ENEL_680_Linear_and_Integer_Programming-1.pdf
- 1. ENEL 680: Applied Optimization for Sustainable Design Linear Programming – Examples and Evaluation Dr. Hatem Abou-Zeid Assistant Professor Electrical and Software Engineering Department Office: ICT 252 Fall 2022
- 2. ENEL 680: Applied Optimization for Sustainable Design Hatem Abou-Zeid Today • Many interesting ideas and posts on Slack! • Gurobi Webinar coming up: Analytics for a Better World - Gurobi • Linear Programming Requirements • There might be a question on that in the mid-term • Overview of non-linear programming • Overview of integer programming • Ideas on multi-objective optimization, evaluation, simulation and it’s importance in your project • Mid-term formulation question
- 3. ENEL 680: Applied Optimization for Sustainable Design Hatem Abou-Zeid Mid term question • Describe and formulate an optimization problem for a sustainability-related design problem. Your formulation should include: • A brief overview of the problem • What are the decision variables – are they integer, linear, mixed? • What is the objective function (mathematically) • What are the constraints (mathematically) • You don’t need to include the constant values, these can be left as symbols as in the standard form in slide-16.
- 4. 4 © McGraw Hill 3.1 Prototype Example Wyndor Glass Co. • Produces windows and glass doors. • Plant 1 makes aluminum frames and hardware. • Plant 2 makes wood frames. • Plant 3 produces glass and assembles products.
- 5. 5 © McGraw Hill Prototype Example 1 Company introducing two new products: • Product 1: 8 feet glass door with aluminum frame. • Product 2: 4 × 6 feet double-hung, wood-framed window. Problem: What mix of products would be most profitable? • Assuming company could sell as much of either product as could be produced.
- 6. 6 © McGraw Hill Prototype Example 2 Products produced in batches of 20. Data needed: • Number of hours of production time available per week in each plant for new products. • Production time used in each plant for each batch of each new product. • Profit per batch of each new product.
- 7. 7 © McGraw Hill Prototype Example 3 TABLE 3.1 Data for the Wyndor Glass Co. problem Plant Production Time per Batch, Hours Product 1: Production Time per Batch, Hours Product 2: Production Time Available per Week, Hours 1 2 3 1 0 3 0 2 2 4 12 18 Profit per batch $3,000 $5,000
- 8. 8 © McGraw Hill Prototype Example 4 • Formulating the model: x1 = number of batches of product 1 produced per week x2 = number of batches of product 2 produced per week Z = total profit per week (thousands of dollars) from producing these two products • From bottom row of Table 3.1 Z = 3x1 + 5x2.
- 9. 9 © McGraw Hill Prototype Example 5 Constraints (see Table 3.1) 1 4 x 2 2 12 x 1 2 3 2 18 x x + 1 0 x 2 0 x Classic example of resource-allocation problem • Most common type of linear programming problem.
- 10. 10 © McGraw Hill Prototype Example 6 Problem can be solved graphically. • Two dimensional graph with x1 and x2 as the axes. • First step: identify values of x1 and x2 permitted by the restrictions. • See Figures 3.1 and Figure 3.2 • Next step: pick a point in the feasible region that maximizes value of Z. • See Figure 3.3
- 11. 11 © McGraw Hill Prototype Example 7 FIGURE 3.1 Shaded area shows values of (x1, x2) allowed by 1 2 1 0, 0, 4 x x x
- 12. 12 © McGraw Hill Prototype Example 8 FIGURE 3.2 Shaded area shows the set of permissible values of (x1, x2), called the feasible region.
- 13. 13 © McGraw Hill Prototype Example 9 FIGURE 3.3 The value of (x1, x2) that maximizes 3x1 + 5x2 is (2, 6).
- 14. 14 © McGraw Hill 3.2 The Linear Programming Model General problem terminology and examples: • Resources: money, particular types of machines, vehicles, or personnel. • Activities: investing in particular projects, advertising in particular media, or shipping from a particular source. Problem involves choosing levels of activities to maximize overall measure of performance.
