaller the increment, the

better the solution obtained. However, the attractiveness of a smaller increment is restricted by

available computer storage.

7. Applicability of dynamic programming to budget allocation in transportation planning is practically

unlimited. In addition to the highway safety improvement program, dynamic progranuuing can be

used to optimize investments in maintenance and construction programs and eventually the entire

transportation department.

LIST OF REFERENCES

L Performance Budgeting System for Highway Maintenance Management, NCHRP Report 131, Highway

Research Board, 1972.

2. Bellman, R., Dynamic Progranuuing, Princeton University Press, 1957.

3. Johnson, M. M., Dare, C. E., and Skinner, H. B., Dynamic Programming of Highway Safety Projects,

Transportation Engineering Jowml, ASCE, November 1971.

4. de Neufville, R., and Mori, Y., Optimal Highway Staging by Dynamic Programming, Transportation

Engineering Jowml, ASCE, February 1970.

5. Funk, M. L. and Tillman, F. A., Optimal Construction Staging by Dynamic Programming, Jowml

of the Highway Division, ASCE, November 1968.

6. Evaluation of Criteria for Safety Improvements on the Highway, Roy Jorgensen and Associates,

Westat Research Analysts, Inc., 1966.

7. Lurie, J. H., and Savage, L. J., Three Problems in Rationing Capital, The Jowml of Business, Vol

28, No. 4, October 1955.

8. Kentucky Highway Budgeting, Bureau of Business Research, College of Commerce, University of

Kentucky, 1963.

9. Budgeting Support for Kentucky Highways by Systems, Bureau of Business Research, College of

Commerce, University of Kentucky, 1963.

I 0. Agent, K. R., Evaluation of the High-Accident Location Spot-Improvement Program in Kentucky,

Kentucky Bureau of Highways, 1973.

11. Zegeer, C. V., Identification of Hazardous Locations on Rural Highways in Kentucky, Kentucky

llureau of Highways, 1974.

14

12. Co"ect: Cost/Benefit Optimization for the Reduction of Roadway Environment Caused Tragedies,

Phase II, State of Alabama Highway Department, 1973.

13. Winfrey, R., Economic Analysis for Highways, 1969.

14. Traffic Safety Memorandum No. 113, National Safety Council, July 1971.

15

APPENDIX A

PROGRAM SOURCE DECK

16

PROGRAM MAIN

C DATE: AI/GUST 5,!974 MA!NOOJO C PROGRAMMER: THIS PROGRAM WAS WRITTEN RY JESSE MAYES, DIVISION OF MAIN0020 C RESEARCH, DEPT. OF TRANS.,COMMONWEALTH OF KY.,533 S. LIMESTONE ST., MAIN0030 C LEXINGTON, KY. PARTS OF THF PROGRAM,!NCLUO!NG THE. DYNAMIC MA]N0040 C PROGRAMMING ALGORITHIJM, HAVF RI'EN ADAPTED FROM A PROGRAM WRITTE~l BY MAIN0050 C THE STATE OF ALABAMA HIGHWAY DEPT., BUREAU OF MAINT ., 1973, SEE MAINOOI>O C REPORT "CORRECT: COST/BFNFFIT OPTIMIZATION FOR THE RFOIICTJON OF ROAOMAJN0070 C ENVIRONMENT CALISED TRAGEDIES''• MA!NOOAO C PURPOSE: THIS PROGRAM CALCIILATFS COSTS AND RfNEFITS FOR EACH MA!N0090 C ALTERNATIVE AT EACH LOCATJ[)N THEN DFTFRMINES THE OPTIMAL SOLUTION MAJN0100 C SET OF ALTERNATIVES TO RF IMPLEMENTED FOR A GIVEN RANGE OF BUDGETS. MA!N0110 C INPUT AND OUTPUT: SEE DIVISION OF RESEARCH REPORT: ''OPTIMAL HJGHWAYMAJN0120 C SAFETY IMPROVEMENTS BY DYNAMIC PROGRAMMING''• MAIN0l30 DIMeNSION ORET(64,40li,NODI64,4011 0 I MENS I ON T I TL 120 I , XLOC 164,5 I , NDE I 641 , C I 64, 111 , ~ 164, 111 , LOC 11>41 NINP = 401

C NINP = NIJMHER OF INCREMfNTS---MAXIMUM BIJDGET EQLJALS N!NPOX!NC NLOC = 64

C NLOC = MAXIMUM NLJMRER OF LOCATIONS INN = 'i JOUTPR = 6

C INN,IOJJTPR = LOCAL INPUT AND OUTPIJT DfVICE NJJMBERS READ( INN,10001 TITL 1000 FORMAT(20A41

WRITE I IOUTPR.l01DI TITL 1010 FORMAT 120X,20A4/////I

READ I !NN,10201 NSTG,XJNC,K1,K2 1020 FORMAT~T1,14 0 T5,F6.0,T1!,2141

CALL COSREN{C, fh XLOC ,LOC ,NDE ,NSTG,NLOC, X INC, INN, ICHJTPR ,KI K l IFIKIK.EQ.11 GO TO 10 CALL DYNAM(C,B,LOC,XLOC,NDE 9 NSTG,XINC,Kl 9 K2,NINP,NLOC,

+ ORFT,NOO,JOUTPRI 10 CONTINUE

CALL FXIT END

MA ll\10140 MA!N01'50 MAINOlhO MAIN0170 MA]N01RO MA!N0190 MAJt\10?..00 MAJN0210 MAJN0220 MA]N0230 MAIN0240 MA!N02'i0 MAIN0260 MAJN0270 My\ IN02AO Ml\_JN0290 MAJN0300 MA!N0310 MAJN0320 MA I NODO MA!N0340 MA!N03'i0

17

SUBROUTINE COSBEN

SUBROIITINE COSREN(PWC,PWA,XLOC,LOC 9 NOE,NSTG,NLOC,XINC,INN,IOIITPR, C:OSROOIO + KIK) COSAOO?O

C THIS SIIBROUTINE CALCIILATFS PRESENT WORTH COSTS AND AENFFITS UlSA0030 C ASSOCIITFD WITH EACH ALTERNATIVE AT EACH LOCATION CoSA0040

DIMENSION XLOCINLOC,5l,SEV(4,8);CSEFI!O,ll),A(8), CoSR0050 + NOEINLOC),PWCINLOC,ll),PWRINLOC,lll,LOCINLOC) COSA006D

REAO(INN,lDDO) CFAT,CINJ,CPOO,RITEIN,RITEGR COSRD07D 1000 FORMAT (RFlO.O) COSRDDAO

WRITFI IDUTPR, 1010) CFAT,CI~IJ 0 CPDO,RATEIN 0 RAHGR COSA0090 1010 FORMliT( 1 MEG UTILITY FATALITY=' ,F7.0, 1 II\I,J\JRY= 1 ,F6.0, 1 PRP DM= 1 COSFHllflO

+ 9 F5.0/ 1 INTEREST RATE = 1 ,F5.3/ 1 EXPONENTIAL GROWTH RATF 1 ,F5.3/COSROllO +II//) CDSAD12D

r THE ABOVE READS IND PRINTS THE BASIC PARAMETERS CONSTANT FOR THF CDSR0130 C ENTIRE PROGRAM CDSR0140

~II(MRER = 1 KIK = 0

C BELOW IS THE INPUT WHICH IS EXECIJTED FOP EACH ACCIDENT LOCATION. 10 READ( INN, 1020) NOl, (XLOC(f\llJMBER,I) ,I=l 9 5) 9 TIME,NMn,NYR 9 NCAU

