MI Preprint SeriesMathematics for Industry

Kyushu University

An indirect search algorithm for

disaster restoration with

precedence and synchronization

constraints

Akifumi Kira, Hidenao Iwane,

Hirokazu Anai, Yutaka Kimura

& Katsuki Fujisawa

MI 2016-11

( Received August 29, 2016 )

Institute of Mathematics for IndustryGraduate School of Mathematics

Kyushu UniversityFukuoka, JAPAN

An indirect search algorithm for disaster restoration with

precedence and synchronization constraints

Akifumi KIRAInstitute of Mathematics for Industry, Kyushu University

744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Hidenao IWANE and Hirokazu ANAIKnowledge Information Processing Laboratory, Fujitsu Laboratories Ltd.

4-1-1 Kamikodanaka, Nakahara-ku, Kawasaki 211-8588, Japan

Yutaka KIMURAFaculty of Systems Science and Technology, Akita Prefectural University

84-4 Aza Ebinokuchi, Tsuchiya, Yurihonjo, Akita 015-0055, Japan

Katsuki FUJISAWAInstitute of Mathematics for Industry, Kyushu University

744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan&

JST CREST4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan

Abstract

When a massive disaster occurs, to repair the damaged part of lifeline networks, plan-ning is needed to appropriately allocate tasks to two or more restoration teams and optimizetheir traveling routes. However, precedence and synchronization constraints make restora-tion teams interdependent of one another, and impede a successful solution by standardlocal search. In this paper, we propose an indirect local search method using the productset of team-wise permutations as an auxiliary search space. It is shown that our methodsuccessfully avoids the interdependence problem induced by the precedence and synchroniza-tion constraints, and that it has the big advantage of non-deteriorating perturbations beingavailable for iterated local search.

KeywordsMathematical optimization, scheduling, vehicle routing problem, indirect local search,linear extensions.

1 Introduction

In the wake of the Great East Japan Earthquake, hands-on research and development is nowrequired to solve concrete social problems. When a massive disaster such as an earthquakeor a typhoon occurs, some damage to service networks is unavoidable (e.g., electricity, waterservice, gas and so on). To repair the damage to the networks as fast as possible, planningis needed to appropriately allocate tasks to two or more restoration teams and optimize their

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traveling routes. Such a scheduling problem is a variant of the vehicle routing problem (VRP)which designs optimal delivery or collection routes from one or several depots to a number ofgeographically scattered customers. There is a wide variety of VRPs and a broad literature onthis class of problems.

The first systemic studies of this problem using VRPs in the context of Japan, appearsto be Watanabe et al. [8, 9]. In these case studies, the problem is formulated as a VRPwith time windows (VRPTW) and standard local search heuristics are then applied. However,Yamashita et al. [10] point out that establishing scheduling techniques for handling precedenceand synchronization constraints has become a very important issue. If electricity is suppliedto spot j via spot i, restoration teams cannot start operating spot j until the restoration ofspot i is completed. Moreover, several teams may participate in restoring a damaged spot. Themain difficulty in dealing with these constraints is that restoration teams are not independentof one another. In other words, a change in one route may affects other routes and may renderall other routes infeasible. This is not usually the case in VRPs. This difficulty, which oftenimpedes a successful solution by standard local search, is called the interdependence problem.See Drexl [3] for a complete review of this problem and a comprehensive collection of other typesof synchronization constraints imposed in VRPs.

A good technique several authors propose for overcoming this situation is the use of indirectsearch (Afifi et al. [1], Derigs and Dohmer [2], Li et al.[5], Nonobe and Ibaraki [6], and others). Inindirect search, given a optimization problem, we first define (i) an auxiliary search space X aux,which is different from the solution space X of the original problem, (ii) a decoder f : X aux → X ,and (iii) search techniques and move types operating on the auxiliary space. Then, a local searchprocedure is executed not on X but on X aux with evaluating search points by the decoder f .Though the computational cost of the decoder is added, it is known that this approach is powerfulfor problems having complicated constraints. Derigs and Dohmer [2] show that the approach ofindirect local search is not only flexible and simple but when applied to the VRP with pickupand delivery and time windows (VRPPDTW) it also gives results which are competitive withstate-of-the-art VRPPDTW-methods by Li and Lim [4], as well as Pankratz [7].

In this paper, we propose an enhanced indirect search method which uses the product setof team-wise permutations as an auxiliary search space, though many successful indirect searchalgorithms use permutations as an auxiliary search space (in the context of VRP, it is permu-tations of customers to be served). It is shown that our method successfully avoids the inter-dependence problem induced by the precedence and synchronization constraints. In addition,when we improve the capability of our algorithm to escape from local optima by using a meta-heuristic technique called iterated local search, we have the big advantage of non-deterioratingperturbations being available.

In Section 2, after describing the main feature of our problem, we define our notation andformulate the mathematical optimization problem. We also summarize the difficulties with theinterdependence problem in our setting. In Section 3, we develop an enhanced indirect searchalgorithm that avoids the interdependence problem induced by the precedence and synchroniza-tion constraints. In Section 4, we perform computational experiments in a practical settingusing a geographic information system (GIS) and geospatial information. The results obtainedfrom numerical examples in this study are then described.

2 The Case Study

2.1 Problem Description

We first enumerate the main feature of our problem.

