A simple variational problem for a moving vehicle - Semantic Scholar
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Operations Research Letters 16 (1994)231-239
A simple variational problem for a moving vehicle Seong-In Kim a'*, In-Chan Choi b aDepartment of lndustrial Engineering, Korea University, Seoul, 136 701 South Korea bDepartment of lndustrial Engineering, The Wichita State University, Wichita, KS 67208, USA Received 15 September 1993; revised 1 June 1994)
Abstract
The problem of determining an optimal path of a moving vehicle, considered in Sherali and Kim (1992), is studied in this paper. We generalize their model by including a general cost structure in the problem and show that a specific, non-trivial cost function in this setting yields an easy solution. This follows from the observation that the variational problem that arises in the model with this particular cost function is the problem of geodesics on the Liouville surface. Through a comparative computational study, we also show that the optimal path for the model with this cost function better approximates the optimal path for the original model in Sherali and Kim than their second-order approximation approach does.
Key words: Variational problem; Geodesics; Liouville surface; Moving vehicle
1. Introduction
In this paper, we consider the problem of determining an optimal path of a moving vehicle, which travels a given region that contains a set of existing facilities. The optimality is with respect to minimizing an appropriately defined total cost that is incurred during the journey of the vehicle. This problem appears in various applications, such as in determining a path of a surveillance aircraft or a patrol car maintaining radio contacts with stations. For motivational thoughts on this problem and its counterpart on the network, readers are referred to [2, 4]. In [4], a model for this and other related problems was constructed based on the squared Euclidean (or rectilinear) distance between the vehicle and the existing facilities over the time period of the journey as a total cost. Three different approaches for solving the problem were provided along with their computational performances. Among the three approaches, Algorithm III(2), a dynamic programming based perturbation approach, was reported to provide the best observed performance in terms of the quality of the solution. This approach utilized the solution of Algorithm II(2), a second-order approximation approach, as an initial solution. The computation time of Algorithm III(2) was reported to be significantly larger, over 100
* Corresponding author. 0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 3 8 - 7
S.-1. Kim, 1.-C. Choi / Operations Research Letters 16 (1994) 231-239
232
times, than the other two approaches. The third approach presented in [4] was the numerical method of solving a second-order ordinary differential equation. Our goal in this paper is twofold. On the one hand, we would like to point out a special case when a model for a moving vehicle becomes particularly easy to deal with, i.e. the case when the variational problem arising from the model becomes easy to solve. To this end, we first extend the model of Sherali and Kim [4] to a larger class that includes a general cost structure, which includes that of [4]. Specifically, the time-varying cost function of the moving vehicle is represented by a nonnegative composite function of separable functions. We then show that in the model with a specific form of nontrivial cost function in this setting, an optimal path can be easily obtained; although, the model with general cost structure can only be handled by numerical methods(see [1, 5]). This follows from the observation that the variational problem associated with the specific model we consider is a problem of geodesics on the Liouville surface. On the other hand, we would like to see how the optimal path of our model with the special cost function approximates that of Sherali and Kim's model with the squared Euclidean distance. As shall be discussed in a later section, the optimal path of our model better approximates that of the Sherali and Kim's model than their Algorithm II(2) does. Thus, it can be used as a better intial solution for Algorithm III(2), or as a real-time solution if a quick and relatively accurate solution is desired. This paper is organized as follows. In Section 2, an extension of the Sherali and Kim's model is presented. Only the relevant points are addressed. For the details of the model of Sherali and Kim, readers are referred to [4]. In Section 3, we show that the special cost function within the general model framework given in Section 2 renders an easy solution procedure in obtaining the optimal solution. Finally, comparative computational results are presented in Section 4.
