Branching in Branch-and-Price

Branching on the exponentially large set of variables γω, ∀ω ∈ Ω, is not a viable option since it would lead to strong asymmetries in the partitioning of the search space. Thus next to variables z_i, ∀i∈F, variables x_u,v,∀(u, v) ∈A, and variables y_k,∀k∈C, we accomplish branching by decisions on assignments between facilities and customers. Integrality on one such assignment between a facility i∈ F and a customer k ∈ C_i can be achieved by adding either branching constraint (4.48) or (4.49) to the model ifP

ω∈˜Ω(k)∩˜Ωiγω is fractional.

X

ω∈Ω(k)∩˜ Ω˜i

γ_ω= 0 (4.48)

X

ω∈Ω(k)∩˜ Ω˜i

γ_ω= 1 (4.49)

For each included branching constraint, we need to consider its dual variable value in the pricing problem when solving a knapsack problem with an item corresponding to an assignment fixed due to an already included branching constraint. Adding such additional terms in the pricing problem eventually modifies an item’s profit but does not affect the structure of the pricing problem, i.e. the approach is robust.

Lemma 15 proves that any solutionS⁰ to the LP relaxation of (dBCP) (denoted by (dBCP)^LP) for which – according to above mentioned branching rules – no further branching can be accomplished represents a feasible solution to CConFL, i.e. even-tually existing pattern variables with fractional values can be replaced by pattern variables with integral values while maintaining all assignments between facilities and customers.

Lemma 15 Consider a solution S⁰ to (dBCP)^LP and an arbitrary facility i ∈ F.

Let Ω⁰ ={ω ∈ ˜Ωi |γ_ω^LP6= 0} denote the set of active patterns for iin S⁰, and C⁰ = {k∈C| ∃ω ∈Ω⁰(k)} denote the set of customers assigned to i in S⁰. Furthermore, assume that P

ω∈Ω⁰(k)γ_ω= 1, ∀k∈C⁰. Then ζ ∈Ω_i exists such that C⁰=C(ζ).

Proof Letζ∈Ω_idenote the single variable replacing all variablesω ∈Ω⁰, i.e.C(ζ) = C⁰. Due to the implicit integrality of each assignment between i and a customer k∈C⁰ we only need to prove thatζ does not violate the capacity constraints. Due to constraints (4.37) the following inequality holds:

4.7 Polyhedral Comparison

In the following, we compare the polyhedra corresponding to the sets of feasible solutions of the LP relaxations to the four models presented in the last sections.

Hereby, we denote by P_dMCFf the polyhedron corresponding to the set of feasible solutions of the linear relaxation of model (dMCFf). Similarly, PdMCFc denotes the polyhedron induced by the LP relaxation of model (dMCF_c),P_dCut those of model (dCut), and P_dBCP the polyhedron corresponding to the LP relaxation of model (dBCP). Furthermore, superscript LP denotes the linear programming relaxation of a model, e.g. (dMCF_f)^LP denotes the LP relaxation of model (dMCF_f).

Lemma 16 (dMCF_f) does not dominate (dMCF_c), i.e. proj_x,y,z(P_dMCFf) * projx,y,z(PdMCFc).

Proof Consider a fractional solutionS⁰ = (R⁰_S, T_S⁰, F_S⁰, C_S⁰, α⁰_S) corresponding to the example given in Figure 4.4. S⁰ can be feasibly described in the LP relaxation of our facility oriented model using the variable values as indicated in the figure, i.e.S⁰ ∈ (dMCF_f)^LP. Here, the corresponding flow to each facility with value ¹₃ is routed over two disjoint paths. For feasible solutions of model (dMCFc)^LP,P

(r,u)∈Aks^k_r,u≥y_k,

∀k∈C, must hold due to the flow conservation constraints. Since P

(r,u)∈Aks^k_r,u ≤ P(r,u)∈Ex_r,u = ¹₃, buty_k= 1, ∀k∈ {0,1}, we conclude that S⁰∈/ (dMCF_c)^LP.