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Precedence relation between pairs of damaged spots If electricity is supplied to spot jvia spot i, then it is necessary to restore spot i before restoring spot j. Namely, a restorationteam cannot start operating spot j, but they may wait at spot j until the restoration of spoti is completed. We denote this by i ≺ j. Notice that (V,≺) is a strictly partially ordered set(or strict poset), where V is the set of damaged spots (and a depot). In our application, wecan assume that the Hasse diagram of the poset is a directed forest (see Figure 8 in Section 4).Moreover, the whole damaged area is divided into plural areas in advance, and each restorationteam has their area of responsibility. For any pair (i, j), i ≺ j implies that spots i and j arelocated in the same area.

Areas of responsibility Each restoration team should give priority to its area of respon-sibility. Namely, they cannot move to another area until their own area is completed. Theseare interpreted as team-wise precedence constraints, and compatible with the precedence con-straints. We use i ≺k j to denote that spot i is in team k’s area of responsibility but spot j isin another area , and (V,≺k) is also a strict poset. The number of restoration teams is greaterthan the number of areas. Hence two or more teams may be allocated in a common area.

Synchronized restoration by two or more teams A spot can be repaired by two or morerestoration teams if each team can help decrease the restoration time of the spot by at least cminutes (otherwise, the spot is repaired by one team). Each team can join the restoration of aspot during the restoration. However, they must stay until the restoration is completed. We setc = 20 in our case study.

Min-max objective The goal is to ensure that entire restoration process is completed asquickly as possible. In other words, we minimize the time required for the team that takes timethe longest to finish their restoration work and return to the depot.

2.2 A MIP formulation

To accurately formulate our problem as a mathematical optimization problem, we first need todefine our notation.

• Let V = {0, 1, . . . , |V|} be the set of all spots to be repaired. “0” represents the depot.

• Let K = {1, 2, . . . , |K|} denote the set of all restoration teams.

• Let P be the set of all triplets (i, j, k) such that team k should respect the precedenceconstraint between spots i and j. Namely,

A := {(i, j) | i ≺ j, i, j ∈ V},Ak := {(i, j) | i ≺k j, i, j ∈ V}, k ∈ K,

P := A×K ∪∪k∈K{(i, j, k) | (i, j) ∈ Ak}.

• c is the threshold for the synchronized restoration: Two or more teams can restore thesame spot if each team can help decrease the restoration time of the spot by at least cminutes.

• dij is the time required to move from spot i to spot j.

• wi represents the volume of work required to complete the restoration of spot i.

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• xkij denotes the binary integer decision variable that equals 1 if and only if team k movesdirectly from spot i to spot j.

• yki denotes the binary integer decision variable that equals 1 if and only if team k visitsspot i.

• bki represents the time point at which team k begin to work at spot i．The value of bki ismeaningful only if yki = 1.

• ei denotes the time point at which the restoration of spot i is completed, where all teamsleave the depot to start the restoration process at e0 = 0.

• t is the time point at which the whole restoration process is completed.

Using the above notation, the desired optimization problem is formulated as follows:

Minimize t

subject to ei + di0 ≤ t, ∀i ∈ V \ {0}, (1a)

yki =∑j∈V

xkij , ∀i ∈ V, ∀k ∈ K, (1b)

ykj =∑i∈V

xkij , ∀j ∈ V, ∀k ∈ K, (1c)

ei ≤ bkj , ∀(i, j, k) ∈ P, (1d)

xkij(ei + dij) ≤ bkj , ∀i,∀j ∈ V, ∀k ∈ K, (1e)∑k∈K

(ei − bki )yki = wi, ∀i ∈ V \ {0}, (1f)

cyki ×∑

g∈K\{k}

ygi ≤ (ei − bki ), ∀i ∈ V \ {0}, ∀k ∈ K, (1g)

xkij ∈ {0, 1}, ∀i,∀j ∈ V, ∀k ∈ K, (1h)

yki ∈ {0, 1}, ∀i ∈ V, ∀k ∈ K, (1i)

0 ≤ bki <∞, ∀i ∈ V, ∀k ∈ K, (1j)

0 ≤ ei <∞, ∀i ∈ V, (1k)

0 ≤ t <∞. (1l)

From the objective function and constraint (1a), we minimize the time required for the teamthat takes time the longest to finish its restoration work and return to the depot. Constraints(1b) and (1c) mean that if team k visits spot i, then team k must move to spot i from anotherexactly once, and must move somewhere from spot i exactly once. The precedence constraintsand the areas of responsibility are considered in (1d). Travel time is considered in (1e). If teamk visits spot j next to spot i, i.e. xkij = 1, then time interval dij is required for traveling betweenthem. Notice that e0 = 0 is automatically satisfied because of the minimization of the objectivefunction. Constraint (1f) ensures that the restoration of spot i is not completed until the totalworking time of the restoration teams working there exceeds wi. From constraint (1g), we have

yki = 1,∑

g∈K\{k}

ygi ≥ 1 =⇒ ei − bki∑g∈K\{k}

ygi≥ c (2)

for every k ∈ K. Therefore, constraint (1g) means that a spot may be served by two or moreteams if each team can help decrease the restoration time by at least c minutes. Moreover, it

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follows from (1g) that bki ≤ ei for all i ∈ V \ {0} and k ∈ K. Hence constraint (1e) eliminatessubtours that do not include the depot.