2. Model formulation We now consider the mathematical modeling of the problem mentioned in Section 1. The formulation is similar to the one given in [4], but our model allows a general cost structure. Suppose that there are m existing facilities located on Ei = (al, bi) for i = 1.... , m on the Cartesian coordinate in a two-dimensional space. Also, suppose that a vehicle moves along the trajectory z(t) = [x(t), y(t)] at a fixed velocity v, starting from the origin and arriving at a destination located at (0, hO). Let the unit incremental cost incurred by the movement of the vehicle on z(t) from t + d t be given as Q(C1,..., Cm), where Ci represents an individual facility i's contribution to the unit incremental cost. Furthermore, let us assume that C~ can be expressed as
(1)
CiEEi, z(t)] =-fi(x(t)) + gi(y(t)),
wheref~(. ) and g~(') are some nonnegative functions associated with facility locations i = 1..... m. Then, the total cost that is incurred by this vehicle following the trajectory z(t) is given as
f Q(C~ ..... C,.) dt,
(2)
where ~ is an arbitrary time frame. Within this framework, the problem for a moving vehicle is then to find an optimal trajectory, z*(t), that minimizes the total cost in (2). Our model formulation allows a general cost structure for the problem, e.g. fffx(t))= e ~x(t~-a`~2, gi(y(t)) = e lytt)-bd, and Q = l o g ~ ( f ( x ( t ) ) + gi(y(t))). If we let j~(x(t)) = Ix(t) - all p and gi(y(t)) = ]y(t) - bil p, i.e.
CiEEi,
z(t)]
= (Ix(t)
--
a,I p
+ ly(t)
--
bi[P),
(3)
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233
and the aggregated cost function Q be the weighted sum of C~, Q(C1, ..., Cm) = Y~i ~_, w~C~,where weight wi is a positive dimensionless quantity, then the resulting problem is the one that is considered in [4]. In Section 3, we consider a specific form of Q,
(4)
Q(c, ..... c,~) = ~=1 wi ci,
where wi >~ 0 and y~wi = 1. A clean form of an integral equation can be obtained as a solution to the variational problem that arises in solving the problem with this cost function. Using this observation, we show that our model with Ci in (3) for p = 2 and Q defined in (4) renders a rather simple solution for the problem of determining an optimal path of a moving vehicle.
3. A simple variational problem
The problem we now consider has the following form, from (1), (2), and (4),
min f~ ff~_ (fi(x(t)) + For simplicity, let y - - y ( x ) . Also, we assume that y(x) is twice continuously differentiable. Noting that v = ds/dt and that ds = xf(dx) 2 + (dy) 2 = x/1 + (y'): dx, the problem of finding y(x) which minimizes (2) can be rewritten as rain Z(x)eZ
fx N/ ~ (f~(x)+ 9i(y(x)))x/1 + (y'(x)) i dx, i= 1
(P)
where Z(x) = [x,y(x)], Z is the feasible set of paths from the origin to the destination (O, hO), and X is a permissible variation in x. Therefore, we want to find a function y(x) that minimizes a functional
J= f f
~ i ~~',1 ( f / ( x ) ' ~ - g g ( y ( x ) ) ) x / l + ( y ' ( x ) )
2 dx,
(5)
with boundary conditions y(0) = 0 and y(O) = hO. By denoting
K(x; y; y') = x~(x) + G(y) x/-( + (y,)2, F(x) = ~ fi(x)
and
G(y) =
i=1
i=1
Oi(y(x)),
we can rewrite expression (5) as ~0
J = J o K (x; y; y') dx =
,Iv(x)
+
+ (y,)2 dx.
(6)
234
S.-I. Kim, I.-C. Choi / Operations Research Letters 16 (1994) 231-239
The above problem is the problem of geodesics on the Liouville surface and it renders the following lemma, which states that the Euler differential equation that is a necessary condition for y(x) to minimize the functional J can be simplified. Lemma 1 (Akhiezer [1]). A function y(x) that solves problem (P) must satisfy the following necessary condition:
+/~
-/~
-
~,
(7)
where ~ and fl are some constants. Proof. The necessary condition for a function y(x) to minimize the functional J of (5) is represented by the Euler equation BK By
d BK dx By'
-
0,
(8)
which is in general of second order. Also, the Legendre condition has to be satisfied, i.e. 02K/By ' By' > 0 for a minimization problem [3]. Checking the Legendre condition first, we see that the condition is satisfied as
32K
x/(F(x) + G(y))
By' By'
(~/1 + (y,)2)3
>0.
