ENGI 5708 Design of Civil Engineering Systems Lecture 10: Sensitivity Analysis Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca
Lecture 10 Objective to understand parameters influencing sensitivity of LP problems 2 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Motivation for Sensitivity Analysis Model Idealization Abstraction of reality, linearization Relationship between variables, constraints, coefficients Scope extent, system hierarchy Quantifiable fact or importance Subjectivity Uncertainty Natural variability or volatility Regulations, economics, resources Mechanisms Poor or incomplete understanding of processes Data Bias, error or sample size 3 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
What is Sensitivity Analysis? Tool for Decision Making Heuristic analysis Trial and error What if scenario analysis Assess importance of possible events (i.e. change in parameter value) and probable outcomes from these changes or variability 4 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Why Conduct Sensitivity Analysis? Rank Assessment Screening tool Identify key elements and parameters Increase or decrease model complexity Focus effort and resources Model advancement or refinement Test optimal solution robustness Identify critical parameters Establish thresholds (upper/lower bounds) Contingency planning Impact to optimal solution or decision making? Set conditions for changes in strategy 5 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Sensitivity Analysis Constraint Equation Coefficient (a mn ) Right-Hand Side Constraints (C m ) Objective Function Coefficient (k n ) Add New Decision Variables Add Constraint Equations Objective Function or Merit Function (Non-$) N ( ) Z = f x = k x N n n n n= 1 n= 1 min or max Constraint Equations M M N ( ) g x = a x, =, C m n mn n m m= 1 m= 1n= 1 6 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Constraint Equation Coefficient Possible Variation Volume, production rate or yield of a process or resource Example 6-01 Clay volume, blending time or storage capacity per unit volume of product Impact Constraint equation slope region Basic feasible solutions F A E D C B Example 6-01 7 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Constraint Equation Coefficient (cont.) Example 6-01 Double HYDIT blending time 5 x + 5 x 50 10 x + 5 x 50 1 2 1 2 Example 6-01 8 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Constraint Equation Coefficient (cont.) Example 6-01 Double FILIT blending time 5x + 5x 50 5x + 10x 50 1 2 1 2 Example 6-01 9 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint Possible Variation Maximum capacity, resource availability, usage or time Example 6-01 Total clay volume, blending time or storage capacity Impact Constraint equation shift region Basic feasible solutions F A E D C B Example 6-01 10 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint (cont.) Example 6-01 Increase total available blending time to 70 hrs 5 x + 5 x 50 5x + 5 x 70 1 2 1 2 Example 6-01 11 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint (cont.) Example 6-01 Decrease total available blending time to 30 hrs 5 x + 5 x 50 5x + 5 x 30 1 2 1 2 Example 6-01 12 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint (cont.) Binding Constraints If binding then RHS limits value of obj. function 2x1+ 4x2 28 5x1+ 5x2 50 x1 8 x2 6 x, x 0 1 2 F A E D Example 6-01 Unique Optimal Solution x 1 = 6; x 2 = 4 C B Z = 140x + 160x 1 2 13 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint (cont.) Binding Constraints Reduced cost coefficients Clay Volume { } B s, x, x, s s s s = Example 6-01 D 3 2 1 4 1 2 3 = 2 2 1+ 5 2 x = 4 s + s 1 1 2 2 1 5 2 1 2 1 = 6 + 1 2 5 2 1 1 4 = 2 + 2 1 5 2 x s s s s s Z = 1480 10s 24s Blending Time 1 2 Unique Optimal Solution x 1 = 6; x 2 = 4 14 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10 F A E D C B
Right-Hand Side Constraint (cont.) Non-Binding Constraints Look at basic variables { } B s, x, x, s s s s D 3 2 1 4 1 2 3 = 2 2 1+ 5 2 x = 4 s + s 1 1 2 2 1 5 2 1 2 1 = 6 + 1 2 5 2 1 1 4 = 2 + 2 1 5 2 x s s s s s Z = 1480 10s 24s HYDIT Storage = Example 6-01 FILIT Storage 1 2 Unique Optimal Solution x 1 = 6; x 2 = 4 15 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10 F A E D Increasing storage capacity has no effect on strategy C B
Right-Hand Side Constraint (cont.) Shadow Prices or Dual Resource Prices Reduced cost coefficients Tolerable range on variability? ll constraint line thru adjacent vertex Clay Volume Z = 1480 10s 24s Blending Time 1 2 16 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10 F A E G D Example 6-01 Unique Optimal Solution x 1 = 6; x 2 = 4 C B H
Right-Hand Side Constraint (cont.) Shadow Prices or Dual Resource Prices 2x1+ 4x2 28 5x + 5x 50 1 2 Clay Volume Blending Time x1 8 x 2 6 HYDIT Storage FILIT Storage Resource Wabash Red Clay Blending Time HYDIT Curing Vat Capacity FILIT Curing Vat Capacity Current RHS Optimal Usage Lower Range 24 m 3 28 m 3 28 m 3 (C 8,2) 50 hr 50 hr 40 hr (E 2,6) 6 m 3 8 m 3 6 m 3 (D 6,4) 4 m 3 6 m 3 4 m 3 (D 6,4) Allowable Decrease Upper Range 4 m 3 32 m 3 (G 4,6) 10 hr 55 hr (H 8,3) Allowable Increase Shadow Price 4 m 3 $10 5 hr $24 2 m 3 Unlimited $0 2 m 3 Unlimited $0 17 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Right-Hand Side Constraint (cont.) Shadow Prices or Dual Resource Shadow price implications If unit price of clay $10/t then purchase If supplier failed to deliver clay then for this range the compensation price is $10/t Vat capacity provides no improved profits Resource Wabash Red Clay Blending Time HYDIT Curing Vat Capacity FILIT Curing Vat Capacity Optimal Usage Allowable Decrease Allowable Increase Shadow Price 28 m 3 4 m 3 4 m 3 $10 50 hr 10 hr 5 hr $24 6 m 3 2 m 3 Unlimited $0 4 m 3 2 m 3 Unlimited $0 18 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Objective Function Possible Variation Unit cost or profit Unit gain or loss Impact Objective function slope Optimal solution 2 1 Increasing HYDIT Profit 3 4 4 Increasing FILIT Profit 1 2 3 19 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
Reading List Pike, R.W. (2001). Optimization for Engineering Systems. http://www.mpri.lsu.edu/bookindex.html Arsham (2007). Linear Programming. http://home.ubalt.edu/ntsbarsh/opre640a/partviii.htm#rplp Arsham (2007). Linear Programming. http://home.ubalt.edu/ntsbarsh/businessstat/opre/partvii.htm#rintrodu 20 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10
References ReVelle, C.S., E.E. Whitlatch, Jr. and J.R. Wright (2004). Civil and Environmental Systems Engineering 2 nd Edition, Pearson Prentice Hall ISBN 0-13-047822-9 21 2008 S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 10