1. 8 - 1 Optimization  Find the best set of decisions for a particular measure of performance  Includes: The goal of finding the best set The algorithms to accomplish this goal

2. 8 - 2 Excel Optimization Software  Solver Standard with Excel  Premium Solver for Education Comes with text – install off text CD More advanced than standard solver Is preferred tool throughout text Click on Premium button in Solver Parameters window to toggle to premium version

3. 8 - 3 Decision Variables  Levers to improve performance  Want to find the best values for the variables  Finding these best values can be challenging Need Solver’s sophisticated software Still relatively easy to construct models beyond Solver’s capabilities

4. 8 - 4 Solver Parameters Window  Target Cell Maximize, minimize, or set equal to target value  Changing cells Decision variables  Constraints Restrictions on decision variables  Should predict outcome before clicking Solve button

5. 8 - 5 Solver Window  ***insert Figure 8.2

6. 8 - 6 Adding Constraints  Click on Add button in Parameters window Use formula cell on leftUse number cell on right

7. 8 - 7 Solver Options Check if decision variables known to be non-negative Scaling discussed later – usually not needed Check unless want to use reports Only used if need intermediate results e.g., for debugging

8. 8 - 8 Formulation  Decision variables What must be decided? Be explicit with units  Objective function What measure compares decision variables? Use only one measure – put in target cell  Constraints What restrictions limit our choice of decision variables?

9. 8 - 9 Constraints  Left-hand-side (LHS) Usually a function  Right-hand-side (RHS) Usually a number (i.e., a parameter)  Three types of constraints LHS <= RHS(LT constraint) LHS >= RHS(GT constraint) LHS = RHS(EQ constraint)

10. 8 - 10 Types of Constraints  LT constraints (LHS<=RHS) Capacities or ceilings  GT constraints (LHS>=RHS) Commitments or thresholds  EQ constraints (LHS=RHS) Material balance Define related variables consistently

11. 8 - 11 A Standard Model Template Is Recommended  Enhances ability to communicate Provides common language Reinforces understanding how models shaped  Improves ability to diagnose errors  Permits scaling more easily

12. 8 - 12 Layout  Organize in modules Decision variables, objective function, constraints  Place decision variables in single row or column Use color or border highlighting  Place objective in single highlighted cell Use SUM or SUMPRODUCT where appropriate  Arrange constraints to make LHS and RHS clear Use SUMPRODUCT for LHS where appropriate

13. 8 - 13 Ranges for Decision Variables and Constraints  Changing cells allows for commas but better to put in one contiguous range  Add Constraint window allows for ranges Group LT, GT, EQ, constraints together Enter as ranges LHS will be matched with RHS in one-to-one correspondence

14. 8 - 14 Results  Optimal values of decision variables Best course of action for the model  Optimal value of objective function Best level of performance possible  Constraint outcomes Constraint is tight or binding if LHS=RHS in LT or GT constraint, otherwise slack

15. 8 - 15 Optimization Solution  Tactical information Plan for decision variables  Strategic information What factors could lead to better levels of performance? Binding constraints are economic factors that restrict the value of the objective

16. 8 - 16 Model Classification  Linear optimization or linear programming Objective and all constraints are linear functions of the decision variables  Nonlinear optimization or nonlinear programming Either objective or a constraint (or both) are nonlinear functions of the decision variables  Techniques for solving linear models are more powerful Use wherever possible

17. 8 - 17 Hill Climbing  Technique used by Solver for nonlinear optimization  Called GRG (Generalized Reduced Gradient) algorithm  Hill climbing in a fog Try to follow steepest path going up After each step, or group of steps, again find steepest path and follow it Stop if no path leads up

18. 8 - 18 Local and Global Optimum  The highest peak is the global optimum What we want to find  Any peak higher than all points around it is a local optimum What the GRG algorithm locates Except in special circumstances, there is no way to guarantee that a local optimum is the global optimum If multiple local optima, then which is found depends on starting point for decision variables – may want to run Solver starting from multiple points

19. 8 - 19 Properties of Linear Functions  Additivity  Proportionality  Divisibility

20. 8 - 20 1.Start with a feasible set of decision variables that corresponds to a corner on a diamond 2.Check to see if a feasible neighboring corner point is better 3.If not, stop; otherwise move to that better neighbor and return to step 2 Guaranteed to converge to the global optimal solution The Simplex Algorithm For Linear Optimization

21. 8 - 21 Integer Values  If fractional decision variables not appropriate  Integer linear programming Implicitly enumerate all possible assignments of integer values – runs simplex algorithm for each Reliable solution but may take a while  Integer nonlinear programming Solver’s solutions cannot be considered reliable

22. 8 - 22 Nonlinear Programming Problems  Revenue maximization Maximize revenue in the presence of a demand curve  Curve fitting Fit a function to observed data points  Economic Order Quantities Trade-off ordering and carrying costs for inventory

23. 8 - 23 Sensitivity Analysis for Nonlinear Programs  Solver Sensitivity Found under Sensitivity Toolkit  Inputs similar to Data Sensitivity tool  Shows effect of parameter changes on optimal value of objective Resolves optimization for each input value

