1. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Optimization Techniques for Civil and Environmental Engineering Systems By Yicheng Wang

2. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu III. Linear Programming (LP) 1. Introdcution to Linear Programming (1) Brief description of LP The development of LP has been ranked among the most important scientific advances of the mid -20 th century. The adjective linear means that all the mathematical functions in an LP model are required to be linear functions. The word programming does not refer here to computer programming; rather, it’s essentially a synonym for planning. The most common type of LP application involves the general problem of allocating limited resources among competing activities in a best possible way. LP has numerous other important applications as well. In fact, any problem whose mathematical model fits the very general format for the linear programming model is a linear programming problem.

3. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (2) Prototype Model in the textbook The Wyndor Glass Co. produces high-quality glass products, including windows and glass doors. It has three plants. Plant 1 produces Aluminum frames Plant 2 produces wood frames Plant 3 produces the glass and assembles the products. The company has decided to produce two new products. Product 1: An 8-foot glass door with aluminum framing Product 2: A 4 x 6 foot double-hung wood framed window Each product will be produced in batches of 20. The production rate is defined as the number of batches produced per week. The company wants to know what the production rate should be in order to maximize their total profit, subject to the restriction imposed by the limited production capacities available in the 3 plants.

4. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu To get the answer, we need to collect the following data. (a) Number of hours of production time available per week in each plant for these two new products. (Most of the time in the 3 plants is already committed to current products, so the available capacity for the 2 new products is quite limited). Number of hours of production time available per week in Plant 1 for the new products: 4 Number of hours of production time available per week in Plant 2 for the new products: 12 Number of hours of production time available per week in Plant 3 for the new products: 18 (b) Number of hours of production time used in each plant for each batch produced of each new product (Product 1 requires some of the production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only Plants 2 and 3). Number of hours of production time used in Plant 1 for each batch produced of Product 1: 1 Number of hours of production time used in Plant 2 for each batch produced of Product 1: 0 Number of hours of production time used in Plant 3 for each batch produced of Product 1: 3 Number of hours of production time used in Plant 1 for each batch produced of Product 2: 0 Number of hours of production time used in Plant 2 for each batch produced of Product 2: 2 Number of hours of production time used in Plant 3 for each batch produced of Product 2: 2

5. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (c) Profit per batch produced of each new product. Profit per batch produced of Product 1: $3,000 Profit per batch produced of Product 2: $5,000 The data collected are summarized in Table 3.1. This is a linear programming problem of the classic product mix type.

6. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Formulation as a Linear Programming Problem To formulate the LP model for this problem, let x 1 = number of batches of product 1 produced per week x 2 = number of batches of product 2 produced per week Z = total profit ( in thousands of dollars) from producing the two new products. Thus, x 1 and x 2 are the decision variables for the model. Using the data of Table 3.1, we obtain (Plant 1:Total production time required)(Production time available) (Plant 2:Total production time required)(Production time available) (Plant 3:Total production time required)(Production time available)

7. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu

8. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (3) Examples of LP models in civil and environmental engineering Example 1: Blending Water Supply

9. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu

10. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Example 2: Cleaning Up the River

11. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu These two functions are as follows.

12. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu

13. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (4) Graphical Solution The Wyndor Glass Co. example is used to illustrate the graphical solution. Fig. 3.1 Shaded area shows values of (x 1, x 2 ) allowed by x 1 ≥ 0, x 2 ≥ 0, x 1 ≤ 4 Fig. 3.2 Shaded area shows values of (x 1, x 2 ), called feasible region

14. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Fig. 3.3 The value of (x 1, x 2 ) that maximize 3x 1 + 5x 2 is (2, 6)

15. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (5) General Form of LP Model The most common type of application of linear programming involves allocating limited resources to activities. Determining this allocation involves choosing the levels of the activities that achieve the best possible value of the overall measure of performance.

16. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu A General Form of Linear Programming Model where The constraints can be any combination of equalities (=) and inequalities ( ) Maximize or Minimize

17. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model Objective Function: The function being maximized or minimized is called the objective function. Constraint: The restrictions of LP Model are referred to as constraints. The first m constraints in the previous model are sometimes called functional constraints. The restrictions x j ≤ 0 are called nonnegativity constraints. Feasible Solution: A feasible solution is a solution for which all the constraints are satisfied. Infeasible Solution: An infeasible solution is a solution for which at least one constraint is violated. A feasible solution is located in the feasible region. An infeasible solution is outside the feasible region. Feasible Region: The feasible region is the collection of all feasible solutions.

18. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model No Feasible Solutions: It is possible for a problem to have no feasible solutions. An Example Fig. 3.4 The Wyndor Glass Co. problem would have no feasible solutions if the constraint 3x 1 + 5x 2 ≤ 50 were added to the problem. In this case, there is no feasible region

19. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model Optimal Solution: An optimal solution is a feasible solution that has the maximum or minimum of the objective function. Multiple Optimal Solutions: It is possible to have more than one optimal solution. An Example Fig. 3.5 The Wyndor Glass Co. problem would have multiple optimal solutions if the objective function were changed to Z = 3x 1 + 2x 2

20. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model Unbounded Objective: If the constraints do not prevent improving the value of the objective function indefinitely in the favorable direction, the LP model is called having an unbounded objective. An Example Fig. 3.6 The Wyndor Glass Co. problem would have no optimal solutions if the only functional constrait were x 1 ≤ 4, because x 2 then could be increased indefinitely in the feasible region without ever reaching the maximum value of Z = 3x 1 + 2x 2

21. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model Corner-Point Feasible (CPF) Solution: A corner-point feasible (CPF) is a solution that lies at a corner of the feasible region. Fig. 3.7 The five dots are the five CPF solutions for the Wyndor Glass Co. problem

22. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Common Terminology for LP Model Relationship between optimal solutions and CPF solutions : Consider any linear programming problem with feasible solutions and a bounded feasible region. The problem must posses CPF solutions and at least one optimal solution. Furthermore, the best CPF solution must be an optimal solution. Therefore, if a problem has exactly one optimal solution, it must be a CPF solution. If the problem has multiple optimal solutions, at least two must be CPF solutions. The prototype model has exactly one optimal solution, (x 1, x 2 )=(2,6), which is a CPF solution (2,6) (4,3) The modified problem has multiple optimal solution, two of these optimal solutions, (2,6) and (4,3), are CPF solutions.

23. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu (5) Standard Form of LP Model In this course, we are going to use the following standard form of linear programming model Algebraic Standard Form of an LP Model

24. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Tabular Standard Form of an LP Model

25. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Matrix Standard Form of an LP Model To help you distinguish between matrices, vectors, and scalars, we use BOLDFACE CAPITAL letters to represent matrices, bold lowercase letters to represent vectors, and italicized letters in ordinary print to represent scalars.

26. University of Colorado at Boulder Yicheng Wang, Phone:303-492-4228, Email:yicheng@colorado.edu Homework (September 1) Chapter 3 (Textbook: Introduction to Operations Research) Problems 3.1-5, 3.2-4, 3.2-5, 3.2-6 The homework is due on September 8 (Tuesday)