u1 ( a1) tk ( I). tk . , , . . , , .
, , .
, . 2. , - . a 1,2,3 () . - u (t), - .
(- , ), R (t), ,
Qk.
0: tk
= ò u (z) dz. (27)
)
H J
A B
E F
C
0 D G tk t
)
.2
, , t1, - ( ) u2 < u1 , a2 < a1 ( 2 DEFC). t4 = t1 + ty , , D u ( 3, a3 > a1 , GHJC) ,
t1 tk tk tk
ò u1 (z) dz + ò u2 (z) dz + ò D u (z) dz = ò u1 (z) dz (28)
0 t1 t1 + ty 0
ty - :
t = ty+tp+tu (29)
t3- ; tp, tu - .
, , , , ty t3 , tp tu. t3 ( ) .
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". . ( ) , .
( ) , .
tp -, [ 5 ], T ,
m
ka * k * åRi Qi
i
tp = T = ----------------------, (30)
k * P * n
k = s*- s - ; H - ; k - ; Pi- , i ; Qi - i - ; m - ; - ( , -), ; k - ; - ; n , n = 1.
() .
, , () t , , . - , , , -, . , . 2. , .
: i=1,2,i*- ; j =1,2, j*- () ; ɣ (ɣ= 1,2, , ɣ *) - ; Sɣ- - ; Sij - i - j - ; R = {rij} , , (); uij - i - j - ; qij - i - ɣ - [0tT] - , ; ti - , m ɤ - ɤ - ; n ɤ - ɤ - ; Mij - - i - j - ;
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πɤ i = (πiɤ 1, , π ɤ i*) - ɤ - ( π ɤ i, - π ɤ i); Δt - , ; [t0, t0 + Δt ] - - , ; ŋ = 1,2, , ŋ* .
, t0 R ɤ q (t0 ) = [ q Ii j ɤ] j - π ɤ i j=1, , π ɤ ij*.
ɤq (t) : ɤ q (t0 ) - , 0
-; ɤq (t0 )- , +
-; ɤ q (t0 ) - , -.
_ 0 +
ɤ q (t 0) = ɤ q (t 0) U ɤ q (t0 ) U ɤ q (t 0) (31)
" , (). , j- (, ) , .. ɤ - j:= const . , , . , , , . ∑∑ rij ≤ R- .
, , : " , ().
uij= const - i - j- - [aij]- i-x ,
óI, 4- ();
a i j : = í -1, I- ;
î 0, i-
( ) " .
( ),
i- j - .
s ɤ j = t ɤ j (s ɤ + M ɤ ji * s ɤ ji ) (32)
ɤ - j -
t ɤ j = q ɤ ji / M ɤ ji 33)
q ɤ ji - i -x j - ɤ -om .
i- j- ɤ -ro
q ɤ ji = M ɤ ji * t ɤ ji. (34)
:
Z = ∑∑ s ɤ j * a ɤ j + ∑s ɤ (z ɤ + t ɤ) >min (35)
∑ t ɤ j * a ɤ j < Tj; j = 1, n; (36)
∑ aji = 1; ɤ = 1, m; (37)
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∑∑ r ɤ j * a ɤ j ≤ R; (38)
N (1 - y ɤ p) + (zp +z ɤ j ) ≥ t ɤ j (a ɤ j + apj - 1); (39)
ɤ, p = 1,m; j = 1,n;
T ɤ - ∑ t ɤ j * a ɤ j ≥ z ɤ ≥ t ɤ; ɤ = 1, m; (40)
zr ≥ z ɤ + ∑ t ɤ j * a ɤ j; ɤ r ~ ɤ = 1,m; (41)
a ɤ j ∈ { 0,1}; y ɤ p ∈ { 0,1}; zɤ ≥ 0. (42)
: S ɤ j- ɤ j - : a ɤ j =1 ɤ j- , a ɤ j =0 - s ɤ - ɤ ; z ɤ - ɤ; t ɤ - ɤ ; t ɤ j - ɤ j - ; Tj - j- : rij - , ɤ j - ; R- ; N - ; yɤ p , p ɤ, yɤ p = 0 ; Tɤ - ɤ; M ɤ ji - .
, , . (36) . (37) . (38) . (39) : ɤ j - , zp - z ɤ ≥ tɤj. (40) . , , (41). zr - , ɤ . , : n ɤ ≥ m ɤ ; ∑∑ rɤj ≤ R - ; πɤij = (π ɤ i j=1, , πɤ ij*) ɤ - i - j . t ɤ =1 ji < t ɤ = 2ji π ɤ = 2ji = 1 π ɤ = 2ji = 2 t ɤ =1 ji; t ɤ =1 ji; [m ɤ i j ] - , .