Lemma 17 (dMCF_c) does not dominate (dMCF_f), i.e. proj_x,y,z(P_dMCFc) * projx,y,z(P_dMCFf).

Proof We consider a fractional solution S⁰⁰ = (R⁰⁰_S, T_S⁰⁰, F_S⁰⁰, C_S⁰⁰, α_S⁰⁰) corresponding to Figure 4.5. Since the capacity constraints as well as all linking constraints are met and the corresponding flow to each of the two customer is routed over two disjoint paths, where each fractional value s^k_u,v is set to ¹₂, S⁰⁰ ∈ (dMCF_c)^LP. For feasible

Chapter 4 The Capacitated Connected Facility Location Problem

z1= ¹₃

z3= ¹₃

y1= 1 y2= 1 y3= 1

xi,k=¹₃,∀i, k∈ {1,2,3}

root noder

facility nodei, withDi= 1,∀i∈ {1,2,3}

customer nodek, withdk= 1,∀k∈ {1,2,3}

s¹u,v= ¹₆

z2= ¹₃

s²u,v= ¹₆

z3=¹₃ s³u,v= ¹₆

z2= ¹₃ z1=¹₃

Figure 4.4: Feasible LP solution of (dMCF_f) which is infeasible for (dMCF_c).

z1 = 1

z2= 1

y1= 1 y2= 1

s^k_u,v=¹₂, ∀k∈ {1,2}

root noder

facility nodei, withDi= 1,∀i∈ {1,2}

customer nodek, withdk= 1,∀k∈ {1,2}

Figure 4.5: Feasible LP solution of (dMCF_c) which is infeasible for (dMCF_f).

4.7 Polyhedral Comparison

Theorem 18 None of the formulations (dMCF_c) and (dMCF_f) dominates the other, i.e. proj_x,y,z(P_dMCFc) * proj_x,y,z(P_dMCFf) and proj_x,y,z(P_dMCFf) * projx,y,z(P_dMCFc).

Proof Theorem 18 immediately follows due to Lemmas 16 and 17.

Lemma 19 (dCut) dominates (dMCFf), i.e. projx,y,z(PdCut)⊆projx,y,z(PdMCFf). Proof (dMCF_f) differs from (dCut) by modeling connections to facilities by multi-commodity flow constraints instead of directed connection cuts (4.31) whereas (dCut) additionally contains directed connection cuts for customers (4.32). The max-flow min-cut theorem [58] implies that for an arbitrary facility i ∈ F with P(u,v)∈δ⁺(W)x^LP_u,v ≥z_i^LP,∀W (V |r∈W∧i /∈W, a feasible flow of valuez_i^LPfrom the root node toiexists; compare [132]. Thus any solution to (dCut)^LPis valid for (dMCF_f)^LP.

Lemma 20 (dCut) dominates (dMCF_c), i.e. proj_x,y,z(P_dCut)⊆proj_x,y,z(P_dMCFc).

Proof (dMCFc) differs from (dCut) by modeling connections to customers by multi-commodity flow constraints instead of directed connection cuts (4.32) whereas (dCut) additionally contains directed connection cuts for facilities. Thus, as for Lemma 19 the max-flow min-cut argument also holds for the flow to customers.

Theorem 21 (dCut) strictly dominates (dMCF_f) and (dMCF_c), i.e.

proj_x,y,z(P_dCut)(proj_x,y,z(P_dMCFf) and proj_x,y,z(P_dCut)(proj_x,y,z(P_dMCFc).

Proof Since none of the multi-commodity flow formulations dominates the other, i.e. proj_x,y,z(P_dMCFc) * proj_x,y,z(P_dMCFf) and proj_x,y,z(P_dMCFf) * proj_x,y,z(P_dMCFc), Theorem 21 follows from Lemmas 19 and 20.

Theorem 22 (dBCP) strictly dominates (dCut), i.e. projx,y,z(P_dBCP) ( proj_x,y,z(P_dCut).