Using the so-called big-M method, nonlinear constraints (1e), (1f), and (1g) are transformedinto linear constraints and we obtain a mixed integer linear programming (MILP) formulationas follows:

Minimize t

subject to (1a), (1b), (1c),

ei ≤ bkj +M(1− ykj ), ∀(i, j, k) ∈ P,ei + dij ≤ bkj +M(1− xkij), ∀i,∀j ∈ V, ∀k ∈ K,ei − bki ≤Myki , ∀i ∈ V \ {0}, ∀k ∈ K,∑k∈K

(ei − bki ) = wi, ∀i ∈ V \ {0}, ∀k ∈ K,

c×∑

g∈K\{k}

ygi ≤ (ei − bki ) +M(1− yki ), ∀i ∈ V \ {0}, ∀k ∈ K,

(1h), (1i), (1j), (1k) and (1l).

However, the problem has many big-M constraints, the structure of the traveling salesmanproblem (TSP), a min-max objective function, and symmetries between restoration teams. Forthis reason, it is thoroughly intractable to find a solution by using a general-purpose MIP solver.

2.3 Interdependence Problem

Let us examine the interdependence problem in the setting laid out in Table 1. Team 1’s route,

Table 1: An instance with nine spots and two teams

Area 1 Area 2

list of spots {1, 2} {3, 4, . . . , 9}list of teams {1} {2}

precedence constraints – 5 ≺ 6, 7 ≺ 9

shown as in Figure 1, does not break the given precedence constraints in itself. The sameapplies to team 2’s route. However, taken as a whole, team 1’s route and team 2’s route dobreak the precedence constraint 5 ≺ 6. Synchronization at spot 8 is also incomplete, because itis prohibited to leave there until the restoration is completed. In other words, whether or not aroute is feasible depends on the other routes.

Case (a)

Case (b)

Figure 1: infeasible schedules

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In case (a), we can render both routes feasible by inserting waiting times into them (seeFigure 2). However, the modified schedule has a longer total duration. From the view point ofoptimization, this is not good. In case (b), by contrast, inserting waiting times never leads afeasible schedule.

Figure 2: Repair of an infeasible schedule by inserting waiting times

3 Proposed Method

In this section, we develop an enhanced indirect search algorithm that avoids the interdependenceproblem induced by the precedence and synchronization constraints. Figure 3 illustrates theconcept of indirect search.

Figure 3: Indirect search

3.1 Linear Extensions and Search Space

Let (V,≪) be the transitive union of (V,≺) and (V,≺k). Namely,

{(i, j) ∈ V×V | i≪ j} = A ∪Ak.

Let Lk denote the set of all linear extensions of (V,≪), where a linear extension σk ∈ Lk is apermutation of spots that is compatible with the precedence constraints and with the team-wiseprecedence constraints for team k’s area of responsibility.

Definition 3.1 (linear extension). A linear extension of a finite poset (Q,≺) is a total orderingσ = σ(1)σ(2) . . . σ(|Q|) of its elements such that i < j whenever σ(i) ≺ σ(j).

Now, we define the search space by

X aux :=∏k∈KLk.

A search point x ∈ X aux is expressed by a |K| × |V| matrix. The following example is a searchpoint for the problem shown in Table 1.

x =

(σ1σ2

)=

(σ1(1) σ1(2) · · · σ1(9)σ2(1) σ2(2) · · · σ2(9)

)=

(1 2 3 5 7 6 8 9 43 7 5 6 8 4 9 1 2

)(3)

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Since the precedence constraints 5 ≺ 6 and 7 ≺ 9 are given, 5 and 7 are located to the leftof 6 and 9 in either row, respectively. Moreover, team k should give priority to their area ofresponsibility. Hence spot 1 and spot 2 are located to the left of the other spots in the first row.Conversely, they are located to the right of the other spots in the second row.

Definition 3.2 (topological ordering). A topological ordering of a directed acyclic graph (DAG)G = (V, E) is a linear ordering of its vertices such that if there is a path form u to v, then uappears before v in the ordering.

We note that (V,≪) constitutes a DAG G = (V,A ∪ Ak). A linear extension of the posetis the same thing as a topological ordering of the DAG. We can obtain an initial search pointx ∈ X aux by a topological ordering method.

3.2 Decoder

Let us define a decoder that relates each search point x ∈ X aux to a feasible schedule f(x) in thefollowing manner. To determine which spot to visit after completing a restoration somewhere,team k examines each spot as a candidate in the order σk(1), σk(2), . . . , σk(|V|). Team k ispermitted to visit spot σk(i) if no team has left for spot σk(i), or if some team(s) has alreadyleft for spot σk(i) but each team operating spot σk(i) can decrease the restoration time by atleast c minutes. Otherwise, team k skips spot σk(i) and examines spot σk(i+ 1). See Figure 4.

x =

(À Á 3 Ä 7 6 Ç È 4Â Æ 5 Å Ç Ã 9 1 2

)Figure 4: Feasible schedule f(x)

To show an implementation of our decoder, we introduce additional variables.

• sk represents the state of team k.

sk =

Initial Team k leaves the depot now and begin the whole processMoving Team k is moving to a spot or the depotWorking Team k is working at a spotWaiting Team k is waiting at a spotCompleted Team k finished their task and arrived at the depot.

• vk ∈ V is the location of team k. If sk = Moving, vk represents the current destination.Otherwise, it represents the place they are currently.

• pk is an index variable that points to an element of σk.

• tk represents the time until their current movement or work is completed.