From Eq. (8), G,(y)x/1 + (y,)2
d y'~/F(x) + G(y)
2x/F(x ) + G(y)
dx
x/1 + (y,)2
~-0.
Rewriting this equation, we obtain
+ ?(yi d x/l+(y,)2
dxL
+_,£(y)] = y'G'(y). x/l+(y)2
l
Integrating the equation, we find the first integral
+ (y)l Thus,
dy= + ~ + ~ dx -- F ~ -- ,B" Introducing a second arbitrary constant c¢ and integrating this equation we obtain the desired result.
[]
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235
In the remainder of the section, we confine our discussion to the model with Ci's as in (3) with p = 2, i.e. Ci = ( x ( t ) - ai) 2 + ( y ( t ) - bi) 2 and Q as in (4). The functional associated with this model becomes
wi((x -- ai) 2 + ( y ( x ) -- bl) 2) x/1 + (y'(x)) 2 dx.
J =
(9)
1
We note that the functional (9) is similar to the one considered in [4] for their model, i.e. wi((x - ai) 2 + ( y ( x ) - bi) 2) x/1 + (y'(x)) 2 dx.
J = i=1
Before we present our main result, we introduce the following notations and two lemmas. Let a = ~wiai,
b = ~wibl,
7x : 2 w,a2~ - ( 2 w ,
a,) 2,
Yy : E wi b2 - ( Z w ,
r ( x ) = ~ wi(x -- a,) 2 = (x -- a) 2 + 7x,
b,) 2,
G ( y ) = ~, wi(y - b,) 2 = (y - b) 2 + 7y.
i-1
i=l
C o o r d i n a t e (a, b) represents the center of gravity. Also, 7x and 7y represent the x-axis and y-axis dispersions of the facilities from the center of gravity, respectively. Since our p r o b l e m is in form (6), the integral equation (7) can be used. In order to use (7), we need the following two lemmas. L e m m a 2 states that if the center of gravity falls within the rectangular region defined by the origin and the destination of the m o v i n g vehicle, then the constant fl should be within a specific range. Lemma2.
I f O 1 for any fl such that - 7y < fl < 7~- Hence, there does not exist fl which satisfies Eq. (14) and thus expression (12) need not be further considered. By rewriting (11) as an explicit function form with c~ = g(0, fl)/f(O, fl), where fl is a solver of Eq. (13), the desired function y(x) can be easily obtained after some simple algebra. [2 We observe that fl that satisfies (13) may not exist if the conditions in Theorem 4 are not satisfied. This case corresponds to the nonexistence of a twice continuously differentiable function that minimizes the functional in (9). For instance, fl that satisfies (13) does not exist when h = 1, 0 = 10, m = 1, E~ = (4, 5), and yi0) = 0. In this example, 7x = 7y = 0 and from L e m m a 2, fl does not exist. Interestingly, Algorithm II(2) in [4] yields y(x) = - 0 . 0 3 3 7 x 2 + 1.337x as an optimal path to this problem with objective function value 51.1 and Algorithm IlI(2) converges to y(x) = 5/4x for 0 ~< x ~< 4 and y(x) = 5/6x + 5/3 for 4 ~< x ~< 10. which is continuous but nondifferentiable at x = 4, with objective function value 51.0. If there exists an optimal path to the problem, it is relatively easy to find the constant fl from (13). In particular, one needs to find zeros of a function of ft. For this, any line search technique such as the bisection method or the N e w t o n - R a p s o n method can be used. Hence, in this case the problem of determining an optimal path of the moving vehicle is as easy as that of finding a root of a function. The following lemma provides a convenient way of obtaining the optimal objective value. Lemma
5 (Akhiezer[1]). The solution (7), which is the general equation of a geodesic, yields
J=
4. C o m p u t a t i o n a l
F~F~))- fi dx +
~
+ fl dy.
results
The main purpose of this section is to show how the optimal path of our model with the square root cost function in (10), SRCF, approximates that of the model of Sherali and Kim with the squared Euclidean distance. If there are a large number of existing facilities spread over a large region, then the fine grid interval for Algorithm III(2), suggested in [4], will result in an expensive computation time unless the initial solution for the algorithm is of good quality. Also, if a real-time approximate solution is sought, then a better solution than that of Algorithm II(2) m a y be desired.
