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Sequencing picking operations and travel time models for man-on-board storage and retrieval warehousing system
International Elsevier
Journal of Production
75
Economics, 29 (1993) 75-88
Sequencing picking operations and travel time models for man-on-board storage and retrieval warehousing system Hark Hwang and Jin Young Song oj’ Industrial Engineering, Korea Advanced Institute Taejon 305-701. South Korea
Department Yusong-Gu, (Received
10 June 1992; accepted
in revised form I September
ofScience and Technology, 373-l Gusung-Dong,
1992)
Abstract This paper deals with the order sequencing problem in a man-on-board storage and retrieval warehousing system which is suitable for storing items of small size and light weight. Considering the operating characteristics of the man-on-board system, a heuristic procedure is presented for the problem of sequencing a given set of retrieval requests. The validity of the procedure is investigated through computer simulation. The results show that the procedure performs better than those from previous studies. We also develop the expected travel time models based on the probabilitic analysis for single and dual commands assuming randomized storage assignment policy.
1. Introduction The unit-load automated storage and retrieval system (AS/RS) is a computer-controlled warehousing system, in which the pallets are stored and retrieved using automatic stacker cranes. The stacker crane travels simultaneously in the horizontal and vertical directions, which is called “Chebyshev travel”. Because a stacker crane is capable of handling one pallet, it operates on a single or dual command basis. In the single command, a single storage or retrieval request is performed, while in the dual command two requests, a storage request followed by a retrieval request, are performed. The man-on-board (MOB) system is a storage and retrieval (S/R) warehousing system which has been used predominatingly for storing and retrieving the items of small size and light weight. In the MOB system, an operator works on the specially designed truck, such as the order-picking truck [l] and the narrow-aisle order picker truck [2]. He usually carries out multi-command which consists of a number of storage or retrieval requests between successive returns to the input and output (I/O) point. Compared to conventional warehousing systems, the MOB system offers a number of benefits such as better space utilization, precise inventory control, shorter lead time, and efficient order picking. In Korea, many manufacturing firms, expecially aero, electric and electronic industries, are utilizing the MOB system. We observe that the travel method of the S/R truck in the MOB system differs from that of the unit-load AS/RS. The MOB system adopts Chebyshev travel for the movement of the S/R truck below a predetermined rack height, but it adopts “rectilinear travel” above that height to ensure the operator’s safety. In rectilinear travel, the Correspondence to: Prof. H. Hwang, Department of Industrial Engineering, Korea Technology, 373-l Gusung-Dong, Yusong-Gu, Taejon 305%701, South Korea.
0925-.5273/93/$06.00
0
1993 Elsevier Science Publishers
B.V. All rights reserved.
Advanced
Institute
of Science and
H. Hwang, J.- Y. Song/Sequencing
76
picking operations
rect~lineor
P2 .
.
.
and travel time models
region
..____.
Chebyshev
re~,ot
Fig. 1. Rack face in the MOB system.
S/R truck can travel in only one direction, that is, horizontal or vertical. We will refer to the above travel method of the S/R truck as “combined travel.” Figure 1 shows the rack face where the I/O point is located at the lower left-hand corner. The height, A, is the predetermined height which divides the rack face into two regions, “Chebyshev region” and “rectilinear region”. The Chebyshev region covers the area below the dotted line Y = A and the rectilinear region covers the area above the line. Let T(Pi, Pj) be the travel time of the S/R truck between points Pi = (Xi, yi) and Pj = (Xj, yj). In the MOB system as shown in Fig. 1, T(Pi, Pj) depends on the locations of the two points and can be expressed as follows: IYl
T(P1,
P2) = Ix1 - x2 I/% +
7V3,
p4)
T(P2,
P3) = max( I x2 - x3 lh
=
mWx3
-x41/~
-
Y2 I/L
ly3 -
Y~I/.L),
(A - Y~)/s,)
+ (y2 -
4/s,,
if y, > A
and
y, > A,
if y, < A
and
y, < A,
if y, > A
and
y, d A,
(1)
where sh and s, are the speeds in horizontal and vertical directions, respectively. Note that if A equals the rack height, T(Pi, Pj) becomes the same as the travel time model usually adopted in the studies of the unit-load AS/RS. There are many factors which have direct effects on the performance of the MOB system. Among them, the order picking (retrieval) is considered as one of the most costly activities. Thus, it becomes the highest priority warehousing activity for productivity improvement. To reduce the order picking time for multi-command, retrieval requests have to be cleverly sequenced. The problem of optimally sequencing a set of the given retrieval requests is certainly the well known traveling salesman problem (TSP). According to Garey and Johnson [3], the TSP under the rectilinear metric and the Chebyshev metric is NP-complete. Considering that order picking sequences have to be determined frequently and mostly in real time using small computers, the development of heuristic solutions of the TSP problem in the MOB system is in order. Thus, this paper presents a heuristic procedure for sequencing picking operations for multicommand in the MOB system. The validity of the procedure is illustrated through comparison with other existing heuristics. Also, the expected travel time models are developed for single and dual commands under randomized storage assignment policy.
2. Heuristic algorithm for sequencing picking operations 2.1. Review of TSP literature For the general TSP, a significant number of heuristic procedures have been developed for solving the TSP. Reviews of the TSP literature can be found in refs. [4, 53. Golden et al. [6]
H. Hwang,
J.- Y. Song/Sequencing
picking operations
and travel time models
77
compared heuristic solution methods for the TSP with respect to their solution qualities and computational efforts. The nearest neighbour heuristic (NNH, see ref. [6]) is a very simple and easy procedure, so it is widely used in industries. The NNH starts with the I/O point as the beginning of a path and then finds the picking point closest to the last point added to the path, and this point is added to the path. The adding procedure is repeated until all points are contained in the path. The last point added is connected to the I/O point. Goetschalckx and Ratliff [7] presented the Chebyshev-convex hull heuristic (CHEB). They used an insertion scheme based on the property of the Chebyshev norm. Therefore, the CHEB procedure can be executed only for Chebyshev metric, and cannot be applied to the MOB system. Bozer et al. [S] developed the l/2 band insertion heuristic (BAND) which proceeds as follows: The center one-half portion of the rack is initially blocked out. In the lower one-fourth unblocked portion, the points are sorted in the increasing x-coordinate direction, while in the upper one-fourth unblocked portion they are sorted in the opposite direction to obtain the partial tour. Subsequently, the remaining points in the blocked region are inserted by using the minimum cost insertion scheme. The minimal insertion is executed following the method given in ref. [9].
2.2. Rectilinear
hull and convex
hull algorithms
Before developing the heuristic procedure, we consider the rectilinear and convex hull algorithms. For the Euclidean norm, Barachet [lo] stated that for the two-dimensional problem, the convex hull of a set of points is the optimal tour if all points of the set lie on the boundary of the convex hull. The length of the sum of segments of the original convex hull of all points is a lower bound on the length of the optimal TSP tour, since it is the optimal tour for the subset of points which lie on the convex hull. The method which uses the convex hull of the points as the initial subtour, has been shown to be a consistently good performer on a variety of test problems [6]. We will use the proceeding argument to construct an initial subtour in our heuristic procedure. The rectilinear hull algorithm is presented by Allison and Noga [9]. The rectilinear hull of the given set S is described in Fig. 2, and may be computed by using a recursive procedure. Consider the set S of n points and let xmax, ymax and xmin, ymin be the points in S with the maximum x, y coordinate and minimum x, y coordinate, respectively. Let H be a circular linked-list
ymax .
Fig. 2. Rectilinear
hull of the set S.
78
H. Hwang, J.-Y. Song/Sequencing
picking operations
and travel time models
containing rectilinear
in order the points (ymin, xmax, ymax, xmin) and set II = n - 4. Then the path of the hull from ymin to xmax may be found using the following procedure PATH:
Procedure BEGIN Step Step Step BEGIN Step
PATH
(S, n, ymin, xmax);
1. Let S1 contain all of the points kc& 2. Let nl = ISi 1; 3. IF izi > 0 THEN 4. Find the point pmaxESi (x pmax-
Xymin)
+
(Yxrnax
such that xk > xymin and yk < y,,,,;
that maximizes -
the function
Ypmax);
Step 5. Insert pmax into the rectilinear hull between S and Si , and decrease n and IZ~by one; Step 6. PATH (S,. n,, ymin, pmax); Step 7. PATH (S,, n,, pmax, xmax) END; END
ymin and xmax, and delete pmax from
It can be observed that the points falling inside the rectangle formed by two extreme points are examined for inclusion into the rectilinear hull. The procedure for computing the other paths is essentially identical except for a simple modifications of the Steps 1 and 4, and corresponding changes in the parameters in Steps 5, 6, and 7. The convex hull algorithm is presented by Akl and Toussaint [ll]. According to the algorithm, once the extreme points are identified, one adds these points to the convex hull and discards all points falling inside the polygon they form. Then one can break the remaining set of points into four regions as shown in Fig. 3. All that remains are to find a convex path from one extreme point to the other in the same region. The remaining points are sorted on their x-coordinate: in ascending order for regions 1 and 2, and descending order for regions 3 and 4. Then the convex path from one extreme point P, to the other extreme point Pb for each region can be found using the following procedure RULE:
vI
region 4 VITICIX , I ;
. .
:
i
/’
‘\\.
,’ ,i
.
c---
\
.
,.I’ /‘_
: I
1 \’
I’
:
s .
/ ’
\. ~pxnlcix ,’ : :
.
region 2
xmg. ..+?+L
region 3
‘\\
-_
region 1 x
Fig. 3. Convex
hull of the set S.
H. Hwang, J.- Y. Song/Sequencing
picking operations
and travel time models
79
Procedure RULE (region i, P,, Pb); Step 1. Starting with one extreme point P,, do (a) and (b) below for every three consecutive points k, k + 1, k + 2 in the region i until the other extreme point Pb is reached. (a) Compute CP = (yk+ 1 - yk) 6 k+2-~k+1)+(~k-~k+l)(~k+2-~k+1)~ (b) If CP 2 0, then move one point forward; else delete point (k + 1) and move one point backward. Step 2. If Step 1 is completed without any deletion, then stop; else repeat Step 1. Note that in almost all cases the rectilinear hull algorithm. 2.3. Combined
hull algorithm
contains
more points than the convex
hull heuristic procedure
The MOB system in this study consists of two rectangular racks, one on each side of a narrow aisle. In the aisle an S/R truck moves from a picking point to the next to carry out a set of the given retrieval requests. Each picking sequence starts and ends at the I/O point located at the lower left-hand corner of the storage rack. We are interested in finding a sequence which minimizes the total picking time. Assuming that the time to handle the items for retrieving operations is constant, the objective is equivalent to minimizing the total travel time of the S/R truck. For the sequencing problem in the MOB system, we propose combined hull heuristic procedure (CHH) which is stated in algorithmic form utilizing the following notations: = the set of retrieval points to be visited by the S/R truck, s = the retrieval point i with coordinates (xi, yi), i = 1, 2, . . . , n, pi = the set of points in the current partial tour, ST xmax = the maximum x-coordinate among xis, = the maximum y-coordinate among yis, ymax = the I/O point with coordinates (0, 0), PO = the rightmost retrieval point on the line Y = 0; if there exists no point other than PO on P, the line Y = 0, set Pa = PO, = the lowest retrieval point on the line X = xmax; if the x-coordinate of the point Pa is pb xmax, set Pb = P,, P, = the highest retrieval point on the line X = xmax; if there exists only one point on the line X = xmax, set P, = Pb, = the rightmost retrieval point on the line Y = ymax; if the y-coordinate of the point P, is pd ymax, set Pd = e, = the leftmost retrieval point on the line Y = ymax; if there exists only one point on the P, line Y = ymax, set P, = Pd, = the highest retrieval point on the line X = 0; if the x-coordinate of the point P, is 0, set Pf Pf = P,; if there exists no point other than PO on the line X = 0, set Pf = PO, = the x-coordinate of the point pj, +, = the y-coordinate of the point pj, yp, region 1 = the triangle formed by the points, P,, Pb and (xmax, 0), if A # 0, region 2 = the rectangle formed by the points, P,, Pd, (xmax, ymax) and (xp,, yp,) if ypc > A, the triangle formed by the points, P,, Pd and (xmax, ymax) if ypc < A, region 3 = the rectangle formed by the points, P,, Pf, (0, ymax) and (xp,, yp,) if yp, 2 A, the triangle formed by the points, P,, Pf and (0, ymax) if yp, < A. The heuristic procedure consists of the following three steps: Step 1. A partial tour is found. (1) Set ST = $.
80
H. Hwang, J.-Y. SongfSequencing
picking operations
and travel time models
(2) Find xmax and ymax. (3) Construct a rectangle with four lines, Y = 0, X = xmax, Y = ymax and X = 0. (4) Starting PO,the points in the rectangle are sorted counterclockwise to obtain a partial tour. (5) SetS=S-S,andn=ISJ. Step 2. The partial tour is updated using the rectilinear and convex hull algorithms. (6) If A is 0, then do (a) and stop. (7) If the region 1 exists, then sort the points in the region on their x-coordinate in ascending order and do (b). (8) If the region 2 is rectangle, then do (c). If the region 2 is triangle, then sort the points in the region on their x-coordinate in descending order and do (b). (9) Repeat (8) for the region 3. (a) BEGIN PATH (S, n, P,, pb); IF n > 0 and xpd < xmax and ypc < ymax THEN PATH (S, n, p,, Pd); IF y1> 0 and xp, > 0 and yp, < ymax THEN PATH (S, n, P,,Pf) END; (b) BEGIN IF (it is region 1) THEN RULE (region 1, P,,Pb) ELSE IF (it is region 2) THEN RULE (region 2, P,, Pd) ELSE RULE (region 3, P,,P,);
s = s - ST; n=ISI END; (c) BEGIN IF (it is region 2) THEN PATH (S, n, P,, Pd) ELSE PATH (S, yt, P,,Pf) END; Step 3. All the remaining points are inserted into the partial tour with the following insertion scheme. (10) If there exists no remaining point in S, then stop. Pj)E ST such that (11) For each point Pk not contained in ST, find two consecutive points (Pi, T(Pi,Pk)+ T(P,,Pj)- T(Pi,Pj)is minimal. (12) Among all the (Pi, Pj,Pk)found in (lo), determine the (Pi*, PI, P:) such that [ T(PT, Pk*)+ T(P:, Pj*)]/T(P*,Pj*) is minimal. (13) Insert Pz into ST between P* and Pj*. (14) Repeat (11) to (13) until all the points are inserted into ST. Figure 4 shows the flow chart of the CHH procedure. To illustrate the combined hull heuristic, an example problem with nine retrieval points is solved. The height of the rack is 30 ft and the length of the rack 60 ft with A = 16 ft. The horizontal speed of the S/R truck is 2 ft/s and the vertical speed 1 ft/s. The set S is given by S = {PI, Pz,P3,P4,Pg,
P6,P,,Pg,P,) = ((20, 5),(39, 26),(23, 2% (3,15),(36,0),(50, 1% (41,20), (50,18), (17, 26)). From these data, the points Pg,P6,Pg,P2,Ps and PO correspond to Pa,Pb,PC,Pd,P,and Pf,respectively. Also, xmax = 50, ymax = 26 and PO = (0,O). Applying step 1, the sequence PO-P5-P6-P8-P2-P9-PO can be identified as the partial tour as shown in Fig. 5. The remaining points are S = {PI, P3,Pa, P,>.In Step 2, we find that the regions 1 and 3 are triangle and the region 2 is rectangle. Utilizing the procedure PATH (S, 4, P8,P2), P, in the region 2 is inserted between P8 and P2.Similarly, P4 is inserted between Ps and PO by the procedure RULE (region 3, Ps,PO).In Step 3, the remaining
H. Hwang, J.-Y. Song/Sequencing picking operations and travel time models
81
/ i
Read (x,y)
(3)
Construct
coordinates
a rectangle
(a)
Use PATH for
(b)
three
Use RULE for
regions
region
1
I
No
(c)
/(lo)
Use PATH for
Any remaining points exist
Compute total
Fig. 4. Flowchart
region
travel
of the CHH
3
No
(b)
(ll)-(14)
time
procedure.
Use RULE for
Insert with the
region
3
all the remaining insertion scheme
I
I
points
82
H. Hwang.
J.- Y. Song/Sequencing reg,on
picking
3
. P3 n
(A=)16
.
2
P2(=Pa)
P9(=Pe)
(ymox=)26
and trace1 time models
reglo” I
I
30
operations
P7
%=P!
. P4 Po(=Pb)
l----z?
.PI
0 P0(=P1)
Fig. 5. Combined
hull heuristic.
points Pi and P3 are added to the current tour. PI is added between PO and P5 and then P3 is generated by the above steps is inserted between Pz and Pg. The final sequence PO-PI-Ps-P6-Ps-P,-Pzp,p,P,-P9-P~-P0 with the total travel time of the S/R truck being 100.5 seconds. 2.4. Evaluation of the CHH procedure This section intends to compare the performance of the CHH procedure with the existing sequencing algorithms, the nearest neighbour heuristic (NNH) and the l/2 band insertion heuristic (BAND) through computer simulation. Three factors are considered which may affect the performance of the algorithms. They are the number N of the picking points in an order, the shape factor h (refer to section 3.1 for the definition) of the rack, and the predetermined height A for the safety of the operator. The following values for each factor are examined: N = 5, 10, 15, 20, 25; b = 0.50, 0.75, 1.00; A = 0, b/4, 2b/4, 3b/4, b. For each combination of the factors, thirty sets of the data are generated, which results in 5 x 3 x 5 x 30 = 2250 replications for each algorithm. We consider a discrete rack with 2500 openings, each of which is represented by a unit square. For N given, the locations of the picking points are obtained utilizing a random number generating function. All computations are executed on personal computer 286/AT and the computation time excludes the time for input and output operations. The program is compiled in Turbo Pascal, version 5.0. Some of the simulation results is provided in Tables 1 and 2. For each combination of A and b with N = 20, Table 1 shows the average travel time based on the results of 30 replications. The overall average travel time is listed in Table 2 for each level of the factors. For instance, the values for each level of b are determined by averaging 5 (levels for N) x 5 (levels for A) x 30 (replications) numbers of the travel time data. The results indicate that the CHH procedure performs better than the other two. The NNH algorithm tends to generate poor solutions though it is simple and easy to use. The pairwise t-test is conducted to analyze the data statistically. With the significance level of c( = 0.05%, the test shows that the CHH procedure performs better than the BAND algorithm. It can be observed that varying the levels of b with A fixed does not have substantial effects while N and A does. All the three algorithms in this study can be classified as construction algorithm. One may attempt to improve it by changing the sequence obtained by the construction algorithm. Golden et al. [6] reported that the composite procedure, a construction procedure followed by an improvement procedure, has been quite effective in obtaining a good solution. A well known improvement procedure due to Lin [12] is based on systematically replacing two or three edges in the current tour with alternative edges. If the resulting tour is better, then the change is made and the procedure is restarted until no improvement is made. The all 3-way interchanges are considered
H. Hwang,
J.- Y. Song/Sequencing
picking
operations
and travel
83
time models
Table 1 The average travel time for the case of N = 20
WA
NNH
BAND
CHH
0.5jo.o 0.5/O. 125 OS/O.25 OS/O.375 OS/O.5
308.80 291.80 253.57 244.33 223.63
269.00 249.50 222.60 210.33 196.77
264.33 246.43 219.60 209.43 194.20
0.75/0.0 0.75/0.1875 0.7yo.375 0.7510.5625 0.7510.75
322.33 273.77 255.53 244.67 221.43
274.87 236.90 221.17 212.33 195.70
267.20 234.73 218.50 208.63 191.57
1.o/o.o 1.0/0.25 l.OlO.5 I .0/0.75 l.O/l.O
313.13 270.53 251.07 237.50 220.67
280.07 246.77 231.70 213.57 196.93
271.13 236.80 224.37 208.83 189.07
Average (N = 20)
262.78
230.55
225.65
a Averaged
over the 450 cases.
Table 2 The overall average travel time for each level of the factors Factors
NNH
BAND
CHH
N= N= N= N = N =
156.53 198.82 233.87 262.78 286.61
149.75 181.93 207.78 230.55 250.87
148.66 179.58 204.38 225.65 246.35
b = 0.50 h = 0.75 h = 1.00
232.03 226.95 218.07
206.58 202.40 198.15
205.17 199.16 193.68
A=0
A=$b A=$b A=$b A=b
269.32 242.73 223.98 208.29 194.28
242.30 217.42 200.42 186.46 174.29
236.88 213.61 197.45 184.93 171.76
Total average”
227.72
204.18
200.92
5 IO 15 20 25
aAveraged over
all the 2250 cases.
only after the all 2-way interchange procedure fails to yield an improvement. Table 3 shows a summary of the overall performance of each heuristic when the improvement schemes are adopted. Each value in the Table corresponds to the average of the results from the 2250 replications. It can be observed that even after the 2 and 3-way improvement routines, the CHH procedure still tends to yield the best solution among the three. Regarding to the computation time, the CHH procedure takes 0.20 s on the average and 0.64 s at the maximum for the case of N = 25.
84
H. Hwang, J.-Y. SongJSequencing
Table 3 Overall performance
of each heuristic
picking operations
and travel time models
Heuristic
Cases
With no improvement With 2-way improvement With 2- and 3-way improvements
NNH
BAND
CHH
221.12 (113.3%) 217.14 (109.0%) 216.21 (108.8%)
204.18 (101.6%) 201.14 (100.9%) 199.20 (100.2%)
200.92 (100.0%) 199.28 (100.0%) 198.77 (100.0%)
3. Probabilitic analysis for the travel time models This section deals with the travel time models of the S/R truck. In developing the expected travel times for both single and dual command cycles, the following assumptions are made: (1) The rack is a rectangular pick face, where the I/O point is located at the lower left-hand corner. (2) The length and height of the rack as well as the S/R truck speeds in the horizontal and vertical directions are known. The S/R truck adopts the combined travel as it moves from a pick point to the next. Randomized storage assignment policy is used. (5) Picking and deposit times associated with loading and unloading are constant and can be ignored. All storage locations are the same size. i;; Only the long-run average behavior of the system is considered.
‘,:I
3.1. Rack normalization We consider the continuous rectangular rack face with known dimensions. Following the work of Bozer and White [13], the rack face will be “normalized” by dividing the horizontal travel time th and the vertical travel time t, by T, where s,, = = “; = H = A = th = t, = t, =
T = b = a =
horizontal travel speed of the S/R truck, vertical travel speed of the S/R truck, rack length, rack height, predetermined height for the operator’s safety, 0 d A < H, L/s,, = time to reach the end of the rack, H/s, = time to reach the top of the rack, A/s, = time to reach the predetermined height, 0 Q t, d t,, Max (t,,, t,) = denormalizing factor, 0 < b < 1, Min (th, t,)/T, t,/T.
The factor, b, has been referred to as the “shape factor” for the rack. It will be assumed without loss of generality that t, d th. Thus, the rack has dimensions 1 x b and 0 < a d b. We will refer to the factor, a, as the “safety factor”.
H. Hwang,
3.2.
Expected
J.-Y.
Song/Sequencing
picking operations
85
and travel time models
single command travel time
We will consider first the single command travel time. Let the storage (or retrieval) point be represented by (x, y) in time, where 0 6 x 6 1 and 0 < y < b. From (1) the travel time T ( PO, P) from I’,, = (0, 0) to P = (x, y) will be
T(Po, P) =
max(x,
Y),
if O
max(x,
a)+y-a,
if a
Let F(z) denote the probability that the travel time from PO to P is less than or equal to z. Assuming the x and y coordinates are independently generated, F(z) can be specified under each of the following three conditions: condition 1: 0 < y < a, condition2:a
y) < 21 = PrCx d
Furthermore, under distributed. Thus, Pr[x
< z] = z,
Pr[y
< z] =
randomized
zlPrCy d 21.
storage,
the coordinate
locations
are assumed
to be uniformly
O
zla,
a
i 1,
Hence,
F,(z) = Therefore
fi(z) =
z2/a,
O
z,
a
the probability
fz’,, i,
density function, fi (z), will be
Odzda, a
Letting E1(SC) denote following is obtained:
the expected
=
s 0
for single command,
the
1
a
E1(SC)
travel time under the first condition
2z2/a dz +
s a
z dz = $a’ + $.
(2)
86
H. Hwang, J.-Y. Song/Sequencing
Under
the second condition,
Fz(z) = Pr[max(x,
we obtain
E,(SC)
i(u + b).
=
=
< z],
adzdb.
(3)
Lastly, under the third condition, F3(z) = Pr[x
and travel time models
a) = a and
a) + y - a d z] = Pr[y
= (z - a)/(b - a),
Similarly,
max(x,
picking operations
+ y - a < z] = Pr[y
max(x,
a) = x and F3(z) can be obtained
as following:
< - x + z + a]
(z - 42/C2(l- 4(b - a)l,
udz
(22 - a - b)/[2(1
b
- a)],
[ - z2 + 2(b - a + 1)z + u2 - b2 - 1]/[2(1
- u)(b - a)],
l
E3(SC) = +(b + 1). Combining found: E(X)
(4)
the results of (2), (3), and (4), the expected
= Er(SC)Pr[O + E3(SC)Pr[u
d y d a] + E,(SC)Pr[a < y < b] Pr[u
travel time E( SC) for single command
dual command
blPrC0 G x G al
< x < 11,
= ( - 2a3 + 3u2b + 3a2 + 3b2 - 6ab + 3b)/(6b). 3.3. Expected
can be
(5)
travel time
A dual command cycle involves two random locations: one representing the storage point, the other representing the retrieval point. Let the two points be represented by PI = (x1, yr) and Pz = (x2, y2) in time, where 0 < x1 < 1, 0 d x2 ,< 1 and 0 d y, < y, < b. Let G(z) denote the probability that the travel time between PI and P2 is less than or equal to z. Assuming the x and y coordinates are independently generated, G(z) can be specified under each of the following three conditions. condition 1: 0 d yl < y2 d a < b, condition 2: 0 d a d y, d y2 d b, condition 3: 0 < y, < a < y2 d b. Letting Gk(z) represent the distribution function corresponding to the condition k, the following results are obtained. Under the first condition, G,(z)=PrCmax(Ix2--~11,
(22 - z2)(2uz
IY~-Y~I)~zI,
- z2)/a2,
O
H. Hwang, J.- Y. Sony/Sequencing
Therefore, g,(z)
the probability
density function,
gl(z), will be
- 6(1/a’
+ l/a)z2 + Bzla,
O
4z3/a2
=
picking operations
a
2 - 22,
Letting E 1(TB) denote the expected travel between time under the first condition, obtained. Ei(TB)
= - $a3
+ ia’
the expected
E,(TB) = ECI x2
II,
-
Xl
Yl
the third condition.
J%CW = ECmW I x2
Therefore,
lY2 -
I +
travel between
time is
*(b - a + 1).
Lastly, we consider
=
the following is
+ 3,
Next, under the second condition,
=
87
and travel time models
-
x1
I, a - yl) +
For y, < y,, y2 -
al,
i(b - a) - &a” + 3a’ + 4. the expected
E(TB) = E1(TB)(a/b)2 = (4a5 -
travel time between + E2(TB)[(b
storage
and retrieval
point, E(TB), will be
- a)/b12 + E3(TB) [2a(b - a)]/b2,
15a4 + 20a3 + lob3 + lob2 - 5a4b + 20a3b - 30a2b)/(30b2).