1020 FORMAT( I4 9 5A4,47X 9 F4.0,12,12, Il) LllC(NIIMBERl = NO! IFIN01)20,1R0,20

20 CONTINUE WRITFI IOUTPR,l030)

1030 FORMAT( lHl) WRITFIIOUTPR,l040)

1040 FORM IT I' REF NO') IFI'ICAIJ,EQ,l) GO TO 30 WRITE( IOUTPR,1050) N01, IXUlC(NUMRFR, I}, 1=1,11}, TIME,NMO,NYR,NCAU

1050 FORMAT{3X 9 14,8X,5A4,//9X 91 ACCIOENT HISTORY 1 ,F4.2 9

+ 1 YEARS. MONTH 1 ,12, 1 ,YFAR t,J2, 1 , 19 11, 1 CAIJSE .. 1 )

GO TO 40 30 WRITFIIOUTPR,l060) t\OOl,(XLOC(NIIMRER,Il,I=l,'i),T!ME,NM0 1 NYR,NCAU

1060 FORMAT{3X,!4,AX,5A4,//9X, 1 ACCIOENT HISTnRY 19 F4.2 9

+ 1 YEARS. MONTH 19 12, 1

9 YEAR 19 12, 1

9 1 ,!1 9 ' CAUSE. 1 )

40 CONTINUE C SECOND CARD INPUT FOR FACH CRITICAL LOCATION (SEVERITIES),

REAr){ INN 9 10 7 0) N 0 2 , ( ( S E V ( I , J l t J = 1 , 4 l , I= 1 , N C A tJ I , A l T 1070 FORMAT I I4,3RFZ.O)

~IALT=ALT/10.+.1 NOEINIIMBER) = NALT

C ROIIT!t\OE TO CHECK CARD SFQIIENCE CODE.

c

IFINOl-NOZl 50,60 1 50 50 WRITE I IOUTPR, !080)N01 ,NO?

!ORO FORMITI' SEQUENCE/CODE NO, ERROR. CHECK ',I'i,' AND' ,15, +' **FXECUT!GN TERMINATFO')

KIK = 1 GO TO 190

60 CONTINUE OIJTPIIT OF SEVERITIES

WRITE I IOUTPR,!090) 1090 FORMAT(/ ' ROADWAY

TOT! =0 TOT2=0 TOT3=0 TOT4=0 DO RO I=l,NCAU

AND TOTILS.

CAliSE TACC NFAT

WRITEIIOUTPR 1 ll00) I,(SFV(I,J),J=l'0 4) 1100 FORMAT 11X 1 I7,Fl2,0,3F6,0)

70 CONTI NIJE TOTl=TOTl+ SEVII,l)

NINJ NPRO I)

COSROl'iO COSR0160 COSR0170 COSAO!RD COSR0190 COSR0200 COSH02)0 COSROZZn CDSR02 30 CDSRD240 COSR02'i0 CDSR02AO COSR0270 COSROZRO COSRD290 COSR0300 COSR03lD COSR0320 COSR0330 COSR0340 COSR03'i0 COSR0360 COSR0370 COSR03RO COSR0390 COSB0400 COSRD410 COSB0420 COSR0430 COSR0440 COSR0450 COSR0460 COSR0470 COSH04RO COSR0490 COSR0'300 COSR0510 COSRO'i20 COSR0530 COSB0540 COSR05'30 COSA0560 COSR0570 COSRO'iRO COSRO'i90 COSR0600

18

TOT2=TOT2+ SEVII 0 2l TOT3=TOT3+ SEVII 0 3) TOT4=TOT4+ SEVIJ,4)

80 CONTINUE WRITEI!OUTPR,lllOl TOT1 0 TOT2 0 TOT3,TOT4

1110 FORMAT{' TOTALS' 0 Fl2.00 3F6,0) C INPUT NFXT SET OF NALT CARDS, ONE FOR EACH ALTERNATIVE

NJ=3HIC AU DO l\0 l=l 0 NALT REAOIINN,lPOl N03,{CSEFIJ 0 ,)) 0 J=l 0 N,I)

1120 FORMATI[4 0 F7.0,F2.0,F5,0o8F3.2l IFIN03-NOll90,l00,90

90 WRITEIIOUTPR,l080l NOlo N03 KIK = l GO TO 190

100 CONTINUE 110 CONTINUE

C OUTPUT OF ALTERNATIVE INFORMATION, WRITE I I 0 UT PR, ll3 0 l I I 0 I= l 0 NC All l

1130 ·FORMAT{/' ALTERNATIVE COST LIFE MAIN COST EFFECT ON ••• •, +8!'5)

C NUMBER COUNT CHECK OF SEVERITIES. 00 120 1=1,NALT WRITEIIDUTPR 0 1140l I 0 ICSEFII 0 Jl 0 J=l,NJl

1140 FO~MATI!7,Fl3.2 0 FH,0 0 F9.2 0 F24.2 0 7F5.2l 120 CONTINUE

C COMPUTATION OF Bill, THE ITH ALTERNATIVE BENEFIT. DO 140 l=loNALT Bill = 0, DO 130 J=l,NCAU ,lEFT = J +3 Rill= Bill+ ICFAT*SEVI.J 0 2l+CIN,I*SFVIJ 0 3l+CPDO*SEV{J,4ll*

+ CSFF I I o JEFTl 130 CONTINUE

COSR06\0 COSB0620 COSB0630 COSR0640 COSR06'50 COSR06AO COSB0670 COSB06RO COSR0690 COSB0700 COSR0710 COSB0720 COSBOBO COSB074D COSR07'5D COSB0760 COSB0770 COSB07R0 COSR0790 COSBOBOO COSHOR\0 COSBOR20 COSBOA30 COSBOA40 COSROB50 COSBOB60 COSBOB70 COSBOBRO COSHOR90 COSR0900 COSR0910 COSB0920 COSB09'l0 COSR0940

140 CONTINUE COSR09'50 C CALCULATION OF BENEFIT/COSTS AND OUTPUT. COSR0960

WR!TEIIOUTPR,1150l COSR0970 1150 FORMATI///5X, 0 BENEFIT/COST ANALYSIS'!!' ALTERNATIVE COST COSH09R0

+ BFNFF!T BENEFIT/COST' l COSB0990 00 150 1=1 0 NALJ CDSBIOOO Blll=BIIl*CSEF{I,2l!TIME COSB10\0 BNCS = Blll!CSEFIIoll COSB1020 WRITEIIDUTPR,1160li,CSFFIIo1loBIIl,BNCS COSB\030

1160 FORMAT I 17 0 F14.2 0 Fl4.2,Fl8.4l COSBI040 150 CONTINUE COSB10'50

WRITEIIOUTPR 0 1170l COSB10AO 1170 FORMAT{ /// 0 BENEFIT/COST ANALYSIS, MAINTENANCI' JNCLliOE0°/l COSB\070

WRITFIJOUTPR 0 1l80l COSBIOR0 !lAO FORMAT{' ALTERNATIVE MAINTENANCE TOTAL COST RENEFJT/COCOSR1090

+ST' ) COSBIIOO DO 160 1=1 0 NALT COSB1110 XMA!N=CSEFII 0 2l*CSEFII 0 3) COSB\120 BNCM = Blll/ICSEFII 0 1l+XMA!Nl COSB\130" TMCST=CSEF11 0 ll+XMAIN COSRll40 WRJTFIJOUTPR 0 1190) l 0 XMAIN 1 TMCST 0 BNCM COSA\150

