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1 CSE 87 QM 72N Network Flows: Generalized Networks Slide Slide 2 Generalized Networks ffl An expansion of the pure network model ffl Multipliers or divisors on the flow of an arc modulate level of flow and the units of measure Outline: ffl Characteristics of generalized networks and their bases ffl Network simplex algorithm for generalized transportation problems ffl Algorithms for transforming a generalized network into a pure network equivalent (if possible) Mathematical Statement of the Problem The capacitated generalized network problem (GN): X Minimize cijxij (i;j)2a to: X subject j2ok dkjxkj X i2ik gikxik = bk; for all k 2 N lij» xij» uij; for all (i; j) 2 A N; A; Ok; Ik; bk; cij; uij; lij, and xij are as defined previously, where gij is the multiplier and dij is the divisor on the flow of arc and (i; j). CSE 87 QM 72N Network Flows: Generalized Networks 2 Slide Slide Corresponding Dual Problem (DGN) Maximize X i2n subject to: + X wibi (lijv ij u ijvij) (i;j)2a dijwi gijwj + v ij v ij» cij; for all (i; j) 2 A Hence we are dual feasible when wi unrestricted;i 2 N vij» ; for all (i; j) 2 A cij cij dijwi + gijwj ; for all (i; j) 2 A: Generalized Transportation Problems GT is the simplest of the many variants of GN. This Problem explores specialized simplex techniques for solving the section uncapacitated version of this model.

2 CSE 87 QM 72N Network Flows: Generalized Networks Slide 5 Slide 6 Mathematical Statement of the Problem The generalized transportation model (GT ) (primal problem): Minimize subject to: The dual problem is given as: Maximize subject to: mx i= nx j= cijxij nx j= xij» ai;i=;:::;m mx i= gijxij = bj;j =;:::;n xij ; for all i; j mx i= airi + nx j= bjkj Ri + gijkj» cij; for all i; j Ri;Kj unrestricted in sign At optimality, cij = cij Ri gijkj ; 8i; j CSE 87 QM 72N Network Flows: Generalized Networks Slide 7 Slide 8 Example GT Problem Primal Formulation Min x +x +5x5 +6x2 +7x2 +8 s.t. x + x + x5 + s = 2 + x2 + + s2 = x2 x + x 2 = 2 x + x2 = x x 25 = 8 xij 2 8/9 / /2 x ij p ij 8 5

3 CSE 87 QM 72N Network Flows: Generalized Networks 5 Slide 9 Dual Formulation Max 2R +R2 +K +K +8K5 s.t. R + 2 K» R + K» R + K5» 5 R2 + K» 6 R2 + K» 7 R K 5» 8 R;R2» K;K;K5 free Slide Problem Characteristics CSE 87 QM 72N Network Flows: Generalized Networks 6 Slide Slide 2 Constraint MatrixAin GT detached-coefficient form: In x2 ::: xn x2 x22 ::: x2n ::: x m x m2 ::: xmn x :::» a :::» a :::» am g2 ::: g m = b g g22 ::: g m2 = b2 g gn g2n ::: gmn = bn Note the Following Characteristics ffl There are two nonzero per column of A, with all remaining values. ffl The eschelon-diagonal pattern denotes a transportation problem. ffl There is a + for each from-node and a multiplier for each to-node in the graphical form.

4 CSE 87 QM 72N Network Flows: Generalized Networks 7 Slide Slide GT and GN Characteristics ffl If the problem has less than full row rank, then it is convertible into an equivalent pure network ffl If the problem has full row rank, It does not have a spanning tree basis. The redundancy property doesnot apply. The duals are unique, and a system of simultaneous must be solved to identify them. equations The basis consists of a forest of quasi-trees, or graphs with a cycle (sometimes called one-trees). single Generalized Transportation Simplex Algorithm. Begin with a basic feasible solution. 2. Determine the values of the dual variables, or node potentials. Price out the nonbasic arcs, and select one with a negative. cost to enter the basis. If none exist, stop. reduced Determine the representation of the incoming arc with respect. the current basis. to Using the representation and current basic variable values, 5. the ratio test to identify the variable to leave the basis perform and the level of flow,, on the incoming arc. Pivot by changing basic flows using the representation and, 6. the leaving arc from the basis, and insert the entering remove arc into the basis at level. Return to step 2. CSE 87 QM 72N Network Flows: Generalized Networks 8 Slide 5 Slide 6 Basis Structure ffl The basis of this class of problems is a forest of quasi-trees The graph of the basis looks like one or more trees, each an extra arc forming a cycle. with If a quasi-tree contains a slack arc, it behaves like a cycle. ffl Within a quasi-tree, arcs are viewed as either: on-loop if they form part or all of the cycle, or off-loop if they are in a subtree hanging from a node in the cycle. GT simplex requires traversal of the quasi-tree cycles in order The compute the unique values of the duals and the representation of to a non-basic arc. Basis for the Example Problem Given the following basis, xb = B@ x x5 x2 x2 CA

5 algorithm specialized to the problem structure simplifies this An calculation. CSE 87 QM 72N Network Flows: Generalized Networks CSE 87 QM 72N Network Flows: Generalized Networks 9 Basis Graph, with x ij ;g ij s x ij p ij /2 How to Determine the Duals? Slide 7 / Slide 9 Note that: ffl We do not have a spanning-tree basis ffl We do not have the redundancy property 8 ffl Duals are unique 2 8/9 ffl We must solve a system of equations to compute them 5 ffl Any basis must satisfy complementary slackness Basis Graph, Re-arranged with c ij ;g ij s Complementary Slackness Equations / /9 / R +=2K = c =K + R2 = c2 Slide 2 Slide 8 7 R2 +8=9K5 = c25 R + K5 = c5 R2 + K = c