Let E(DC) denote
the expected
travel time for a complete
dual command
cycle. Then,
E(DC) = 2 E(SC) + E(TB). Hence, E(DC) = (4a5 - 15a4 + 20a3 + 40b3 + 40b2 - 5a4b + 30a2b2 - 60ab2)/(30b2). Note that the results of the expected travel times for single and dual commands system are the same as those obtained by Bozer and White [13] for a = b.
in the MOB
4. Conclusions In the MOB system, the operator works on the platform of the S/R truck and handles a number of retrieval (or storage) requests between successive returns to the I/O point. The system efficiency is greatly affected by how these requests are sequenced. The sequencing problem can be formulated as the TSP problem whose optimal solution can be hardly found for large problems within the computational time limitation. Thus, this paper presents the combined hull heuristic procedure which consists of three steps. Through computer simulation, we show that the procedure generates solutions which are better than those by existing algorithms. The computational experiences
88
H. Hwang, .I.- Y. Song/Sequencing
picking operations
and travel time models
indicate that the procedure can be easily implemented on microcomputer for the real time use. Additionally, with the application of probabilitic analysis the expected travel time models of the S/R truck are developed for single and dual commands under randomized storage assignment policy. References Cl1 Apple, J.M., 1972. Material Handling Systems Design. Wiley, New York. Tompkins, J.A. and White, J.A., 1984. Facilities Planning, Wiley, New York. A Guide to the Theory of NP-completeness. F:; Carey, M.R. and Johnson, D.S., 1979. Computers and Intractability: W.H. Freeman and Co. c41 Bellmore, M. and Nemhauser, G.L., 1968. The traveling salesman problem: A survey. Oper. Res., 16: 538. CSI Parker, R.G. and Rardin, R.L., 1983. The traveling salesman problem: An update of research. Nav. Res. Logist. Q,, 30: 69. [61 Golden, B., Bodin. L., Doyle, T. and Stewart, W., 1980. Approximate traveling salesman algorithms. Oper. Res., 28: 694. M. and Ratliff, H.D., 1988. Sequencing picking operations in a man-aboard order picking system. c71 Goetschalckx, Material Flow, 4: 255. PI Bozer, Y.A., Schorn, E.C. and Sharp, G.P., 1990. Geometric approaches to solve the Chebyshev traveling salesman problem. HE Trans., 22: 238. c91 Allison, D.S. and Noga, M.T., 1984. The L, traveling salesman problem. Inf. Proc. Letters, 18: 195. Cl01 Barachet, L.L., 1957. Graphic solution of the traveling salesman problem. Oper. Res., 5: 841. Cl11 Akl, S.G. and Toussaint, G.T., 1978. A fast convex hull algorithm. Inf. Proc. Letters, 7: 219. Cl21 Lin, S., 1965. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44: 2245. systems. IIE Trans., 16: 329. Cl31 Bozer, Y.A. and White, J.A., 1984. Travel-time models for automated storage/retrieval
Journal of Production
75
Economics, 29 (1993) 75-88
Sequencing picking operations and travel time models for man-on-board storage and retrieval warehousing system Hark Hwang and Jin Young Song oj’ Industrial Engineering, Korea Advanced Institute Taejon 305-701. South Korea
Department Yusong-Gu, (Received
10 June 1992; accepted
in revised form I September
ofScience and Technology, 373-l Gusung-Dong,
1992)
Abstract This paper deals with the order sequencing problem in a man-on-board storage and retrieval warehousing system which is suitable for storing items of small size and light weight. Considering the operating characteristics of the man-on-board system, a heuristic procedure is presented for the problem of sequencing a given set of retrieval requests. The validity of the procedure is investigated through computer simulation. The results show that the procedure performs better than those from previous studies. We also develop the expected travel time models based on the probabilitic analysis for single and dual commands assuming randomized storage assignment policy.
1. Introduction The unit-load automated storage and retrieval system (AS/RS) is a computer-controlled warehousing system, in which the pallets are stored and retrieved using automatic stacker cranes. The stacker crane travels simultaneously in the horizontal and vertical directions, which is called “Chebyshev travel”. Because a stacker crane is capable of handling one pallet, it operates on a single or dual command basis. In the single command, a single storage or retrieval request is performed, while in the dual command two requests, a storage request followed by a retrieval request, are performed. The man-on-board (MOB) system is a storage and retrieval (S/R) warehousing system which has been used predominatingly for storing and retrieving the items of small size and light weight. In the MOB system, an operator works on the specially designed truck, such as the order-picking truck [l] and the narrow-aisle order picker truck [2]. He usually carries out multi-command which consists of a number of storage or retrieval requests between successive returns to the input and output (I/O) point. Compared to conventional warehousing systems, the MOB system offers a number of benefits such as better space utilization, precise inventory control, shorter lead time, and efficient order picking. In Korea, many manufacturing firms, expecially aero, electric and electronic industries, are utilizing the MOB system. We observe that the travel method of the S/R truck in the MOB system differs from that of the unit-load AS/RS. The MOB system adopts Chebyshev travel for the movement of the S/R truck below a predetermined rack height, but it adopts “rectilinear travel” above that height to ensure the operator’s safety. In rectilinear travel, the Correspondence to: Prof. H. Hwang, Department of Industrial Engineering, Korea Technology, 373-l Gusung-Dong, Yusong-Gu, Taejon 305%701, South Korea.
0925-.5273/93/$06.00
0
1993 Elsevier Science Publishers
B.V. All rights reserved.
Advanced
Institute
of Science and
H. Hwang, J.- Y. Song/Sequencing
76
picking operations
rect~lineor
P2 .
.
.
and travel time models
region
..____.
Chebyshev
re~,ot
Fig. 1. Rack face in the MOB system.
S/R truck can travel in only one direction, that is, horizontal or vertical. We will refer to the above travel method of the S/R truck as “combined travel.” Figure 1 shows the rack face where the I/O point is located at the lower left-hand corner. The height, A, is the predetermined height which divides the rack face into two regions, “Chebyshev region” and “rectilinear region”. The Chebyshev region covers the area below the dotted line Y = A and the rectilinear region covers the area above the line. Let T(Pi, Pj) be the travel time of the S/R truck between points Pi = (Xi, yi) and Pj = (Xj, yj). In the MOB system as shown in Fig. 1, T(Pi, Pj) depends on the locations of the two points and can be expressed as follows: IYl
T(P1,
P2) = Ix1 - x2 I/% +
7V3,
p4)
T(P2,
P3) = max( I x2 - x3 lh
=
mWx3
-x41/~
-
Y2 I/L
ly3 -
Y~I/.L),
(A - Y~)/s,)
+ (y2 -
4/s,,
if y, > A
and
y, > A,
if y, < A
and
y, < A,
if y, > A
and
y, d A,
(1)
where sh and s, are the speeds in horizontal and vertical directions, respectively. Note that if A equals the rack height, T(Pi, Pj) becomes the same as the travel time model usually adopted in the studies of the unit-load AS/RS. There are many factors which have direct effects on the performance of the MOB system. Among them, the order picking (retrieval) is considered as one of the most costly activities. Thus, it becomes the highest priority warehousing activity for productivity improvement. To reduce the order picking time for multi-command, retrieval requests have to be cleverly sequenced. The problem of optimally sequencing a set of the given retrieval requests is certainly the well known traveling salesman problem (TSP). According to Garey and Johnson [3], the TSP under the rectilinear metric and the Chebyshev metric is NP-complete. Considering that order picking sequences have to be determined frequently and mostly in real time using small computers, the development of heuristic solutions of the TSP problem in the MOB system is in order. Thus, this paper presents a heuristic procedure for sequencing picking operations for multicommand in the MOB system. The validity of the procedure is illustrated through comparison with other existing heuristics. Also, the expected travel time models are developed for single and dual commands under randomized storage assignment policy.
