36106 Managerial Decision Modeling Linear Decision Models: Part II

Transcription

1 Managerial Decision Modeling Linear Decision Models: Part II Kipp Martin University of Chicago Booth School of Business January 20, 2014

2 Reading and Excel Files Reading (Powell and Baker): Sections Section 9.8 Files used in this lecture: simplelp.xlsx

3 Lecture Outline Constrained Optimization Optimization Categories Optimization Concepts Solution Report Vocabulary

4 Constrained Optimization Google the word optimization and you get over 62 million links, Googling optimal gives about 70 million links. Words such as optimal and optimization mean different things to different people. In this class they take on a precise meaning. We will devote a lot of time this quarter to building optimization models and implementing them in Excel. Before we can build some neat applications and learn some really useful stuff, we need a few basic math concepts. You need to understand the concepts: objective function, constraint, feasible, variable, parameter, and optimal.

5 Constrained Optimization Everyone is familiar with the idea of an equation. Here are two examples 2x 1 + 7x 2 = 3 (1) 5x 2 1 2x 1 x 2 + 5x 1 3x 2 = (2) The first equation is a linear equation, the second equation is a quadratic equation. You are also familiar with the idea of an inequality. 2x 1 + 7x 2 3 (3) 5x 2 1 2x 1 x 2 + 5x 1 3x (4) Again, the first inequality is linear while the second is nonlinear.

6 Constrained Optimization In this class (and in the world of optimization) equations and inequalities are called constraints or objectives. Example 1: in the cash flow matching problem, there was an equation =B6+C6+Price_Bnd1*Num_Bnd1+Price_Bnd2*Num_Bnd2 that represents the objective of minimizing the initial cash requirement. Example 2: in the cash flow matching problem there was an inequality =Num_Bnd1*Int_Bnd1+Num_Bnd2*Int_Bnd2+ C7*(1+Sav_Int)-Sav_Acct-Cash_Req that is a constraint saying sources of cash must meet or exceed uses of cash in period 2.

7 Constrained Optimization Every constrained optimization problem has an objective function that is maximized or minimized subject to constraints. In the cash flow matching problem the objective is to minimize the initial amount of cash required subject to constraints that require the sources of cash to meet the uses of cash. In the Markowitz portfolio optimization we study later the objective is to minimize the variablity of a stock portfolio subject to a constraint on the required return of the portfolio. In the airline revenue management problem we study later the objective is to maximize revenue subject to constraints of the seats available in various categories. Definition: the variables in the constraints and objective are called decision variables.

8 Constrained Optimization In this class you will learn how to take business problems and frame the problem in terms of constraints and objectives. First, we start by seeing what a problem with an objective function and constraints look like. Don t worry about the application for now this is slight modification of a standard MATLAB example. max x + 2y 2x + 2y 4 2x 4y 2 2x + y 8 2x + y 2 y 7

9 Constrained Optimization Here is the feasible region of a simple two-variable constrained optimization problem. Attempting a What-If analysis by picking feasible points is simply not efficient!

10 Constrained Optimization Feasible Solution: A point is a feasible solution if and only if it satisfies every constraint. Going three for five or even four for five does not suffice. Is (5, 3) a feasible point? How can you test this? Is (3, 5) a feasible point? How can you test this? Feasible Region: The set of feasible points.

11 Constrained Optimization In this example, there are an infinite number of feasible solutions. We want the best or optimal solution. A solution is an optimal solution to a maximization problem if there is no feasible solution that has a strictly larger objective function value. A solution is an optimal solution to a minimization problem if there is no feasible solution that has a strictly smaller objective function value.

12 Constrained Optimization In practice, optimization problems may involve thousands or even millions of variables and constraints. These large problems are solved using optimization software codes which are often called solvers. The Excel Solver is one such example. There are numerous optimization solvers, both commercial and open source. Solvers work by finding a feasible point and then move in an improving direction (where the objective gets bigger in case of a maximization). Then a step is taken in the direction of improvement and the process repeats. Finding improving directions and moving in the direction of improvement can be mathematically complex. We won t worry about this. We let Excel worry about this and do all the work.