S.-I. Kim, I.-C. Choi / Operations Research Letters 16 (1994) 231 239
238
Table 1 Objective function values for the squared Euclidean cost function w = (wl, w2, (0,1,0.1, w3, w4) 0.1,0.7)
(0.1,0.1, 0.7,0.1)
(0.1,0.7, 0.1,0.1)
(0.7,0.1, 0.1,0.1)
(0.25,0.25, 0.25,0.25)
(0.03,0.03, 0.03,0.91)
h = 1/3
AlglI AlgllI SRCF a
760.1 759.4 759.6
827.0 822.7 n/e b
514.5 513.6 514.0
588.9 588.6 588.8
684.6 684.2 684.6
779.6 774.1 774.5
h = 2/3
Alg 1I Alg III SRCF
962.5 944.7 947.9
865.5 865.4 865.4
634.4 634.1 634.1
859.8 859.4 859.6
842.6 841.8 842.3
999.4 953.7 956.1
h= 1
AlglI AlgllI SRCF
1553.5 1500.9 1508.2
1155.1 1147.7 1148.6
1073.1 1069.6 1070.1
1552.6 1550.5 1551.4
1345.9 1339.7 1342.2
1631.2 1523.0 1527.6
a SRCF: Square root cost function model. b The center of gravity fell outside the region.
~5
13.
/' /
11.
,I if "
0J
m/ ii
/" i
o
0
patl'),l~y ,'~lg II .................+ j
9
,./ i
.......................... i ..................
11
13
15
x coordinate •
Existing facilities
•
Center-of-Gravity
Fig. 1. Optimal vs. approximated paths h = 1; w = (0.03, 0.03, 0.03, 0.91).
Six problem sets, each containing three problems, were generated from the four data points used in I-4]. These poin ~,esented coordinates of the existing facilities and they were (4, 2), (8, 5), (11, 8) and (13, 2). Different destinations (O,hO) were used by setting 0 = 15 and h = 1/3, 2/3, and 1. The weights used in our computation ,¢ere taken from (0.1,0.1,0.1, 0.7), equal weights, and (0.03, 0.03, 0.03, 0.91).
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239
Our computational experiment is summarized in Table 1. As shown in the table, the optimal paths of our model with the square root cost function in (10) consistently provided better objective function values than those of Algorithm II(2) for all problems but one with w = (0.1, 0.1, 0.7, 0.1) and h = ½. In this case, the center of gravity fell outside the rectilinear region and thus a solution could not be obtained by using Theorem 4. Table 1 also clearly indicates a pattern in which the optimal path of our model with the square root cost function performed far better than Algorithm II(2) did, i.e. when facilities were clustered in a region and/or the weight was heavily biased toward one specific facility location. The computation times required by Algorithm II(2) and SRCF were negligibly small (in tenths of a second) and comparable to each other since they both essentially utilized a root finding algorithm. Their computation times were much smaller than those required by Algorithm III(2) (in teens of seconds), which employed a perturbation approach with a dynamic programming technique for solving several shortest path problems. Three different paths obtained by using the three approaches for the problem with w = (0.03, 0.03,0.03,0.91) and h = 1 are shown in Fig. 1.
Acknowledgement The authors would like to thank the anonymous referee for his helpful comments.
References [1] N.I. Akhiezer, The Calculus of Variations, Harwood Academic Publishers, Chur, Switzerland, 1988. [2] S. Kim, H.D. Sherali and H. Park, "Minisum and minimax paths of a moving facility on a network", Comput. Oper. Res. 19(2), 123 131 0992). [3] I.P. Petrov, Variational Methods in Optimum Control Theory, Translated by M.D. Friedman, Academic Press, New York, 1968. [4] H.D. Sherali and S. Kim, "Variational problems for determining optimal paths of a moving facility", Transportation Sci. 26(4), 330 345 (1992). [5] R. Weinstock, Calculus of Variations with Applications to Physics and Engineering, McGraw-Hill, New York, 1952.
