COST SAVING APPROACH USING SOLUTION ALGORITHMS

Angelina AnaniKwame Awuah-offei, PhD

2

OUTLINE

• Introduction

• Motivation

• Objectives

• Solution Methodology

• Results & Discussion

• Summary

• Future Work

3

INTRODUCTION

Brown et al.2012 applied mixed integer programming (MILP) to:– Determine the optimal mixture of aggregates and binder that

minimizes cost

– Ensure the optimal aggregate proportions in the mixture are technically feasible.

– Solved using IBM ILOG CPLEX Optimizer

4

MOTIVATION

• Optimization software for solving MILP problems (e.g. CPLEX, LINDO etc.) are expensive

• These software contain algorithms to solve general optimization problems and are not tailored to solve this particular a problem

5

OBJECTIVE

• The objective of this study is to develop a novel solution algorithm to the HMA optimization problem presented by Brown et al. (2012)

• Negate the need for expensive commercial packages.

6

The Optimization Problem

• Cost minimization model

• cj, cRAP and cVB are the unit costs ($/ton) of aggregate stockpile j, RAP and

binder respectively.

• Subject to:

percentage, gradation, maximum particles size, binder temperature, total

binder, technological, lower and upper bound constraints

7

The Optimization Problem

Constraints & Variables

• Thirty-five (35) constraints - 24 gradation constraints,1 percentage

constraint, 5 Bailey ratio constraints , 5 temperature constraints.

• Ten (10) binary constraints - technological constraints

• Nine (9) decision variables - 5 continuous variable, 4 binary variables

• Continuous variable is the percentage of aggregate stockpile , in the mix

• Binary variable such that 1 if a bin is used and 0, otherwise

8

BRANCH & BOUND• Solution algorithm used to solve integer

and discrete problems

• Divides the problem into sub-problems and solves them.

• Define policies to find optimal solutions without complete enumeration

• Policies include : node selection, variable selection, pruning, bounding function and termination criteria

Fig 1. Poole et al. 2010

9

SOLUTION METHODOLOGY

• Node selection policy: best first policy

• Variable selection policy: In their natural order

• Bounding function: the LP relaxation

• Terminating criterion: The incumbent solution is within 0.2% of the best bounding function

10

VALIDATION

• Contractor had to design a 12.5 mm HMA mix for Washington State

Department of Transportation (WSDOT) projects.

• The contractor submitted an aggregate blend of 22, 73, and 5% of 3/4 in. × #4,

3/8 in. × 0 and sand, respectively

• The percentage of binder was 5.2 % with PG grade of PG64-28.

• Cost of the 3/4 in. × #4, 3/8 in. × 0, and sand material are $8.50, $7.50, and

$6.00/Mg, respectively.

• The contractor did not include RAP in the mix design.

11

INPUT DATA

12

INPUT DATA

13

RESULTS & DISCUSSION

• Algorithm replicated the aggregate and asphalt ratios

• Aggregate ratios were still within recommended ranges

• The computational time using CPLEX was 3.44 seconds.

• After 12 iterations, the branch and bound algorithm took 2.29 seconds to find a solution

Material Contractor

 

LP 

 

CPLEX

Branch &

Bound

3/4 in × #4 

ratio22.00 22.00 22.02 22.02

3/8 in × #4 

ratio73.00 72.85 72.94 72.94

Sand ratio 5.00 5.02 5.04 5.04

RAP ratio 0.00 0.14 0.00 0.00

VB ratio 5.20 5.03 5.04 5.04

14

GRAPHIC USER INTERFACE

15

SUMMARY

• Demonstrate the use of a solution algorithms as a cost saving approach to solving optimization problems

• Algorithm replicated the aggregate and asphalt ratios

• Branch and bound algorithm outperformed CPLEX by 33% for this specific problem

• Incorporated into a software package with an easy-to-use graphical user interface

16

FUTURE WORK

• Future work will incorporate solving the different optimization

problems with the developed branch and bound algorithm as a

performance measure against commercial software (e.g. CPLEX).

• Different size problems will be solved with different number of

constraints.

• The effect of the number of variables, equality constraints and

inequality constraints on the efficiency of the developed

algorithm will be analyzed.

17