A Generalized Framework for Optimization with Risk Damaris - - PowerPoint PPT Presentation

A Generalized Framework for Optimization with Risk

Damaris Zipperer & Andrew Brown

Agenda

Problem Overview of Results Methodology Review Further Applications Wrap-Up

Background

Problem

• Long term forecasts
• Underlying risk
• Trying to optimize in volatile environments

Building a Schedule

Task 1 Task 4 Task 2 Task 6 Task 9 Task 7 Task 5 Task 3 Task 8

So, which problem do we fix: cost or coverage?

Cost vs. Coverage

Cost ($): Labor Cost (Including Extra Costs Due to Forecast Inaccuracy) Coverage (%): The chosen solution’s performance against the actualized schedule

Current Schedule Optimization isn’t… Optimal

Results Overview

Deterministic Optimization Solution: ~62% Coverage, $770,000 Cost Risk-Integrated Optimization Solution Range of Cost/Coverage Options At Equivalent Cost ($770,000), 84% Coverage

Creating a Deterministic Solution

Duration Description Total Cost ($) Unit Cost/day ($) Block of Hours One worker for one full day 81 81.00 6 Month One worker for 130 Work days 8668 66.68 7 Month One worker for 152 Work days 9574 62.99 8 Month One worker for 174 Work days 10942 62.89 9 Month One worker for 195 Work days 12309 63.12 10 Month One worker for 217 Work days 13677 63.03 11 Month One worker for 239 Work days 15045 62.95 12 Month One worker for 260 Work days 15858 60.99

Creating a Deterministic Solution

Object ctive Funct ction:

! ! 𝑦#,%

& #'(

𝑑%

* %'(

;

where m = total contract types and n = total schedule days

Co Costs for each contract type (j) are denoted as:

𝑑%

Deci cision var ariab ables of the model, x, planned hires, by

day (i) and contract type (j) ∶ 𝑦#,% Cu Cumulat ative Hires matrix: 𝑧#,% Co Constrai aints: 𝑦#,% ≥ 0 𝑧#,% ≥ 0 ! 𝑧#,%

* %'(

≥ ℎ𝑓𝑏𝑒𝑑𝑝𝑣𝑜𝑢 𝑠𝑓𝑟𝑣𝑗𝑠𝑓𝑛𝑓𝑜𝑢 𝑏𝑢 𝑒𝑏𝑧 𝑗

Defining How and Why Schedules Change

Risk Par aram ameterizat ation

• Which task types move
• What is the probability that they move
• Which direction do they move
• Are there interdependencies
• Are there tensions that must be preserved

Using Historic c Dat ata

Simulation Tasks

Assign risk coverage values Optimize each schedule Simulate different schedules 1.Import and analyze Output portfolio of

• ptions

Assign new requirement s Weigh the risk coverage impact 1.Understan d the resulting statistics

Using Risk to Simulate Schedule Iterations

Our simulator takes in risk parameters, an input schedule, and iterates…

Risk Coverage Assessment

5 10 15 20 25 30 35 40 1 29 57 85 113 141 169 197 225 253 281 309 337 365 393 421 449 477 505 533 561 589 617 645 673 701 729 757 785 813 841 869 897 925 953 981 1009 1037 1065 1093 1121 1149 Headcount Schedule Day

95th Percentile Set Coverage, Sample Schedule

The risk coverage for each simulated schedule was then calculated as follows: 𝑆𝑗𝑡𝑙𝐷𝑝𝑤B = 1 −

∑ GHI#J#H&JKL,M

LNOPQRSPT LNU

∑ VWXYZL

Risk Statistics

Bottom-Up Risk Integration

𝑸𝒔𝒑𝒄 =

GH`#aHbZcdHaWeHfgW`HZcd (hh%fgW`HZcd

; where BaseCov = coverage achieved by input schedule. 𝑶𝒇𝒙𝑰𝑫𝑺𝒇𝒓𝒋 = 𝑨 ∗ 𝐵𝑤𝑕𝐸𝑓𝑔𝑇𝑢𝐸𝑓𝑤# + 𝐵𝑤𝑕𝐸𝑓𝑔𝐵𝑤𝑕# + 𝐶𝑏𝑡𝑓𝑇𝑑ℎ𝑓𝑒#

Top-Down Risk Integration

𝐻𝑏𝑞 𝑄𝑓𝑠𝑑𝑓𝑜𝑢𝑏𝑕𝑓 = (𝑧 − 𝑦 ) 𝑧 ⁄ This gap percentage was used to determine the percentile value to utilize across list of daily average deficiencies

Results

Cost-Coverage Frontier

700,000 750,000 800,000 850,000 900,000 950,000 1,000,000 55% 65% 75% 85% 95%

Co Cost Co Coverag age

Co Cost Co Coverag age F Frontier – Me Method 1 1

Base Optimization

Solution

750,000 755,000 760,000 765,000 770,000 775,000 780,000 785,000 790,000 55% 60% 65% 70% 75% 80% 85% 90%

Co Cost Schedule Co Coverag age

Co Cost-Co Coverag age F Frontier – Me Method 2 2

Base Optimization

Solution

Coverage Assessment & Prediction

0% 20% 40% 60% 80% 100% 5 10 15 20 Coverage Schedule Iteration

Simulat ated v

• vs. Act

Actual al Co Coverag age – Me Method 1

Actual Coverage Predicted Coverage 0% 20% 40% 60% 80% 100% 5 10 15 20 Coverage Schedule iteration

Simulat ated v

• vs. Act

Actual al Co Coverag age – Me Method 2 2

Actual Coverage Simulated Coverage

Generalized Framework

Re-determine Coverage, Benchmark Costs Remove Coverage Not Exploiting Profit Gains Develop New Requirements, Optimize Develop Strategically Appropriate Method Measure Base Set Coverage and Deficiency Statistics Optimize Individual Scenarios Simulate Scenarios with Risk Parameters Measure Risk Parameters

Further Applications

• Genetic Algorithms approach
• Applications for other S.C. functions
• Other Industries
• Method for situations without prior history

Any Questions?