2. Heuristic algorithm for sequencing picking operations 2.1. Review of TSP literature For the general TSP, a significant number of heuristic procedures have been developed for solving the TSP. Reviews of the TSP literature can be found in refs. [4, 53. Golden et al. [6]
H. Hwang,
J.- Y. Song/Sequencing
picking operations
and travel time models
77
compared heuristic solution methods for the TSP with respect to their solution qualities and computational efforts. The nearest neighbour heuristic (NNH, see ref. [6]) is a very simple and easy procedure, so it is widely used in industries. The NNH starts with the I/O point as the beginning of a path and then finds the picking point closest to the last point added to the path, and this point is added to the path. The adding procedure is repeated until all points are contained in the path. The last point added is connected to the I/O point. Goetschalckx and Ratliff [7] presented the Chebyshev-convex hull heuristic (CHEB). They used an insertion scheme based on the property of the Chebyshev norm. Therefore, the CHEB procedure can be executed only for Chebyshev metric, and cannot be applied to the MOB system. Bozer et al. [S] developed the l/2 band insertion heuristic (BAND) which proceeds as follows: The center one-half portion of the rack is initially blocked out. In the lower one-fourth unblocked portion, the points are sorted in the increasing x-coordinate direction, while in the upper one-fourth unblocked portion they are sorted in the opposite direction to obtain the partial tour. Subsequently, the remaining points in the blocked region are inserted by using the minimum cost insertion scheme. The minimal insertion is executed following the method given in ref. [9].
2.2. Rectilinear
hull and convex
hull algorithms
Before developing the heuristic procedure, we consider the rectilinear and convex hull algorithms. For the Euclidean norm, Barachet [lo] stated that for the two-dimensional problem, the convex hull of a set of points is the optimal tour if all points of the set lie on the boundary of the convex hull. The length of the sum of segments of the original convex hull of all points is a lower bound on the length of the optimal TSP tour, since it is the optimal tour for the subset of points which lie on the convex hull. The method which uses the convex hull of the points as the initial subtour, has been shown to be a consistently good performer on a variety of test problems [6]. We will use the proceeding argument to construct an initial subtour in our heuristic procedure. The rectilinear hull algorithm is presented by Allison and Noga [9]. The rectilinear hull of the given set S is described in Fig. 2, and may be computed by using a recursive procedure. Consider the set S of n points and let xmax, ymax and xmin, ymin be the points in S with the maximum x, y coordinate and minimum x, y coordinate, respectively. Let H be a circular linked-list
ymax .
Fig. 2. Rectilinear
hull of the set S.
78
H. Hwang, J.-Y. Song/Sequencing
picking operations
and travel time models
containing rectilinear
in order the points (ymin, xmax, ymax, xmin) and set II = n - 4. Then the path of the hull from ymin to xmax may be found using the following procedure PATH:
Procedure BEGIN Step Step Step BEGIN Step
PATH
(S, n, ymin, xmax);
1. Let S1 contain all of the points kc& 2. Let nl = ISi 1; 3. IF izi > 0 THEN 4. Find the point pmaxESi (x pmax-
Xymin)
+
(Yxrnax
such that xk > xymin and yk < y,,,,;
that maximizes -
the function
Ypmax);
Step 5. Insert pmax into the rectilinear hull between S and Si , and decrease n and IZ~by one; Step 6. PATH (S,. n,, ymin, pmax); Step 7. PATH (S,, n,, pmax, xmax) END; END
ymin and xmax, and delete pmax from
It can be observed that the points falling inside the rectangle formed by two extreme points are examined for inclusion into the rectilinear hull. The procedure for computing the other paths is essentially identical except for a simple modifications of the Steps 1 and 4, and corresponding changes in the parameters in Steps 5, 6, and 7. The convex hull algorithm is presented by Akl and Toussaint [ll]. According to the algorithm, once the extreme points are identified, one adds these points to the convex hull and discards all points falling inside the polygon they form. Then one can break the remaining set of points into four regions as shown in Fig. 3. All that remains are to find a convex path from one extreme point to the other in the same region. The remaining points are sorted on their x-coordinate: in ascending order for regions 1 and 2, and descending order for regions 3 and 4. Then the convex path from one extreme point P, to the other extreme point Pb for each region can be found using the following procedure RULE:
vI
region 4 VITICIX , I ;
. .
:
i
/’
‘\\.
,’ ,i
.
c---
\
.
,.I’ /‘_
: I
1 \’
I’
:
s .
/ ’
\. ~pxnlcix ,’ : :
.
region 2
xmg. ..+?+L
region 3
‘\\
-_
region 1 x
Fig. 3. Convex
hull of the set S.
H. Hwang, J.- Y. Song/Sequencing
picking operations
and travel time models
79
Procedure RULE (region i, P,, Pb); Step 1. Starting with one extreme point P,, do (a) and (b) below for every three consecutive points k, k + 1, k + 2 in the region i until the other extreme point Pb is reached. (a) Compute CP = (yk+ 1 - yk) 6 k+2-~k+1)+(~k-~k+l)(~k+2-~k+1)~ (b) If CP 2 0, then move one point forward; else delete point (k + 1) and move one point backward. Step 2. If Step 1 is completed without any deletion, then stop; else repeat Step 1. Note that in almost all cases the rectilinear hull algorithm. 2.3. Combined
hull algorithm
contains
more points than the convex
hull heuristic procedure
The MOB system in this study consists of two rectangular racks, one on each side of a narrow aisle. In the aisle an S/R truck moves from a picking point to the next to carry out a set of the given retrieval requests. Each picking sequence starts and ends at the I/O point located at the lower left-hand corner of the storage rack. We are interested in finding a sequence which minimizes the total picking time. Assuming that the time to handle the items for retrieving operations is constant, the objective is equivalent to minimizing the total travel time of the S/R truck. For the sequencing problem in the MOB system, we propose combined hull heuristic procedure (CHH) which is stated in algorithmic form utilizing the following notations: = the set of retrieval points to be visited by the S/R truck, s = the retrieval point i with coordinates (xi, yi), i = 1, 2, . . . , n, pi = the set of points in the current partial tour, ST xmax = the maximum x-coordinate among xis, = the maximum y-coordinate among yis, ymax = the I/O point with coordinates (0, 0), PO = the rightmost retrieval point on the line Y = 0; if there exists no point other than PO on P, the line Y = 0, set Pa = PO, = the lowest retrieval point on the line X = xmax; if the x-coordinate of the point Pa is pb xmax, set Pb = P,, P, = the highest retrieval point on the line X = xmax; if there exists only one point on the line X = xmax, set P, = Pb, = the rightmost retrieval point on the line Y = ymax; if the y-coordinate of the point P, is pd ymax, set Pd = e, = the leftmost retrieval point on the line Y = ymax; if there exists only one point on the P, line Y = ymax, set P, = Pd, = the highest retrieval point on the line X = 0; if the x-coordinate of the point P, is 0, set Pf Pf = P,; if there exists no point other than PO on the line X = 0, set Pf = PO, = the x-coordinate of the point pj, +, = the y-coordinate of the point pj, yp, region 1 = the triangle formed by the points, P,, Pb and (xmax, 0), if A # 0, region 2 = the rectangle formed by the points, P,, Pd, (xmax, ymax) and (xp,, yp,) if ypc > A, the triangle formed by the points, P,, Pd and (xmax, ymax) if ypc < A, region 3 = the rectangle formed by the points, P,, Pf, (0, ymax) and (xp,, yp,) if yp, 2 A, the triangle formed by the points, P,, Pf and (0, ymax) if yp, < A. The heuristic procedure consists of the following three steps: Step 1. A partial tour is found. (1) Set ST = $.
