f(x) x S (optimal solution) f(x ) (optimal value) f(x) (1) 3 GLPK glpsol -m -d -m glpsol -h -m -d -o -y --simplex ( ) --interior --min --max --check -
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- うまじ かんざとばる
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1 GLPK by GLPK mmg.at.infoseek.co.jp/mmg/glpk/ : update 1 GLPK GNU Linear Programming Kit GNU LP/MIP ILOG AMPL(A Mathematical Programming Language) (optimization problem) X S X f : X R f ( x S f(x) (1) x S (2) f (objective function) S (feasible region) (2) (constraint condition) x S f(x ) 1
2 f(x) x S (optimal solution) f(x ) (optimal value) f(x) (1) 3 GLPK glpsol -m -d -m glpsol -h -m -d -o -y --simplex ( ) --interior --min --max --check --nomip MIP LP --wmps MPS ( ) --wlpt CPLEX ( ) --wtxt ( ) glpsol -m model1.mod -d data1.dat -o result1.sol --wlpt cplexform.txt --min result1.sol CPLEX cplexform.txt 4 GLPK.mod 2
3 .dat 5 (set) GLPK set 5.1 set {, }; ) Node set Node; 5.2 := )Node 1,2,3,4 set Node := ; 6 (parameter) 0 GLPK param 6.1 param {, }; 3
4 ) Cost i i Node param Cost{Node}; ) 1) param Cost := ; 2) param Cost[1] := 5; param Cost[2] := 3; param Cost[3] := 6; param Cost[4] := 9; ) ( ) 4
![2 3 3 6 4 9; 2) param Cost[1] := 5; param](/docs-images/49/20192541/images/page_4.jpg "Cost[2] := 3; param Cost[3] := 6; param")
5 1 p 10 1 q 20 2 p 15 2 q 23 3 p 12 3 q (tr) tr (transpose matrix) 1) 2) tr 1) param Val: p q := ) param Val(tr): 1 2 3:= p q ) ( )
6 1) param ArcResourceFC := ; 2) param ArcResourceFC[1,2,1] := 10; param ArcResourceFC[1,2,2] := 15; param ArcResourceFC[1,3,1] := 13; param ArcResourceFC[1,3,2] := 17; ) param : ) param: LT CT:=
![ArcResourceFC[1,3,1] := 13; param ArcResourceFC[1,3,2] := 17; ) param : ) 1](/docs-images/49/20192541/images/page_6.jpg "2 1 1 1 3 1 2 2 1 2 6 1 3 1 1 1 4 1 3 2 1 2 8 param: LT CT:= 1 2 1 1 1 3 1")
7 ; 7 (variable) 0 GLPK var var {, } )2 Flow i,j i Node, j Node GLPK var Flow{Node, Node}; ) x Z + integer binary 0-1 / set x integer, >=0; 8 (constraint condition) GLPK s.t. subject to s.t. {, } ) r Res x ir M i Node 7
8 s.t. COND1{i in Node}: sum{r in Res}x[i,r] <= M; in sum (objective function) (maximize) (minimize) GLPK maximize minimize minimize : ) : min. Cost i x i i Node minimize OBJ: sum{i in Node} Cost[i] * x[i]; 10 GLPK GNU * / less A > B A-B, A < B 0 div A/B mod A/B ** ˆ A B 8
9 10.2 / A < B A < B A <= B A B A > B A > B A >= B A B A <> B A! = B A B A in B A B A not in B A! in B A / B A within B A B if( ) then else ) t 1 X t 1 t = 1 0 if(t!=1) X[t-1] else : )i > j (i,j) Arc w ij (i > j) sum{(i,j) in Arc : i > j} w[i,j] within within ) 9
![1 if( ) then else ) t 1 X t 1 t = 1 0 if(t!=1) X[t-1] else 0 10.2.](/docs-images/49/20192541/images/page_9.jpg "2 : )i > j (i,j) Arc w ij (i > j) sum{(i,j) in Arc : i > j} w[i,j] 10.")
10 set A; set K within A; K A set A := ; set K := 5; 10.3 A union B A B A diff B A/B A simdiff B A B A inter B A B A cross B A B a.. b [a,b] setof setof{(, ) in } (, ) ) ARP( (i,j), (r), (p) 3 ) (i,j) (p) 2 AP set AP := setof{(i,j,r,p) in ARP}(i,j,p); 10.4 ) param p default 9999; p
![. b [a,b] 10.3.](/docs-images/49/20192541/images/page_10.jpg "1 setof setof{(, ) in } (, ) ) ARP( (i,j), (r), (p) 3 ) (i,j) (p) 2 AP set")
11 10.5 display display ; display 11
12 11 AMPL AMPL GLPK AMPL AMPL GLPK AMPL ampl ampl: AMPL model data solve option solver expand show display reset quit exit let commands AMPL AMPL m1.mod model m1.mod; AMPL commands option solver cplex; model m1.mod; data d1.dat; solve; com1.cms commands com1.cms; expand 12
13 reset; quit; MINOS( ) MIP MIP option solver cplex; CPLEX( ) AMPL 13
14 unbounded (or badly scaled) problem ( or ) must have * subscripts rather than # *,# * # 1 1 ) set Node; set Arc{Node, Node}; var X{Arc}; X Node2 14
15 12.3 ( or )[ ] out of domain ( ) 1 {1..N} 2 GLPK param T; # set Period := 1..T; # param T:=10; # 15
16 param T; # set Period; # param T:=10; # set Period := 1..T; # 12.4 syntax error in set statementset = := = := VB param 12.5 operand preceding = has invalid type if 16
17 x > 0 y = 1 if if 17
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106 4 4.1 1 25.1 25.4 20.4 17.9 21.2 23.1 26.2 1 24 12 14 18 36 42 24 10 5 15 120 30 15 20 10 25 35 20 18 30 12 4.1 7 min. z = 602.5x 1 + 305.0x 2 + 2
105 4 0 1? 1 LP 0 1 4.1 4.1.1 (intger programming problem) 1 0.5 x 1 = 447.7 448 / / 2 1.1.2 1. 2. 1000 3. 40 4. 20 106 4 4.1 1 25.1 25.4 20.4 17.9 21.2 23.1 26.2 1 24 12 14 18 36 42 24 10 5 15 120 30
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n=360 28.6% 34.4% 36.9% n=360 2.5% 17.8% 19.2% n=64 0.8% 0.3% n=69 1.7% 3.6% 0.6% 1.4% 1.9% < > n=218 1.4% 5.6% 3.1% 60.6% 0.6% 6.9% 10.8% 6.4% 10.3% 33.1% 1.4% 3.6% 1.1% 0.0% 3.1% n=360 0% 50%
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