1190 FORMATII7 0 F23.2 0 Fl4.2 0 F16,4) CDSA11AO 160 CONTJNIIE COSR1170

WRITEtlOUTPR,l200l COSB11RO 1200 FORMAT{ Ill' BENEFIT/COST ANALYSIS, MAINTENANCE INCLUDED ***PRECOSR1190

+SENT WORTH METHOD***'/) COSB\200 WRITEI!OUTPR 0 1210l COSB!ZlO

1210 FORMAT{' ALTERNATIVE MAINTENANCE TOTAL COST BENEFIT CDSB1220 +BENEFIT/COST') COSB\230

DO 170 l=l 0 NALT COSBl240 LIFE = CSEFI 10 2) COSAI250 X= ll.+RATE!Nl**LIFE C~SB~A~

19

PWF = (X-1.1/(RATEIN*XI COSB1270 Y = (1.+RATEGRI/(1,+RATEINI CDSB12AO PWEXGR = CY**(LIFE+11-1.1/(Y-1.1- 1 C0$81290 PWC(NUMBER,11 • 0 COSB1300 PWBCNLIMBER,ll = 0 COSA13!0 PWMAIN = PWF*CSEF(!,31 CQSB1320 PWC(NIJMBER.I+ll = CSEF(l,!l + PWMAIN COSB1330 C ROUND PRESENT WORTH COSTS TO NEAREST INCREMENT CDSA1340 IPWC = PWC(NUMBER,I+11/XINC + .5 CDSB1350 PWC(NUMBER,I+11 • IPWC*XINC COSA13AO PWBCNIIMBER,!+11 = PWEXGR*RC II/LIFE CoSB1370 PWBC • PWR!NlJMBER,I+li/PWCCNUMRER,I+ll CDSB13AO WR ITF (I OUTPR, 1220 I I, PWMA IN, PWC (NUMBER, I+ 11 , P~B CNUMRER, I +11 0 PWBC COSB 1390 1220 FORMATCI7,F23.2,F!4,2,F11,2,5X,F11.21 COSBJ400 170 CONTINIJE COSR1410 NUMBER = NUMBER + 1 COSA1420 GO TO 10 COSRI430 180 CONTINUE COSAI440 NIJMRER = NUMBER - 1 COSRI4o0 !F(NUMBER.EO.NSTGI GO TO !90 COSB14AO WRITE(JOUTPR,12301 COSRI470 1230 FORMATC•l 1 ,40('*'>'' WARf\.ilNG 1

9 40{ 1 * 1 )//) COSR14AO WRITE(IOUTPR,12401 NUMAER,NSTG CoSRI490 1240 FORMAT!' '•'NUMBER OF LOCATIONS READ= •,!3/' •,•NUMBER DF LOCATIOCOSB1500 +NS EXPECTED= •,!31 COSR1510 190 CONTINUE CDSB1520 RETURN

ENO COSRl'i30 COSRI540

20

SUBROUTINE DYNAM

SUBROUTINE DYNAM(C 0 8 0 LOC 0 XLOC,NDE,NSTG 0 XINC,K1,K2 1 NINP,NLOC, DYNA0010 + ORET 0 N00 0 !0UTPRI OYNA0020

C THIS SIIBROUTINE USES "DYNAMIC P~ROGRAMMING" TO FIND THE OPTIMAL OYNA0030 C SOLUTION SET ALTERNATIVES (ONE AT EACH LOCATION) GIVEN COSTS, OYNI004D C BENEFITS AND I RANGE OF AliOGETS. THE ALGOR ITHUM IS RASED ON WORK BYOV'IAOO~O

C RICHARD aELLMAN (DYNAMIC PRDGRAMMING,1957) DYNAOOhO DIMENSION ORETINLOC,NINPI,NODINLOC,NINP),NDEINLOC), OYNA0070

+ CINLDC,11),R(NLOC,11) 0 RI111 0 XLOCINLOC,51,LOCINLOC) OYNAOOAO IST•O 0YNA0090 VRET•O.O DYNA0100 WRITFIIOUTPR,1130) WRITE( IOUTPR,lOOO)

1000 FORMAT(' •,40( '*'I ,•PARAMETER VALUES 00 40( '*' IIIII)

WRITEIIDUTPR 0 1010)

OV'll 0110 OYNA0120 OYNA0130 OYNA0140

1010 FORMAT{' 1 ,?.7X,lR(I-1) 9 1QtJTPlJT' 9 18( 1 - 1 1l OYNAOl"iO WRITEIIOUTP~,1020) NSTG,XINC,K1 0 K2 DYNA01h0

1020 FORMITI5X,'LOCATIONS---INCREMENT---LOWER LIMIT---INCREMENTS PER STDYNA0170 + E P 1 , I , 3 X, I 9, 3 X, F 12 • 2 , I 9, lOX , I 9, II , 1 -----LO CAT I ON---AlTERNATIVES 1 ) nYf\1 A 0 1 8-0

DO 10 1=1,NSTG DYNA0190 WRITE I IOUTPR 0 1030) LOC (I I ,NOEl I I OYNA0200

\030 FORMATI7X,!5,!10) DYNA0210 10 CONT!NliE DYN10220

WRITFIIDUTPR,1040) OYNA0230 !040 FORMATI'1°,30I'*'I,'LOCITill'tS,ALTERNATIVES,Cl1STS AND BENEFITS', OYNA0240

+ 301 '''') ////) DY"A02o0 WRITEI!DliTPR,l050) OYNA0260

1050 FORMAT llH ,•--LOCATION---LOCATION NAME--------ALT-NlJM--------COSTDYNA0270 +-----RETURN-----B/C RATIO') OYNA02AO

C FIND THE OPTIMAL ALTERNATIVE AT THE 1-TH LOCATION WITH J INCREMENTS OYNA0290 C AVAILABLE OYNA0300

DO 140 I•!,NSTG DYNA03!0 NDEC•NOEI I )+1 DYNA0320 R(1)=0, DYNA0330 on 20 !C•Z,NOEC OYNA0340

20 RIICI • BlloiCI DYNA0350 00 30 IC•Z,NDEC OYNA0360 ICM1 • IC-1 OYNA0370 BCRAT = RIICI/Cil,!C) DYNA03AO WRITFIIOUTPR,1060) LOCI!),(XLOCII 0 J),J•!,5),!CMI,CII,!C) 0 RIICI, DYNA0390

+ RCRAT OYNA0400 1060 FORMATI!9,5X,5A4,!6,3X,F11.0 0 F11.0,4X 0 Fl0.2) OY"A0410

30 CONTI"IIE DYNA0420 1070 FORMATISF!O.O)

00 130 J•1 0 NINP C INCREMENT BUDGET

XIN•IJ-li*XINC OliM•-1 000000000000. NOEC•NOEII 1+1

C DETERMINE THE REST ALTERNATIVE--NODII,JI--AT 1-TH LOCATION GIVEN C J-1 INCREMENTS TO SPENO ON LOCATION 1 THRll LOCATION I-----YIELDING C A RETliR" OF--DRETII,JI--

00 120 K•1,NDEC CALL XDUTII, IST,X!N,K,KICK,XINC,C,NLOC) IFIKJCKI50,50 0 40

40 GO TO 120 '00 CONTI NilE

IF(I-1160,60 0 70 hO TEST•RIK)

GO Tn RO 70 TEST•RIKI+DRETII-!,ISTI

OY!\IA0430 DYNA0440 OYNA0450 OYNA0460 DYNA0470 OYNA04A() OYNA0490 OYNA0500 OYfl..lAOSlO OYNA0520 OYJ\IA0'530 OYNA0540 f1YNAOS'50 DYNA05hn OYNA0'070 OYNA05RO OYNAO"igO OYNAOhDO

21

GO TO 80 AO IFC(OUM-TEST))90,l00,!00 90 OUM=TEST

DRET(!,J)=DUM NOD(!,J)=K

100 GO TO l!O 110 CONTINUE PO CONTINUE 130 CONTINUE 140 CONTINUE

!PAGE = 0 C WRITE MAIN BUDGET OUTPUT HEADING

WRITF(IOUTPR,1D8D) lOBO FORMAT('l 1

9 90(t>:=')/ 1 1 ,37{'* 1 ), 1 RlJDGET OliTPtJT 1 ,37( 1 >:<1)/ 1

+ 90( '*' )/////) DO 16n M=Kl,NINP,K2 J=M XIN=(J-1)*XINC IPAGF = !PAGE + 1 IF! IPAGE.NE.1) WRITE( IOUTPR,1130)