80
H. Hwang, J.-Y. SongfSequencing
picking operations
and travel time models
(2) Find xmax and ymax. (3) Construct a rectangle with four lines, Y = 0, X = xmax, Y = ymax and X = 0. (4) Starting PO,the points in the rectangle are sorted counterclockwise to obtain a partial tour. (5) SetS=S-S,andn=ISJ. Step 2. The partial tour is updated using the rectilinear and convex hull algorithms. (6) If A is 0, then do (a) and stop. (7) If the region 1 exists, then sort the points in the region on their x-coordinate in ascending order and do (b). (8) If the region 2 is rectangle, then do (c). If the region 2 is triangle, then sort the points in the region on their x-coordinate in descending order and do (b). (9) Repeat (8) for the region 3. (a) BEGIN PATH (S, n, P,, pb); IF n > 0 and xpd < xmax and ypc < ymax THEN PATH (S, n, p,, Pd); IF y1> 0 and xp, > 0 and yp, < ymax THEN PATH (S, n, P,,Pf) END; (b) BEGIN IF (it is region 1) THEN RULE (region 1, P,,Pb) ELSE IF (it is region 2) THEN RULE (region 2, P,, Pd) ELSE RULE (region 3, P,,P,);
s = s - ST; n=ISI END; (c) BEGIN IF (it is region 2) THEN PATH (S, n, P,, Pd) ELSE PATH (S, yt, P,,Pf) END; Step 3. All the remaining points are inserted into the partial tour with the following insertion scheme. (10) If there exists no remaining point in S, then stop. Pj)E ST such that (11) For each point Pk not contained in ST, find two consecutive points (Pi, T(Pi,Pk)+ T(P,,Pj)- T(Pi,Pj)is minimal. (12) Among all the (Pi, Pj,Pk)found in (lo), determine the (Pi*, PI, P:) such that [ T(PT, Pk*)+ T(P:, Pj*)]/T(P*,Pj*) is minimal. (13) Insert Pz into ST between P* and Pj*. (14) Repeat (11) to (13) until all the points are inserted into ST. Figure 4 shows the flow chart of the CHH procedure. To illustrate the combined hull heuristic, an example problem with nine retrieval points is solved. The height of the rack is 30 ft and the length of the rack 60 ft with A = 16 ft. The horizontal speed of the S/R truck is 2 ft/s and the vertical speed 1 ft/s. The set S is given by S = {PI, Pz,P3,P4,Pg,
P6,P,,Pg,P,) = ((20, 5),(39, 26),(23, 2% (3,15),(36,0),(50, 1% (41,20), (50,18), (17, 26)). From these data, the points Pg,P6,Pg,P2,Ps and PO correspond to Pa,Pb,PC,Pd,P,and Pf,respectively. Also, xmax = 50, ymax = 26 and PO = (0,O). Applying step 1, the sequence PO-P5-P6-P8-P2-P9-PO can be identified as the partial tour as shown in Fig. 5. The remaining points are S = {PI, P3,Pa, P,>.In Step 2, we find that the regions 1 and 3 are triangle and the region 2 is rectangle. Utilizing the procedure PATH (S, 4, P8,P2), P, in the region 2 is inserted between P8 and P2.Similarly, P4 is inserted between Ps and PO by the procedure RULE (region 3, Ps,PO).In Step 3, the remaining
H. Hwang, J.-Y. Song/Sequencing picking operations and travel time models
81
/ i
Read (x,y)
(3)
Construct
coordinates
a rectangle
(a)
Use PATH for
(b)
three
Use RULE for
regions
region
1
I
No
(c)
/(lo)
Use PATH for
Any remaining points exist
Compute total
Fig. 4. Flowchart
region
travel
of the CHH
3
No
(b)
(ll)-(14)
time
procedure.
Use RULE for
Insert with the
region
3
all the remaining insertion scheme
I
I
points
82
H. Hwang.
J.- Y. Song/Sequencing reg,on
picking
3
. P3 n
(A=)16
.
2
P2(=Pa)
P9(=Pe)
(ymox=)26
and trace1 time models
reglo” I
I
30
operations
P7
%=P!
. P4 Po(=Pb)
l----z?
.PI
0 P0(=P1)
Fig. 5. Combined
hull heuristic.
points Pi and P3 are added to the current tour. PI is added between PO and P5 and then P3 is generated by the above steps is inserted between Pz and Pg. The final sequence PO-PI-Ps-P6-Ps-P,-Pzp,p,P,-P9-P~-P0 with the total travel time of the S/R truck being 100.5 seconds. 2.4. Evaluation of the CHH procedure This section intends to compare the performance of the CHH procedure with the existing sequencing algorithms, the nearest neighbour heuristic (NNH) and the l/2 band insertion heuristic (BAND) through computer simulation. Three factors are considered which may affect the performance of the algorithms. They are the number N of the picking points in an order, the shape factor h (refer to section 3.1 for the definition) of the rack, and the predetermined height A for the safety of the operator. The following values for each factor are examined: N = 5, 10, 15, 20, 25; b = 0.50, 0.75, 1.00; A = 0, b/4, 2b/4, 3b/4, b. For each combination of the factors, thirty sets of the data are generated, which results in 5 x 3 x 5 x 30 = 2250 replications for each algorithm. We consider a discrete rack with 2500 openings, each of which is represented by a unit square. For N given, the locations of the picking points are obtained utilizing a random number generating function. All computations are executed on personal computer 286/AT and the computation time excludes the time for input and output operations. The program is compiled in Turbo Pascal, version 5.0. Some of the simulation results is provided in Tables 1 and 2. For each combination of A and b with N = 20, Table 1 shows the average travel time based on the results of 30 replications. The overall average travel time is listed in Table 2 for each level of the factors. For instance, the values for each level of b are determined by averaging 5 (levels for N) x 5 (levels for A) x 30 (replications) numbers of the travel time data. The results indicate that the CHH procedure performs better than the other two. The NNH algorithm tends to generate poor solutions though it is simple and easy to use. The pairwise t-test is conducted to analyze the data statistically. With the significance level of c( = 0.05%, the test shows that the CHH procedure performs better than the BAND algorithm. It can be observed that varying the levels of b with A fixed does not have substantial effects while N and A does. All the three algorithms in this study can be classified as construction algorithm. One may attempt to improve it by changing the sequence obtained by the construction algorithm. Golden et al. [6] reported that the composite procedure, a construction procedure followed by an improvement procedure, has been quite effective in obtaining a good solution. A well known improvement procedure due to Lin [12] is based on systematically replacing two or three edges in the current tour with alternative edges. If the resulting tour is better, then the change is made and the procedure is restarted until no improvement is made. The all 3-way interchanges are considered
H. Hwang,
J.- Y. Song/Sequencing
picking
operations
and travel
83
time models
Table 1 The average travel time for the case of N = 20
WA
NNH
BAND
CHH
0.5jo.o 0.5/O. 125 OS/O.25 OS/O.375 OS/O.5
308.80 291.80 253.57 244.33 223.63
269.00 249.50 222.60 210.33 196.77
264.33 246.43 219.60 209.43 194.20
0.75/0.0 0.75/0.1875 0.7yo.375 0.7510.5625 0.7510.75
322.33 273.77 255.53 244.67 221.43
274.87 236.90 221.17 212.33 195.70
267.20 234.73 218.50 208.63 191.57
1.o/o.o 1.0/0.25 l.OlO.5 I .0/0.75 l.O/l.O
313.13 270.53 251.07 237.50 220.67
280.07 246.77 231.70 213.57 196.93
271.13 236.80 224.37 208.83 189.07
Average (N = 20)
262.78
230.55
225.65
a Averaged
over the 450 cases.