- 15. 15 © McGraw Hill The Linear Programming Model 1 TABLE 3.2 Common terminology for linear programming Prototype Example General Problem Production capacities of plants 3 plants Resources m resources Production of products Activities 2 products Production rate of product j, xj Activities n activities Level of activity j, xj Profit Z Overall measure of performance Z
- 16. 16 © McGraw Hill The Linear Programming Model 2 • Standard form: 1 1 2 2 Maximize n n Z c x c x c x = + + + 11 1 12 2 1 1 21 1 22 2 2 2 1 1 2 2 1 2 subject to the restrictions and 0, 0 0. n n n n m m mn n m n a x a x a x b a x a x a x b a x a x a x b x x x + + + + + + + + +
- 17. 17 © McGraw Hill The Linear Programming Model 3 Other legitimate forms: • Minimizing (rather than maximizing) the objective function. • Functional constraints with greater-than-or-equal-to inequality. • Some functional constraints in equation form. • Some decision variables may be negative.
- 18. 18 © McGraw Hill The Linear Programming Model 4 Feasible solution: • Solution for which all constraints are satisfied. • Might not exist for a given problem. Infeasible solution: • Solution for which at least one constraint is violated. Optimal solution: • Has most favorable value of objective function. • Might not exist for a given problem.
- 19. 19 © McGraw Hill The Linear Programming Model 5 Corner-point feasible (CPF) solution: • Solution that lies at the corner of the feasible region. Linear programming problem with feasible solutions and bounded feasible region: • Must have CPF solutions and optimal solution(s). • Best CPF solution must be an optimal solution.
- 20. 20 © McGraw Hill The Linear Programming Model 5 Corner-point feasible (CPF) solution: • Solution that lies at the corner of the feasible region. Linear programming problem with feasible solutions and bounded feasible region: • Must have CPF solutions and optimal solution(s). • Best CPF solution must be an optimal solution.
- 21. 21 © McGraw Hill 3.3 Assumptions of Linear Programming Proportionality assumption: • The contribution of each activity to the value of the objective function (or left-hand side of a functional constraint) is proportional to the level of the activity. • If assumption does not hold, one must use nonlinear programming (Chapter 13).
- 22. 22 © McGraw Hill Assumptions of Linear Programming 1 Additivity: • Every function in a linear programming model is the sum of the individual contributions of the activities. Divisibility: • Decision variables in a linear programming model may have any values. • Including noninteger values. • Assumes activities can be run at fractional values.
- 23. 23 © McGraw Hill Assumptions of Linear Programming 2 Certainty: • Value assigned to each parameter of a linear programming model is assumed to be a known constant. • Seldom satisfied precisely in real applications. • Sensitivity analysis used.
- 24. 24 © McGraw Hill For the mid-term – you may be required to solve graphically • A type of cost-benefit trade-off problem. FIGURE 3.12 Graphical solution for the design of Mary’s radiation therapy.
- 25. 25 © McGraw Hill Graphical Illustration of Nonlinear Programming Problems 1 Maximize 1 2 3 5 , z x x = + subject to 1 4 x 2 2 1 2 9 5 216 x x + and 1 2 0, 0 x x
- 26. 26 © McGraw Hill Graphical Illustration of Nonlinear Programming Problems 3 Maximize 2 2 1 1 2 2 126 9 182 13 z x x x x = − + − subject to 1 4 x 2 2 12 x 1 2 3 2 18 x x + and 1 2 0, 0 x x
- 27. 27 Copyright 2020 © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Frederick S. Hillier ∎ Gerald J. Lieberman Chapter 12 Integer Programming
- 28. 28 © McGraw Hill Introduction 1 The integer programming (IP) problem is a restricted form of the linear programming problem. • Decision variables must have integer values. • The divisibility assumption does not hold. The mixed integer programming (MIP) variant. • Only some of the variables must have integer values.
- 29. 29 © McGraw Hill Introduction 2 Binary integer programming (BIP). • Two possible integer values, 0 and 1. • Example: yes or no decisions.
