ENGI 5708 Design of Civil Engineering Systems

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1 ENGI 5708 Design of Civil Engineering Systems Lecture 04: Graphical Solution Methods Part 1 Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca

2 Lecture 04 Objective To examine the solution of linear programming (LP) problems using graphical solution methods S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

3 Characteristics of LP Problems Objective Function Decision variables What decision are to be made? Linear behaviour 1 st degree polynomial Variables are added or subtracted Continuous variables Find optimal solution Extrema (minimum or maximum) y = cx 1 1+ cx 2 2+ cx 3 3+ cnxn S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

4 Characteristics of LP Problems (cont.) Constraint Equations Defines relationships Decision variables parameters Defines requirements or bound limits Natural, physical or practical considerations Linear behaviour Left-hand side 1 st degree polynomial Right-hand side constants Closed form only, =, or expressions ax+ ax + ax + ax = b i i i in n i S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

5 Graphical Solution LP Problems Method Plot constraint equations Define solution space Plot objective function Find optimal solution Advantage Simple, visual method Illustrates basic LP concepts Limitations Decision variables < 4 variables Problem idealization Limited characterization of reality S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

6 Example Objective Function y Constraint Equations x = 2x 1 2 x1 6 1 Objective Function (y) Control Variable x 1 Feasible Region Solution to objective function that satisfies constraint equations Basic feasible solution located at vertices Minimum Maximum Feasible Region S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

7 Graphical Solution Can Be Complex Objective Function Global maximum but where? Local extrema Global minimum but where? Decision Variable (x 2 ) Decision Variable (x 1 ) Ref: MATLAB (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

8 Example 4-02 A company produces two varieties of a product. Variety A has a profit per unit of 2.00 and variety B has a profit per unit of Demand for variety A is at most four units per day. Production constraints are such that at most 10 hours can be worked per day. One unit of variety A takes one hour to produce but one unit of variety B takes two hours to produce. Ten square metres of space is available to store one day's production and one unit of variety A requires two square metres whilst one unit of variety B requires one square metre. Formulate the problem of deciding how much to produce per day as a linear program and solve it graphically. Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

9 Example 4-02 (cont.) A company produces two varieties of a product. Variety A has a profit per unit of 2.00 and variety B has a profit per unit of Demand for variety A is at most four units per day. Production constraints are such that at most 10 hours can be worked per day. One unit of variety A takes one hour to produce but one unit of variety B takes two hours to produce. Ten square metres of space is available to store one day's production and one unit of variety A requires two square metres whilst one unit of variety B requires one square metre. Formulate the problem of deciding how much to produce per day as a linear program and solve it graphically. Decision Variables x 1 = number of units of variety A produced x 2 = number of units of variety B produced Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

10 Example 4-02 (cont.) A company produces two varieties of a product. Variety A has a profit per unit of 2.00 and variety B has a profit per unit of Demand for variety A is at most four units per day. Production constraints are such that at most 10 hours can be worked per day. One unit of variety A takes one hour to produce but one unit of variety B takes two hours to produce. Ten square metres of space is available to store one day's production and one unit of variety A requires two square metres whilst one unit of variety B requires one square metre. Formulate the problem of deciding how much to produce per day as a linear program and solve it graphically. Objective Function Maximize z = 2x + 3x 1 2 Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

11 Example 4-02 (cont.) A company produces two varieties of a product. Variety A has a profit per unit of 2.00 and variety B has a profit per unit of Demand for variety A is at most four units per day. Production constraints are such that at most 10 hours can be worked per day. One unit of variety A takes one hour to produce but one unit of variety B takes two hours to produce. Ten square metres of space is available to store one day's production and one unit of variety A requires two square metres whilst one unit of variety B requires one square metre. Formulate the problem of deciding how much to produce per day as a linear program and solve it graphically. Constraint Equations x + 2x 10hr day Production time Space available x 2 x1 x2 m day 1 4 units day Demand variety A Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

12 Example 4-02 (cont.) Constraint Equation Inequality 2x + x x1 4 x1 0 x + 2x Non-Negativity Constraint x 2 0 Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

13 Example 4-02 (cont.) Feasible Region 2x + x x1 4 Feasible Region x2 0 x + 2x Non-Negativity Constraint x 1 0 Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

14 Example 4-02 (cont.) Vertex Points Constraint Equations Satisfied Strict Equality Feasible Region Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

15 Example 4-02 (cont.) Feasible Region Interior points Boundary points Vertex points Basic feasible solutions Optimal solution(s) Boundary Points Interior Points Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

16 Example 4-02 (cont.) Optimal Solution x 1 = x 2 = 10/3 z = 2x + 3x = Objective Function Increasing Profit Isolines Ref: Beasley (2007) S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

17 Linear Programming Outcomes Unique Optimum Intersection of objective function and feasible space is a single point Feasible Region Unique Optimal Solution S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

18 Linear Programming Outcomes Alternate Optima Intersection of objective function and feasible space is a line segment Feasible Region Alternate Optima S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

19 Linear Programming Outcomes No Feasible Solution Over constraint Conflicting constraints or error S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

20 Linear Programming Outcomes Unbounded Solution Under constraint Conflicting constraints or error Unbounded Feasible Region S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

21 Reading List Arsham (2007). Graphical Solution Method. mlp Pike (2001). Chapter IV. Concepts and Geometric Interpretation S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

22 References Beasley (2007). lass2q.html MATLAB (2007). ReVelle, C.S., E.E. Whitlatch, Jr. and J.R. Wright (2004). Civil and Environmental Systems Engineering 2 nd Edition, Pearson Prentice Hall ISBN S. Kenny, Ph.D., P.Eng. ENGI 5708 Civil Engineering Systems Lecture 04

ENGI 5708 Design of Civil Engineering Systems

ENGI 5708 Design of Civil Engineering Systems Lecture 09: Characteristics of Simplex Algorithm Solutions Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial