Production and Supply Chain Management Logistics. Paolo Detti Department of Information Engeneering and Mathematical Sciences University of Siena

Transcription

1 Producton and Supply Chan Management Logstcs Paolo Dett Department of Informaton Engeneerng and Mathematcal Scences Unversty of Sena

2 Convergence and complexty of the algorthm

3 Convergence of the algorthm Durng the smplex algorthm the selecton of the leavng arc s a crtcal step that may affect the algorthm convergence In case of degenerate pvot teraton the algorthm mght not termnate fnetely. A pvot teraton s the operaton of movng from a feasble spannng tree structure to another. A pvot teraton s degenerate (and the tree s degenerate) when the maxmum flow that can be sent around the pvot cycle s 0. Ths may occur only when the spannng tree s degenerate (some arc of the tree s restrcted).

4 Convergence of the algorthm The convergence can be assured by usng a specal class of spannng trees called strongly feasble spannng trees Equvalent defntons: Let us arbtrarly choose the root node of the spannng tree (e.g. node ). A spannng tree T s strongly feasble f every tree arc wth flow equal to the lower bound s upward pontng the root node and f every tree arc wth flow equal to the upper bound s downward pontng the root node. A spannng tree T s strongly feasble f we can send a postve amount of flow from any node to the root along the tree arcs wthout volatng any arc bound

5 Equvalent defntons: Let us arbtrarly choose the root node of the spannng tree (e.g. node ). A spannng tree T s strongly feasble f every tree arc wth flow equals to the lower bound s upward pontng and f every tree arc wth flow equals to the upper bound s downward pontng the root node. A spannng tree T s strongly feasble f we can send a postve amount of flow from any node to the root along the tree arcs wthout volatng any arc bound (,3) (0,) 6 (x, u ), l =0 not strongly feasble tree (0,) (,) (3,) 3 4 (6,6) (,4) (,)

6 Obtanng an ntal strongly feasble spannng tree structure T,L,U (when all arcs have 0 lower bounds) Let us suppose that Between each par of nodes a drect path exsts wth nfnte maxmum capacty: Gven a network G=(V,E), let us add n G dummy arcs (,) and (,) for each node n V\{}, wth nfnte upper bound and cost (no optmal soluton exsts usng one of these arcs) An ntal spannng tree structure T,L,U can be obtaned as follows For each suppler (a()>0) or transt (a()=b()=0) node add n T the arc (,) wth flow equal to a() For each demand (b()>0) node add n T the arc (,) wth flow equal to b() Add n the set L all other arcs of G The set U s empty Note that, a nondegenerate spannng tree s strognly feasble

7 Durng a pvot operaton, the movng from strongly feasble spannng tree to another can be performed by the followng selecton rule Leavng arc selecton rule: Let the apex node be the frst common acenstor of nodes that are endponts of the enterng arc, n the paths connectng these nodes wth the root node (,3) 6 (0,) (,) 9 0 (x, u ), l =0 (,3) (0,) (0,) (,) Apex node Pvot cycle (degenerate)

8 Durng a pvot operaton, the movng from strongly feasble spannng tree to another can be performed by the followng selecton rule Leavng arc selecton rule: The leavng arc s the last blockng arc found n traversng the pvot cycle along ts orentaton startng at the apex node (,3) 6 (0,) (,) 9 0 (x, u ), l =0 (,3) (0,) (0,) (,) Apex node Pvot cycle (degenerate)

9 Leavng arc selecton rule: The leavng arc s the last blockng arc found n traversng the pvot cycle along ts orentaton startng at the apex node (.e., the frst common acenstor of nodes that are endponts of the enterng arc, n the paths connectng these nodes wth the root node) (x, u ) (,3) (0,) (0,) (,) Apex node Pvot cycle (degenerate) Blockng arcs: (,3) e (7,) Selected arc: (7,) The flow n the cycle does not change

10 The new spannng tree s strongly feasble. In fact: (x, u ) (,3) W Apex node W 9 0 k (0,) l 6 (,) Arcs n the segment W : snce (,7) s the last blockng arc, n the new spannng tree, each node n W can send a postve amount of flow to the root Arcs n the segment W : ) If the pvot cycle was not degenerate, then postve flow has been sent to the arcs of W. Such a flow can be re-sent backward toward the root; ) If the pvot cycle was degenerate, then no flow changed n the tree. Hence, snce the prevous tree was strongly feasble, the new tree s strongly feasble, too (.e., a postve amount of flow could be sent from nodes n W to the root). c.d.d.