Now, our decoder Decoder() can be implemented as follows:

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Initialize t← 0, K′ ← K;Initialize w′

i ← wi, Ki ← ∅ for all i ∈ V;Initialize sk ← Initial, vk ← 0, pk ← 1, tk ← 0 for all k ∈ K;while K′ = ∅ do

Execute Algorithm 1;Execute Algorithm 2;

return t;

Function Decoder(parameters: x ∈ X aux)

Two subroutines are shown in Algorithms 1 and 2.

forall the k ∈ K′ such that tk = 0 doswitch the value of sk do

case Working or Initial /* evk = t */

Declare a local integer variable i;Set i← vk;

♯ while pk ≤ |V|, and Visitable(arguments: k, σk(pk)) returns false dopk ← pk + 1;

if pk ≤ |V| thenvk ← σk(pk), Kvk ← Kvk ∪ {k}; /* ykvk = 1 */

pk ← pk + 1;

elsevk ← 0; /* go back to the depot */

sk ← Moving, tk ← di,vk ; /* xki,vk = 1 */

case Movingif vk = 0 then

sk ← Completed, K′ ← K′ \ {k};else

sk ←Waiting;

case Waiting or Completed(Do nothing)

forall the k ∈ K′ such that sk = Waiting doif the work at spot vk can be started then

// We can judge this in O(1) time under the assumption that the

Hasse diagram of (V,≪) is given as a directed forest

sk ←Working; /* bkvk = t */

Algorithm 1: Subroutine for determining next action

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forall the k ∈ K′ such that sk = Working dotk ← w′

vk/nvk , where ni := #{g ∈ Ki | sg = Working};

Declare a local floating-point variable tmin;tmin ← min{tk | k ∈ K′ such that sk = Moving or Working};forall the k ∈ K′ such that sk = Moving or Working do

tk ← tk − tmin;if sk = Working then

w′vk← w′

vk− tmin;

t← t+ tmin;

Algorithm 2: Subroutine for updating tk

if Kj = ∅ then /* no team has left for spot j */

return true;else

// Decide whether each team can decrease the restoration time by at

least c. Suppose that there is no waiting time at spot j.Declare local floating-point variables {τg}g∈Kj∪{k} and set

τg ←

dvk,j if g = k0 if g = k, sg = Waiting or Workingtg if g = k, sg = Moving

for all g ∈ Kj ∪ {k};Sort {τg}g∈Kj∪{k} as τ ′1, τ

′2, . . . , τ

′m in such a way that τ ′1 ≤ τ ′2 ≤ · · · ≤ τ ′m, where

m := |Kj ∪ {k}|;Declare a local floating-point variable w;

Set w ← w′j −

m−1∑g=1

g × (τg+1 − τg);

if wm−1 −

wm ≥ c then

return true;else

return false;

Function Visitable(parameters: k ∈ K, j ∈ V)

In Function Visitable, we decide whether team k can join the restoration of spot j underthe assumption that there is no waiting time due to the precedence constraints. In other words,we decide it by a sufficient condition for which each team can help decrease the restoration timeof the spot by at least c minutes. On the other hand, deciding it accurately with consideringthe waiting time is impractical. If there exists a spot i such that i ≺ j, the waiting time atspot j depends on ei. However, the final value of ei cannot be found when Function Vis-itable(arguments: k, j) is called from line ♯ of Algorithm 1, because it will be updated ifanother team joins the restoration of spot i. Therefore, wheter the team k can join the restora-tion of spot j depends on whether another team can join the restoration of spot i. In thissituation, one possible way for deciding it accurately is to use a back-track search method, but

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it will take an exponential time. Hence, our way using the sufficient condistion is more suitablefor efficiency.

Theorem 3.1. If we consider the additional constraint∑k∈K

yki ≤ 3, ∀i ∈ V \ {0},

then Function Decoder runs in O(|K||V|) time.

Proof. Algorithm 2 runs in O(|K|) time. Moreover, under the additional constraints, we have∑i∈V

∑k∈K

yki ≤ 3|V|.

This implies that the main while loop in Function Decoder is repeated O(V) times. Hence,it takes O(|K||V|) time to execute the loop of Algorithm 2. On the other hand, Algorithm 1does not necessary run in O(|K|) time, because Function Visitable may be called repeatedlyin line ♯. However, Function Visitable is called exactly once for each (k, j) ∈ K × V in themain while roop in Function Decoder. Therefore, it also takes O(|K||V|) time to execute theloop of Algorithm 1.

3.3 Neighbourhood

We shall define a move type operating on X aux.

Definition 3.3 (intra-swap and its feasibility). For any σk = σk(1)σk(2) · · ·σk(|V|) ∈ Lk, anintra-swap move is a transposition (i j) that swaps σk(i) and σk(j) for some i, j ∈ {1, 2, . . . , |V|}.Namely,

σk ◦ (i j) = σk(1) · · ·σk(j) · · ·σk(i) · · ·σk(|V|).

An intra-swap move is said to be feasible if σ′k := σk ◦ (i j) ∈ Lk.

Lemma 3.1. An intra-swap (i j) operating a linear extension σk = σk(1)σk(2) · · ·σk(|V|) ∈ Lkis feasible if and only if

σk(i) ⊀ σk(h) i < h ≤ j,σk(h) ⊀ σk(j) i < h < j,σk(i) ⊀k σk(j).

Proof. The result follows directly from the definition of Lk.

From Lemma 3.1, we can judge whether an intra-swap move is feasible or not in O(|V|)time. We define the intra-swap neighbourhood N : X aux → 2X

auxin the following manner:

N (x) :=∪k∈K

{x′ = (σ−k, σ

′k) |σ′

k = σk ◦ (i j) ∈ Lk, i, j = 1, 2, . . . , |V|},

x = (σ1, σ2, . . . , σ|V|) ∈ X ,

where (σ−k, σ′k) := (σ1, . . . , σk−1, σ

′k, σk+1, . . . , σN ).