Chapter 4 The Capacitated Connected Facility Location Problem

arc (u, v) with LP valuexu,v= 1 arc (u, v) with LP valuexu,v= 0.75 z1= 1 y1= 0.75

y2= 0.75

root noder

facility node 1 withD1= 3,f1 = 1.1

customer nodek, withdk= 2,pk= 1,∀k∈ {1,2}

Figure 4.6: Feasible LP solution of (dCut) which is infeasible for (dBCP).

Proof Consider a fractional solutionS⁰according to the example given in Figure 4.6 assuming zero costs for all included arcs. As can be easily seen S⁰ is valid for (dCut)^LP. For describing S⁰ in the space of (dBCP), each assignment patternω can only contain one of the customers. However, since those patterns do not pay off – i.e. the collected profit is smaller than the facilities’ opening costs f₁ – ω /∈Ω and thusS⁰ ∈/(dBCP)^LP.

Now, we consider a solution S^bcp ∈ (dBCP)^LP and denote by γω^bcp, ∀ω ∈ Ω, x^bcpu,v,

∀(u, v)∈A,z_i^bcp,∀i∈F, andy_k^bcp,∀k∈C, the values of all variables ofS^bcp. Using equations (4.50)–(4.53) we transform these values to the space of (dCut), where superscript cut denotes a value with respect to (dCut)^LPandS^cutthe corresponding solution to (dCut)^LP.

z_i^cut=z^bcp_i ∀i∈F (4.50)

y_k^cut=y_k^bcp ∀k∈C (4.51)

x^cut_u,v =x^bcp_u,v ∀(u, v)∈A (4.52) x^cut_i,k = X

ω∈Ωi∩Ω(k)

γ_ω^bcp ∀i∈F, ∀k∈C_i (4.53)

To show that S^cut ∈ (dCut)^LP and thus (dBCP)^LP ⊆ (dCut)^LP we consider each set of constraints from (dCut) in turn. S^cut obviously does not violate constraints (4.27), since (4.39) identically models them in (dBCP). Validity of constraints (4.28) follows from above mentioned transformation rules and constraints (4.37):

x^cut_i,k = X

ω∈Ωi∩Ω(k)

γ_ω^bcp≤ X

ω∈Ωi

γ_ω^bcp ≤z^bcp_i =z^cut_i .

Using our transformation rules and constraints (4.38) the following inequality en-sures that S^cutdoes not violate constraints (4.29):

4.8 Lagrangian Decomposition

Using constraints (4.37) and the fact that the total demand of a single patternω∈Ωi

does not exceed the maximum assignable demand of its facilityi∈ F, the validity of the capacity constraints (4.30) is ensured as follows:

X

Since directed connection cuts for facilities are identically included in both formu-lations and the validity of customer connection cuts (4.32) immediately follows by substitutingP

ω∈Ωi∩Ω(k)γ_ωbyx_i,kin the customer connection cuts (4.41) of (dBCP), we conclude thatS^cut∈(dCut)^LP.

4.8 Lagrangian Decomposition

Since Lagrangian relaxation based approaches have proven to be quite successful for the Steiner tree problem [11] as well as for the capacitated facility location problem [93] and CConFL is composed of these two problems it is quite natural to decompose CConFL by means of Lagrangian relaxation. Model (4.54)–(4.62) which we will relax in the following, is a more abstractly written, undirected variant of model (dMCFc). As previously, binary variablesxe,∀e∈E⁰, indicate if an edgeeis part of the solution, variablesz_i ∈ {0,1},∀i∈F, specify if a facilityiis opened, and variablesy_k∈ {0,1},∀k∈C, if a customerkis feasibly assigned to an open facility.

Similarly to the flow variables of model (dMCFc), we use variables s^k_e ∈ {0,1},

∀k∈C, ∀e∈E⁰, to indicate if an edge e∈E⁰ is part of the unique path from the root node r to a connected customer k. Finally P_k ∈ {0,1}^|E0| denotes the set of incidence vectors corresponding to these simple paths fromr tok∈C using exactly one assignment edge (i, k)∈E⁰\E.

