Minimizing ‘cut and fill’ costs in roadmaking

Minimizing 'cut and fill' costs in roadmaking M Trypia

Once the horizontal alignment o f a road has been defined, its vertical alignment has to be determined. A mixedinteger programming model is presented that determines the height o f the road at specified points over its length, so that the total earthwork cost is minimized, and constraints referring to the desired gradient o f the road are satisfied.

The design of a road from A to B is a complex problem, as social, economic and aesthetic factors are involved, which all have to be taken into consideration. The designer has to determine a route with safe and comfortable driving conditions and balance the construction cost with that of the user. The multitude of factors involved has to be valued to produce a socially and aesthetically acceptable solution. Some cannot be evaluated quantitatively, in which case the decision makers have to make a choice. Once the horizontal alignment has been determined, the most economic vertical profile has to be found. The model proposed here can be used to find the heights of the road at specified points along its length to minimize the total construction cost, while fulfilling certain conditions, such as the desired gradient of the road and the predetermined height of the road at certain points.

MODEL

I

Cost

I

I I

I

ci2

li

xi2

hi

xi~

ui

Figure I. The cost function o f excavation and filling at point i

If hi is the height of the ground at point/, the final height of the road is either (hi + Xil ) or (h i - xi2), where xi2 represents the height of excavation, and x i l >1 O, xi2 >10. Let u i >t 0 and/i I> 0 denote the upper and lower bounds on the final height of the road, respectively (if such bounds do not exist then u i and I i can be taken equal to any large number M). The final height of the road r i at point i, can be expressed by one of the following conditions : either ri = hi + xil

A grid is defined of points 1, 2 , . . . , n through which the road passes. The density of this grid of pointsis decided by the engineer and the distance between any two points could vary depending on the type of ground at which any two consecutive points are located. For example, the number of the points that are specified along a section of the road on highly anomalous ground could be much larger than the number of points corresponding to another section of the road where the ground is less anomalous. Let the distance between two points i and i+1 be ~v', and the cost function of excavation and filling at point i be of the simple form of Figure 1, i.e. the cost of filling/unit is cil and the cost of excavation/unit is ci2. The cost of filling and excavation at point i depends on the type of the ground there, the availability of machinery and equipment etc.

The Athens Graduate School of Economics and Business Science, Greece

0 ~
(1)

Xi2 = 0 ri = hi - xi2

1

xil = 0 I

(I])

0 <<~Xi2 <~ li

Condition (1) holds for filling at point i, and condition (I[) for excavation at point i. This dichotomy can be expressed as one condition, if a binary variable 6i, ai ~/ 0, / 1r is ./ introduced O<~xi] <<.Ui8 i

(1)

O<<-xi2 <-Ii (1 -$i)

(2)

Relations (1) and (2) express that • if8 i = 0, then x i l = 0 and O ~ x i 2 <~/i, in which case there is excavation at point i, • if 8 i =/, then 0 ~
0010-4485/79/060337-03 $02.00 © 1979 IPC Business Press

337

The binary variable'6i ensures that either Xil ~ 0 and

EXAMPLE

xi2 = O, or Xil = 0, xi2 ~ O. The final height r i of the

A road to be constructed between A and B has already had the horizontal alignment determined. Five points along the road's length are selected, where the length of the intervals between these points are X, ,X2 ,?%,~4, respectively, and it is necessary to find the heights of the road so that the earthwork cost is minimized.-the natural height of the ground at these points is h l , h E , h 3 , h 4 , h s , with the corresponding earthwork cost given in Table 1. The upper and lower bounds of the height of the road at these points is u~,/1 at point 1, u2,/2 at point 2 etc, and the maximum allowed gradient is s. To find the heights of the road at the 5 specified points for minimum earthwork cost, a mixed-integer programming problem is formulated

road at point i can then be expressed as

(3)

r i = h i + Xil -- xi2

and the construction cost at point i as (4)

Cil Xil + Ci2 Xi2

Finding the heights of the road at n specified points 4 i = 1 , . . . , n, so that the total construction cost is minimized, can be formulated as a mixed-integer programming problem by minimizing /7

/7

Z = ]~

(Cil Xil + C i 2 X i 2 ) + ]~

i=1 subject to

i=1

(5)

0.6 i

Xil < Ui 6 i

(6)

x;2~
(7)

a,~ (0,z) As the road should not be characterized by 'hills' and 'valleys', it is necessary to introduce a gradient constraint, so that the slope of the road lies between acceptable limits. The heights of the road at the points, i, i+I are related as either ri+ 1 >1 r i or ri+ 1 ~ r i. For every pair of consecutive points i, i+1 two constraints are introduced (ri+ 1 -- ri) / h i <~ S

(8)

(--ri+ I + ri) / ?ti <~ S

(9)

where s (s/> O) denotes an acceptable value of the gradient, and hi (hi/> O) is the distance between the two points i and > i+1 .Depending onthe relation between r i arid ri+ 1 (r i ~ ri+ 1 ),

one of the constraints [(9) or (8)] is 'active', forcing the slope of the road to lie within acceptable limits, and the other constraint remains 'inactive'. For example ifri+l > r i, then constraint (8)is 'active' and (9)is 'inactive'; ifri+ 1 < r i (9) is active, (8) inactive.

