Graph Optimization - Lab 1

Transcription

1 Graph Optimization - Lab 1 ORLAB - Operations Research Laboratory Stefano Gualandi October 24, 2011

2 What you do during lab sessions Data Math Model Algorithms Solver Solution(s) Modeling Interpretation Problem Strategy

3 What you do during lab sessions Data Math Model Algorithms Solver Solution(s) Modeling Interpretation Problem Strategy Check this out:

4 WARNINGS Warning 1: Lab sessions complement exercise sessions Warning 2: read carefully the error messages! Common pitfalls: unbounded variables missing constraints missunderstood constraints wrong quantification in the summation of constraints wrong objective function

5 A short introduction to AMPL In AMPL, models, data, and scripts are written in three separate text (ASCII) files: FILENAME.mod: it contains the model of the problem, i.e., variables, constraints, and objective function; FILENAME.dat: it contains the data specifing an instance of the problem, i.e., the coefficients of the constraints and objective function FILENAME.run: it contains script commands that typically select the solver, set parameters, solve the problem, and display the solution in a format of your choice. It can be used to implement basic algorithms. As editor, you can use Notepad, Emacs (with syntax highlight), or Gusek (based on SciiTech).

6 AMPL keywords set: defines set of elements param: defines constant parameters like constants, vectors, or matrices var: defines continuous variables. They can be defined also as integer binary sum: defines a linear sum subject to: defines a family of constraints minimize or maximize: defines the objective function Check this out:

7 AMPL support for scripting let...: change the value of a set or parameter for {set expression}...: iterate over the elements of a set if <condition> then... else...: check conditional repeat {... } while (condition): loop while a given condition is verifies repeat {... } until (condition): loop until a given condition is verifies Uniform(a,b): draw a random number in the interval [a...b] Uniform01(): draw a random number in the interval [0...1] Check this out:

8 Invoking AMPL commands You have two options: 1. Use the AMPL shell, and type the following commands (good for learning the basic steps): ampl: option solver cplex; ampl: model YOUR PATH/YOUR FILE.mod; ampl: data YOUR PATH/YOUR FILE.dat; ampl: solve; ampl: display x; 2. Use the script command file by DOS or LINUX shell (good for rapid math modeling) % ampl YOUR FILE.mod YOUR FILE.dat YOUR FILE.run;

9 Example 1: Cabled Networks

10 Example 2: Wireless Networks

11 Dealing with Directed Graphs param n > 0; set N := 1..n; set A within N cross N; # Number of nodes # Set of nodes # Set of arcs # Forward and Backward star of each node set FS {i in N} := setof {j in N: (i,j) in A} (i,j); set BS {i in N} := setof {j in N: (j,i) in A} (j,i); # In and Out degree of each vertex param in_degree {i in N} := card(bs[i]); param out_degree {i in N} := card(fs[i]); # Example of node weight vector param w {N} default 1; # Example of arc capacity vector param u {A} default 1;

12 Graph examples data; param n := 5; param: A : u := ;

13 Print a Directed Graph printf "\nforward STARS:\n"; for {i in N} { printf "node %d: ", i; for {(j,l) in FS[i]} printf "(%d,%d)[u=%d] ", j, l, u[j,l]; printf "\n"; } printf "\nbackward STARS:\n"; for {i in N} { printf "node %d: ", i; for {(j,l) in BS[i]} printf "(%d,%d)[u=%d] ", j, l, u[j,l]; printf "\n"; } printf "\ndegrees \n"; display out_degree, in_degree; Command line: ampl directed-graphs.mod digraph-1.dat print-digraph.run

14 Warm up (or Weak up!) Start with the exercise 1 on the handouts

15 Exercise 3: kruskal.run repeat { let MinCostEdges := {(i,j) in F: c[i,j] == min{(u,v) in F} c[u,v]}; let F := F diff MinCostEdges; for {(i,j) in MinCostEdges} { if cc[i] == 0 then { if cc[j] == 0 then { # Case 1: neither vertex is in the tree. let cc[i] := max{v in N} cc[v] + 1; let cc[j] := cc[i]; let T := T union {(i,j)}; } else { # Case 2: j is in the tree, but i isn t. let cc[i] := cc[j]; let T := T union {(i,j)}; } } else { if cc[j] == 0 then { # Case 3: i is in the tree, but j isn t. let cc[j] := cc[i]; let T := T union {(i,j)}; } else if cc[i] <> cc[j] then { # Case 4: i and j are in different connect for {v in N: cc[v] == cc[j]} let cc[v] := cc[i]; let T := T union {(i,j)}; } else printf "Adding edge (%d,%d) would create a cycle.\n\n",i,j; } } } while (card(t) < n - 1); printf "A minimum MST has cost %d.\n",sum{(i,j) in T} c[i,j];

16 Challenge 2: FarmVille

17 Challenge 2: FarmVille You begin the game with a 12 by 12 piece of land. That gives you 144 one by one squares in which to plant crops. To simplify the exercise, we have the following assumptions 1. Restrict my attention to the four crops initially available (strawberries, eggplant, wheat, and soybeans); 2. Assume crops will be harvested as soon as they are ripe (that is, ignore wilting); 3. Disregard the random bonuses and gifts that show up every now and then. The cost column includes the cost of plowing the land (15 coins) plus the cost of purchasing the seeds. Because the harvest times of all four crops are multiples of four hours, we can think in terms of 4-hour periods.

18 Challenge 2: FarmVille You initially have 200 coins, one ripe and one half-grown square of strawberries, one ripe and one half-grown square of eggplant. By harvesting the two ripe squares, your fortune grows to 323 coins. We also go ahead and clean (delete) the two half-grown squares so that we can begin with a clean slate. Question: What is the largest amount of cash one can obtain in 6 days? (need to rest on the 7th day).

19 Links and Lab sessions web site: http: //www-dimat.unipv.it/~gualandi/graph-optimization/ stefano.gualandi [at] unipv.it