ó I, n ɤ - ;
m ɤ ji = í - I, ;
î 0, n ɤ - .
ɤ q (t0 ) n ɤ < m ɤ .. Δ t ɤ =1 < Δ t ɤ =s < Δ t ɤ =n < m ɤ ji t0 ɤ - .. m I=I* =0. , .. rij = m ɤ ji , m ɤ , rij , ɤ - .. rij m - ɤ - j- . t0 ∑ r ɤ ji. .. Rij - ∑ r ɤ ji =0 , ɤ q (t) . , Δ Rij = Rij - ∑ r ɤ ji - ().
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t , .. Δ Rij = { rij } j - .
j - - ɤ d ɤ = min, mɤ .j- { rij }ti ɤ - , .. { rij } = m ɤ ji . {rij }ti , .. { rij }ti = O { ɤ q* (t0) } = { q ɤ I * } = O.
j -
q ɤ ji / t ɤ = u ij (43)
ti , .. [ ti, T ] [ 0,T ],
å qi j ɤ / (å qi j ɤ / Mij = u(n), (44)
i j ɤ ɤ [ti,T]
_
u(p) = (å qi j ɤ - å qi j ɤ) / (T ti). (45)
[ti,T] [0,T] [0,ti]
, , () , -. , . , , , , .
.
Δ t. e - , R = { rij }. - , [ t0, t0, + Δ t ] t - rij ɤq (t) = { qɤji }, , . , { qɤji } to, . { q ɤ ji }.
ɤ q (t) . -, .. { q ɤ ji } [ t0, t0, + Δ t ] t0, n ɤ < m ɤ - .. t ɤ ɤ t ɤ t0 , .. , δd1 ji = 1
δd1 ɤji = 0 - ).
n ɤ < m ɤ [ t0, t0, + Δ t ], å q1 ɤ ji .
- η -
å Δqij m ɤ =å Δqij m ɤ - å Δqij n ɤ
[ t0 η ] [ 0,T ] [0, t0 η ]
ri j , m- ɤ -ro pij mɤ /. å pij mɤ, /.
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[rij], pij mɤ - å Δqɤji η - [ t0, t0, + Δ t ]
{ri j }1 = å Δqij m ɤ / pij m ɤ * Δ t, (46)
[ t0, t0, + Δ t ]
, .
{ R } = { rij } ≥{ rij m ɤ } { rij m ɤ } (47)
η - , ..
{ R } = { rij } <{ rij m ɤ } { rij m ɤ }1 (48)
[ t0, t0, + Δ t ]
, , , .
, , 120 , , 10 - 1,2,3 4- (0,5 ). , , , (0,5 ). , ( ) ()
{ rij m ɤ }1 ≤ 0,5 * { rij m ɤ } / η. (49)
, .
Z = tp = å Δqij m ɤ / å rij m ɤ pij = min (50)
ij ij
å rij ≥ å rij m ɤ + å(rij m ɤ)1; (51)
ij i ij
å rij m ɤ ≤ 0,5 å rij m ɤ / η = B; (52)
ij ij
å (rij mɤ)1 > B, .
Δqij m ɤ ≥ 0; (53)
I =1,2,I*; j=1,2,j*; m ɤ =1,2,; m ɤ *; ɤ =1,2, ɤ *;
(51) - . : (52) - ; (53) - ; (2854) - ; (55)- .
. rij, a. : rij (t) t; τij - qij mɤ . t [ t0, t0, + Δ t ].
rij = å d (qij m ɤ t) δq ji m ɤ + å b (qij m ɤ). (54) qji m ɤ ɤ q (t) t0 - τij(q ji m ɤ) ≤t0
(2854) ɤ q (t) [ t0, t0, + Δ t ]. (54) , t0 - τij(qjim ɤ). t0, , [t0, t0, + Δ t], , [t0, + Δ t]; b (q ji m ɤ), t: = 1, q ji m ɤ rij t, b (q ji m ɤ), t: = 0, . : (t - t q ji m ɤ) ɤ τij(q ji mɤ), b (q ji mɤ t) = 1; (t - t q ji m ɤ) ≥ τij(q ji m ɤ), b (q ji mɤ t) = 0. , (54) t { rij}, . (54) t t0, .. . : rij (t) ≤ r ij (a), r ij(a) - a. ɤ0q (t) ɤ 0q (t). ɤ 0q (t) , -. , ɤ0q(t). ɤ+q(t) - qji mɤ -. qji mɤ . , ɤ +q(t) .
, , .