Table 1. Earthwork costs Point

C32 X32 + C41 X41 + C42 X42 + C51 XSl + C52 X52

Subject to

for i=1 , . . . , n

Xil,Xi2~O

Minimizez--Cll XII +CI2 X12 4-C21 X21 "1"C22 X22 4-C31 X31 4-

Xll ~ U l ( ~ l

X12 E l 1 (]--61)

x21 ~
x22 ~<12 (1-65)

x31 ~
x32 ~
X41 ~ U 4 6 4

X42 ~ / 4 (]--64)

X51 < U 5 6 5

X52

~
Gradient constraints Between point 1 and point 2

(h2 +x~, -x~2) - ( h , +xu - x n )

IX,
= -(h2 +x21 -x22)+(h, +Xl, - x,2)

IX1 ~
(r2--q)lX, = (-r2+r,)lX,

Similarly, gradient constraints are introduced for the pairs of points (2, 3), (3, 4) (4, 5) Other forms of cost functions could be associated with various points i, along the length of the road. For example, a cost function which is often encountered, is of the form in Figure 2. In this case, an excavation deeper than mi m incurs a constant term d i in the cost. Three continuous variables x i l , xi2 , xi3 are introduced

Cost of filling/unit

Cost of excavation/unit

1

C,1

el2

O<~xil <.ui

2

c2,

622

O~xi2

3

c 3,

c 32

0 ~ xi3 <~ ( l i - m i )

4

C41

C42

5

Csl

c52

~m i

The height of the road r i would be, in this case, either r ~ h i + X i l or r f = h i - x i 2

xi2 = xi3 = 0

(I)

Xil = xi3 =0

(11)

or r f = h i - - m i - - x i 3 Xil = O, Xi2 = m i

~ lI "~ i

.,~ose Xi2

Case (/)

c i = Cil Xi]

Case (11)

Ci ~- Ci2 Xi2

Case(Ill)

ci = ci2 xi2 + di + ci3 x i3 = ci2 m i + di + ci3 xi3

IIl

m i

To express the final height of the road at poing i, r i and the corresponding cost ci, when the above 'trichotomy'is present, two binary variables 6i1,6i2 are introduced

II

hi

Figure 2. A c o m m o n c o s t f u n c t i o n

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the corresponding earth work cost at point i would be

Cost

}di

"

(111)

xtl

ui

Xil
(]0)

xi2 + xi3 ~ l/ (1--6il)

(1 |)

Xi2 ~ m i 6i2

(12)

computer-aided design

xi2 <~mi ( 1 - 6 i l )

(13)

xi2 >~ m i 6i2

xi3 <~ (/i--mi) 6i2

(14)

Xi2 <~m i (1--6il)

Xil , xi2 , xi3 ~ 0

xi3 <~ (li--mi) ~i2

If6i1=1,6i2 =0 : O<~xil <~ui, xi2=xi3=O I f 6 i l = 0 , 6i2=0 : relation (10) b e c o m e s : x i l = O

Case I Case II

Similarly xi2 + xi3 <~ I i

for (11 )

xi2 >10

for (12)

xi2 <~mi

for (13)

Xi3 <~ 0

for (14)

(r4!--ri)/~i={(hi+l + x(/+l )l--X(i+l )2 --x(i+l )3) -

(& + Xi2--Xi2

-- X i 3 ) ~ I ?ti <~ s

( - r i + l + ri) / ~ki = - ( h ( i + l )1 + x(i+l )1 - x(i+l )2 --X(i+1).3) + (hi + Xil - xi2 - xi3) } / hi <~s Xil, xi2 , xi3 >~ 0

~il, 6i2 e (0, I)

for i = 1 , . . . , n

Xll = xi3 =O, O ~ x i 2 ~ m i

Case III If 6il =0, 6i2=1 Xi1:0

for (10)

xi2 + xi 3 <~ li

for (1 1 )

Xi 2 >~m i

for (12)

xi 2 <~mi - - * xi2 = rn i

for (13)

Xi 3 <~ (li--mi)

for (14)

Xil =0, xi2=mi, 0 <~ Xi3 <~ (li--mi)

There is no possibility of having both binary variables 6il , 6i2 at the value 1, because if 6il =6i2=1 , xi2 >~ m i

for (12)

for (13) The analysis above of the use of the two binary variables 6 i l and 6i2 leads to the statement that the height of the road at point i, ri, is xi2 <~ 0

r i = h i + Xil - x i 2 --xi3

and the corresponding cost ci c i =Cil Xil + ci2 xi2 + ci3 xi3 + d i 6i2

The Mixed Integer Programming model minimizes n

z= ~

i=1

Cil Xil + ci2 xi2 + ci3 xi3 + d i 6i2

Su bject to Xil <~ ui 6il xi2 + xi3 <~ 1i (1--6il)

volume 11 number 6 november 1979

The inputs to the model are: • cost coefficients Cil , ci2 , ci3 i=1 , . . . ,n, • values of the steps of the cost functions di, • values of u i, li, rn i, • desired gradient of the road s, • required and predetermined heights of the road at certain points. Many mixed-integer programming computer packages of varying quality and speed are available that can give a solution to the above model. This solution can help the design engineer by providing the values of the heights of the road at various points leading to the minimum construction cost. The solution can be considered as an initial vertical alignment, and can be a good starting point for the subsequent work of the design engineer.

REFERENCES Calogero, V 'A new method in road design: polynomial alignment' CAD Vol 1 (January 1969) Calogero, V and Athanasoulis, CoC, 'A computer technique for route planning']. Inst. Highway Eng. (April 1973) Start, JP. 'The optimization of road layout by computer methods' Proc. Inst. Civil Eng. Part 2 (March 1973) Dennett, D W 'Highway route selection' P.T.R.C., Annual meeting (7-11 July 1975) Trypia, MN 'A nonbinary tree search for the 0-1 Linear Programming problem' New Zealand Op. Res. Vol 5 (1977) p35

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