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1 CSE QM N Network Flows: Transshipment Problem Slide Slide Transshipment Networks The most general pure network is the transshipment network, an extension of the transportation model that permits intermediate transshipment nodes in addition to pure source and pure sink nodes. Figure : Example transshipment problem CSE QM N Network Flows: Transshipment Problem Slide Slide Outline Rormally define this class of models Characterize their bases present a specialized simplex algorithm data structures and computer implementations. Formulation Definitions N the set of problem nodes A the set of problem arcs Ok fj N :(k; j) Ag the to-nodes of arcs out of node k Ik fi N :(i; k) Ag, the from-nodes of arcs into node k bk the supply (if positive) or demand at node k

2 CSE QM N Network Flows: Transshipment Problem Slide Slide Minimum Cost Network Flow Problem A mathematical formulation of the uncapacitated primal network problem (PN): Minimize cijxij (i;j)a subject to: xkj iik jok Maximize in subject to: Hence we are dual feasible when xik = bk; for all k N xij 0; for all (i; j) A Dual Problem (DN) biwi wi wj» cij; for all (i; j) A wi unrestricted;i N cij cij wi + wj 0; for all (i; j) A: CSE QM N Network Flows: Transshipment Problem Slide Slide PN Formulation of the Example Min : 0 Node x x x x x x x x x x b = 00 = 0 = 0 = 0 = 0 = 0 = 0 DN Formulation of the Example Max : Arc w w w w w w w x :» 0 x :» x :» x :» x :» x :» x :» x :» x :» x :»

3 CSE QM N Network Flows: Transshipment Problem Slide Slide 0 Capacitated Transshipment Networks Most applications take thecapacitated form: Minimize cijxij (i;j)a subject to: xkj iik jok Maximize in subject to: xik = bk; for all k N () 0» xij» uij; for all (i; j) A () Dual Formulation + wibi uijvij (i;j)a wi wj + vij» cij; for all (i; j) A wi unrestricted;i N vij» 0; for all (i; j) A CSE QM N Network Flows: Transshipment Problem Slide Slide Lower Bounds Note that we can translate variables with lower bounds to this form Ifanarc(i; j) with flow x 0 ij is restricted as: easily. we can define yielding `0ij» x 0 ij» u 0 ij xij x 0 ij `0ij 0» xij» uij uij u 0 ij `0ij. This transformation changes the RHS such where that bi b 0 i `0ij and b j b 0 j + `0ij: Example Transformation Consider the arc (; ) with cost c,» x 0» 0, and 0 = b0 =0: b The transformed problem has: 0» x» ;b = ;b =+; the objective is to minimize P (i;j)a c ijxij +c. and The last objective function term is a constant which canbe during transformed problem optimization. ignored

4 CSE QM N Network Flows: Transshipment Problem Slide Slide Characteristics of Transshipment Networks Since the transportation problem is a special, bipartite case of the transshipment problem, many of the general properties of pure networks have been discussed previously. Enumerated below are the key characteristics of pure networks needed for a presentation of the network simplex algorithm. Properties Different from Transportation. The constraint matrix has one + and one entry for each arc (instead of two + entries).. Because of property, the reduced cost is defined differently: cij cij wi + wj:. Because of properties and, all duals that change during a pivot change by a constant amount (±cij of the incoming arc).. Since the problem is not bipartite, information regarding the orientation of basic arcs must be maintained.. The simplex algorithm's ratio test must be modified to take into account property.. The problem is difficult to display in tableau form. CSE QM N Network Flows: Transshipment Problem Slide Slide Properties in Common with Transportation. The basic arcs form a spanning tree of the problem nodes.. The basis matrix is triangular.. Solutions are integral, when the RHS is integral.. There is one redundant primal constraint and, therefore, one redundant dualvariable.. The complementary slackness conditions take the same form. A solution is optimal if: cij = 0; for all basic arcs (i; j) B cij 0; for all nonbasic arcs at 0 (i; j) L : cij» 0; for all nonbasic arcs at their upper bound: (i; j) U Capacitated Network Simplex Algorithm The specialization of the primal simplex method for transshipment pure networks is a minor expansion of the algorithm we have used for capacitated transportation problems. The notation developed below is consistent with the linear programming review and transportation problem.

