A probabilistic programming model for blending aggregates

A probabilistic prog Iramming m.odel for blending aggregates Mario T. Tabucanon,

Pakorn Adulbhan and Stephen S. Y. Chen

Industrial Engineering and Management Division, Asian Institute of Technology, PO Box 2754, Bangkok, Thailand (Received August I9 79; revised November

I9 79)

The paper introduces the formulation of a probabilistic programming model to find the optimum mix proportion of aggregates to meet the specific grading requirement in order to minimize the cost which consists of the material cost and the expected penalty cost. The model is probabilistic since the gradation, which is the major parameter, is a random variable. A linear programming model is first formulated. llsing the LP solution as initial value, a direct search technique is then employed to solve the problem. The model is expected to be applicable to any problem of aggregates blending. In this paper, however, the mixing aggregates of an asphalt mixing plant is exemplified to test the applicability of the model.

Introduction

(decimal), which has to satisfy the following condition:

To determine the optimum mix proportion of two or more aggregates to meet the specific grading requirement is a common problem in construction works. Several methods have been developed to solve this problem. Usually the graphical method is used to find the starting point, followed by a trial-anderror method for solving it.’ If the number of aggregates increases, the manual method becomes complicated. Furthermore, the aggregates are of different costs, for example, the finer aggregate needs more processing to produce it, and accordingly the cost is higher. The existing methods cannot reasonably produce the optimum solution to minimize the total cost. To develop an optimization model taking the above factors into consideration is thus necessary. The optimum mix proportion is determined on the basis of minimizing the unit cost which includes the unit aggregate material cost and the expected penalty cost due to the rejection of defective product. Here the proportion of the aggregates is the decision variable.

Development

of the model

The basic model describing the unit material cost to produce one unit product comprised of k materials is expressed as follows:

cm=

2 qxj

(1)

j=1

where: C, = unit material cost of combined aggregate; Cj = unit material cost of aggregate j; k = number of kinds of aggregates used; Xi = mix proportion of aggregate j used 0307-904x/80/040257-1 l/$02.00 0 1980 IPC Business Press

i$

Xi = l

Since the variation of aggregate grading is significant2 the expected penalty cost has to be considered, which is due to the rejection of defective product whose grading fails to meet the specification. The probability that the combined passing percentage of sieve i lies between within the corresponding specific limits is given as: Pi = Prob [LSLi < Yi < US&]

for i = 1,2, . . . , YI (3)

where: LSLi = lower limit of specification i; CrSLi = upper limit of specification i; Yi = combined passing percentage of sieve i, which is a random variable distributed by a probability density function g( Yi); n = number of grading specifications. In equation (3), Yi is a convolution of the passing percentage of sieve i of all aggregates: Yi = i

aijXj

(4)

j=l

where: aij = passing percentage of sieve i of aggregate j, which is an independent random variable distributed by a probability density function fii(aij). The probability density function of the passing percentage is found2 to be a normal distribution with the parameters mean ~ii and standard deviation oij, i.e., aij N(,+, 0;). The random variable Yi could be proved to

Appl. Math. Modelling,

1980, Vol 4, August

257

A probabilistic

programming

model for blending

follow a normal distribution

aggregates: M. T. Tabucanon et al.

with the mean:

k

I-lYi =

c i-q+ j=1

and the standard deviation:

=(jl

OYj

Oijx,)‘:

The defective probability of a product, defined by any passing percentage of combined aggregate outside its corresponding specific limits, is given as: P’=l-

ii Pi i=l

= 1-

?I Prob[LSLj

< Yi < USLi]

(7)

i=l

The penalty cost is dependent on the penalty system stated in the contract between the client and the construction company. The simplest penalty system currently used is that the defective probability of product constitutes the percentage of the bid price that is deducted from the final payment. The defective probability of product is estimated from random sampling and inspection by the client. The expected penalty cost is expressed as:

used to solve the problem. The procedure is divided into two phases; (1) finding the initial point and (2) searching the minimum point. In order to reduce the iterative time for searching the global minimum point, an initial point close to the global minimum should be provided. If each gradation of the combined grading is moved near to the centre of the specified limits, the defective probability will be reduced and the expected penalty cost of equation (9) is decreased. Furthermore, the global minimum cost is composed of a near minimum unit material cost and some expected penalty cost, hence the deterministic linear programming could be used to determine the initial point by minimizing the unit material cost with the constraint that the mean of each combined aggregate grading is located within the specification limits. The linear programming problem is given by : Minimize :

c, = f cjxj

(11)

j=l

subject to

:

k C

fiijXj 2 LSLi

i=l,2,...,n

&jXj G USLj

i= 1,2,...,n

j=l k C

(12)

j=1

E[C,] = C,

1~ f

Prob[LSLi

< Yi < Usl;i]