Operations Research Letters 16 (1994)231-239
A simple variational problem for a moving vehicle Seong-In Kim a'*, In-Chan Choi b aDepartment of lndustrial Engineering, Korea University, Seoul, 136 701 South Korea bDepartment of lndustrial Engineering, The Wichita State University, Wichita, KS 67208, USA Received 15 September 1993; revised 1 June 1994)
Abstract
The problem of determining an optimal path of a moving vehicle, considered in Sherali and Kim (1992), is studied in this paper. We generalize their model by including a general cost structure in the problem and show that a specific, non-trivial cost function in this setting yields an easy solution. This follows from the observation that the variational problem that arises in the model with this particular cost function is the problem of geodesics on the Liouville surface. Through a comparative computational study, we also show that the optimal path for the model with this cost function better approximates the optimal path for the original model in Sherali and Kim than their second-order approximation approach does.
Key words: Variational problem; Geodesics; Liouville surface; Moving vehicle
1. Introduction
In this paper, we consider the problem of determining an optimal path of a moving vehicle, which travels a given region that contains a set of existing facilities. The optimality is with respect to minimizing an appropriately defined total cost that is incurred during the journey of the vehicle. This problem appears in various applications, such as in determining a path of a surveillance aircraft or a patrol car maintaining radio contacts with stations. For motivational thoughts on this problem and its counterpart on the network, readers are referred to [2, 4]. In [4], a model for this and other related problems was constructed based on the squared Euclidean (or rectilinear) distance between the vehicle and the existing facilities over the time period of the journey as a total cost. Three different approaches for solving the problem were provided along with their computational performances. Among the three approaches, Algorithm III(2), a dynamic programming based perturbation approach, was reported to provide the best observed performance in terms of the quality of the solution. This approach utilized the solution of Algorithm II(2), a second-order approximation approach, as an initial solution. The computation time of Algorithm III(2) was reported to be significantly larger, over 100
* Corresponding author. 0167-6377/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0 1 6 7 - 6 3 7 7 ( 9 4 ) 0 0 0 3 8 - 7
S.-1. Kim, 1.-C. Choi / Operations Research Letters 16 (1994) 231-239
232
times, than the other two approaches. The third approach presented in [4] was the numerical method of solving a second-order ordinary differential equation. Our goal in this paper is twofold. On the one hand, we would like to point out a special case when a model for a moving vehicle becomes particularly easy to deal with, i.e. the case when the variational problem arising from the model becomes easy to solve. To this end, we first extend the model of Sherali and Kim [4] to a larger class that includes a general cost structure, which includes that of [4]. Specifically, the time-varying cost function of the moving vehicle is represented by a nonnegative composite function of separable functions. We then show that in the model with a specific form of nontrivial cost function in this setting, an optimal path can be easily obtained; although, the model with general cost structure can only be handled by numerical methods(see [1, 5]). This follows from the observation that the variational problem associated with the specific model we consider is a problem of geodesics on the Liouville surface. On the other hand, we would like to see how the optimal path of our model with the special cost function approximates that of Sherali and Kim's model with the squared Euclidean distance. As shall be discussed in a later section, the optimal path of our model better approximates that of the Sherali and Kim's model than their Algorithm II(2) does. Thus, it can be used as a better intial solution for Algorithm III(2), or as a real-time solution if a quick and relatively accurate solution is desired. This paper is organized as follows. In Section 2, an extension of the Sherali and Kim's model is presented. Only the relevant points are addressed. For the details of the model of Sherali and Kim, readers are referred to [4]. In Section 3, we show that the special cost function within the general model framework given in Section 2 renders an easy solution procedure in obtaining the optimal solution. Finally, comparative computational results are presented in Section 4.