C WRITE INDIVIDIJAL BUDGET OUTPIJT HEADING WRITECIOUTPR,1090)

1090 FORMAT( 1 '• 15X,If:WDGET LOCATION = 1 ,4X, 1 LOCATION NAME + 1 ,4X, 1 ALT-NUM 1 ,5X,•CnST 1 ,6X,IRETURN 1 ,4X, 1 ACClJM RETURN')

WRITE( IOUTPR,1100) XIN 1100_ FORMAT( 1 0 1 ,6X,Fl5.2)

TOTCST = 0 TnTRTN = 0 no ISO L=I,NSTG ·I=NSTG+I-L K=NOO(!,J) KK=NOO( !,,1)-1 TOTCST = TOTCST + CC!,K) TOTRTN = TOTRTN + RCI,K)

C WRITE 1-TH LOCATION INFORMATION---TOTAL BUDGET OF M INCREMENTS WRITE( IOUTPR,l110) LOC(!) ,CXLOC( I ,_IJ) ,JJ=1,S) ,KK,C(!,K),

+ R(I,K),nRET(I,.J) 1110 FORMAT(' 1 9 24X,I4,9X,5A4,~X,I4,2Fl2.0 9 4X,Fl2.0)

CALL xnUT(I,IST,XIN,K,KICK,XINC,C,NlflC) J=IST XIN = XIN-C( !,K)

1'50 CO~ITINUE

r; WRITE TOTALS WRITE( IOUTPR,ll20) TOTCST,TDTRTN,ORETINSTG,M)

1120 FORMAT( 1 0' ,29( '*' ) 9 1 TOTALS •,29( 1 •:• 1 ) ,2Fl?..0,4X,Fl2.0)

lAO CONT !NilE 170 \4RIH( IOIITPR, 1130)

ll30 FORMAT( '1') lAO CONTINUE

RETURN END

OYNA0610 OYNA0620 OYNA0630 OYNA0640 DYNA0650 OYNA0660 DYNA0670 OYNA06RO OYNA06qo DYNA0700 OYNA0710 OYNA072D nYMA0730

•• DYNA0740 DYNA07'i0 OYNAD76D OYNA0770 OYNA07AO DYNAOHO OYNA0800 DY~IAOA!O OYNAOAZO DYNAOA30 DYNA0840 OYNA0850 DYNAOR60 DYNA0870 OYNAOAAO. 0Yfi.IAOR90 OYJ\IA0900 0Yfi.!A0910 nYJ\IA09ZO OYNA093D OYNA0940 OYI\!A09SO OYNA0960 OYNA0970 OYNA09AO f)YJ\IA0990 OYNAIODO OYNAlOlO DYNA\020 nYr'IIA 1030 nYNA1040 OYNA1050 flYNA l OM OYNA1070 OYJ\ItdORO OYNA1090 DYNAllOO OYNA1ll0

22

r. c c c c c

SUBROUTINE XOUT

SUBROlJTINf XOUT( 11IST,Xlr\.1,KrKICK,XINC,C,NLOCl

THIS SIJBROUTINE CALCIJLAHS THE' OUTPUT STAH NUMRER RESULTING FROM THE INPUT X!N AND SAFETY MEASIJRF K, IT ALSO DETERMINES THE COST OF A PARTICIJLAR SAFFTY MEASURE CORRESPONDING TO STAGE I.

DIMENSION CINLDC, lll OUT~ X IN-C I I ,K) IF ( OtlT l 10, 20,20

10 KICK~! I ST ~ 1 GO TO 30

~0 KICK~o IST~IotlT/XINC) + 1.5

30 RETIJRN END

XntlTOOlO XfJIJTOO?O XntJT003() )(()(JT()0t+0

xnproosn xntnnohn XntJl007n XniiTOORO XntJTII090 XOIJTO l on XOIITOl lfl XntJTOl ?0 Xrli!T0130 xr-n JT n 11.- n xntJTOl'?O XCII !TOlAr) XO!IT017(l

23

APPENDIX II

V ARIAIILE LIST

24

V AIUABLE LIST

The following is a description of the variables used in the main program and in subroutines COSBEN,

DYNAM, and XOUT. Variables preceded by * are part of the input data.

VARIABLE LIST FOR MAIN

INN Device number for local card reader (specify in MAIN)

IOUTPR Device number for local printer (specify in MAIN)

*TITL(K) Title of run

*XLOC(N,K) Alphanumeric array containing location name for N-th location

*LOC(N ,K) Integer array containing reference number for N-th location

*NDE(N) Integer array containing number of alternatives at N-th location

NLOC Maximum number of locations to be considered.

*NSTG Number of locations

*XINC Increment size

*XINP Number of increments into which budget is divided

*Kl Starting budget for printout (in number of increments + I)

*K2 Budget printout intervals (in number of increments)

C(N,I) Cost of l·th alternative at N-th location

B(N,l) Benefit of I-th alternative at N-th location

25

V ARIAIILE LIST FOR COSBEN

The fol!OV<1ng variables are stored and kept throughout tbe entire program execution:

NUMBER Number of locations

XLOC(N ,K) See variable list for MAIN

INN See variable list for MAIN

IOUTPR See variable list for MAIN

KIK See variable list for MAIN

LOC(N ,K) See variable list for MAIN

NDE(N) See variable list for MAIN

*RATEIN Present interest rate (decimal)

*RATEGR Present traffic volume growth rate (decimal)

PWC(N,I) Present worth cost (including exponential growth factor) for 1-tb alternative at N-tb location

PWB(N,I) Present worth benefit (including exponential growth factor) for 1-th alternative at N-tb

location.

The following variables pertain to each location and tbe values are destroyed after cost-benefit

calcnlations are made:

*SEV(J,I) Real array containing tbe following accident history for tbe 1-tb alternative:

26

SEV(l,I) --- Total number of accidents

SEV(2,1) -- Number of fatal accidents

SEV(3,1) -- Number of nonfatal injury accidents

SEV( 4,!) - Number of property damage only accidents

*CSEF(J,I) Real array containing the following cost and effect data for I-th alternative:

B(l)

XMAIN

BNCS

BNCM

FWBC

*ALT

NALT

*NCAU

*TIME

*NMO

CSEF(l ,I) -- Initial cost

CSEF(2,1) -- Life (years)

CSEF(3,1) - Maintenance cost per year

CSEF(4,I) -- Effect (percent reduction) on cause (J - 3); J = 4, 5, ---

Real array containing total benefit for the I-th alternative (calculated neglecting economic

and volume growth factors.)

Total maintenance cost for the 1-th alternative (calculated neglecting economic and volume

growth factors)

Benefit-cost ratio for the I-th alternative (calculated neglecting economic and volume

growth factors) excluding maintenance.


growth factors) including maintenance.


growth factors) including maintenance and using an exponential growth rate t"llct<lr and

the present worth method of calculating costs and benefits

Number of alternatives

Number of alternatives

Number of accident causes

Time period of accident history (years)

Month of investigation

*NYR Year of investigation

LIFE Life (years)

VARIABLE LIST FOR DYNAM

The following variables are described in the variable list for MAIN. All are passed as arguments

into DYNAM:

*C

*B

*LOC

*XLOC

*NDE

*NSTG

*XINC

*Kl

*K2

*NINP

*NLOC

The following variables are used for calculations at the I-th stage:

K

1ST

NDEC

R(K)

BCRAT

Stage of investigation

Alternative at Location I being considered

Number of increments that would remain if K~th alternative, Location I, were chosen

at Stage I

Number of alternatives + 1 (Location I)

Return from K-th alternative (Location I)

Benefit-cost ratio for K-th alternative (Location I)

28

XIN

NOD(I,J)

ORET(I,J)

TEST

DUM

KICK

TOTCST

TOTRTN

Variable budget ((J - I) increments)