- 30. 30 © McGraw Hill 12.1 Prototype Example California Manufacturing Co. • Considers building new factory in Los Angeles or San Francisco, or both. • Considers building one new warehouse. • Choice of location restricted to city where factory is being built. • Objective: find the feasible combination of alternatives that maximizes the net present value.
- 31. 31 © McGraw Hill Prototype Example 1 • Formulating the model. ( ) 1 if decision is yes, 1, 2, 3, 4 . 0 if decision is no, Let total net present value of these decisions. = = = j j x j j Z TABLE 12.1 Data for the California Manufacturing Co. example Decision Number Yes-or-No Question Decision Variable Net Present Value Capital Required 1 Build factory in Los Angeles? x1 $9 million $6 million 2 Build factory in San Francisco? x2 $5 million $3 million 3 Build warehouse in Los Angeles? x3 $6 million $5 million 4 Build warehouse in San Francisco? x4 $4 million $2 million Capital available: $10 million California Manufacturing Co. • Considers building new factory in Los Angeles or San Francisco, or both. • Considers building one new warehouse. • Choice of location restricted to city where factory is being built. • Objective: find the feasible combination of alternatives that maximizes the net present value.
- 32. 32 © McGraw Hill Prototype Example 2 • Net present value given by: 1 2 3 4 9 5 6 4 = + + + Z x x x x • Model constraints. 1 2 3 4 3 4 6 3 5 2 10 1 + + + + x x x x x x • Decisions 3 and 4 are contingent on decisions 1 and 2. 3 1 4 2 x x x x
- 33. 33 © McGraw Hill Prototype Example 3 1 2 3 4 1 2 3 4 3 4 1 3 2 4 Maximize 9 5 6 4 , subject to 6 3 5 2 10 1 0 0 1 0 and isinteger, for 1, 2, 3, 4. = + + + + + + + − + − + = j j j Z x x x x x x x x x x x x x x x x x j
- 34. 34 © McGraw Hill Prototype Example 4 Software options for pure or mixed BIP models. • Excel. • LINGO/LINDO. • MPL/Solvers.
- 35. 35 © McGraw Hill 12.2 Some BIP Applications Investment analysis. • Decisions: whether to make certain fixed investments. • Used in capital budgeting decisions. • Upgrading a nation’s defense force. Site selection. • Decisions: For each potential site, should it be selected to be the location of a certain new facility.
- 36. 36 © McGraw Hill Some BIP Applications 1 Designing a production and distribution network. • Types of decisions. • Should a certain plant remain open? • Should a certain site be selected for a new plant? • Should a certain distribution center remain open? • Should a certain site be selected for a new distribution center?
- 37. 37 © McGraw Hill Some BIP Applications 2 Dispatching shipments. • Decision: should a certain route be selected for one of the trucks? • Can also select several factors simultaneously. • Certain route. • Certain size of truck. • Certain time period for departure. Scheduling interrelated activities. • Decision: should a certain activity begin in a certain time period?
- 38. 38 © McGraw Hill Some BIP Applications 3 Airline applications. • Decision types. • Should a certain type of aircraft be assigned to a certain flight leg? • Should a certain sequence of flight legs be assigned to a crew? • Reassignment of crews to flights to adjust to weather delays, etc.
- 39. 39 © McGraw Hill 12.3 Using Binary Variables to Deal with Fixed Charges The fixed charge problem. • Setup cost incurred to undertake an activity. • Total cost equals variable cost plus fixed charge. ( ) 1 Minimize , subject to the originalconstraints, plus 0 and is binary, for 1, 2,..., . n j j j j j j j j Z c x k y x My y j n = = + − =
- 40. 40 © McGraw Hill 12.4 A Binary Representation of General Integer Variables Problem with mostly binary variables and a few integer variables. • Cannot directly use an efficient BIP algorithm. • Substitute the binary representation for each of the general integer variables. • See Pages 472 to 473 in the text for formulation.