2. Model formulation We now consider the mathematical modeling of the problem mentioned in Section 1. The formulation is similar to the one given in [4], but our model allows a general cost structure. Suppose that there are m existing facilities located on Ei = (al, bi) for i = 1.... , m on the Cartesian coordinate in a two-dimensional space. Also, suppose that a vehicle moves along the trajectory z(t) = [x(t), y(t)] at a fixed velocity v, starting from the origin and arriving at a destination located at (0, hO). Let the unit incremental cost incurred by the movement of the vehicle on z(t) from t + d t be given as Q(C1,..., Cm), where Ci represents an individual facility i's contribution to the unit incremental cost. Furthermore, let us assume that C~ can be expressed as
(1)
CiEEi, z(t)] =-fi(x(t)) + gi(y(t)),
wheref~(. ) and g~(') are some nonnegative functions associated with facility locations i = 1..... m. Then, the total cost that is incurred by this vehicle following the trajectory z(t) is given as
f Q(C~ ..... C,.) dt,
(2)
where ~ is an arbitrary time frame. Within this framework, the problem for a moving vehicle is then to find an optimal trajectory, z*(t), that minimizes the total cost in (2). Our model formulation allows a general cost structure for the problem, e.g. fffx(t))= e ~x(t~-a`~2, gi(y(t)) = e lytt)-bd, and Q = l o g ~ ( f ( x ( t ) ) + gi(y(t))). If we let j~(x(t)) = Ix(t) - all p and gi(y(t)) = ]y(t) - bil p, i.e.
CiEEi,
z(t)]
= (Ix(t)
--
a,I p
+ ly(t)
--
bi[P),
(3)
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233
and the aggregated cost function Q be the weighted sum of C~, Q(C1, ..., Cm) = Y~i ~_, w~C~,where weight wi is a positive dimensionless quantity, then the resulting problem is the one that is considered in [4]. In Section 3, we consider a specific form of Q,
(4)
Q(c, ..... c,~) = ~=1 wi ci,
where wi >~ 0 and y~wi = 1. A clean form of an integral equation can be obtained as a solution to the variational problem that arises in solving the problem with this cost function. Using this observation, we show that our model with Ci in (3) for p = 2 and Q defined in (4) renders a rather simple solution for the problem of determining an optimal path of a moving vehicle.
3. A simple variational problem
The problem we now consider has the following form, from (1), (2), and (4),
min f~ ff~_ (fi(x(t)) + For simplicity, let y - - y ( x ) . Also, we assume that y(x) is twice continuously differentiable. Noting that v = ds/dt and that ds = xf(dx) 2 + (dy) 2 = x/1 + (y'): dx, the problem of finding y(x) which minimizes (2) can be rewritten as rain Z(x)eZ
fx N/ ~ (f~(x)+ 9i(y(x)))x/1 + (y'(x)) i dx, i= 1
(P)
where Z(x) = [x,y(x)], Z is the feasible set of paths from the origin to the destination (O, hO), and X is a permissible variation in x. Therefore, we want to find a function y(x) that minimizes a functional
J= f f
~ i ~~',1 ( f / ( x ) ' ~ - g g ( y ( x ) ) ) x / l + ( y ' ( x ) )
2 dx,
(5)
with boundary conditions y(0) = 0 and y(O) = hO. By denoting
K(x; y; y') = x~(x) + G(y) x/-( + (y,)2, F(x) = ~ fi(x)
and
G(y) =
i=1
i=1
Oi(y(x)),
we can rewrite expression (5) as ~0
J = J o K (x; y; y') dx =
,Iv(x)
+
+ (y,)2 dx.
(6)
234
S.-I. Kim, I.-C. Choi / Operations Research Letters 16 (1994) 231-239
The above problem is the problem of geodesics on the Liouville surface and it renders the following lemma, which states that the Euler differential equation that is a necessary condition for y(x) to minimize the functional J can be simplified. Lemma 1 (Akhiezer [1]). A function y(x) that solves problem (P) must satisfy the following necessary condition:
+/~
-/~
-
~,
(7)
where ~ and fl are some constants. Proof. The necessary condition for a function y(x) to minimize the functional J of (5) is represented by the Euler equation BK By
d BK dx By'
-
0,
(8)
which is in general of second order. Also, the Legendre condition has to be satisfied, i.e. 02K/By ' By' > 0 for a minimization problem [3]. Checking the Legendre condition first, we see that the condition is satisfied as
32K
x/(F(x) + G(y))
By' By'
(~/1 + (y,)2)3
>0.
From Eq. (8), G,(y)x/1 + (y,)2
d y'~/F(x) + G(y)
2x/F(x ) + G(y)
dx
x/1 + (y,)2
~-0.