Suppose that x = (σ1, σ2, . . . , σ|K|) ∈ X is the provisional solution. We first choose k ∈ K,select i, j ∈ V, and check whether the intra-swap (i j) operating σk is feasible. If it is feasible,then the intra-swap move is applied to x and the new search point x′ is evaluated by the decoder.If the objective value of f(x′) is better than that of f(x), then we update the provisional solutionby x← x′. Figure 5 shows that the schedule is improved from f(x) to f(x′) by swapping σ2(2)and σ2(6). A procedure for searching for a better schedule is shown in Algorithm IndirectSearch.

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x =

(À Á 3 Ä 7 6 Ç È 4

Â Æ 5 Å Ç Ã 9 1 2

)

x′ =

(À Á 3 5 Æ 6 8 È 4Â Ã Ä Å Ç 7 È 1 2

)Figure 5: An example of an improvement from an intra-swap operation

Declare floating-point variables vmin and v;vmin ← Call Evaluate(arguments: x);

♮ forall the x′ ∈ N (x) dov ← Call Evaluate(arguments: x′);if v < vmin then

vmin ← v, x← x′;go to Line ♮;

return x;

Function IndirectSearch(parameters: x ∈ X )

3.4 Iterated Indirect Local Search

In this subsection, we improve the capability of our algorithm to escape from local optima byusing a metaheuristic technique. In multi-start local search, a number of initial solutions aregenerated, and local search procedure is applied to each of them. Finally, the best solutionobtained in the entire search is output. In iterated local search, which is an effective variantof multi-start local search, the initial solutions for local search are generated by perturbatinggood solutions found in the previous search. On this point, we have the big advantage of non-deteriorating perturbations being available.

Definition 3.4 (value-invariant swap). An intra-swap move is called to be value-invariant if itconverts x ∈ X aux to x′ ∈ N (x) such that the objective values of x and x′ are the same eachother.

Our iterated indirect local search algorithm is summarized as follows.

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for m← 1 to m dofor n← 1 to n do

♭ Choose an value-invariant feasible intra-swap randomly, and add it to x;

x← Call IndirectSearch(arguments: x);

return x;

Function IteratedSearch(parameters: x ∈ X , m ∈ Z+, n ∈ Z+)

4 Computational Results

To test our algorithms in a practical setting, we generate a virtual instance on a real map. Adepot and 57 damaged spots are located in Minami ward, Fukuoka city, Fukuoka prefecture,Japan. The authors depict them in Figure 6. The volumes of work {wi} and their precedenceconstraints are depicted in Figure 7 and in Figure 8, respectively. We suppose that there areeight restoration teams in total, and that their areas of responsibility are the whole damagedarea. In other words, we do not consider team-wise precedence constraints1. For each pair(i, j) in V × V, we set dij to the minimum travel time (min) from i to j along the real roadnetwork. The regulations of traffic such as one-way roads and restricted turns are also taken intoconsideration. To do this, we solve the shortest path problem, in advance, by using GIS softwareArcGIS for Desktop ver. 10.4 (Esri Inc., USA) and ArcGIS Data Collection Road Network 2015(Esri Japan Corporation).

(C) Esri Japan

depot

Figure 6: Locations of damage spots

1Rather, this situation is computationally harder for our algorithm, because team-wise precedence constraintsreduce the size of N (x) for every x ∈ X aux.

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(C) Esri Japan

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Figure 7: The volumes of work {wi} Figure 8: Hasse diagram of (V,≺)

In our computational experiments, we first generate a topological ordering σ of the Hassediagram of (V,≺), and use x0 = (σ, σ, . . . , σ) ∈ X aux as an initial search point. Then weexecute Function IteratedSearch with arguments: (x, m, n) = (x0, 1000, 1000) repeatedlywith changing the seed for random numbers used in line ♭ of the function. We implemented thealgorithm using C++ Language and executed it on a desktop PC with Intel R⃝ CoreTM i7-3770Kprocessor of 3.4 GHz and 16 GB memory installed. The results are shown in Table 2 and Figure9. Routes 1 to 8 output by trial C are depicted in Figures 10 and 11. The spots that are servedby two or more teams are highlighted in yellow.

Table 2: Computational Resultsobjective value (min) CPU time (sec) [# of calls to Function Decoder ]

seed initial sol. final sol. 1st iteration 1000 iterations final update

A 655.889 452.000 0.298 [11,691] 71.739 [8,253,466] 33.754 [3,853,235]B ↓ 451.000 0.678 [40,122] 75.119 [8,397,206] 61.158 [6,799,271]C ↓ 449.000 0.187 [ 7,199] 74.202 [8,312,991] 45.523 [5,110,908]

We note that our auxiliary search space, the product set of team-wise permutations (linearextensions) is larger than the set of permutations that is used in the usual indirect search.Therefore, it seems reasonable that our algorithm should be able to find a good sub-optimalsolution. Estimating the optimization gap is quite hard in our challenging problem, but Figure9 clearly shows that our indirect search has a high local search ability and that its iterationwith non-deteriorating perturbations is useful to escape from local optima. The final solution,found in trial C, achieved a 13 % decrease of the objective value, compared with the output bya existing solver developed in Fujitsu group2.

In each trial, 1000 iterations finished in about 75 second, and Function Decoder is calledmore than eight million times. This computational efficiency is mainly because of the O(|K||V||)implementation of the decoder. It would be able to quickly present the most recent plan thatresponds to changing conditions, including the extent of an area affected by a disaster and thepace of recovery work.