Integer array containing best alternative from I-th Location given (J - I) increments to

spend at 1-th stage

Real array containing optimum return for spending (J - I) increments at 1-th stage

Return at 1-th stage from K-th alternative plus optimum return for remaining budget at

(I - 1)-th stage

Maximum value of TEST

Integer containing "0" if there is insufficient budget left to do K-th alternative (Location

I)

Total cost of chosen improvements

Total return from chosen improvements

VARIABLE UST FOR XOUT

The following variables are described in DYNAM. All are passed as arguments into XOUT:

I

IST

XIN

K

KICK

*XINC

•c *NINP

OUT Budget that would remain if K-th alternative, Location I, were chosen at Stage I

29

APPENDIX C

PROGRAM FWW CHART

30

c---------"-~--------,

I NJ:NP•401 I L---------- --------'

' ' ' NJ:NP • NUMBP.~ 6F I NCR BM£1/TS- --MAX IMUK BUDGET F.QUAI.S ~TNP*XlNC

c--~~--------,

I Nl.OC ~ b4 I L-----------~

' I ' NLOC MAXI~!IM NIJMBH~

OF LOCATJ:ONS

' ' I G_J r---------------,

I ~N ~ :, I

' lOUTPF ~ 6 I

l~N, J()IITPI' • LO~A L INPIJ'r A~D OU'l'PUT D~VlCE NUMBEI-:~

I I

' "" / nAD 1'<'0~ ULV /

/ l~~ / / VIA fUk~A·r

1 Tn<lO / / IN'l"O TIIC' Ll.;T /

LIST 1'ITI,

/ fJiliTE 1"0 OEV / I IOUTP!I /

I VIA fORMAT / / 1010 I

/_ PR(JM TJIE LIS'!" I

~orr. O?

,.IS']" TITL

PROGRAM MAIN

r------------------> o

' ' ' "" ' --~----- I I READ FROM DEV I

I --- fiiti 1 / VIA FOTI/!A'r I

I 1020 1 I INTO T!IE LIST I

' I

' ' ' ' ' ' ' I I ~UTE 0~ I .

LIST • NSTG, XINC, K1, K2 • * * ••••

I

' I ' I I

'" ' c-------~~----, I I I ll I 14 I COSBEN I I· I (C,B,XI.OC,LOC, I 10 1 NDE,NSTG,NI.OC, I 11 1 xruc,.;;wN, 1 I I IOU'l"P~,K!K) I t_______ _________ _. I

I I

' ' L---------->

T~UE

c----• ' I ' I

KI K • f.Q, 1

I I FALSE I I

' ' I I I I I I I I 13 I r---------------, I I " I 19 OYNAI! I j. (C,B,LOC,XLOC, I jO ND~,NSTG,U~c, H I !1 K1,K2.~lNP, I I NLOC,OI!J':T,NOD, H I I IDUTPR) ) L------------------1 I I L---------> I

10 I NOTE 14 • * * • CONTINU~

* EXIT

CHL EXIT

. ' .

31

I C05BEN I

Q2,10--->rt

'l'IHS SU~~OOTHIE CALCIJLATF.S PRR5ENf ~OH'ill COS'l'.'i AND BP.~fPI'fS

~SSOCIATED WITII EACH AL'hi<NATlVE ~T EACII LOUTIO~

/ N~AIJ fROI'I I UN

/ VU PORNAT I lODD

I I~l'O THL LBT

' '

0' D~V I

I I

I I

1 non 02 ••• * *

usr ~ CFhT, tHi.J, cpnn,

SUIIROliTJNt; C08Bt;N

1i h fUN, :<AT f,(; R r------------------>o •• * * •• ~ •

0'

1 -•RI1'E TO UEV I / lfJUTI'<, I

I Vl,\ f"<lh~AT I 1 l•_ll·l I

I '""~ !'Ill lb'1 /

'<0'1 I l•fl

:.1ST Cf',\': t:H.l,

<,\".FD, ••• * ••

"I<D1'~ Td1 ~A!ilC P•\IIA~;.T~i'S t:lJNSj'ANT rn~ ~.~~~

' ' ' ' ' ' ' ' ' ' ' "' ' r-------------------,

I NIJ~BF.R 1 ! I I I KB - 0 ! L-----------------J

' ' I

BF.LOI." .LS TH~ INPUT W~!CH IS HECIJTED FOR ~ACil ACCIDENT LOCAUON.

I Oi. 14--->1

' ' ' ' ' ' ' ' I I

' ' I

' '" I

0. I ---------------- !

I READ FROM DEV I / IN~ I

/ VIA FO~MAT I I 10ZO I

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N01'R 01 ! * •• "

LIST N01, (X LOC { NU~BER, I) , I ~ 1,51, TIME, NMO, NY~, NC~U

**.*"***

I I I I I I I I

oo I .----------~~

I LOC(NOHBER) - N01 l "-------------J

I I I I

I I L---------"

L) ~

101 • ---------------------> ~

I (~/•I I I I I I

20 ! NOTE 10 . " .. ClJNTIIID~

" I I

I W~ITJ:: TO PEV I / IOIITPR I

I VH FONHAT I I 1030 I

" I I

I WRIU TO D~Y I I IOOTPH I

I vu fORMAT I 1 1040 I

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HC~O • EQ. ' ·~ . I I

'~~ I

• !FALSE ' I "' I I '" I I I '"

I II RITE TO DEV I I IOUTPR I

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NOTE 15 ... LIST N01,

(XLOC {NOH!lEI!, I), I - 1,5), TINE, 1!110, HHI, NCAU •••• * •• "' ..

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• • • ~0

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CONTINtiE .... *

n r------------~,

I HUIIEER HU~B~k - !

I ' > L--------------'

'" TRUE .----· NOHBBil • BQ.

I I I I

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NSTG +

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J NSTG : ····.;··· '''~-~O~;~g•->: NOTE 22

'~

..... COIITIH!IE

• • * "'

• EliT

32

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I WRITE TO DEV / I JD!JTPR I

/ VH fORMAT I 1 1060 I

/ FRO~ T~f LI~'l' /

NOTE 02

LIST tm1, (lLOC(NU~DEII,l), J:" 1,5), TI,~E, ti~O, NY~, NC:AO

~******** I

Qq,15--->l qQ I l!OTE OJ ...

CON~'INUE

I I I

SECOND CHD Irlri!T FOR I:ACH Cflll'ICH LOCATION (SJ:V~Pl'riRS)

I I

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m

.-----r----------, I HAU ~ ALT/10. + I I ,1 I I I I NDEJNU,~SER) = I I NUT I L------------------J

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CON'l'!NOE * • * * •••

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I I ' 12

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I I I NOTE H •• * •• * ...

BEGIN DO LOOP BO I • 1, NCAU * ···-+-.---.. -.-~

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/ VIA fOR~AT I I 1 1oo I

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ToTJ "' TOTJ -- SEV iT;:w-

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1 j I I I I I

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L------.-------iifDOFOO'-. LOOP?

I I I I I I j I I I I

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F ruiirffl" 1 / VU POHRA'l' /

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' I N01'll 1~ .. --..-.---.:--.--..- -.. ,...--.; . * LIST "' TOT1, -.,-~-TOT3, T0T4 ••• ~ * * ••••

' I j

IIIPOT HEIT SET OI' MALT C'.i.!1ifS; O!!E--Jioli EACH ALTEBUTIVE

I I l n

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BEGIN 00 LlrD"P 110 I"' 1, IIALT ....... " +

I IIUD PIIOH DEV / I 111M I

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/ INTO THE LIST /

HO'l:E 25 ... LIST "' WOJ,

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I I I HOTB 28 ... ·~····-LIST ~01, 1103 * ......... - ...