24. 8 - 24 Linear Programming Problems  Allocation models Maximize objective (e.g., profit) subject to LT constraints on capacity  Covering models Minimize objective (e.g., cost) subject to GT constraints on required coverage  Blending models Mix materials with different properties to find best blend  Network models Describe patterns of flow in a connected system

25. 8 - 25 Model Scaling  Consider scaling parameters to appear in thousands or millions  Saves work in data entry – decreases errors  Spreadsheet looks less crowded  Helps with Solver algorithms Value of objective, constraints, and decision variables should not differ from each other by more than a factor of 1000, at most 10,000  Can always display model output on separate sheet with separate units

26. 8 - 26 Automatic Scaling  Use if scaling problems difficult to avoid  Consider when: Solver claims linearity conditions not met in a model that is definitely linear Solver reports convergence but not that the optimality conditions are satisfied in a nonlinear model  Preferable for model-builder to do the scaling

27. 8 - 27 Sensitivity Analysis for Linear Programs  A distinct pattern to the change in the optimal solution when varying a coefficient in the objective function  In some interval around the base case No change in optimal decisions Objective will change if decision variable is positive  Outside this interval a different set of values is optimal for decision variables

28. 8 - 28 Shadow Prices  Improvement in objective function from a unit increase (or decrease) in RHS of constraint  Presented in third column of Solver Table from Solver Sensitivity  Break-even price where attractive to acquire more of a scarce resource

29. 8 - 29 Sensitivity Analysis for Binding Capacity Constraints  A distinct pattern in sensitivity tables when varying availability of scare resource  In some interval around the base case: Marginal value (shadow price) of capacity remains constant Some variables change linearly with capacity Others remain the same  Below this interval the value decreases and eventually reaches zero

30. 8 - 30 Patterns in Linear Programming Solutions  The optimal solution tells a “story” about a pattern of economic priorities Leads to more convincing explanations for solutions Can anticipate answers to “what-if” questions Provides a level of understanding that enhances decision making  After optimization should always try to discern the qualitative pattern in the solution

31. 8 - 31 Constructing Patterns  Decision variables Which are positive and which are zero?  Constraints Which are binding and which are not?  “Construct” the optimal solution from the given parameters Determine one variable at a time Can be interpreted as a list of priorities which reveal the economic forces at work

32. 8 - 32 Defining Patterns  Qualitative description  Pattern should be complete and unambiguous Leads to full solution Always leads to same solution  Ask where shadow prices come from Should be able to trace the incremental changes to derive shadow price of constraint

33. 8 - 33 Integer Programming  Fractional solutions are less important for strategic rather than tactical implications of model  Solver constraints window allows choices of int and bin for any variable int: variable must be an integer bin: variable must be binary, i.e, 0 or 1  Tolerance parameter under options Distance away from the optimal solution given solution allowed to be Default is 5% (quite large) – allows for fast solutions

34. 8 - 34 Binary Variables  All or nothing variables (0 or 1)  Represent go/no-go decisions  PROJ i = Indicator for Project i =  Can use to represent structural or policy relationships Select at least m of the possible projects Select at most n of the possible projects 1 accept Project i 0 reject Project i

35. 8 - 35 Logical Relationships with Binary 0/1 Variables 1. K out of N Projects Constraint Example: Must Choose at least one of Projects 5, 6, 7 2. Mutually Exclusive Constraints Example: Can’t select both Project 1 and Project 3 3. Contingency Constraints Example: Project 4 cannot be done unless Project 2 is done PROJ 1 + PROJ 3  1 PROJ 4  PROJ 2 or PROJ 4 - PROJ 2  0 PROJ 5 + PROJ 6 + PROJ 7  1

36. 8 - 36 Excel Use  The logical functions in Excel (IF, AND, OR, etc.) can express logical relationships  Such functions are nonlinear  Models with IF statements will often stop at local optimum  Using binary variables is the preferred approach

37. 8 - 37 Fixed Costs  F – fixed cost  c – per unit cost  x – units of activity (regular variable)  y – go/no go of activity (binary variable)  Cost = Fy + cx  Constraint: x <= My or x - My <= 0 M = upper bound on x x =0 if y=0 (can’t have activity if no-go)

38. 8 - 38 Threshold Levels  A decision variable must be at least as large as a specified minimum m, else it is zero  x – units of activity (regular variable)  y – go/no go of activity (binary variable)  Constraints: x >= my or x – my >=0 x <= My or x - My <= 0 M = upper bound on x x >= m if y=1 (must meet minimum level if go) x = 0 if y=0 (can’t have activity if no-go)

39. 8 - 39 Guidelines for Optimization Modeling  Follow a standard form whenever possible  Enter cell references in Solver windows Keep numerical values in cells  Use linear models in preference to nonlinear  Use the weak form of a constraint Give the model maximum flexibility

40. 8 - 40 Guidelines for Optimization Modeling (Continued)  Explore some feasible (and infeasible) possibilities as a way of debugging  Test intuition and suggest hypotheses before running Solver  Identify the patterns of economic priorities that appear in the solution

41. 8 - 41 Summary  Optimization: what values of the decision variables lead to the best possible value of the objective?  Excel Solver: Collection of procedures Linear, (linear) integer, non-linear programs  Steps: Formulating, Solving, and Interpreting Results  Results still need to be interpreted in the real world Look for the pattern, or economic priorities, in solution