2This existing method is a greedy construction method, though the detail cannot be publishable due to businesssecret.

13

(a) Trial A (50 iterations) (b) Trial A (1000 iterations)

(c) Trial B (50 iterations) (d) Trial B (1000 iterations)

(e) Trial C (50 iterations) (f) Trial C (1000 iterations)

Figure 9: Effect of the iterated local search

14

The authors think that the large auxiliary search space, the linear time implementation ofthe decoder, the iterated local search using non-deteriorating perturbations are responsible forthe successful results.

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(a) Route 1

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(b) Route 2

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(c) Route 3

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(d) Route 4

Figure 10: Routes 1 to 4 output by Trial C

15

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(a) Route 5

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(b) Route 6

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(c) Route 7

(C) Esri Japan, Sumitomo Electric Industries, Ltd.

(d) Route 8

Figure 11: Routes 5 to 8 output by Trial C

5 Conclusion and future work

Based on the concept of the indirect search, we have proposed a simple local search method usingthe product set of team-wise permutations as an auxiliary search space. We have demonstratedthat this new method successfully avoids the interdependence problem induced by the prece-dence and synchronization constraints, and that it has the big advantage of non-deterioratingperturbations being available for iterated local search. The authors believe that this approachwill also be useful for other scheduling problems. We would like to investigate applications toother problems as challenges in the future.

16

Acknowledgments

The authors wish to thank Dr. Hidefumi Kawasaki for leading us to this joint research. Weare deeply grateful to Dr. Shunji Umetani for his valuable advice regarding this investigation.Akifumi Kira was supported in part by JSPS KAKENHI Grant Number 26730010. YutakaKimura was supported by JSPS KAKENHI Grant Number 26400207. Katsuki Fujisawa wassupported by JST CREST and JSPS KAKENHI Grant Number 16H01707.

References

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[2] Ulrich Derigs and Thomas Dohmer. Indirect search for the vehicle routing problem withpickup and delivery and time windows. OR Spectrum, Vol. 30, No. 1, pp. 149–165, 2008.

[3] Michael Drexl. Synchronization in vehicle routing-a survey of vrps with multiple synchro-nization constraints. Transportation Science, Vol. 46, No. 3, pp. 297–316, 2012.

[4] Haibing Li and Andrew Lim. A metaheuristic for the pickup and delivery problem withtime windows. International Journal on Artificial Intelligence Tools, Vol. 12, No. 02, pp.173–186, 2003.

[5] Yanzhi Li, Andrew Lim, and Brian Rodrigues. Manpower allocation with time windowsand job-teaming constraints. Naval Research Logistics (NRL), Vol. 52, No. 4, pp. 302–311,2005.

[6] Koji Nonobe and Toshihide Ibaraki. Formulation and tabu search algorithm for the resourceconstrained project scheduling problem. In Essays and surveys in metaheuristics, pp. 557–588. Springer, 2002.

[7] Giselher Pankratz. A grouping genetic algorithm for the pickup and delivery problem withtime windows. Or Spectrum, Vol. 27, No. 1, pp. 21–41, 2005.

[8] Isamu Watanabe, Tokoro Kenichi, and Uemura Satoshi. An optimal rout planning methodfor emergency restoration work in natural disasters. CRIEPI Research Report, No. R07025,2008. (in Japanese without abstract).

[9] Isamu Watanabe, Tokoro Kenichi, and Uemura Satoshi. An optimal allocation method ofrestoration teams for emergency restoration work. CRIEPI Research Report, No. R08008,2009. (in Japanese without abstract).

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17

List of MI Preprint Series, Kyushu University

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MI2008-1 Takahiro ITO, Shuichi INOKUCHI & Yoshihiro MIZOGUCHIAbstract collision systems simulated by cellular automata

MI2008-2 Eiji ONODERAThe intial value problem for a third-order dispersive flow into compact almost Her-mitian manifolds

MI2008-3 Hiroaki KIDOOn isosceles sets in the 4-dimensional Euclidean space

MI2008-4 Hirofumi NOTSUNumerical computations of cavity flow problems by a pressure stabilized characteristic-curve finite element scheme

MI2008-5 Yoshiyasu OZEKITorsion points of abelian varieties with values in nfinite extensions over a p-adic field

MI2008-6 Yoshiyuki TOMIYAMALifting Galois representations over arbitrary number fields

MI2008-7 Takehiro HIROTSU & Setsuo TANIGUCHIThe random walk model revisited

MI2008-8 Silvia GANDY, Masaaki KANNO, Hirokazu ANAI & Kazuhiro YOKOYAMAOptimizing a particular real root of a polynomial by a special cylindrical algebraicdecomposition

MI2008-9 Kazufumi KIMOTO, Sho MATSUMOTO & Masato WAKAYAMAAlpha-determinant cyclic modules and Jacobi polynomials

MI2008-10 Sangyeol LEE & Hiroki MASUDAJarque-Bera Normality Test for the Driving Levy Process of a Discretely ObservedUnivariate SDE

MI2008-11 Hiroyuki CHIHARA & Eiji ONODERAA third order dispersive flow for closed curves into almost Hermitian manifolds

MI2008-12 Takehiko KINOSHITA, Kouji HASHIMOTO and Mitsuhiro T. NAKAOOn the L2 a priori error estimates to the finite element solution of elliptic problemswith singular adjoint operator