(8)

i=l

i

where: C, = bid price or penalty cost due to rejection. Considering the unit material cost and the expected penalty cost, the probabilistic programming model is given as: Minimize Z = C,,, + E [C,]

5 CjXj +Cp (1 -

=

j=l

2

Prob [LSLi < Yi < USLi]

j=l

(9) Subject to: $J CZijXj> LSLi j=l

i=l,2,...,n

k C

LlijXj f USLi

i=l,2,...,n

(10)

j=l k

cxj=1 j=l

forj=1,2,...,k

x>o

where aij is random variable distributed by a normal distribution. The unit penalty cost, C,, is generally higher than the unit material cost. Equation (9) has been proved to be a convex function3 since increasing or decreasing any variable Xl from the optimum value will increase the expected penalty cost.

Solution procedure The sectioning search method (alternatively called one-ata-time method) described by Beveridge and Schechter4 is

258

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Xj=l

_

j=l Xj>O,

j=l,2,...,k

The simplex method is used to solve the linear programming problem. The optimum solution is taken to be the initial value of the direct search method which will be discussed later. However, during the iterative procedure of the simplex method there is a possibility that no negative nonbasic variable exists, but there are still one or more artificial variables with positive value in the basis. The interpretation of such an occurrence is that there is no feasible solution in the original linear programming problem. Since linear programming is used to obtain an initial value, the procedure cannot be terminated at this stage. In this case, in order to obtain a feasible solution, the original linear programming should be modified by subtracting the positive values of the artificial variables in the last iterative step from the constraint which has the corresponding artificial variable in the original linear programming tableau. This generates a new linear programming problem with a feasible solution. The new problem will be solved by the simplex method. After the linear programming problem has been solved, the sectioning search method is used to search the global minimum. The algorithm proceeds as follows: (1) Pick the optimum solution of linear programming as the initial value. (2) An incremental value AX, is assigned to each independent variable Xi. These incremental values control the size of the exploratory move and they are represented by a small fraction of the initial value. (3) Evaluate the objective function, equation (9) at the initial point. (4) The variables from X1 to Xk_, are held constant. The objective function is thus reduced to a function of only the

A probabilistic

programming

model for blending

variable X,. Evaluate the objective function at every increment AX, on both sides of the initial value of X, until the minimum is found. Since there is a constraint for the sum of the proportion Xi to be equal to 1, it must be ensured that after increasing X, by AX,, each Xi value must be adjusted to keep the same proportion with the sum being kept at 1. (5) The variables from X1 to Xk-2 are held constant. The value of Xk_, is then moved a distance AX,_, . Step (4) is then performed. The procedure is thus repeated until the minimum value of the objective function with the new value of X,_, has been found. (6) The same logic is performed varying each value of xj (j = 1,2,. . . ) k) until the minimum value of the objective function with the new value of Xi has been found. If the incremental value AXi is less than a predetermined value, then the approximate minimum has been found. The result is printed and the operation is terminated. Otherwise reduce the incremental value by a factor, pick the minimum point that has been found as the new initial point and return to step (3). A logical diagram illustrating the above procedure is shown in Figure 1.

aggregates: M. T. Tabucanon et al.

Yes

Modify LP problem

Pick LPoptimum solution as initial value to search for global minimum of objective function: Mm.Z-~~~tCp.P’ &

I

Ptck incremental values AXj, 1 *I, ..,k

I

t

Model application IEvaluate

A mix of cold aggregates in an asphalt mixing plant is presented as an illustrative case. The mean and standard deviation of each grading are estimated by the sample mean and standard deviation of repeated sieve analysis respectively (see Table 1). The grading curves are shown in Figure 2. These aggregates are used to produce Type IV-b densegraded (following the classification system of the Asphalt Institute) asphalt concrete. The grading specification is shown in Table 2. The average market price per ton (T) delivered is chosen to be the cost of aggregates. The respective prices are:

I

j

j

Cr = 83$/T

Intermediate

C, = 100$/T

size aggregate

Natural sand

ca=

Mineral filler

C, = 550$/T

function

/beenachieved-when\

I

Coarse size aggregate

objective

-. iental

1

rue

1

68$/T

TYes

The bidding price is chosen to be the penalty cost: CP = 650$/T. The probabilistic programming model of equations (9) and (10) is used to find the optimal mix proportion with accuracy within l%, as follows:

Prrnt the optimum

Figure

X1 = 0.286

1

Logical diagram to solve problem

X* = 0.018 Xa = 0.660 X, = 0.036 The combined grading is shown in Table 3 and Figure 3. The defective probability is P’= 0.0136. Thus, Material cost Expected penalty cost

90.19$/T 8.80$/T 98.99$/T

The illustrative case used 32.83 s of computer time for finding the initial point by the linear programming and found the optimum solution by 52 iterations on an IBM 370 MODEL 145 computer system.

Conclusion In this paper, a probabilistic programming model is developed for finding the global optimum solution by combining the aggregates at the minimum cost. From the example discussed the computation appears to be economically acceptable. The case where the costs of aggregates are also random variables (since they are subject to market price), and the case of more complex penalty systems will be recommended for future studies.

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Math.

Modelling,

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4, August

259

A probabilistic programming model for blending aggregates: M. T, Tabucanon et al. Table

1

Grading

of aggregates Aggregate 1

3

2

Coarse aggregate

Intermediate aggregate

4 Mineral filler

Sand

Sieve size

Mean

SD.

Mean

S.D.

Mean

S.D.

Mean

SD.

314” 112” 31%” No. 4 No. 8 No. 30 No. 50 No. 100 No. 200

100.0 70.0 31.2 3.0 0.0 0.0 0.0 0.0 0.0

0.0 2.10 0.89 0.30 0.0 0.0 0.0 0.0 0.0

100.0 100.0 48.0 9.9 2.4 0.0 0.0 0.0 0.0

0.0 0.0 1.80 0.70 0.53 0.0 0.0 0.0 0.0

100.0 100.0 100.0 97.6 62.9 35.0 19.5 9.0 3.8

0.0 0.0 0.0 0.56 1.62 0.95 1.44 0.88 0.41

100.0 100.0 100.0 100.0 100.0 100.0 100.0 98.2 80.5

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.25 0.55

Tab/e 3

Combined

grading Passing percentage

Sieve size 314” 112” 318” No. 4 No. a No. 30 No. 50 No. 100 No. 200

200

100

50

30

8

Sieve

6

openrng,

I,,

B$@

, g

Table 2

,I/

100-100 80-I 00 70-90 50-70 35-50 18-29 13-23 8-16 4-10

80

I

BZ$G

60

2:

ii84

Grading chart for cold aggregates

Grading specification Specification limit passing percentage

314” 112” 31%” No. 4 No. 8 No. 30 No. 50 No. 100 No. 200

100 80-100 70-90 50-70 35-50 18-29 13-23 8-16 4-10

References The Asphalt Institute.‘Mix Design Methods for Asphalt Concrete and Other Hot-Mix Types, MS-2 (4th edn), Maryland, 1974 Huculak, N. A. In Proc. Assoc. Asphalt Paving Technol. 1968, 37,288

260

0.0 0.5997 0.2563 0.3798 1.0698 0.6273 0.9509 0.5812 0.2715

In (jog scale)

Sieve size

2

100.0 91.4 79.4 69.1 45.2 26.7 16.5 9.5 5.4

6 06

0 Figure 2

1

S.D.

4

US sieve number

I

Specification limits

Mean

Awl.

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Modelling,

1980,

Vol 4, August

0 200

100

50

30

8

4

US sieve number Sieve opening,

8 8

B88Q 060 0000

“0 6

Figure 3 Grading chart of job-mix programming model

In (log scale)

q9m.Y 0000 0 66 formula

cy 0

v 0

9”O oo-

from probabilistic

Beale, E. M. L. J. Roy. Stat. sot. 1955, 17, 173 Beveridge, G. S. G. and Schechter, R. S. ‘Optimization: Theory and Practice’, McGraw-Hill, New York, 1970 Chen, S. S. Y. hf.Eng. Thesis, Asian Institute of Technology, Bangkok, Thailand (1979) Stephens, J. E. In Proc. Assoc. Asphalt Paving Technol. 1968, 37,265

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