Cost saving approach using solution algorithms
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Using solution algorithms as a cost saving approach to general optimization problems. The case study presented is to solve Hot Mix Asphalt blends.
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Cost saving approach using solution algorithms
- 1. COST SAVING APPROACH USING SOLUTION ALGORITHMS Angelina Anani Kwame Awuah-offei, PhD
- 2. OUTLINE β’ Introduction β’ Motivation β’ Objectives β’ Solution Methodology β’ Results & Discussion β’ Summary β’ Future Work 2
- 3. INTRODUCTION Brown et al.2012 applied mixed integer programming (MILP) to: β Determine the optimal mixture of aggregates and binder that minimizes cost β Ensure the optimal aggregate proportions in the mixture are technically feasible. 1 Min π§ = 100 π ππ π₯π + π π π΄π π π π΄π + π ππ΅ π ππ΅ π=1 β Solved using IBM ILOG CPLEX Optimizer 3
- 4. MOTIVATION β’ Optimization software for solving MILP problems (e.g. CPLEX, LINDO etc.) are expensive β’ These software contain algorithms to solve general optimization problems and are not tailored to solve this particular a problem 4
- 5. OBJECTIVE β’ The objective of this study is to develop a novel solution algorithm to the HMA optimization problem presented by Brown et al. (2012) β’ Negate the need for expensive commercial packages. 5
- 6. The Optimization Problem β’ Cost minimization model π Min π = πππ π ππ ππ + π πΉπ¨π· π· πΉπ¨π· + π π½π© π· π½π© π=π β’ cj, cRAP and cVB are the unit costs ($/ton) of aggregate stockpile j, RAP and binder respectively. β’ Subject to: ο§ percentage, gradation, maximum particles size, binder temperature, total binder, technological, lower and upper bound constraints 6
- 7. The Optimization Problem Constraints & Variables β’ Thirty-five (35) constraints - 24 gradation constraints,1 percentage constraint, 5 Bailey ratio constraints , 5 temperature constraints. β’ Ten (10) binary constraints - technological constraints β’ Nine (9) decision variables - 5 continuous variable, 4 binary variables β’ Continuous variable is the percentage of aggregate stockpile , in the mix β’ Binary variable such that 1 if a bin is used and 0, otherwise 7
- 8. BRANCH & BOUND β’ Solution algorithm used to solve integer and discrete problems β’ Divides the problem into sub-problems and solves them. β’ Define policies to find optimal solutions without complete enumeration β’ Policies include : node selection, variable selection, pruning, bounding function and termination criteria Fig 1. Poole et al. 2010 8
- 9. SOLUTION METHODOLOGY β’ Node selection policy: best first policy β’ Variable selection policy: In their natural order β’ Bounding function: the LP relaxation β’ Terminating criterion: The incumbent solution is within 0.2% of the best bounding function 9
- 10. VALIDATION β’ Contractor had to design a 12.5 mm HMA mix for Washington State Department of Transportation (WSDOT) projects. β’ The contractor submitted an aggregate blend of 22, 73, and 5% of 3/4 in. Γ #4, 3/8 in. Γ 0 and sand, respectively β’ The percentage of binder was 5.2 % with PG grade of PG64-28. β’ Cost of the 3/4 in. Γ #4, 3/8 in. Γ 0, and sand material are $8.50, $7.50, and $6.00/Mg, respectively. β’ The contractor did not include RAP in the mix design. 10
- 11. INPUT DATA 11
- 12. INPUT DATA 12
- 13. RESULTS & DISCUSSION Branch & β’ Algorithm replicated the aggregate and asphalt ratios Material β’ Aggregate ratios were still within recommended ranges 3/4 in Γ #4 Contractor LP CPLEX Bound 22.00 22.00 22.02 22.02 73.00 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 ratio β’ The computational time using CPLEX was 3.44 seconds. 3/8 in Γ #4 ratio β’ After 12 iterations, the branch and bound algorithm took 2.29 seconds to find a solution 13
- 15. SUMMARY β’ Demonstrate the use of a solution algorithms as a cost saving approach to solving optimization problems β’ Algorithm replicated the aggregate and asphalt ratios β’ Branch and bound algorithm outperformed CPLEX by 33% for this specific problem β’ Incorporated into a software package with an easy-to-use graphical user interface 15
- 16. FUTURE WORK β’ 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). β’ Different size problems will be solved with different number of constraints. β’ The effect of the number of variables, equality constraints and inequality constraints on the efficiency of the developed algorithm will be analyzed. 16
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