11 Convergence of the algorthm Each non-degenerate pvot teraton strclty decreases the obectve functon value (of the amount c kl δ) where (k,l) s the enterng arc and δ>0 s the flow augmentaton around the cycle. Then, the number of nondegenerate pvot s fnte. It can be proved that, when strongly feasble spannng trees are consdered the number of consecutve degenerate pvot teratons between any two nondegenerate pvot teratons s fntely bounded.

12 Convergence of the algorthm Let us suppose that the enterng arc (k,l) has c kl <0, and flow x kl =l kl =0. Let us suppose that the pvot cycle s degenerate. Snce the spannng tree was strongly feasble, we can send a postve amount of flow from any node to the root (node ) along the tree arcs wthout volatng any arc bound (x, u ) (,3) (0,) (0,) k l (,) Apex node As consequence, n a degenerate pvot cycle, the blockng leavng arc must be between the apex node and node k (.e., the arcs between node l and the apex node are not blockng). In the example, the leavng arc s (7,).

13 Convergence of the algorthm Note that, the removal of arc (7,) produces a partton of the tree nto two subtrees: T ={,,3,4,,6,8,0} and T ={7,9}. (x, u ) (,3) (0,) (0,) k l (,) Let us compute the new node potentals: c ',, π = c π( ) + ( ) = 0 (, ) T The condton π( ) = 0 mples that potentals of nodes n T do not change, whle potentals of nodes n T only reduce of the amount c kl = c 9,0. In fact, snce c 9,0 <0: before addng the arc (9,0): π(9) = c 9,0 + π(0) c ' 9,0 after: c ' 9,0 = c 9,0 π(9) + π(0) = 0 π(9) = c 9,0 + π(0) before addng the arc (9,0): π(7) = c 9,7 + π(9) = c 9,7 + c 9,0 + π(0) c ' 9,0 after: π(7) = c 9,7 + π(9) = c 9,7 + c 9,0 + π(0)

14 Convergence of the algorthm Hence, n a degenerate cycle wth c kl <0, node potentals ether decrease or reman the same. Snce the node potental cannot be smaller than -nc (), wth n= V, and C = max (, ) A { c, }, t follows that the number of degenerate cycles s fnte. (x, u ) (,3) 6 (0,) (,) 9 0 (0,) k l () It can be proved that f π( ) = 0, then the absolute value of the node potental k corresponds to the cost of sendng unt flow from the rooot node to node k along arcs of the tree. Such a cost cannot be greater than nc.

15 Convergence of the algorthm The same result s obtaned when the reduced cost c kl of the enterng arc (k,l) s postve. (x, u ) (,3) 6 (,) 9 0 (,) k l

16 Convergence of the algorthm Summarzng: - Each non-degenerate pvot teraton strclty decreases the obectve functon value (of the amount c kl δ) where (k,l) s the enterng arc and δ>0 s the flow augmentaton around the cycle. Hence, the number of nondegenerate pvot teratons s fnte. - When strongly feasble spannng trees are consdered, the number of consecutve degenerate pvot teratons between any two nondegenerate pvot teratons s fnte, too. Hence: when strongly feasble spannng trees are consdered, the network smplex algorthm converges to an optmal soluton n a fnte number of teratons.

17 Complexty of the algorthm The computatonal complexty of the algorthm s not polynomal n nstance s dmeson (equal to m+n wth m= A and n= V ). However, polynomal varants exst, such as the dual algorthm, wth polynomal complexty O(m 3 log n) Furthermore, many polynomal algorthms exst for solvng mnmum cost network problems: