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- きよたつ ごちょう
- 2 years ago
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4 2014
5 i Mobile Testing System Optimizing Test Information and Movement Distance Yoshimitsu Miyasawa Abstract Knowledge is embedded in various situations rather than existing on its own. Moreover, knowledge is known to be acquired in conjunction with past experiences. These observations suggest that mobile technologies enhance learning in situations. Some effective learning support systems using mobile technologies have been developed. However, assessment methods for these learning styles have not been discussed sufficiently. Effective assessment is expected to be embedded in the same situations. For this purpose, mobile testing, which provides test items on a mobile device in the field, is known to be effective. However, accurate assessment of student s ability requires a sufficient number of responses to items. The mobile testing might not be sufficient to provide enough test items because it sometimes consumes time for movement in the field. Therefore, mobile testing must use a more efficient test item presentation strategy. A more effective presentation method is known as Computerized Adaptive Testing (CAT). In fact, CAT progressively estimates the examinee s ability from the answer history and uses an item bank to present test items that maximize the amount of item information with regard to an ability estimate of the user. However, tradi-
6 ii tional adaptive testing does not guarantee optimization of the examinee s movement distance during testing in the field. The purpose of this study is to improve the accuracy of estimating an examinee s ability for tests in the field. Therefore, we propose a mobile CAT system that optimizes both test information and movement distance in the field. For this purpose, we use the traveling purchaser problem (TPP), an optimization problem using graph theory. Furthermore, we provided some simulation and actual experiments to demonstrate more accurate measuring performance of the proposed method.
7 iii Traveling Purchaser Problem (TPP) (1) (2) (3)
8 iv
9 v 3.3 Traveling Purchaser Problem Traveling Purchaser Problem(TPP) TPP Traveling Purchaser Problem
10 vi 2.1 a i b i c i b i
11 vii
12 viii (** 1% ) (** 1% * 5% )
13 1 1 [1] [2] [3][4] [5] RFID(Radio Frequency IDentification ) QR [6][7][8][9] Santos [10] GPS. Huang [11]
14 2 RFID (Computerized adaptive testing:cat) [12, 13, 14, 15, 16] 2,, ,,
15 3 3,,, Traveling Salesman Problem Traveling Purchaser Problem (TPP) TPP, TPP, TPP, TPP
16 [1] [2] [6][3][7][8][9][4][5] [10, 11] (Computerized adaptive testing:cat) [12, 13, 14, 15, 16]
![[6][3][7][8][9][4][5]](/docs-images/57/20071567/images/page_16.jpg "[10, 11] (Computerized")
17
18 6 2.2 Hwang QR [6]. Chiou [3]. Ogata QR [7]. Hwang RFID(Radio Frequency Identification) [8]. Chu RFID [9]. Chu [4]Chen [5] Santos GPS(Global Positioning System) [10]. Huang
19 2.3 7 RFID [11]. 2.3 CBT(Computer Based Testing) [12, 13, 14, 15, 16] [17][18] [19, 20, 21]
20 8 i = 1,..., I k = 1,..., K k i k u ik i k 1 0 U i = {u i1,..., u ik } θ (, ) i 3 1 p(u i = 1 θ) = c i + (1 c i ) 1 + exp[ 1.7a i (θ b i )] (2.1) a i [0, ) i b i (, ) i c i [0, 1] i (2.1) c i = 0 2 a i = 1 1 (2.1) (2.2) (2.3) item response functionirf (2.1) a i 3 a i 1 a i (2.2) b i 3
![7a i (θ b i )] (2.1) a i [0, ) i b i (, ) i c i [0, 1] i (2.](/docs-images/57/20071567/images/page_20.jpg "1) c i = 0 2 a i = 1 1 (2.1) (2.2) (2.3) item response functionirf (2.")
21 a i b i 3 12 b i 0.5 (2.3) c i 3 c i 3 k 1 u i1,..., u ik 1
22 b i L(θ u i1... u ik 1 ) k 1 j=1 {exp[a ij (θ b ij )]} u i j 1 + exp[a ij (θ b ij )] (2.2) J ui1...u ik 1 (θ) θ L(θ u i 1... u ik 1 ) (2.3) k 1
23 c i I Ui1...U ik 1 (θ) E[J ui1...u ik 1 (θ)] (2.4) = k 1 j=1 [p (U ij = 1 θ)] 2 p(u ij θ)[1 p(u ij = 1 θ)] p (U ij = 1 θ) θ p (U ij = 1 θ) (2.5) (2.4) 2
24 (2.4) 0.5 [22] u i1,..., u ik 1 θ = ˆθ ui1,...,u ik 1 k
25 i k = arg max{i U1,...,U k 1,U j (ˆθ ui1,...,u ik 1 ) : j R k } (2.6) j R k = {1,..., I}\{i 1,..., i k 1 } (2.5) 0 k [19] [20] [23] k g(θ) g(θ u 1j,, u kj ) = L(θ u 1j,, u kj )g(θ) (L(θ u1j,, u kj )g(θ)) dθ (2.7) L(θ u 1j,, u kj ) k ˆθ EAP(expected a posteriori) [21]
26 ˆθ = θ g(θ u 1j,, u kj )dθ : θ (, ) (2.8)
27 k 2.7 Android Maps API Google GPS 2.8 TEST
28 16 Web Android [18] [6][3][7][8][9][4][5] [10][11] RFID QR ID GPS
29 GPS RFID QR ID [11]2 3 4
30 RFID
31 ID XML(Extensible Markup Language)
32 20 要 素 内 容 例 活 用 例 出 題 位 置 緯 度 経 度,RFIDやQR コードのIDによって 管 理 , ( 緯 度, 経 度 ) 公 園 で 受 検 者 の 現 在 位 置 に 応 じ たテスト 項 目 を 出 題 期 間 開 始 の 年 月 日 と 終 了 の 年 月 日 によって 管 理 6 月 1 日 ~8 月 1 日 星 座 のように 季 節 によって 変 化 するものを 時 期 に 応 じて 出 題 時 間 帯 開 始 の 時 間 と 終 了 の 時 間 によって 管 理 13 時 00 分 ~15 時 00 分 アサガオのように 時 間 によって 姿 が 変 わる 草 花 を 時 間 に 応 じて 出 題 天 候 晴 れ, 曇 り, 雨. 雪 の 中 か ら 選 ぶ 晴 れ 星 座 のように 曇 りでは 出 題 不 可 能 なものを 制 御 対 象 物 対 象 物 に 関 する 情 報 ( 名 称 や 位 置, 方 向 )を 管 理 北 極 星, 北 誤 答 時 に 北 極 星 といった 星 座 の 方 向 を 示 す 事 ができる RFID ID
33 要 素 具 体 的 な 内 容 項 目 の 出 題 位 置 期 間 時 間 帯 天 候 対 象 物 GPS RFID QRコード 緯 度, 経 度 ID 情 報 QRコードの 情 報 出 題 開 始 の 年 月 日 出 題 終 了 の 年 月 日 出 題 開 始 の 時 間 出 題 終 了 の 時 間 出 題 可 能 な 天 候 名 称 位 置 方 向
34 22 <?xml version= 1.0 encoding= SHIFT-JIS?> <itembank> <item> <question> この 仏 像 が 遭 った 災 害 は 何 か? 仏 像 に 痕 跡 があります </question > ( 一 部, 省 略 ) <location> <gps> </gps> </location> <latitude> </latitude> <longitude> </longitude> <object> <name> 仏 </name> <location> <gps> <latitude> </latitude> <longitude> </longitude> </gps> </location> <angle>315</angle> </object> <area> 寺 </area> </item> </itembank>
35 <?xml version= 1.0 encoding= SHIFT-JIS?> <itembank> <item> <question> この 標 識 の 意 味 は? </question > ( 一 部, 省 略 ) <location> <RFID> <ID>2002</ID> </RFID> </location> <area> 大 学 </area> </item> </itembank>
36 24 b j b j b j b j b i θ [23] b i b θ τ θ, τ b g(θ τ θ ) θ g(τ θ ) g(b τ b ) b g(τ b ) g(θ, b, τ b ) U
37 g(θ, b, τ b U) L(U θ, b)g(θ, b, τ b ) L(U θ, b)g(θ τ θ )g(τ θ )g(b τ b )g(τ b ) (2.9) L(U θ, b) = I i=1 j=1 J [P u ij ij Q (1 u ij) ij ] D ij (2.10) P ij j i ( (2.1))Q ij D ij 1 0 b i θ [23] g(τ θ, b, τ b U) L(U θ, b)g(θ, b, τ b )dθ L(U b, τ θ )g(τ θ )g(b τ b )g(τ b ) (2.11) EM [23]
38 26 仮 想 環 境 で 解 答 した 項 目 実 際 の 観 光 地 で 解 答 した 項 目 実 際 の 観 光 地 と 仮 想 環 境 で 解 答 した ユーザー 実 際 の 観 光 地 と 仮 想 環 境 で 解 答 したユーザーの 反 応 データ 仮 想 環 境 でのみ 解 答 したユーザー 仮 想 環 境 でのみ 解 答 したユーザー の 反 応 データ 欠 測 値 2.14 b i 2. 3.
39 (2.1)
40 % NacEMR-8
41
42
43
44
45
46
47 () A B C D %
48 Computer Based TestingCBT (1)
49 (** 1% ) ** 286(1810) 96.3(30.0) ** 55(17.8) 36.4(14.2) 1400(761) 1827(716), (2) (3),
50 38
51 Linden [24, 25, 26] 2 1 2
52 40 Traveling Purchaser Problem(TPP ) TPP Traveling Salesman Problem [27, 28, 29, 30, 31, 32, 33]. TPP TPP TPP n O(n!) TPP O(2n 2 n ) [34] [35]TPP TPP TPP n I n = I O(I!) O(2 2 ) 1. 2.
53 Linden [24] [24, 25, 26] Shadow Test Approach (STA) k Maximize w = I I U1,...,U k 1 (ˆθ ui1,...,u ik 1 )x i (3.1) i=1
54 42 I exp(β i )x i t tot (3.2) i=1 1 i, x i = 0. (3.3) ˆθ k 1 k 1 t tot τ (, ) i t i [36]. f(t i ; τ, α i, β i ) = α { i exp 1 t i 2π 2 [α i(ln t i (β i τ))] 2} (3.4) β i (, ) i α i (0, ) i t i τ (, ) M τ l = 0 (3.5) l=1 (3.4) t i β i τ τ (3.5) 0 β i
55 3.3 Traveling Purchaser Problem Traveling Purchaser Problem Traveling Purchaser Problem(TPP) Traveling Purchaser Problem(TPP) TPP TSP TPP TPP S := {v 1,..., v n } K := {p 1,..., p m } S 0 := {S o}o G = (V, E) V := S 0 E := {[v i, v j ] : v i, v j V, i < j} p k b k v i v j d ij v i p k
56 44 z ik = 1, z ik = 0 v i v j x ij = 1 x ij = 0 Minimize w = v i S b k z ik + D d ij x ij (3.6) p k K (i,j) L TPP TPP TPP S := {v 1,..., v n }, K := {p 1,..., p I } S 0 := {S o}o p k b k p k t k v i v j d ij T Maximize w = v i S b k z ik D d ij x ij (3.7) p k K (i,j) L t k z ik + d ij x ij < T (3.8) v i S p k K (i,j) L (3.7) (3.8)
57 3.3 Traveling Purchaser Problem 45 n I n = I TPP [37] I O(I!) O(2I 2 I ) [34] [35] TPP TPP 1 1 Main OptCalculator SearchPath Main OptCalculator SearchPath Main S
58 46 t 0 OptCalculator Svt DP S DP DP 3.1 0, 1, 3, 4 S 134 v , 1, 3, 4 DP[S][v] 4 0 DP 0, 3, 1, 4 OptCalculator DP -INF -INF OptCalculator OptCalculator S t OptCalculator OptCalculator DP SearchPath OptCalculator DP DP
59 3.3 Traveling Purchaser Problem 47 {0} {0, 1} {0, 2} {0, 3} {0, 4} {0, n} {0, 1, 2} {0, 1, 3} {0, 3, 1} {0, 3, 2} {0, 1, 3, 2} {0, 1, 3, 4} {0, 3, 1, 4} {0, 1, 3, 4, 2} {0, 1, 3, 4, 2} {0, 1, 3, 4, 2, 0} {0, 1, 3, 4, 2, 0} 3.1 SearchPath S DP[S][v] v v S v SearchPath TPP Traveling Purchaser Problem TPP
60 I O(2I 2 I ) I [24][38] [12][13][14][15][16] a j N(1.7, ) b j N(0, )
61 = 0 Yes N o 3.2 t j N(30, 10 2 ) x U(0, 300) y U(0, 300) (2.1) 5.
62 (2) (6) ()
63 [24][38] [24][38]
64
65 ( 2.1 ) () 3.5 [24][38]
66
67 TPP [24][38] a i, b i, t i, x i, y i
68
69
70 :5:
71 TPP
72 60 1 (1) (2) 3.6 TPP TPP TPP I O(I!) O(2I 2 I ) (1)
73 (2) (3)
74 62 1 Require: Ensure: function Main S = 0 v = start t = 0 OP T MOBILET EST ING(S, v, t) SearchP ath() end function function OptCalculator(S, v, t) DP[S][v] if t > MaxTime then return -INF then end if if t + DistanceTime[v][0] > MaxTime then return DP[S][v] = -INF then end if DP[S][v] OptimumScore = 0 for all i in Itembank do if i / S then then T ime i = t + DistanceT ime[v][i] + ResponceT ime[i] S i := S i T mpevalu = OP T MOBILET EST ING(S i, i, T imei) +Info[i] (DistanceT ime[v][i] + ResponceT ime[i]) D(D << 1) if OptimumScore < T mpevalu then then OptimumScore := T mpevalu end if end if end forreturn OptimumScore end function function SearchPath DP OptimumP ath[0] = 0 S = 0 for all i in P athlength do OptimumScore = 0 OptimumItem = 0 for all j in ItemNum do S j := S {j} T mpevalu = dp[s j ][j] + Info[i] (DistanceT ime[v][i] + ResponceT ime[i]) D(D << 1)) if OptimumScore < T mpevalu then then OptimumScore = T mpevalu OptimumItem = j end if S := S {j} OptimumP ath[i] = OptimumItem end for end forreturn OptimumP ath end function
75 [s] [24][38] [12][13][14][15][16] (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.4 2 ) ( ) ( ) (0.0 2 ) ( )
76 [s] [24][38] [12][13][14][15][16] (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( )
77 [s] [24][38] [12][13][14][15][16] (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (4.2 2 ) ( ) (0.0 2 ) ( )
78 [s] [24][38] (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) ( ) (0.0 2 ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) (0.0 2 ) ( ) (0.0 2 ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) ( ) ( ) ( ) ( ) ( ) (0.0 2 ) (0.0 2 ) ( ) ( ) ( ) ( ) (0.0 2 )
79 [s] [24][38] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.6 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
80 [s] [24][38] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
81 [s] [24][38] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
82 [s] [24][38] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.1 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.8 2 ) ( ) ( ) ( ) ( ) ( ) ( )
83 (** 1% * 5% ) [24][38] ** 4.24( ) 2.03( ) []* 110( ) 238( ) * 4.0( ) 5.8( ) ** 18.2( ) 9.0( ) (1)** 4.8( ) 3.4( ) (2)* 4.6( ) 3.2( )
84 (1), (2) (3),
85 73 3 Traveling Purchaser Problem (TPP) TPP TPP TPP n O(n!) TPP O(2n 2 n ) [34] [35] TPP TPP TPP n I n = I O(I!) O(2I 2 I )
86
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