Rewriting this equation, we obtain
+ ?(yi d x/l+(y,)2
dxL
+_,£(y)] = y'G'(y). x/l+(y)2
l
Integrating the equation, we find the first integral
+ (y)l Thus,
dy= + ~ + ~ dx -- F ~ -- ,B" Introducing a second arbitrary constant c¢ and integrating this equation we obtain the desired result.
[]
S.-I. Kim, I.-C. Choi / Operations Research Letters 16 (1994) 231 239
235
In the remainder of the section, we confine our discussion to the model with Ci's as in (3) with p = 2, i.e. Ci = ( x ( t ) - ai) 2 + ( y ( t ) - bi) 2 and Q as in (4). The functional associated with this model becomes
wi((x -- ai) 2 + ( y ( x ) -- bl) 2) x/1 + (y'(x)) 2 dx.
J =
(9)
1
We note that the functional (9) is similar to the one considered in [4] for their model, i.e. wi((x - ai) 2 + ( y ( x ) - bi) 2) x/1 + (y'(x)) 2 dx.
J = i=1
Before we present our main result, we introduce the following notations and two lemmas. Let a = ~wiai,
b = ~wibl,
7x : 2 w,a2~ - ( 2 w ,
a,) 2,
Yy : E wi b2 - ( Z w ,
r ( x ) = ~ wi(x -- a,) 2 = (x -- a) 2 + 7x,
b,) 2,
G ( y ) = ~, wi(y - b,) 2 = (y - b) 2 + 7y.
i-1
i=l
C o o r d i n a t e (a, b) represents the center of gravity. Also, 7x and 7y represent the x-axis and y-axis dispersions of the facilities from the center of gravity, respectively. Since our p r o b l e m is in form (6), the integral equation (7) can be used. In order to use (7), we need the following two lemmas. L e m m a 2 states that if the center of gravity falls within the rectangular region defined by the origin and the destination of the m o v i n g vehicle, then the constant fl should be within a specific range. Lemma2.
I f O 1 for any fl such that - 7y < fl < 7~- Hence, there does not exist fl which satisfies Eq. (14) and thus expression (12) need not be further considered. By rewriting (11) as an explicit function form with c~ = g(0, fl)/f(O, fl), where fl is a solver of Eq. (13), the desired function y(x) can be easily obtained after some simple algebra. [2 We observe that fl that satisfies (13) may not exist if the conditions in Theorem 4 are not satisfied. This case corresponds to the nonexistence of a twice continuously differentiable function that minimizes the functional in (9). For instance, fl that satisfies (13) does not exist when h = 1, 0 = 10, m = 1, E~ = (4, 5), and yi0) = 0. In this example, 7x = 7y = 0 and from L e m m a 2, fl does not exist. Interestingly, Algorithm II(2) in [4] yields y(x) = - 0 . 0 3 3 7 x 2 + 1.337x as an optimal path to this problem with objective function value 51.1 and Algorithm IlI(2) converges to y(x) = 5/4x for 0 ~< x ~< 4 and y(x) = 5/6x + 5/3 for 4 ~< x ~< 10. which is continuous but nondifferentiable at x = 4, with objective function value 51.0. If there exists an optimal path to the problem, it is relatively easy to find the constant fl from (13). In particular, one needs to find zeros of a function of ft. For this, any line search technique such as the bisection method or the N e w t o n - R a p s o n method can be used. Hence, in this case the problem of determining an optimal path of the moving vehicle is as easy as that of finding a root of a function. The following lemma provides a convenient way of obtaining the optimal objective value. Lemma
5 (Akhiezer[1]). The solution (7), which is the general equation of a geodesic, yields
J=
4. C o m p u t a t i o n a l
F~F~))- fi dx +
~
+ fl dy.
results
The main purpose of this section is to show how the optimal path of our model with the square root cost function in (10), SRCF, approximates that of the model of Sherali and Kim with the squared Euclidean distance. If there are a large number of existing facilities spread over a large region, then the fine grid interval for Algorithm III(2), suggested in [4], will result in an expensive computation time unless the initial solution for the algorithm is of good quality. Also, if a real-time approximate solution is sought, then a better solution than that of Algorithm II(2) m a y be desired.