----;r====o'=':"= 2~" I KIK- 1 I ~ '

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I I I W~ITE TO DEV /

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/ 1210 1

) r~-----~--->o I I I fl<f ) NOTE Ou I ,.--------~

"' ~ + ,-.- .--1<--.----.----,.- j f rr~t· n 1 DEGIN DO LOOP ) ) PWC (NU~BER,l + I

170 I ~ 1, NALT * I I 1)1IINC + .5 I ******•*** I I I I I I P~C(NU~IIEII,l + I

I<~----~--)---, I 11 ~ IPIIC"'XlNI' I 0'5" 1 I r

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"' .-----~------~

Y = (1. + I ~ATEC.R)/(1. I

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I I I I I

I [ I I I I I I I I I I I

'" r----_c_ __ ~ I PWHC I PWB(HUMBEII,l I L__11L!',!I_!_Cjll_ll~BER,I + I L ___ ~ _ _____!

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/ 1220 I FROM THE LIST / ~~~~MUM~~_:_~;

[ -----------

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I / PIIC(NUMBEH,I + I 1) CSEF{I,1) + I J!W!!_A!!l ~-"CC-~~==

I I I

HOUND PRESENT IIORTH '::OSTS TO Nf.!REST

I I I 1 I I

I I L----------'

1 I I I I I I I I I I [ I I I I

' I I MOTE * ,._ ,._ •• ,. ••

LIST " I, PWIIAIM, P_W_(;(~UIIB~R,l - --D-.-PWH(Nu!!a~R,t

1), PWBC

* * • "' "' "'

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I "' ~· RND Of "' LUUP1

. I YES

J I I I I I "

~Bil'Ei ;;:-!lUrt~Eli'"+CJ I J I ~--------'

~. 06 •

... 10

35

SUBROUTINE DYNAM

0{. Ll---)11 r-----------------> II

T'llS ~lli<:•ciT'nNO USE:O "JlYI:A~IC f·iUJI;Cht.lf~li"

JJ FUH ;'il~ O~I'J,~,\L

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()ti 'OllH ~y

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r------------------, I 'L~'I • 0 I I I 1 v;-~;- v.o I "-------------------'

I 1 ,,.r rr~ T·J o~v ;

I lOUTh< / I VH FOnAT /

1 1130 I

I :.n'l'~ Ti1 IJ~~ I I !OIJTPi' I

/ vu i'0''~/,'i / / IGL•O /

'" I I

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I o!i!l'F. 711 ~C:I'

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------------ I / >.liTH ~o ti!:V / I

I IOIII'PK / I I VH POh-~AT / I

1 111.10 / I I !·iUP T.ll- J.l:it I I

-------------- I I I

11•11~ G" I

Ll.~" J,,JC ({),

~ •• :;~;:1'~ •. I I I I I I

I 10 * lU

1-:.iD 111-· ~,;

!.00~?

I~ L~

I I I I I I

I I I .ur;.: IO llEV I

I fO!I'Hli 1 / VI;. FOHH'l' /

/ 1J40 /

I I I -~rt,; TO Ui.\' I

I WU'f?R / I VH l-'li<~AI' /

I 1J~U I

I I I

i'BJ '"10 l1P'I'HAL ALj'~!CIAriVO AI 'fcirl 1-'!'11 LOCA'ilO~ oltll J !KC<~I'.P.'Ij'S

I1VA TLi•~l.c:

'lli'.'E 13

Jl'_,;r,; ue LUll~

14!1 I 1, .'IST!i

I 11,.11--->1

I '" r-------------------, 1 .u~c • NJ<.(ll ~ 1 I I I I ;r (1) • G, I L------------------'

:n;r:r ~o Lo8~ 2U JC ~ L, IUZC * .......

I I<----------,

lfl 1 1b l ,------------------, I

~--:~~~~-=-~~~~~=!_____~ ! I

' I n I

I I

:;e 1 ~~J IH D<l •----'

LOOP?

l r:~

IL,; ctl or. Ln•lf IV lC ~ 2, ~U~C

• ~ * ....... .

/I'J.U 1

36

~-~-oq, T!J•-->u I 1 a 1 I r-------------------, 1 1 H''11 ~ rc - 1 1 I I I I I flCI•AT ~ I I I ~'(IC)IC'(I,IC) I I C-----~-----------J

I I I I I I I I I

02

I <:>I'i 0: "J'G DEV I I IUTI?~ I

I V'A fUR~H I I 1'100 I

I F~o·1 •riiY u~·;· I I -----~--------

1 I I I I I I I I I I I I I I I

I I I ~OT~ OJ

• ~ " * .. Ll.'il' U:JL (1), (XT.•JC (T ,J) ,.1

1,5), lC.-1, C(l,IC), '(H'),

JL>A'I

'

1 ~o C----->" i:N:J ,)j' ~n

LUO!'?

~,. ~('(>r

J.lf ~ 0

1: : T ;i• I

11 or.---> 1 I

r-------------------, I •:·. (•! I I 1)"';·.· I I I I I I 1 G1' JL~r-'> 1 • 1

I I I 'I:'~'-" ',.)j. (! J + 1 I L-------------------'

')};•' ···l< .\l, ,. ,, , : l 'i.,- -,,u I' , . -- ,\T u c.\:· r•.: 1:I ~ "i •; J-1 1 -:c.,., ....

,. ~ T'J,. C·t'- ·: ( '• ,I)--

1 I

._.:' r . .,

;j" .l" l.LI'l~

12) l' ,._,

l 1, "'•·- --> 1

' I' 1'1 , •. .,, ,, I ,, '

r------------------ >n

1'1

(-IC•) :ILL'

1'1

••• 1~0

r--~-------------->" I ~o 1 ,;or~ 1~

I *•~••~ 1 co:;n~UE . ..

I')

' - ' *---~---------~- --><::

·----------~-,

I 'l'tSI' • ~ (K) t I I o~l:'T (L - l,IS·q I

(-10) L--~----~---~-~

,, • 11. 0_1.

r-------------------, I 'l'r.;;•; • -i (K) I ••• llO

L---··-------------J

• 11. <11.

37

I ,,., 1 •1•--> 1

I Ql

(~ jt) ,-----* p fl.~ - 1"~J"i')

I I

')~

,-----;~~-=-~~~--~ I I 1 oewr (I,J) n1:~' 1 I I I >ma(I,,J) I

I <-----------------~

I I •-----------> I

10<1 I OJ

,, , 1Hl

"" I

ll.OJ--->1 I

·---------'---'' . ' '

I 10,11--->1

I 121)

""

'

'-~~ or oo L•JOP7

I YES I I I I I

!

':~;, <H DlJ to on

I I I I I I I I I I

'" I ~-, I

I I I I I I

I • 1U • I • oa • I

I I I I I I I I

~o I ··-, I

I I I I I I

I H~ • Hl • I

"" " ;:ou "' "" L()l}~?

I YES

I I I I I I

•. ·)6 • l I I I I

oc •-,

I I I

' ,.

'"

I I I I I I I I I I I I I

·-----------------,

I I I I I I I L _____ ~:~~~-~ o __ _1

I I I

Oo/l'!'~ ~AI~ ~UDGJ::T

OIUPU1' HEADING I I I

I I I I I I I

J9 I -------------- I

I I I I I I I I I I

/ ;;J<I'f~ TO D~V I / lOU'r?.< /

/ VH fDKMNr I 1 1 u~u 1

I I I 10n * •••

>0 I ' ' I

I I I I

A~t;I~ U() LOOP

160 .1 • Kl, H~P,

" I I

l<.Ocl--->1 I I 11 I

~-------;-:-~-------~ : I I I I UN ~ (J - I I 1 1) •XI tiC I I I I I I !~Ao.;E ~ IPAGE + 1 I I •--------------------' I

I I I I .. ____________ ,

fALSE • ~ r---* IPAGE .~E. 1

I I I ' I I TRUE I I I I I I I I I I I 1 13 I ----------·-1 /- I

. .l I ~Hn to OEV I I I IOUTPR_ / ) / VIA FD~HT I I I UJO I l -----------1 I ·-----:>!