13 Historical Notes 13 Historically, the first constrained optimization problems were diet problems solved for the US military in the late 1930s and early 1940s. Booth (before we were Booth) professor George Stigler and 1982 Nobel Laureate in economics was one of the first to solve this problem. See His original problem had 77 potential foods and 9 nutrient requirements. Here is what you should eat. Food Annual Quantities Annual Cost Wheat Flour 370 lb. $13.33 Evaporated Milk 57 cans $ 3.84 Cabbage 111 lb. $ 4.11 Spinach 23 lb. $ 1.85 Dried Navy Beans 285 lb. $16.80 Total Annual Cost $ Non-vegetarians should consume beef liver.

14 Historical Notes Stigler did not actually find the optimal solution, but rather employed a heuristic to find a near optimal solution. Big Breakthrough: George Dantzig found a way to solve constrained optimization problems when the constraints and objective function are linear. He showed that if there was an optimal solution then there was an optimal extreme point solution. Extreme point: a point on the boundary of the feasible region where the constraints intersect. Dantzig s extreme point result led to a very effective solution algorithm for linear programs (linear objective and linear constraints), called the Simplex Algorithm. It is now possible to solve linear programs with quite literally millions of variables and constraints.

15 Types of Constrained Optimization Models The Generic Model: max(min)f (x 1, x 2,..., x n ) g 1 (x 1, x 2,..., x n ) b 1 g 2 (x 1, x 2,..., x n ) b 2... g m (x 1, x 2,..., x n ) b m Optimization problems fall into categories depending upon the structure of the objective function and constraints. It is critical to understand the structure of the problem you are trying to solve! We will see why this is so shortly.

16 Types of Constrained Optimization Models Linear Program: The objective function and all constraints are linear. The easiest type of optimization problem to solve. Initially most business problems solved were linear. Nonlinear Program: At least one constraint or the objective function is a nonlinear function. Convex Nonlinear Programs Nonconvex Nonlinear Programs Don t worry about nonconvex versus convex for now, just linear versus nonlinear. 2x 1 + 5x 2 2x 1 + 5x 1 x 2 2x1 2 + x 2 x 1 /x 2 is linear is nonlinear is nonlinear is nonlinear

17 Types of Constrained Optimization Models Integer Program: at least one variable must be either binary (0/1) or a general integer (we buy an integral number of bonds). The 0/1 variables are incredibly useful to model logical conditions (more later). 1. Linear Integer Programs the objective function and constraints are linear, but some or all variables are restricted to be integer. 2. Nonlinear Integer Programs at least one constraint is nonlinear or the objective function is nonlinear and some or all variables are restricted to be integer. This is an argle bargle! Stochastic Program: A least one parameter is NOT deterministic, i.e. it is stochastic. In practice, stochastic parameters are often replaced with their expected values. This can be very dangerous! Hence we later move and RiskSolver.

18 18 Types of Constrained Optimization Models When working with solver it is important to know the kind of optimization problem you have. Use the Select a Solving Method judiciously.

19 Types of Constrained Optimization Models Model Category Solver Solving Method Linear program Simplex LP Integer linear program Simplex LP Convex nonlinear program GRG Nonlinear Nonconvex nonlinear program GRG Nonlinear Integer nonlinear program Evolutionary Stochastic RiskSolver Solving a stochastic, nonlinear, nonconvex, integer programming model you are kidding, right?

20 Types of Constrained Optimization Models 20 A Few Excel Hints: 1. Assume you have a formula in cell A2 which is = A1 B1. At first glance, this is a nonlinear expression. However, if for example, cell A1 has = 7, then Excel will make this substitution into A2 and the formula in A2 is really =7*B1 which is linear in B1. After substitutions, ideally there are no cells with nonlinear relationships involving adjustable cells! 2. Functions are dangerous! Consider the following use of the IF function in mpfphotfudge.xlsx. =Int_Bnd1*Num_Bnd1*IF(Mat_Bnd1>=Period,1,0) This is fine since the function arguments (inputs) are parameters! This would be nonlinear, and even require the Evolutionary solver, if an argument involved an adjustable cell. Be careful of using functions that use adjustable cells unless the function is a linear function such as SUM.

21 21 Types of Constrained Optimization Models Scaling: This is bit on the techie side, but Solver can have problems when you have numbers of substantially different magnitudes. Indeed, you may get an error message about nonlinear cells even when they are linear.