5 CSE QM N Network Flows: Transshipment Problem Slide Slide The set B denotes the set of jnj basic arcs L and U respectively denote the sets of nonbasic arcs with flow their lower and upper bounds at Once the flows on the nonbasic arcs have been set, the flows on basic arcs are uniquely determined by (). the This assignment oows is a basic solution and is denoted by triplet (B;L;U) the If () is also satisfied, (B;L;U) is a basic feasible solution. A feasible basis is an optimal basis if: cij = 0; for each arc (i; j) B; () cij 0; for each arc (i; j) L; () cij» 0; for each arc (i; j) U; () Algorithm Steps. Initialization. Identify an initial entended basic feasible solution.. Pricing. A nonbasic arc (k; l) which violates () or () is selected to be the incoming arc. If no such arc exists, the algorithm terminated with the current basis being optimal if it contains no artificial arcs with positive flow; otherwise the problem is infeasible.. Ratio test. Adding arc (k; l) to the basis forms a unique basis cycle, consisting of (k; l) and the basis equivalent path, C(k; l). (a) If (k; l) L, its flow will increase by, causing those arcs in C(k; l) oriented away froml to increase, and the arcs oriented towards l to decrease, in flow by the same amount. (b) Else, (k; l) U, itsflow will decrease from its upper bound CSE QM N Network Flows: Transshipment Problem 0 Slide Slide 0 by, causing the arcs in C(k; l) oriented away from k to increase and those oriented dowards k to decrease in flow. The flow change = min f ; ; g, and the arc (p; q) associated with that minimum is the blocking arc, where (a) = ukl; (b) =minfxijj(i; j) C(k; l) and decreasesg, and (c) = minfuij xijj(i; j) C(k; l) and increasesg.. Basis update The flows in the basis cycle are adjusted by ±, depending on orientation. If the incoming arc is the blocking arc, it remains nonbasic, changes L=U set membership. but Otherwise (k; l) becomes a member of B, arc(p; q) becomes of L or U, and the duals are updated using the amember relation cij = wi wj for each basic arc (i; j). Return to step. Example Problem + + (0,) (0,) (0,) (0,)

6 CSE QM N Network Flows: Transshipment Problem Slide Slide. Starting Solution Basis Tree for Advanced Start CSE QM N Network Flows: Transshipment Problem Slide Slide Alternative, All-Artificial Start. Checking for Optimality. Set duals such that cij wi + wj = 0 for all basic arcs Arc orientation is important wi = wj + cij wj = wi cij. Check eachnonbasic 0 x = x = x = x = x,0 = x,0 = 0 x,0 =0 x = x =0

7 CSE QM N Network Flows: Transshipment Problem Slide Slide Basis with Dual Variable Values Pricing Nonbasics CSE QM N Network Flows: Transshipment Problem Slide Slide. Determine Leaving Arc. Change of Basis Operation M = 00 M = 00 w = c = c = c = 00 c = c,0 =00 0 c,0 = 00 c,0 = c = c = w = 0 c = w = - c = w = - c = 00 w = c = w0 = - 0 w = c,0 = 00 c,0 = w = - c = c = w = w = x = x = x = x = x = x,0 = x,0 = x,0 =0 x = 0 x = x =0 x = x = x,0 = 0 x,0 = x,0 =0 x = x =0

8 CSE QM N Network Flows: Transshipment Problem Slide Slide 0 Basis : Updated Duals Basis : Pricing Nonbasics CSE QM N Network Flows: Transshipment Problem Slide Slide Basis : Ratio Test Basis c = c = c =00 c =00 c = c = c,0 =00 0 c,0 = c,0 =00 c = c = c = c = c,0 =00 0 c,0 = c,0 =00 c = c = x = x = x = x = x,0 = 0 x,0 = x,0 =0 x = x =0 0

9 CSE QM N Network Flows: Transshipment Problem Slide Slide Basis Basis CSE QM N Network Flows: Transshipment Problem Slide Slide Basis Basis