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Car pooling problem is to design feasible paths employees pick up colleagues while driving to/from work to minimize the number of private cars. I propose a technique to improve satisfaction for pairs of servers and clients using their preferences.
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電子情報通信学会論文誌 2020/11 Vol. J103–A No. 11
通勤者間の嗜好を考慮する Car Pooling 問題
橋上 英宜a)
(正員)
Car Pooling Problem with Preference for Each Employee
Hidenobu HASHIKAMIa),Member
ソフトコンピューティング,流山市
Soft Computing, 2–7–9 Minami-nagareyama, Nagareyama-shi, 270–0163 Japan
a) E-mail: hashikami@softcomp.jp
DOI:10.14923/transfunj.2020JAL2013
あらまし Car pooling 問題は通勤車両数の削減を目
的とし,同乗者を送迎する運転者の実行可能経路を決
定する問題である.通勤者間の嗜好を考慮することに
より,運転者と同乗者の組み合わせに対する満足度を
高める手法を提案する.
キーワード Car pooling 問題,通勤者間の嗜好
1. ま え が き
カープールとは同地域から同方向への複数の通勤者
が自家用車で送迎し合うことである.交通量を減らし
道路渋滞の緩和,CO2の削減を図れる.総務省統計局
の平成 22 年調査によれば,自家用車による通勤・通学
者は 2,634.8 万人46.5%と交通手段別割合で最も
く,大規模事業所・生産工場のある地域などでは,朝
夕の時間帯を中心に深刻な渋滞が発生していると報告
されている [1].対策の一つとしてカープールが注目
れており,海外では,既にビジネス化されている [2]
最近では総合自動車部品メーカー Robert Bosch が企
業・大学・自治体向けのカープールサービスのスター
トアップ Splitting Fares Inc. を買収し,市場に参入し
[3].国内では試用が始まっており,ネッツトヨタと
富士通が試乗車などの遊休車両を利用した従業員向け
乗合通勤サービスの運用を開始した [4]
Car pooling 問題(car pooling problemCPP)は通
勤車両数の削減を目的とし,同乗者を送迎する運転者
の実行可能経路を決定する問題である.出勤と退勤の
問題は異なり,それぞれ定式化する必要がある.本論
文では出勤の問題のみ扱う.
CPP は通勤者の要求を制約に与える vehicle routing
problem [5] 特殊な場合と えられ,NP 難で
[6]Baldacci らは出勤の CPP に対して,通勤者を
同乗させることで発生する迂回の許容範囲内で,運転
者の総走行距離と乗車できなかった同乗者のペナル
ティの和を最小化する整数計画問題と,CPP を集合分
割問題として扱いラグランジュ緩和法と列生成法によ
る解法を提案し[6]Bruck らは自家用車の利用者に
公共交通の利用者を加えた実社会に即したモデルを提
案し,CO2削減の可能性を示した [7]
これらの先行研究は CPP の問題提起と解決策を示
す有用な研究である.しかしながら経路の効率面のみ
着目しており利用者の心理的側面を考慮していない.
そのためサービス実装した際利用者の満足度は低く,
サービス継続率も低いと想定される.地方部の低密度
居地域におけるライドシェアの実証研究では,「同乗
者の気兼ね」はライドシェアが成立しない一因になっ
ており,サービス継続の障害になりうると報告されて
いる [8].そこで運転者と同乗者の組み合わせに対す
る満足度を高めるために,通勤者間の嗜好を考慮する
CPP を提案する.運転者と同乗者の組み合わせは一意
に定まる.毎日同じ相手と一緒に通勤することになる
ため,より相性が重要になる.ここで嗜好とは同性,
部署の同僚,はたまた同じ趣味をもっているなどの通
勤者同士の安心感や楽しさ,気兼ねさの低減につなが
るよう設計する機能の一つである.本論文では基本的
CPP を定式化した後,提案する CPP を定式化する.
計算機実験を通じて提案手法の有効性を示す.
2. 提 案 手 法
2. 1 基本的な CPP の定式化
道中で通勤者を同乗させ,勤務地に向かう運転者の
実行可能経路を決定する.いま,勤務地や同乗者との
待ち合わせ場所を頂点とし,それらの頂点間の道路を
辺とするグラフ G=(V,E)を考える.ここで,V
頂点集合であり,勤務地を 0と表すと,n人の通勤者
集合 V={1, . . . , n}を加えて V={0,. . . ,n}={0} V
とする.また,Vは運転者server)集合 Vsと同乗者
client)集合 Vcに分割して,V=VsVcとする.E
は辺集合であり,頂点 iVから頂点 jVに向かう
有向辺 (ij) ∈ Eには,iから jの距離 di j 0
与えられているものとする.無向グラフはその辺を互
いに向きの異なる二つの有向辺で置き換えられるの
で,以下 Gを有向グラフとする.CPP は,運転
は他の運転者を乗せられないため,運転者間の辺集合
EVsVs×VsEから除き,E=E\EVsとする有
向グラフ G=(V,E)を再定義する.
運転者 kVsの経PkEkの車両の乗車定員
Qk1kの最大運転時間 Tk0iから jの移動時
tij,(i,j) ∈ Eとし,以下を仮定する.
各車両の乗車数は定員以下であるとする.すな
わち,|Pk|Qk,kVsであるとする.
kの総運転時間は Tkを超えないものとする.
274 電子情報通信学会論文誌 A Vol. J103–A No. 11 pp. 274–277 ©一般社団法人電子情報通信学会 2020
レター
Gの辺に変数 xk
ij ∈ {0,1},(i,j) ∈ E,kVsを付与
して,運転者 kが同乗者,勤務地 i,j間を移動する場
合を 1,その他を 0と定義する.また任意の頂点に変
yi∈ {0,1},iVcを付与して,同乗者 iが乗車しな
かった場合を 1,その他を 0と定義し,CPP を整数計
画問題に定式化する.
min
kVs
(i,j)∈ E
dij xk
ij +
iVc
αdi0yi(1)
s.t.
j{0}∪Vc
xk
k j =1,kVs,(2)
j{k}∪Vc
xk
j0=1,kVs,(3)
j{k}∪Vc
xk
ji
j{0}∪Vc
xk
ij =0,
iVc,kVs,(4)
xk
ij =0,(i,j) ∈ E,i=j,kVs,(5)
(i,j)∈ E
xk
ij Qk,kVs,(6)
(i,j)∈ E
ti j xk
ij Tk,kVs,(7)
uk
i+1Qk(1xk
i j )uk
j,
1uk
iQk,
i,jV,i,j,kVs,(8)
kVs
j{0}∪Vc
xk
ij +yi=1,iVc,(9)
xk
ij ∈ {0,1},(i,j) ∈ E,kVs,(10)
yi∈ {0,1},iVc.(11)
(1) は目的関数であり,第 1項は運転者の総走行
距離,第 2項は乗車できなかった同乗者のペナルティ
である.αはペナルティ項の重みである.式 (2) は運
転者の自宅から出発すること,式 (3) は勤務地に到着
することを表す.式 (4) は同乗者にいずれか 1台の車
両が一度だけ訪問することを表す.ただし,運転者に
訪問されない同乗者の場合,左辺の変数は全て 0にな
る.式 (5) は同乗者自身の辺を車両は通らないことを
表す.式 (6) は車両の乗車数は定員以下であることを
表す.式 (7) は総運転時間が最大運転時間以下である
ことを表す.式 (8) は勤務地と結ばれない一部の同乗
者だけの巡回経路を除去するための制約(部分巡回路
除去制約)であり,ここでは Miller-Tucker-Zemlin
約を応用する [9].式 (9) は同乗者が運転者に訪問され
るか車両に乗らないかを表す.式 (10)(11) は変数の定
義である.
2. 2 通勤者間の嗜好を考慮する CPP の定式化
サービス内で通勤者に嗜好の全体集合 Lを提示し,
通勤者 iVに任意の嗜好集合 liL,iVを選択し
てもらう.嗜好とは性別,部署,趣味などの通勤者同
士の安心感や楽しさ,気兼ねさの低減につながるよう
設計する機能である.
通勤者間の嗜好を考慮すCPP を整数計画問題に
定式化する.まず通勤者 iと通勤者 jの嗜好の共起度
sij =
|lilj|
min(| li|,|lj|) ∈ [0,1],i,j,i,jV(12)
を求める.ここで,通勤者 iと勤務地 0の共起度 si0
通勤者 i自身の共起度 sii 0とする.si j の計算に
は差集合の影響が小さくなる Simpson 係数を用いる.
次に sij CPP の目的関数に組み込み,嗜好の共起度
が高い通勤者の組み合わせが選ばれやすくなるように
し,加えて運転者の総走行距離と乗車できない同乗者
のペナルティが最小になる経路を決定する.すなわち
目的関数を
min
kVs
(i,j)∈ E
(1βsi j )dij xk
ij +
iVc
αdi0yi
(13)
とする.ここで,βは通勤者間の嗜好の共起度をどの
ぐらい反映させるかの重みである.制約条件は式 (2)
から (11) とする.
3. 計算機実験
実験データとして,大規模事業所のある地方都市で
の利用を想定して生成した通勤者データを用いた.運
転者数を 10 名,同乗者数20 名,通勤者と勤務地の
住所を同一郵便区番号の圏内,乗車定員を 5名,最大
運転時間を 3,600 秒とした.また通勤者の嗜好は五つ
の元から確率 0.5で選ばれた集合とする.表 1に実験
データの一部を示す.
数値評価及び主観評価について述べる.提案手法の
パラメータとしてペナルティ項の重み α1.0,通勤
者間の嗜好 の共起度の重β0.5また1.0とし
た.評価指標として運転者と同乗者の組み合わせの成
立率(matching rate: %m| {yi|yi=0,iVc} | /
|Vc|,通勤者の総 離削 率(total distance reduc-
tion rate: %tdr1(i,j)∈P
k,kVsdij iVdk0,運
275
電子情報通信学会論文誌 2020/11 Vol. J103–A No. 11
1勤務地,通勤者の実験データの一部
Table 1 Experimental pseudo data of workspace and employees.
ID Kind Postcode Address Capacity Max ride time Preference
0 workspace 471-8571 1 Toyotacho, Toyota, Aichi
1 server 471-0808 2003-3-15 Takagami, Toyota, Aichi 5 3,600 {M, W, Eng. Div., tennis}
.
.
.
10 server 471-0846 4-16-1 Tashirocho, Toyota, Aichi 5 3,600 {tennis}
11 client 471-0003 3-14-7 Iwatakicho, Toyota, Aichi {W, tennis, fishing}
.
.
.
30 client 471-0029 2-10 Sakuramachi, Toyota, Aichi {W, Eng. Div.}
1実行可能経路の比較.(a) 勤務地,運転者,同乗者をプロットした地図,(b) 基本的
CPP(c) 提案 CPPβ=0.5(d) 提案 CPPβ=1.0
Fig. 1 Comparison of feasible paths. (a) Map plotted workspace, server, and client, (b) basic CPP,
(c) proposal with β=0.5, (d) proposal with β=1.0.
転者の移動時間増加率(ride time increase rate: %rti
(i,j)∈ P
k,kVstij kVstk0,組み合わせに対する満
足度(通勤者間の嗜好の共起度平均)employee satis-
faction rate: %esµ({Jij | {i,j} P2(Vk)}) を用いた.
ここで,Jij i j 間の Jaccard 係数 |lilj| /| lilj|∈
[0,1],li,ljLP2(Vk)は運転者 kとその車両に乗
る同乗者の 集合 VkV2- 合せ(例えば運転者
1が同乗者 11 12 を車両に乗せる場合,P2(V1)
{{1,11},{1,12},{11,12}})である.表 2に数値比較を
示す.全ての手法で運転者と同乗者の組み合わせは成
立した.提案 CPP は基本的な CPP に比べて,総距離
と移動時間の増加を抑えつつ,満足度が高くなってい
ることを確認できる.図 1に実行可能経路の比較を示
す.(a) は勤務地(オフィスのアイコン),運転者(車
のアイコン)同乗者(人のアイコン)をプロットした
地図である.(b) では運転者は道中で他の通勤者を同
乗させながら短経路で勤務地に向かっていることを確
認できる.(c) では運転者は嗜好の共起度が高い通勤
276
レター
2基本的な CPP 提案 CPP の数値比較 [%]%m
運転者と同乗者の組み合わせの成立率,%tdr は通
勤者の総距離削減率,%rti は運転者の移動時間増加
率,%es は組み合わせに対する満足度
Table 2 Comparison for basic CPP and proposal [%]. %m is
matching rate, %tdr is total distance reduction rate, %rti
is ride time increase rate, and %es is employee satisfac-
tion rate.
Basic CPP Proposal
β=0.5β=1.0
%m 100.0 100.0 100.0
%tdr 49.5 46.4 30.7
%rti 139.5 144.5 182.2
%es 29.4 46.2 56.7
3提案 CPP のマッチング精度 [%]
Table 3 Matchingaccuracy, recall, and precision of proposal [%].
Proposal
β=0.5β=1.0
accuracy 54.8 0.0
recall 69.4 29.6
precision 65.7 23.1
者を同乗させ,迂回を抑えつつ,短経路で勤務地に向
かっていることを確認できる.(d) では数人の運転者
の迂回距離が長くなってしまっている.
次に提案 CPP のマッチング精度について述べる.評
価指標として正解率accuracy再現率recall適合
率(precision)を用いた [10].各指標は基本的な CPP
に対する値,つまり [10] intended result に基本的な
CPP で求めた通勤者の組み合わせを用いた.表 3に結
果を示す.βが大きくなるにつれて,精度が下がる傾
向にあることがわかる.
なお,整数計画ソルバとして Google OR-Tools を用
いた.ソルバを用いて CPP を解くための計算環境は
CPUIntel®Xeon®2.00GHzGPUNVIDIA GK210GL
[Tesla K80],メモリ:12.0GiBOSUbuntu 18.04.3
プログラム言語:Python 3.6.9 である.基本的 CPP
対してのソルバが終了するまでに要した計算時間は
19.9 秒,ソルバが終了するまでの分枝限定法の反復回
数は 22,923,分枝限定ノード数は 689 だった.提案
CPP の計算時間は 23.1 秒,反復回数は 45,042,ノー
ド数は 1,664 だった.
4. むすび
本論文では,運転者と同乗者の組み合わせに対する
満足度を高めるために,通勤者間の嗜好を考慮する
CPP を提案した.計算機実験を通じて提案手法の有
効性を確認した.今後の課題は,部分巡回路除去制約
の一つである単品種フロー定式化による処理時間の改
善,単独運転者の同乗者化の問題定義,退勤の CPP
定式化である.
文 献
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Article
It has been observed by many people that a striking number of quite diverse mathematical problems can be formulated as problems in integer programming, that is, linear programming problems in which some or all of the variables are required to assume integral values. This fact is rendered quite interesting by recent research on such problems, notably by R. E. Gomory [2, 3], which gives promise of yielding efficient computational techniques for their solution. The present paper provides yet another example of the versatility of integer programming as a mathematical modeling device by representing a generalization of the well-known “Travelling Salesman Problem” in integer programming terms. The authors have developed several such models, of which the one presented here is the most efficient in terms of generality, number of variables, and number of constraints. This model is due to the second author [4] and was presented briefly at the Symposium on Combinatorial Problems held at Princeton University, April 1960, sponsored by SIAM and IBM. The problem treated is: (1) A salesman is required to visit each of n cities, indexed by 1, … , n . He leaves from a “base city” indexed by 0, visits each of the n other cities exactly once, and returns to city 0. During his travels he must return to 0 exactly t times, including his final return (here t may be allowed to vary), and he must visit no more than p cities in one tour. (By a tour we mean a succession of visits to cities without stopping at city 0.) It is required to find such an itinerary which minimizes the total distance traveled by the salesman. Note that if t is fixed, then for the problem to have a solution we must have tp ≧ n . For t = 1, p ≧ n , we have the standard traveling salesman problem. Let d ij ( i ≠ j = 0, 1, … , n ) be the distance covered in traveling from city i to city j . The following integer programming problem will be shown to be equivalent to (1): (2) Minimize the linear form ∑ 0≦ i ≠ j ≦ n ∑ d ij x ij over the set determined by the relations ∑ n i =0 i ≠ j x ij = 1 ( j = 1, … , n ) ∑ n j =0 j ≠ i x ij = 1 ( i = 1, … , n ) u i - u j + px ij ≦ p - 1 (1 ≦ i ≠ j ≦ n ) where the x ij are non-negative integers and the u i ( i = 1, …, n ) are arbitrary real numbers. (We shall see that it is permissible to restrict the u i to be non-negative integers as well.) If t is fixed it is necessary to add the additional relation: ∑ n u =1 x i 0 = t Note that the constraints require that x ij = 0 or 1, so that a natural correspondence between these two problems exists if the x ij are interpreted as follows: The salesman proceeds from city i to city j if and only if x ij = 1. Under this correspondence the form to be minimized in (2) is the total distance to be traveled by the salesman in (1), so the burden of proof is to show that the two feasible sets correspond; i.e., a feasible solution to (2) has x ij which do define a legitimate itinerary in (1), and, conversely a legitimate itinerary in (1) defines x ij , which, together with appropriate u i , satisfy the constraints of (2). Consider a feasible solution to (2). The number of returns to city 0 is given by ∑ n i =1 x i 0 . The constraints of the form ∑ x ij = 1, all x ij non-negative integers, represent the conditions that each city (other than zero) is visited exactly once. The u i play a role similar to node potentials in a network and the inequalities involving them serve to eliminate tours that do not begin and end at city 0 and tours that visit more than p cities. Consider any x r 0 r 1 = 1 ( r 1 ≠ 0). There exists a unique r 2 such that x r 1 r 2 = 1. Unless r 2 = 0, there is a unique r 3 with x r 2 r 3 = 1. We proceed in this fashion until some r j = 0. This must happen since the alternative is that at some point we reach an r k = r j , j + 1 < k . Since none of the r 's are zero we have u r i - u r i + 1 + px r i r i + 1 ≦ p - 1 or u r i - u r i + 1 ≦ - 1. Summing from i = j to k - 1, we have u r j - u r k = 0 ≦ j + 1 - k , which is a contradiction. Thus all tours include city 0. It remains to observe that no tours is of length greater than p . Suppose such a tour exists, x 0 r 1 , x r 1 r 2 , … , x r p r p +1 = 1 with all r i ≠ 0. Then, as before, u r 1 - u r p +1 ≦ - p or u r p +1 - u r 1 ≧ p . But we have u r p +1 - u r 1 + px r p +1 r 1 ≦ p - 1 or u r p +1 - u r 1 ≦ p (1 - x r p +1 r 1 ) - 1 ≦ p - 1, which is a contradiction. Conversely, if the x ij correspond to a legitimate itinerary, it is clear that the u i can be adjusted so that u i = j if city i is the j th city visited in the tour which includes city i , for we then have u i - u j = - 1 if x ij = 1, and always u i - u j ≦ p - 1. The above integer program involves n ² + n constraints (if t is not fixed) in n ² + 2 n variables. Since the inequality form of constraint is fundamental for integer programming calculations, one may eliminate 2 n variables, say the x i 0 and x 0 j , by means of the equation constraints and produce an equivalent problem with n ² + n inequalities and n ² variables. The currently known integer programming procedures are sufficiently regular in their behavior to cast doubt on the heuristic value of machine experiments with our model. However, it seems appropriate to report the results of the five machine experiments we have conducted so far. The solution procedure used was the all-integer algorithm of R. E. Gomory [3] without the ranking procedure he describes. The first three experiments were simple model verification tests on a four-city standard traveling salesman problem with distance matrix [ 20 23 4 30 7 27 25 5 25 3 21 26 ] The first experiment was with a model, now obsolete, using roughly twice as many constraints and variables as the current model (for this problem, 28 constraints in 21 variables). The machine was halted after 4000 pivot steps had failed to produce a solution. The second experiment used the earlier model with the x i 0 and x 0 j eliminated, resulting in a 28-constraint, 15-variable problem. Here the machine produced the optimal solution in 41 pivot steps. The third experiment used the current formulation with the x i 0 and x 0 j eliminated, yielding 13 constraints and 9 variables. The optimal solution was reached in 7 pivot steps. The fourth and fifth experiments were used on a standard ten-city problem, due to Barachet, solved by Dantzig, Johnson and Fulkerson [1]. The current formulation was used, yielding 91 constraints in 81 variables. The fifth problem differed from the fourth only in that the ordering of the rows was altered to attempt to introduce more favorable pivot choices. In each case the machine was stopped after over 250 pivot steps had failed to produce the solution. In each case the last 100 pivot steps had failed to change the value of the objective function. It seems hopeful that more efficient integer programming procedures now under development will yield a satisfactory algorithmic solution to the traveling salesman problem, when applied to this model. In any case, the model serves to illustrate how problems of this sort may be succinctly formulated in integer programming terms.