S.-I. Kim, I.-C. Choi / Operations Research Letters 16 (1994) 231 239
238
Table 1 Objective function values for the squared Euclidean cost function w = (wl, w2, (0,1,0.1, w3, w4) 0.1,0.7)
(0.1,0.1, 0.7,0.1)
(0.1,0.7, 0.1,0.1)
(0.7,0.1, 0.1,0.1)
(0.25,0.25, 0.25,0.25)
(0.03,0.03, 0.03,0.91)
h = 1/3
AlglI AlgllI SRCF a
760.1 759.4 759.6
827.0 822.7 n/e b
514.5 513.6 514.0
588.9 588.6 588.8
684.6 684.2 684.6
779.6 774.1 774.5
h = 2/3
Alg 1I Alg III SRCF
962.5 944.7 947.9
865.5 865.4 865.4
634.4 634.1 634.1
859.8 859.4 859.6
842.6 841.8 842.3
999.4 953.7 956.1
h= 1
AlglI AlgllI SRCF
1553.5 1500.9 1508.2
1155.1 1147.7 1148.6
1073.1 1069.6 1070.1
1552.6 1550.5 1551.4
1345.9 1339.7 1342.2
1631.2 1523.0 1527.6
a SRCF: Square root cost function model. b The center of gravity fell outside the region.
~5
13.
/' /
11.
,I if "
0J
m/ ii
/" i
o
0
patl'),l~y ,'~lg II .................+ j
9
,./ i
.......................... i ..................
11
13
15
x coordinate •
Existing facilities
•
Center-of-Gravity
Fig. 1. Optimal vs. approximated paths h = 1; w = (0.03, 0.03, 0.03, 0.91).
Six problem sets, each containing three problems, were generated from the four data points used in I-4]. These poin ~,esented coordinates of the existing facilities and they were (4, 2), (8, 5), (11, 8) and (13, 2). Different destinations (O,hO) were used by setting 0 = 15 and h = 1/3, 2/3, and 1. The weights used in our computation ,¢ere taken from (0.1,0.1,0.1, 0.7), equal weights, and (0.03, 0.03, 0.03, 0.91).
S.-I. Kim, L-C. Choi / Operations Research Letters 16 (1994) 231-239
239
Our computational experiment is summarized in Table 1. As shown in the table, the optimal paths of our model with the square root cost function in (10) consistently provided better objective function values than those of Algorithm II(2) for all problems but one with w = (0.1, 0.1, 0.7, 0.1) and h = ½. In this case, the center of gravity fell outside the rectilinear region and thus a solution could not be obtained by using Theorem 4. Table 1 also clearly indicates a pattern in which the optimal path of our model with the square root cost function performed far better than Algorithm II(2) did, i.e. when facilities were clustered in a region and/or the weight was heavily biased toward one specific facility location. The computation times required by Algorithm II(2) and SRCF were negligibly small (in tenths of a second) and comparable to each other since they both essentially utilized a root finding algorithm. Their computation times were much smaller than those required by Algorithm III(2) (in teens of seconds), which employed a perturbation approach with a dynamic programming technique for solving several shortest path problems. Three different paths obtained by using the three approaches for the problem with w = (0.03, 0.03,0.03,0.91) and h = 1 are shown in Fig. 1.
Acknowledgement The authors would like to thank the anonymous referee for his helpful comments.
References [1] N.I. Akhiezer, The Calculus of Variations, Harwood Academic Publishers, Chur, Switzerland, 1988. [2] S. Kim, H.D. Sherali and H. Park, "Minisum and minimax paths of a moving facility on a network", Comput. Oper. Res. 19(2), 123 131 0992). [3] I.P. Petrov, Variational Methods in Optimum Control Theory, Translated by M.D. Friedman, Academic Press, New York, 1968. [4] H.D. Sherali and S. Kim, "Variational problems for determining optimal paths of a moving facility", Transportation Sci. 26(4), 330 345 (1992). [5] R. Weinstock, Calculus of Variations with Applications to Physics and Engineering, McGraw-Hill, New York, 1952.