Chapter 4 The Capacitated Connected Facility Location Problem

Note that we usex_e in the objective function (4.54) as well as in inequalities (4.55) and (4.60) when considering graph as well as assignment edges, whilex_i,k is used to denote assignment edges only in inequalities (4.57) and (4.58).

We relax inequalities (4.55) linking variablessandxin a classical Lagrangian fashion by adding corresponding terms weighted with nonnegative Lagrangian multipliers π_k,e ≥ 0, ∀k ∈ C, ∀e ∈ E⁰, to the objective function. This yields the parameter-ized model (LD(π)). See for example [18] for a general introduction to Lagrangian relaxation.

(LD(π)) decomposes into independent subproblems (LD_s,y(π)) for determining vari-ables s^k_e, ∀k ∈ C, ∀e ∈ E⁰, and y_k, ∀k ∈ C, subproblem (LDx(π)) for determin-ing variables x_e, ∀e ∈ E, and subproblem (LD_x,z(π)) to determine variables x_e,

4.8 Lagrangian Decomposition

(LDs,y(π)) consists of|C|independent shortest path problems. Thus it can be solved for customer k ∈ C by computing the cheapest path with respect to edge costs π_k,e from the root to customer node k which includes exactly one assignment edge (i, k)∈E⁰\E, i.e. we need to determine the corresponding incidence vectorq ∈Pk. If the total costs of this path are smaller than the customers prizep_k,y_k as well as the corresponding path variabless^k_e,∀e∈E⁰ |q_e= 1, are set to one. Since, all edge costsπ_k,e are nonnegative we use|C|runs of Dijkstras’ algorithm [50], resulting in a total time-complexity ofO(|C|(|E|+|V|log|V|)) for solving (LDs,y(π)) when using a binary heap implementation of Dijkstras’ algorithm.

(LDx(π)) min X (LD_x(π)), can be trivially solved by inspection in time O(|C||E|). Variables x_e,

∀e∈E, are set to one if ce<P

Chapter 4 The Capacitated Connected Facility Location Problem

Model (LD_x,z(π)) resembles |F|0–1 knapsack problems, one for each facilityi∈F. In such a knapsack problem for facility i ∈ F, we are given the total knapsack capacity Di, and one item for each potential assignment e= (i, k) ∈ E⁰\E, with profit P

k∈Cπ_k,e −c⁰_e and weight d_k. Obviously, we can neglect all items with negative or zero profit. Letχ^∗_i denote the optimal solution to the knapsack problem of facility i ∈ F, and o(χ^∗_i) the according objective value (i.e. the total profit). zi

and all variables x_e corresponding to items used in χ^∗_i are set to one if o(χ^∗_i) >

f_i. Although the knapsack problem is weakly N P-hard [69], several algorithms capable of solving large instances relatively quickly are known, see e.g. [104, 147].

In our implementation we use the Combo algorithm¹ of Martello et al. [136]. Since (LD_x,z(π)) does not possess the integrality property, we may be able to determine better lower bounds than by a simpler LP relaxation of model (4.54)–(4.62).

In the Lagrangian dual problem, we aim to maximize the resulting lower bound by determining optimal Lagrangian multipliers π^∗. Since this maximization problem is convex and piecewise linear, we can approximately solve it using subgradient-like methods. We use the volume algorithm [13], which is an extension of the classic sub-gradient method [65], for solving the Lagrangian dual. While the volume algorithm has been reported to be more efficient in a number of other applications [11, 86], it sometimes might converge too quickly – see e.g. [29] – thus leading to poorer lower bounds than other subgradient-based algorithms. However, preliminary tests in our scenario indicated that the volume algorithm usually yields better lower bounds than the classic subgradient method. Therefore, even though a comparison of vari-ous existing methods for solving the Lagrangian dual would be interesting in future, we decided to focus on using the volume algorithm.

Im Dokument SolvingTwoNetworkDesignProblemsbyMixedIntegerProgrammingandHybridOptimizationMethods DISSERTATION (Seite 124-132)