MI2008-13 Jacques FARAUT and Masato WAKAYAMAHermitian symmetric spaces of tube type and multivariate Meixner-Pollaczek poly-nomials

MI2008-14 Takashi NAKAMURARiemann zeta-values, Euler polynomials and the best constant of Sobolev inequality

MI2008-15 Takashi NAKAMURASome topics related to Hurwitz-Lerch zeta functions

MI2009-1 Yasuhide FUKUMOTOGlobal time evolution of viscous vortex rings

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MI2009-3 Hidetoshi MATSUI & Sadanori KONISHIVariable selection for functional regression model via the L1 regularization

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MI2009-5 Toshiro HIRANOUCHI & Yuichiro TAGUCHIIFlat modules and Groebner bases over truncated discrete valuation rings

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MI2009-10 Shingo SAITOGeneralisation of Mack’s formula for claims reserving with arbitrary exponents forthe variance assumption

MI2009-11 Kenji KAJIWARA, Masanobu KANEKO, Atsushi NOBE & Teruhisa TSUDAUltradiscretization of a solvable two-dimensional chaotic map associated with theHesse cubic curve

MI2009-12 Tetsu MASUDAHypergeometric τ-functions of the q-Painleve system of type E

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MI2009-13 Hidenao IWANE, Hitoshi YANAMI, Hirokazu ANAI & Kazuhiro YOKOYAMAA Practical Implementation of a Symbolic-Numeric Cylindrical Algebraic Decompo-sition for Quantifier Elimination

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MI2009-17 Masato WAKAYAMA & Keitaro YAMAMOTONon-linear algebraic differential equations satisfied by certain family of elliptic func-tions

MI2009-18 Me Me NAING & Yasuhide FUKUMOTOLocal Instability of an Elliptical Flow Subjected to a Coriolis Force

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MI2009-21 Hiroshi YOSHIDA, Yoshihiro MIWA & Masanobu KANEKOElliptic curves and Fibonacci numbers arising from Lindenmayer system with sym-bolic computations

MI2009-22 Eiji ONODERAA remark on the global existence of a third order dispersive flow into locally Hermi-tian symmetric spaces

MI2009-23 Stjepan LUGOMER & Yasuhide FUKUMOTOGeneration of ribbons, helicoids and complex scherk surface in laser-matter Interac-tions

MI2009-24 Yu KAWAKAMIRecent progress in value distribution of the hyperbolic Gauss map

MI2009-25 Takehiko KINOSHITA & Mitsuhiro T. NAKAOOn very accurate enclosure of the optimal constant in the a priori error estimatesfor H2

0 -projection

MI2009-26 Manabu YOSHIDARamification of local fields and Fontaine’s property (Pm)

MI2009-27 Yu KAWAKAMIValue distribution of the hyperbolic Gauss maps for flat fronts in hyperbolic three-space

MI2009-28 Masahisa TABATANumerical simulation of fluid movement in an hourglass by an energy-stable finiteelement scheme

MI2009-29 Yoshiyuki KAGEI & Yasunori MAEKAWAAsymptotic behaviors of solutions to evolution equations in the presence of transla-tion and scaling invariance

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MI2009-35 Yosuke MIZUYAMA, Takamasa SHINDE, Masahisa TABATA & Daisuke TAGAMIFinite element computation for scattering problems of micro-hologram using DtNmap

MI2009-36 Reiichiro KAWAI & Hiroki MASUDAExact simulation of finite variation tempered stable Ornstein-Uhlenbeck processes

MI2009-37 Hiroki MASUDAOn statistical aspects in calibrating a geometric skewed stable asset price model

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MI2010-2 Reiichiro KAWAI & Hiroki MASUDAInfinite variation tempered stable Ornstein-Uhlenbeck processes with discrete obser-vations

MI2010-3 Kei HIROSE, Shuichi KAWANO, Daisuke MIIKE & Sadanori KONISHIHyper-parameter selection in Bayesian structural equation models

MI2010-4 Nobuyuki IKEDA & Setsuo TANIGUCHIThe Ito-Nisio theorem, quadratic Wiener functionals, and 1-solitons

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MI2010-6 Shuichi KAWANO, Toshihiro MISUMI & Sadanori KONISHISemi-supervised logistic discrimination via graph-based regularization

MI2010-7 Teruhisa TSUDAUC hierarchy and monodromy preserving deformation

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MI2010-18 Toshimitsu TAKAESUScaling limits for the system of semi-relativistic particles coupled to a scalar bosefield

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MI2010-22 Toshimitsu TAKAESUA Generalized Scaling Limit and its Application to the Semi-Relativistic ParticlesSystem Coupled to a Bose Field with Removing Ultraviolet Cutoffs

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MI2010-24 Toshimitsu TAKAESUA Hardy’s Uncertainty Principle Lemma inWeak Commutation Relations of Heisenberg-Lie Algebra

MI2010-25 Toshimitsu TAKAESUOn the Essential Self-Adjointness of Anti-Commutative Operators

MI2010-26 Reiichiro KAWAI & Hiroki MASUDAOn the local asymptotic behavior of the likelihood function for Meixner Levy pro-cesses under high-frequency sampling

MI2010-27 Chikashi ARITA & Daichi YANAGISAWAExclusive Queueing Process with Discrete Time

MI2010-28 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAMotion and Backlund transformations of discrete plane curves

MI2010-29 Takanori YASUDA, Masaya YASUDA, Takeshi SHIMOYAMA & Jun KOGUREOn the Number of the Pairing-friendly Curves