22 Types of Constrained Optimization Models 22 If you have selected Simplex LP and are getting nonlinear error messages, and you are convinced your model is linear, then check Use Automatic Scaling.

23 Optimization Concepts 23 Something new to worry about in the nonlinear world the software finds a local optimum. First, the global concept. A point x is a global maximum of f (x) if and only if f (x) f (x) for all x A point x is a global minimum of f (x) if and only if f (x) f (x) for all x

24 Optimization Concepts 24 Is there a global maximum? Is there a global minimum?

25 Basic Concepts 25 Now the local concept. A point x is a local maximum of f (x) if and only if there are no other feasible solutions with a larger objective function value in the immediate neighborhood. A point x is a local minimum of f (x) if and only if there are no other feasible solutions with a smaller objective function value in the immediate neighborhood.

26 Basic Concepts 26 What are the local maxima? What are the local minima?

27 Basic Concepts 27 A nice function f (x, y) = x 2 y 2. This is called concave. It is bowl shaped down. 0 Z Y X

28 Basic Concepts 28 A nice function f (x, y) = x 2 + y 2. This is called convex. It is bowl shaped up. 40 Z Y X

29 29 Basic Concepts A seriously mean function: f (X, Y ) = 3(1 X ) 2 e ( X 2 (Y +1) 2 ) 10(X /5 X 3 Y 5 )e ( X 2 Y 2 ) e ( (X +1) 2 Y 2 ) /3. Z Y 0 X 2-2

30 Basic Concepts 30 5 Z Y X

31 Model and Solution Tradeoff 31 Linear programs are the easiest to solve. However, most real problems usually have some nonlinearities and discrete aspects.

32 Model and Solution Tradeoff Even now, the most commonly solved problems are deterministic mixed-integer linear programs. Is the world linear? I don t think so! Question: So why all the solving of mixed integer linear programs? Answer: Say s Law!

33 Solution Report When you solve an optimization problem using a solver, you get a solution report. This solution report will indicate whether or not it found an optimal solution. If an optimal solution is found, the solution report will give the optimal objective function value and will give the optimal values of the decision variables. The solution report also provides other information given in Chapter 9, in particular Appendix 9.1.

34 34 Solution Report Go back to the simple LP we graphed earlier (simplelp.xlsx.). Convert to max x + 2y 2x + 2y 4 2x 4y 2 2x + y 8 2x + y 2 y 7 max x + 2y 2x 2y 4 2x 4y 2 2x y 8 2x + y 2 y 7

35 Solution Report 35 Solver generates a solution report.

36 36 Solution Report See the spreadsheet Answer Report 1

37 The constraints Solution Report 2x y 8 y 7 are tight or binding constraints since they are satisfied as strict equalities in the optimal solution. There is zero unused resource or in optimization lingo the slack is zero. However the other constraints 2x 2y 4 2x 4y 2 2x + y 2 are not tight and have positive slack of 25, 15, and 6, respectively. The concept of a tight constraint is used later when we discuss sensitivity analysis.

38 Solution Report Note also that the optimal solution of x = 7.5 and y = 7 with optimal solution value of 21.5 is: on the boundary of the feasible region, and it is at the intersection of two or more constraints a corner point Linear programming software only searches for extreme point solutions.

39 Solution Report The slack values are calculated at the optimal solution of x = 7.5 and y = 7 as follows: 2x 2y = 2(7.5) 2(7) = 29 2x 4y = 2(7.5) 4(7) = 13 2x + y = 2(7.5) + 7 = 8 Since the respective right-hand-sides of these constraints are -4, 2, and -2, the slacks are 25, 15, and 6, respectively.

40 Solution Possibilities A constrained optimization problem either has an optimal solution or it does not. If it has an optimal solution, then it has a unique optimal solution or alternative optima. What is the graphical interpretation of alternative optima? If it does not have an optimal solution, then in the linear case it is unbounded or infeasible.

41 Vocabulary Objective Function Constraint Decision Variable Nonnegativity Constraints Feasible Point Feasible Region Optimal Solution Slack and surplus

42 Vocabulary 42 Linear program Nonlinear program Integer program Stochastic program Infeasible problem Unbounded problem Alternative optima Redundant constraint Tight or binding constraint