6 CSE 87 QM 72N Network Flows: Generalized Networks Slide 2 Slide 22 Computing Duals for A Basis Quasi-Tree Iftheloopisaslack arc, or self-loop, the dual is the cost of the. Otherwise, select any node on the loop, say node. slack. 2. Temporarily set R =, and traverse the loop computing: ffl Temporary duals, using Ri + gijkj = cij. ffl The loop factor defined as the ratio: ffl R(), the recomputed value for R upon traversal of the loop. Compute the true R = R() ( R=F), and the other loop duals can. determined by traversing the loop again. be. Any off-loop duals may be computed via Ri + gijkj = cij. Computing Representations of Nonbasic Arcs ffl Consider increasing the flow on a nonbasic arc from to. The representation of this nonbasic variable corresponds to a ffl of changes in the basic arcs' flows that are needed to vector this flow increase while maintaining feasibility accommodate of flow at the nodes). (conservation Representations in pure networks are found by following ffl paths from each end of the incoming arc to the predecessor node, noting the changes required in the basic arcs intersection encountered. In generalized networks, the approach is the same, although ffl computation is required because of the multipliers and more the basis structure. CSE 87 QM 72N Network Flows: Generalized Networks 2 Slide 2 Slide 2 Computing Representation in GNs Determine the off-loop flow adjustments, since these can be. directly. Create vectors of flow reductions for the computed arcs, for example V and V2. This will usually result in basic supply" or demand" on a loop node. unallocated For each unallocated loop supply" or demand" solve the 2. equations associated with the loop to determine simultaneous flow changes on the loop arcs. Create vectors of flow the for the basic arcs in each case, as necessary, for reductions example V and V. Combine the results of the previous steps to form the. For example, ~ Aij = V + V2 + V + V. representation. now use the example in Figure and an associated basis shown We Figure to compute the representation of arc (,). in Figure : Generalized transportation problem R = Q g ij; for arcs traversed in the reverse" direction Q gij; for arcs traversed in the forward" direction F 2 8/9 / /2 x ij p ij 8 5

7 is on-loop, so no off-loop adjustments are to be made From-node = ). Due to the supply" of = atnode,we increase flow on (V by =, resulting in a supply" of = atnode2. The (2,) decreases to the basis variables then are: following B@ ; Decreases = B@ CA CA = V 2 has a demand" of. The set of equations to be solved to Node the adjustments needed are: determine a demand" a b i at loop node i, the decreae in flow Given the arc out of i in the direction of R()'s calculation on a In this definition, an on-loop demand" is a positive value and an on-loop b i ( F R ) if i is an origin node i =g ji b F R ) if i is a destination node ( this flow, the rest of the loop arc flow adjustments Given be determined by traversing the loop. can on clockwise arc (,) is = =. Substituting around decrease CSE 87 QM 72N Network Flows: Generalized Networks CSE 87 QM 72N Network Flows: Generalized Networks /2 On-Loop Adjustments /9 / 7 =2x + =x2 = (node ) x2 + = (node 2) Slide 25 Slide 27 8=9 + x5 = (node 5) x + x5 = (node 2: demand") Instead of solving the equations, we can use the following shortcut. Figure 2: Generalized transportation basis supply" is a negative value. Off-Loop Adjustments (clockwise or counterclockwise) is Slide 28 Slide 26 x x2 x B = For our example, at origin node with a demand" of, the x5 x2 =

8 CSE 87 QM 72N Network Flows: Generalized Networks 5 Slide 29 Slide the loop yields the following vector of flow decreases. xb = B@ x x2 x5 x2 CA ; Decreases = B@ 9=2 9=2 CA = V origin node 2 has a supply" of, hence a demand" of Similarly, So the clockwise arc (2,5)'s reduction in flow is = = =. =. Substituting around the loop yields the following vector of flow decreases. xb = B@ x x2 x5 x2 CA ; Decreases = B@ 8=9 = 8=9 CA = V CSE 87 QM 72N Network Flows: Generalized Networks 6 Slide Slide 2 Putting It All Together representation A now computed V V2 + V + V. The ~ can be as + 8=9 9=9 = 9=2 9=6 B x = C A =B@ +B@ A C +B@ C A x x2 x5 x2 C A : = 9=2 8=9 Performing the Ratio Test 7=2 The ratio test is performed for the elements of A that are positive: b Ξ ~ A = B@ 8 CA Ξ 9=9 9=6 7=2 289 CA = na 8=9 = :97 na 72=28 = 2:57 So x2 has the minimum ratio and the flow change = 8= CA C A = ~A

9 CSE 87 QM 72N Network Flows: Generalized Networks 7 Slide Performing the Change of Basis new set of flows is computes as b ~A. Then the outgoing The is removed from the basis and the incoming arc added. arc

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D1 D2 D3 - 50 CSE 8374 QM 721N Network Flows: Transportation Problem 1 Slide 1 Slide 2 The Transportation Problem The uncapacitated transportation problem is one of the simplest of the pure network models, provides