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MI2010-31 Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling and spike detection via Gaussian basis expansions

MI2010-32 Nobutaka NAKAZONOHypergeometric τ functions of the q-Painleve systems of type (A2 +A1)

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MI2010-33 Yoshiyuki KAGEIGlobal existence of solutions to the compressible Navier-Stokes equation aroundparallel flows

MI2010-34 Nobushige KUROKAWA, Masato WAKAYAMA & Yoshinori YAMASAKIMilnor-Selberg zeta functions and zeta regularizations

MI2010-35 Kissani PERERA & Yoshihiro MIZOGUCHILaplacian energy of directed graphs and minimizing maximum outdegree algorithms

MI2010-36 Takanori YASUDACAP representations of inner forms of Sp(4) with respect to Klingen parabolic sub-group

MI2010-37 Chikashi ARITA & Andreas SCHADSCHNEIDERDynamical analysis of the exclusive queueing process

MI2011-1 Yasuhide FUKUMOTO& Alexander B. SAMOKHINSingular electromagnetic modes in an anisotropic medium

MI2011-2 Hiroki KONDO, Shingo SAITO & Setsuo TANIGUCHIAsymptotic tail dependence of the normal copula

MI2011-3 Takehiro HIROTSU, Hiroki KONDO, Shingo SAITO, Takuya SATO, Tatsushi TANAKA& Setsuo TANIGUCHIAnderson-Darling test and the Malliavin calculus

MI2011-4 Hiroshi INOUE, Shohei TATEISHI & Sadanori KONISHINonlinear regression modeling via Compressed Sensing

MI2011-5 Hiroshi INOUEImplications in Compressed Sensing and the Restricted Isometry Property

MI2011-6 Daeju KIM & Sadanori KONISHIPredictive information criterion for nonlinear regression model based on basis ex-pansion methods

MI2011-7 Shohei TATEISHI, Chiaki KINJYO & Sadanori KONISHIGroup variable selection via relevance vector machine

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MI2011-9 Chikashi ARITA, Arvind AYYER, Kirone MALLICK & Sylvain PROLHACRecursive structures in the multispecies TASEP

MI2011-10 Kazunori YASUTAKEOn projective space bundle with nef normalized tautological line bundle

MI2011-11 Hisashi ANDO, Mike HAY, Kenji KAJIWARA & Tetsu MASUDAAn explicit formula for the discrete power function associated with circle patternsof Schramm type

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MI2011-13 Vladimır CHALUPECKY & Adrian MUNTEANSemi-discrete finite difference multiscale scheme for a concrete corrosion model: ap-proximation estimates and convergence

MI2011-14 Jun-ichi INOGUCHI, Kenji KAJIWARA, Nozomu MATSUURA & Yasuhiro OHTAExplicit solutions to the semi-discrete modified KdV equation and motion of discreteplane curves

MI2011-15 Hiroshi INOUEA generalization of restricted isometry property and applications to compressed sens-ing

MI2011-16 Yu KAWAKAMIA ramification theorem for the ratio of canonical forms of flat surfaces in hyperbolicthree-space

MI2011-17 Naoyuki KAMIYAMAMatroid intersection with priority constraints

MI2012-1 Kazufumi KIMOTO & Masato WAKAYAMASpectrum of non-commutative harmonic oscillators and residual modular forms

MI2012-2 Hiroki MASUDAMighty convergence of the Gaussian quasi-likelihood random fields for ergodic Levydriven SDE observed at high frequency

MI2012-3 Hiroshi INOUEA Weak RIP of theory of compressed sensing and LASSO

MI2012-4 Yasuhide FUKUMOTO & Youich MIEHamiltonian bifurcation theory for a rotating flow subject to elliptic straining field

MI2012-5 Yu KAWAKAMIOn the maximal number of exceptional values of Gauss maps for various classes ofsurfaces

MI2012-6 Marcio GAMEIRO, Yasuaki HIRAOKA, Shunsuke IZUMI, Miroslav KRAMAR,Konstantin MISCHAIKOW & Vidit NANDATopological Measurement of Protein Compressibility via Persistence Diagrams

MI2012-7 Nobutaka NAKAZONO & Seiji NISHIOKA

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MI2012-8 Naoyuki KAMIYAMAA new approach to the Pareto stable matching problem

MI2012-9 Jan BREZINA & Yoshiyuki KAGEISpectral properties of the linearized compressible Navier-Stokes equation aroundtime-periodic parallel flow

MI2012-10 Jan BREZINAAsymptotic behavior of solutions to the compressible Navier-Stokes equation arounda time-periodic parallel flow

MI2012-11 Daeju KIM, Shuichi KAWANO & Yoshiyuki NINOMIYAAdaptive basis expansion via the extended fused lasso

MI2012-12 Masato WAKAYAMAOn simplicity of the lowest eigenvalue of non-commutative harmonic oscillators

MI2012-13 Masatoshi OKITAOn the convergence rates for the compressibleNavier- Stokes equations with potential force

MI2013-1 Abuduwaili PAERHATI & Yasuhide FUKUMOTOA Counter-example to Thomson-Tait-Chetayev’s Theorem

MI2013-2 Yasuhide FUKUMOTO & Hirofumi SAKUMAA unified view of topological invariants of barotropic and baroclinic fluids and theirapplication to formal stability analysis of three-dimensional ideal gas flows

MI2013-3 Hiroki MASUDAAsymptotics for functionals of self-normalized residuals of discretely observed stochas-tic processes

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MI2014-6 This paper was withdrawn by the authors.

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