I •RITt.: lM!llVl!lllAL bODGET OUTPUT HEADING

I I I '"

I I I ~RlTE TO DEV I

I IOUT!'R I I VH POR~AT I

1 1090 1

I OHTE TO D~V I I IOUTPN I

I VIA FORMAT I 1 1100 1

I FhO~ TilE l.lSr I

. ' ' LI~T

' .. NOTE . ' "" ' ' '

n ,-------~

J ',rOTCST ~ 0 J

I I l TOTRTN ~ 0 l '-------~

I I I NOTE * .....

BEGIN DO LOOP 150 L ~ 1, NSTG

* • * ...... * I

12.05->t I

"

r----------• II RSTG+I-1 I I I I X = NOD(l,J) I I I I XK ". NOD(I,J) - 1 I I I I IOTCST ~ TO·rCST I I C (I, K) I •-------~

'" r-----------, I TOUtH ~ 'WTRTN + I

~-----~~_j I I I

ORlTE 1-Til LOCATION l N FORMATION---TOTAL B!IDG~1' OF H INCN~MENTS

I I I I I

112.01

38

11, 21---)(]

I "' I ~·~I'I"E TO U1V I

I IOUTPI! I I VIA F<Pl~.AT I

1 1110 I I Ff<OH Jll~ U!)r I

UO'!'F. G2

J.I,;1 J,UC (I), (~r.or (1 ,.1.11 ,J.l • 1, ~I , 1:~, C (I ,KI , f;(l,KI, o,d:T(I,J) ~ . .. .. . . . . .

c-------------------, 11 I II I~ I i:OU'I' J. I (l, I:i'l', .nN, ;;, 10 I f:lLO,HN~·,c,

11 1 :we) '-------------------'

r-------------------1

1 ,J • I ~;'I' I

' ' 1 u~•t!'l- I I r (I,~~ I '-------------------"'

ii'J

'''' <IC no •-, LUOr? I

. ' ' 1 Y F.:; , 11 I , J<;

