1. Optimization for Network Planning Includes slide materials developed by Wayne D. Grover, John Doucette, Dave Morley © Wayne D. Grover 2002, 2003 E E 681 - Module 3 (version for book website)

2. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 2 Outline Optimization –Mathematical Programming –Linear Programming (LP) –Formulating LP Problems –Solving LP Problems –Integer Programming –Solving MIP Problems –Algebraic Expression of LP/IP Problems Mesh-Restoration Concept –Terminology –Spare Capacity Sharing –Spare Capacity Placement (SCP) –SCP Integer Programming problem

3. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 3 Mathematical Programming 1 T.W. Knowles, Management Science: Building and Using Models, Irwin, 1989. Maximize (or minimize): Subject to: Constraints Objective Definition –“A Mathematical Programming Model is a mathematical decision model for planning (programming) decisions that optimize an objective function and satisfy limitations imposed by mathematical constraints.” 1 General Symbolic Model … … where are the decision variables.

4. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 4 Mathematical Programming Types of Mathematical Programs: –Linear Programs (LP): the objective and constraint functions are linear and the decision variables are continuous. –Integer (Linear) Programs (IP): one or more of the decision variables are restricted to integer values only and the functions are linear. Pure IP: all decision variables are integer. Mixed IP (MIP): some decision variables are integer, others are continuous. 1/0 MIP: some or all decision variables are further restricted to be valued either “1” or “0”. –Nonlinear Programs: one or more of the functions is not linear.

5. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 5 Linear Programming Maximize: Subject to: Constraints Objective Bounds … where are the model parameters. General Symbolic Form …

6. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 6 Linear Programming General Restrictions –All decision variables must be nonnegative, –Constant terms cannot appear on the LHS of a constraint. –No variable can appear on the RHS of a constraint. –No variable can appear more than once in a function, i.e. objective or constraint. Steps for Formulating LP Models –Construct a verbal model. –Define the decision variables. –Construct the symbolic model.

7. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 7 Formulating LP Problems 2 from, R. Fourer, D. Gay, B. Kernighan, AMPL, Boyd & Fraser, 1993, pp. 2-10. Tons/Profit/ hour ton Bands200$25 Coils140$30 Maximum tons:Bands6,000 Coils4,000 An example 2 –A steel company must decide how to allocate production time on a rolling mill. The mill takes unfinished slabs of steel as input and can produce either of two products: bands and coils. The products come off the mill at different rates and also have different profitabilities: –The weekly production that can be justified based on current and forecast orders are:

8. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 8 Formulating LP Problems Maximize:total profit Subject to:total number of production hours  40 tons of bands produced  6,000 tons of coils produced  4,000 An example (cont’d) –The question facing the company is: If 40 hours of production time are available, how many tons of bands and coils should be produced to bring in the greatest total profit? Constructing the Verbal model –Put the objective and constraints into words. –For constraints, use the form {a verbal description of the LHS} {a relationship} {an RHS constant}

9. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 9 Defining the Decision Variables –X B number of tons of bands produced. –X C number of tons of coils produced. Construct the Symbolic Model Formulating LP Problems Maximize: Subject to:

10. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 10 Solving LP Problems Bands 0 02000400060008000 Coils 2000 4000 6000 Constraints Feasible region 0 02000400060008000 Bands Coils 2000 4000 6000 220K 192K 120K Profit Optimal solution Hours Graphical Solution Method

11. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 11 Solving LP Problems Unique Optimal SolutionAlternate Optimal Solutions No Feasible Solution Unbounded Optimal Solution 4 Possible Outcomes

12. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 12 Solving LP Problems Simplex method –Efficient algorithm to solve LP problems by performing matrix operations on the LP Tableau. –Developed by George Dantzig (1947). –Can be used to solve small LP problems by hand. AMPL and CPLEX –AMPL: modeling language (and software) for designing large and complex LP/IP problems. –CPLEX: software package (“solver”) to solve large and complex LP/IP problems. Sub-Optimal Algorithms (Heuristics) –Simulated annealing. –Genetic algorithms. –Tabu search. –Many others, often very specific to the type of problem.

13. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 13 Integer Programming Maximize: Subject to: integer Convert Example to Integer Program –Assume that orders for bands and coils are placed (and filled) in 1,000s of pounds only. –Although feasible region is greatly reduced, problem becomes much more difficult. New Symbolic Model –Let the new decision variables be the number of 1000 pound “units” or orders of bands and coils.

14. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 14 Integer Programming 0 0 246 8 2 4 6 Feasible integer solutions Bands Coils $185K Optimal integer solution ($185K) Graphical Solution Method

15. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 15 Solving MIP Problems Branch-and-Bound Procedure –The solution space consists of a tree of LP subproblems, in which each integer variable is either fixed or its integrality constraint is “relaxed.” –The root node of the tree is the LP relaxation of the problem, i.e. all integer variables are relaxed. –The relaxation can result in an all integer solution, or a fractional solution (some decision variables are non-integer). –If the solution of the relaxation has fractional-valued integer variables, a fractional variable is selected for branching and two new subproblems are generated, each with more restrictive bounds on the branching variable. –The subproblems can result in an all integer solution, an infeasible problem or another fractional solution. –If the solution is fractional, the process is repeated. –Branches are “fathomed” if the subproblem is infeasible, the objective value is worse than the current best integer solution or the subproblem gives an integer solution.

16. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 16 Solving MIP Problems 0 12 36 45 Bounds 0<=XB<=6 0<=XC<=1 Solution Obj. = 180K XB = 6.00 XC = 1.00 Bounds 6<=XB<=6 2<=XC<=4 Solution *Infeasible Bounds 0<=XB<=5 3<=XC<=4 Solution Obj. = 183K XB = 3.00 XC = 3.71 Bounds 0<=XB<=6 0<=XC<=4 Solution Obj. = 192K XB = 6.00 XC = 1.40 Bounds 0<=XB<=6 2<=XC<=4 Solution Obj. = 189K XB = 5.14 XC = 2.00 Bounds 0<=XB<=5 2<=XC<=4 Solution Obj. = 189K XB = 5.00 XC = 2.10 Bounds 0<=XB<=5 2<=XC<=2 Solution Obj. = 185K XB = 5.00 XC = 2.00 1 2 3 4 5 6 8 7 9 10 Branch-and-Bound Tree (Example)

17. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 17 Algebraic Expression of LP/IP Problems Why use it? –Most LP/IP problems are quite large and it becomes very cumbersome to describe them by explicitly giving each linear function, equality, and inequality in full. –It is desirable to model problems in a more general fashion (e.g. give an IP for optimally designing a mesh-restorable network in general as opposed to doing so for a specific network).

18. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 18 Algebraic Expression of LP/IP Problems Basic Production Model (Revisited) \Problem name: prob.lp Maximize 25 XB + 30 XC Subject To 0.005 XB + 0.007143 XC <= 40 Bounds 0 <= XB <= 6000 0 <= XC <= 4000 End Maximize: Subject to: Given: a set of products hours available at the mill tons per hour of product j, for each profit per ton of product j, for each maximum tons of product j, for each Define variables: tons of product j to be made, for each Algebraic ModelOriginal Model

19. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 19 Example: LP/IP for Mesh-Restoration Design Networks are Inherently Mesh-Like Distributed mesh-restoration exploits network connectivity to allow sharing of redundancy. Level(3) North American Network

20. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 20 Spare Capacity Sharing Consider 2 different failure scenarios X X –Restoration is allowed to follow multiple distinct routes. –Restoration route for both failure scenarios have several spans in common. –Spare capacity on each span contributes to restorability of many spans.

21. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 21 Spare Capacity Placement (SCP) Optimal Design –Objective is to find least costly way to place sufficient spare capacity on a network such that all spans are fully restorable. –Can we use LP/IP? –Reference: M. Herzberg and S. J. Bye, “An Optimal Spare-Capacity Assignment Model for Survivable Networks with Hop Limits”, IEEE Globecom’94, 1994 Integer Programming Approach –Objective Function: Minimize Cost of Spare Capacity Placement –Constraints: Each possible span failure has enough restoration flow for full restoration. Enough spare capacity exists on each span to accommodate restoration flows.

22. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 22 SCP Integer Programming Problem Parameters (Inputs) –C j :Cost of each unit of capacity on span j –L i : Target Restoration level for span i (L i = 1 assumed) –S: Number of spans in the network –P i :Number of eligible routes for restoration of span i –w j :Number of working links (capacity units) on span j –  i,j p :Equal to 1 if p th eligible route for span i uses span j Variables (Outputs) –f i p :Restoration flow assigned to pth route for span i –s j :Number of spare capacity units placed on span j

23. E E 681 Lecture #3 © Wayne D. Grover 2002, 2003 23 SCP Integer Programming Problem Objective Function: Subject To: –1. –2.