COST SAVING APPROACH USING SOLUTION ALGORITHMS
Angelina AnaniKwame Awuah-offei, PhD
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OUTLINE
• Introduction
• Motivation
• Objectives
• Solution Methodology
• Results & Discussion
• Summary
• Future Work
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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
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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
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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.
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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
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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
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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
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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
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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.
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INPUT DATA
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INPUT DATA
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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
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GRAPHIC USER INTERFACE
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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
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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.
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