I

' ' ' I

;,;nJTI ·r·l'l'ALS

' ' I 'Jb

I ~RI'lF. ·ro UiiV I I IOU7PE> I

I VJA l"UlL~A·: I I 11LO I

I FE0:1 Til~ t.IS1' I

'!OTE 07

LJ:S'l' • 'tOTCST, TU'.'i<rN,

* ~~~;T ~~~-r~·;> *

[H)

EN:' ''" no •-, ~OOP? I

I

' . IHS • 11 1 • 11 I I I

'

c-------------->u 170 I

, _________ , I / I •.;nl'rE ·~o DU I

I IOU'l'l'l• I 1 Vlt1 t'u~:H:- I

I 11 JO I

' I l~U I XOTE 1<)

• * •• * t:uNri~llE

39

SUliROUTIN~; XOUT

),O!If

1U • .,~*-->o

'['liS ,:ICI,,Uii'~Hf,

CALl"JLA ~1.5 'lll8 ,);)iJ'IIi'

!I~SU! l'lClG f~O.~ r,,. ·c;rll'i' Xll• A:D 5!.1-0:1'~

~ ~/, SiJ!i' I'l' ~I.SO DHI ~~;~

COST Of ,\ <'E<rlCIJLil~

s~n.n H:.~»Ul<l' C()f PC:-if(HJJ)J!~'' 'tO

,;rAr.i·. 1,

r-------------·------, I <'II.' ~ ;;u - I I r (l, ~~ I , ______________ J

icl/+) OIT l ._--------------------.. -->11

r---------------, <'rCK (F I

' ,_, T~T I

'" ,------------------, I ~ Ir « 1 I

' ' I Lh I

(OU'l'/AT'1l) 1, ~ I

' ,----------------->I I h I ~~

' I EXI c

' ' ' '----.. ---------' I

' ' ' , __________ ,

40

APPENDIX D

INPUT CODING INSTRUCfiONS

41

INPUT CODING INSTRUCTIONS

The following is a description of the input required to use the program presented in APPENDIX

A. It should be pointed out that the input and output device numbers, INN and IOUTPR, respectively,

must be defmed in MAIN. Also in MAIN, the following dimensions must be specified: the dimension

of NDE and LOC must be the same as the first dimension of XLOC, C, B, ORET and NOD; all of

these dimensions are equal to NLOC. The second dimension of ORET and NOD must be equal to NINP.

The variables NLOC and NINP correspond to the maximum number of locations and budget increments,

respectively, and must be defmed in MAIN. Any capitalized term refers to the variable exactly as found

in the program. All integer quantities must be right-adjusted. Real numbers should be punched with

a decimal or right-adjusted.

CARD I (Type A)Title Card

I. Title of run: TITL

In Columns 1-80 place any alphanumeric symbols desired

CARD 2 (Type B) Printout Card

I. Number of locations: NSTG

In Columns 1-4 place the number of locations actually being considered (integer number)

II. Size of increment: XINC

In Columns 5-10 place the size of the increments into which the budget is divided (real number)

III. Starting budget increment: Kl

In Columns 11-14 place the number of increments (+ !) corresponding to the first budget

desired printed out (integer number)

IV. Budget increments: K2

In Columns 15-18 place the number of increments between successive budgets desired printed

out (integer number)

CARD 3 (Type C) Accident Cost Card

I. Cost of fatality accident: CFAT

42

In Columns 1-10 place the average cost of a fatal accident (rea] number)

II. Cost of nonfatal injury accident: CJNJ

In Columns 11-20 place the average cost of a nonfatal injury accident (real number)

III. Cost of property damage only accident: CPDO

In Columns 21-30 place the average. cost of a property damage only accident (real number)

IV. Interest rate: RATEIN

In Columns 31-40 place the .present available interest rate (real number)

V. Exponential growth rate: RATEGR

In Columns 41-5 0 place the expected traffic volume growth rate (real number)

Note: Card types D, B, and F are repeated for each location.

CARD 4 (Type D)Location Card

I. Location reference number: LOC

In Columns 1-4 place location reference number (integer number)

II. Location name: XLOC

In Columns 5-68 place the alphanumeric name associated with the location

Ill. Leave Columns 69-71 blank

N. Time period of accident history

In Columns 72-75 place the time period (in years) of the accident history (real number)

V. Present date

In Columns 76-77 place month (integer number)

In Columns 78-79 place two last digits of year (integer number)

VI. Number of causes: NCAU

In Column 80 place the number of accident causes (integer number)

CARD 5 (Type E) Severity Card

I. Location reference number: XLOC

In Columns 1-4 place location reference number (integer number); this should be the same

as on Card 4

43

II. Severities for CAUSE 1 (real number, right-adjusted)

In Columns 5-6 place number of accidents attributed to CAUSE 1

In Columns 7-8 place number of fatal accidents attributed to CAUSE 1

In Columns 9-10 place number of nonfatal injury accidents attributed to CAUSE I

In Columns 11-12 place number of property damage only accidents attributed to CAUSE I

Ill. Repeat II for CAUSE 2, CAUSE 3, etc., continuing on same card; use integer fields of two,

i.e., Columns 13-14, Columns 15-16, etc.

Note: Maximum of eight causes

IV. Number of alternates: ALTR

In Columns inunediately following last CAUSE place tbe number of alternatives (Real number,

right-adjusted)

Note: Maximum of ten alternatives.

CARD 6 (Type F) Alternative Description Card

I. Location reference number: LOC

In Columns 1-4 place location reference number (integer' number);

tbis number should be tbe same as on Cards 4 and 5

II. Cost

In Columns 5-11 place initial cost of alternative (real number)

III. Life

In Columns 12-13 place estimated life (in years) of alternative (integer number)

IV. Maintenance cost

In Columns 14-18 place estimated maintenance cost per year of alternative (real number)

V. Effect on CAUSE 1

In Columns 19-21 place the fractional reduction of CAUSE I by implementation of alternative

(real number)

VI. Repeat V for CAUSE I, CAUSE 2, etc. continuing on tbe same card using Columns 22-24,

Columns 25-27, etc.

Reveat Card type F for each alternative at given location. Last card of data deck MUST be blank.

44

APPENDIX E

SAMPLE INPUT AND OUTPUT

45

.., "'

SAMPLE OF INPUT DATA

II 21 31 41

TEST I RUN I l1YNAM!C PRIDGRAMM lNG

'""· 0 i " ., 450DD. 2100. ~oo. ID .10 10.04 163-25 10.9 I 4 I 21 l 10 10 100. ol2

REPEAT CARD TYI"ES D, E, AND F FOR EACH LOCATIOIII LAST CARD IS BLAIIIK

51 61 71

1.0

SOl COLUMN NO.

lh91

CARD TYPE A CARD TYPE B CARD TYPE C

CARD TYPE D) LOCATION CARD TYPE E NO. 1 CARD TYPE F

SAMPLE OF OUTPUT

TFST RUN 1 DYNAMIC PROGRAMMING

NEG UTILITY fATALITY• 45000. INJURY• 2700. PRP OM= 400. INTEREST RATE = 0.100

REF NO 1

EXPUN~NTIAL GROWTH RATl • 0.0~0

COST-BENEFIT OUTPUT FOR LOCATION I (see Table 2 for summary of all 61 locations)

63-25-10.9

ACCIDENT HISTORY 1.00 YfARS. MONTH l,YEAK 69, 1 CAUSE.

ROADWAY CAU~r 1

TOlALS

TACt 4. 4.

ALTERNATIVE COST 1 1000.00

NFAT NINJ 1. 2. 1. 2.

NPRO 2 .• " ,.

LIFE MAIN COST 10. 100.00

BENEFIT/COST ANILYSI~

1 (ie02

ALTERNATIVE COST 1 I 000.00

HNlfl 1 HJ24C. vO

btNlFil/CCST 1C•.24l0

BENEF'Il/COST ANALYSIS, MAlNTFNAtKL. INCLUDED

Al HRNA T I VI: 1

MA I NT EfM'''C t lOOO.LiO

TCTAL COST 2000.CO

LFNEF lT /COST 5.1200

BENEF!l/CO~l ANALYSIS, Mt.II•TENANCE !1KLUlJfO ***PRFSENT WUKTH MFTHCtD***

AlllkNATlVt 1

MAlNllhlM>iC!: f] ".'<5

TOTAL COST l5oc.c,o

HNH IT 761'!'.84

bU\EF IT/COST 5.0R

*************•*****•********PARAMEllR VALUES*********************~******

----------------DUTPUT-----------------LLCATION~,---INCRFHicNT---l.O\ifR LIMlT--INCkH1tNTS PER STEP

Dl 2S0.00 41 40

47

ALTERNATIVES SELECTED FOR $50,000 BUDGET

BUDGET LOCATION LOCATION NAM~ ALT-NUM COST RETURN ACCUM RETURN

50000 .. 00 61 63-25-10 .. 0 0 o. o. 868692 ..

60 22-60-26 .. 2 0 o. o. 868692@

59 30-60-16 .. 4 2 500 .. 3338 .. 868692.,

•• 82-31W-lo2 1 500 .. 1921 .. 865354 ..

57 51-4l-2o .. o 1 500 .. 10792 .. 863433 ..

56 a2-3lw-2 .. 2 0 o. o. 852640 ..

55 102-25-11 .. 9 1 500 .. 3934 .. 852640 ..

54 63-25-10 .. 4 7 2250 .. 19645 .. B4B707e

53 47-31W-23.,3 0 o. o. 829062 ..

52 1o-usoo-u .. 3 5 1750., 6021 .. 829062.,

51 79-641-12 .. 5 1 1000 .. 26532 .. 821040 ..

50 79-641-12 .. !1 3 1000 .. 11152 .. 794506 ..

49 10-60-9&2 3 750~ 5860 .. 777356.,

48 30-60-15e0 0 o. o. 771496 ..

47 t:I2-31W-1 .. 0 0 o. o. 771496.,

46 ti2-IJS60-l2 .. 7 0 o. o. 17149be ., az-usoo-12 .. 1 I 750., 9097 .. 771496G

44 70-60-11 .. 4 0 o. o. 762399 ..

43 51-60-20 .. 3 l 75(JG 86411 .. 762399.,

42 7Q-641-13 .. 0 0 o. o. 675988.,

41 10-60-8 .. 3 0 o. o. 675988B

40 82-60-12 .. 3 l 500 .. 11078 .. 675988 ..

39 120-60-12~6 I 500 .. 9008 .. 664910~

30 84-68-18 .. 5 1 1000. 34486 .. 6 55902 ~

37 82-31W-l .. l l 500 .. 17168 .. 621416 ..

3t> 051-41-20 .. 0 I 500 .. 4699 .. 6042480

30 51-41-20 .. 0 0 o. o. 599549~

34 41-115-155.,6 1 4000 .. 152094 .. 599549 ..

33 63-25-10 .. 9 0 o. o. 447455 ..

3Z 5 6-lb:iKTP-130 .. 1 1 1500., 11111~ 447455~

" 72-641-8 .. ~ 3 2000 .. 56100 .. 435743 ..

30 54-4lA-12 .. 2 3 75(, .. 8513 .. 379644 ..

29 63-25-10 .. 8 0 o. o. :nu31~

28 73-62-16 .. 9 3 3000~ l2286e 371131 ..

27 73-45-o .. s 0 o. o. 358846e ,. 13-62-17 .. 8 0 o. o. 358846 ..

25 41-22-11 .. 3 4 1000 .. 11530 .. 3581.!46 ..

" ll4-31W-16 .. l 0 o. o. 347316 ..

23 30-54-12 .. 6 1 750 .. 4290 .. 347316 ..

22 4 7-31\<j'-26 .. o 1 2250 .. 24654 .. 343026 ..

2l 37-l;:_7-8.,J 0 o. o. 318372 ..

zo 106-6-3 .. 2 0 o. o. 31e:n2 ..

19 b3-25-16 .. 0 0 o. o. 318372,.

18 !l4-31W-16&2 3 750 .. 4071 .. 318372 ..

n 24-68-9 .. 1 0 o. o. 314301 ..

16 54-41A-12 .. 4 3 750 .. 18040 .. 314301 ..

15 79-641-18 .. 9 1 750., 3548 .. 296261 ..

14 39:-171-63 .. 7 1 4500e 26029 .. 292713 ..

13 30-60-4 .. 1 0 o. o. 266684 ..

12 7<:-641-8 .. 5 1 750 .. 40588 .. 266684 ..

11 30-b(J-15 .. 2 3 5000., 3H!b6o 22b095.,

10 1UZ-25-54 0 o. o. 194229 ..

• 102-25-9 .. 2 0 o. o. 194-229.,

8 20-51-1 .. 1 0 o. o. 194229.,

7 4&-421-14 .. 4 0 o. o. 194229 ..

b 73-4-5-7 .. 2 3 750 .. 6726 .. 194229 ..

' 73-45-bmS 0 o. o. 187504 ..

4 54-4-U-12 .. 0 I 500 .. 9792 .. 187504.,

3 30-b0-15 .. 2 I 2000 .. 11385 .. 177712 ..

2 30-60-15 .. 1 I 4250 .. 158707 .. 166327 ..

1 63-ZS-10 .. 9 1 1500 .. 7620 .. 7620 ..

**************** TOTALS ***************************** 50000., 868692 .. 868692m

48

optimal highway safety improvement investments by dynamic

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