Table 2 The overall average travel time for each level of the factors Factors
NNH
BAND
CHH
N= N= N= N = N =
156.53 198.82 233.87 262.78 286.61
149.75 181.93 207.78 230.55 250.87
148.66 179.58 204.38 225.65 246.35
b = 0.50 h = 0.75 h = 1.00
232.03 226.95 218.07
206.58 202.40 198.15
205.17 199.16 193.68
A=0
A=$b A=$b A=$b A=b
269.32 242.73 223.98 208.29 194.28
242.30 217.42 200.42 186.46 174.29
236.88 213.61 197.45 184.93 171.76
Total average”
227.72
204.18
200.92
5 IO 15 20 25
aAveraged over
all the 2250 cases.
only after the all 2-way interchange procedure fails to yield an improvement. Table 3 shows a summary of the overall performance of each heuristic when the improvement schemes are adopted. Each value in the Table corresponds to the average of the results from the 2250 replications. It can be observed that even after the 2 and 3-way improvement routines, the CHH procedure still tends to yield the best solution among the three. Regarding to the computation time, the CHH procedure takes 0.20 s on the average and 0.64 s at the maximum for the case of N = 25.
84
H. Hwang, J.-Y. SongJSequencing
Table 3 Overall performance
of each heuristic
picking operations
and travel time models
Heuristic
Cases
With no improvement With 2-way improvement With 2- and 3-way improvements
NNH
BAND
CHH
221.12 (113.3%) 217.14 (109.0%) 216.21 (108.8%)
204.18 (101.6%) 201.14 (100.9%) 199.20 (100.2%)
200.92 (100.0%) 199.28 (100.0%) 198.77 (100.0%)
3. Probabilitic analysis for the travel time models This section deals with the travel time models of the S/R truck. In developing the expected travel times for both single and dual command cycles, the following assumptions are made: (1) The rack is a rectangular pick face, where the I/O point is located at the lower left-hand corner. (2) The length and height of the rack as well as the S/R truck speeds in the horizontal and vertical directions are known. The S/R truck adopts the combined travel as it moves from a pick point to the next. Randomized storage assignment policy is used. (5) Picking and deposit times associated with loading and unloading are constant and can be ignored. All storage locations are the same size. i;; Only the long-run average behavior of the system is considered.
‘,:I
3.1. Rack normalization We consider the continuous rectangular rack face with known dimensions. Following the work of Bozer and White [13], the rack face will be “normalized” by dividing the horizontal travel time th and the vertical travel time t, by T, where s,, = = “; = H = A = th = t, = t, =
T = b = a =
horizontal travel speed of the S/R truck, vertical travel speed of the S/R truck, rack length, rack height, predetermined height for the operator’s safety, 0 d A < H, L/s,, = time to reach the end of the rack, H/s, = time to reach the top of the rack, A/s, = time to reach the predetermined height, 0 Q t, d t,, Max (t,,, t,) = denormalizing factor, 0 < b < 1, Min (th, t,)/T, t,/T.
The factor, b, has been referred to as the “shape factor” for the rack. It will be assumed without loss of generality that t, d th. Thus, the rack has dimensions 1 x b and 0 < a d b. We will refer to the factor, a, as the “safety factor”.
H. Hwang,
3.2.
Expected
J.-Y.
Song/Sequencing
picking operations
85
and travel time models
single command travel time
We will consider first the single command travel time. Let the storage (or retrieval) point be represented by (x, y) in time, where 0 6 x 6 1 and 0 < y < b. From (1) the travel time T ( PO, P) from I’,, = (0, 0) to P = (x, y) will be
T(Po, P) =
max(x,
Y),
if O
max(x,
a)+y-a,
if a
Let F(z) denote the probability that the travel time from PO to P is less than or equal to z. Assuming the x and y coordinates are independently generated, F(z) can be specified under each of the following three conditions: condition 1: 0 < y < a, condition2:a
y) < 21 = PrCx d
Furthermore, under distributed. Thus, Pr[x
< z] = z,
Pr[y
< z] =
randomized
zlPrCy d 21.
storage,
the coordinate
locations
are assumed
to be uniformly
O
zla,
a
i 1,
Hence,
F,(z) = Therefore
fi(z) =
z2/a,
O
z,
a
the probability
fz’,, i,
density function, fi (z), will be
Odzda, a
Letting E1(SC) denote following is obtained:
the expected
=
s 0
for single command,
the
1
a
E1(SC)
travel time under the first condition
2z2/a dz +
s a
z dz = $a’ + $.
(2)
86
H. Hwang, J.-Y. Song/Sequencing
Under
the second condition,
Fz(z) = Pr[max(x,
we obtain
E,(SC)
i(u + b).
=
=
< z],
adzdb.
(3)
Lastly, under the third condition, F3(z) = Pr[x
and travel time models
a) = a and
a) + y - a d z] = Pr[y
= (z - a)/(b - a),
Similarly,
max(x,
picking operations
+ y - a < z] = Pr[y
max(x,
a) = x and F3(z) can be obtained
as following:
< - x + z + a]
(z - 42/C2(l- 4(b - a)l,
udz
(22 - a - b)/[2(1
b
- a)],
[ - z2 + 2(b - a + 1)z + u2 - b2 - 1]/[2(1
- u)(b - a)],
l
E3(SC) = +(b + 1). Combining found: E(X)
(4)
the results of (2), (3), and (4), the expected
= Er(SC)Pr[O + E3(SC)Pr[u
d y d a] + E,(SC)Pr[a < y < b] Pr[u
travel time E( SC) for single command
dual command
blPrC0 G x G al
< x < 11,
= ( - 2a3 + 3u2b + 3a2 + 3b2 - 6ab + 3b)/(6b). 3.3. Expected
can be
(5)
travel time
A dual command cycle involves two random locations: one representing the storage point, the other representing the retrieval point. Let the two points be represented by PI = (x1, yr) and Pz = (x2, y2) in time, where 0 < x1 < 1, 0 d x2 ,< 1 and 0 d y, < y, < b. Let G(z) denote the probability that the travel time between PI and P2 is less than or equal to z. Assuming the x and y coordinates are independently generated, G(z) can be specified under each of the following three conditions. condition 1: 0 d yl < y2 d a < b, condition 2: 0 d a d y, d y2 d b, condition 3: 0 < y, < a < y2 d b. Letting Gk(z) represent the distribution function corresponding to the condition k, the following results are obtained. Under the first condition, G,(z)=PrCmax(Ix2--~11,
(22 - z2)(2uz
IY~-Y~I)~zI,
- z2)/a2,
O
H. Hwang, J.- Y. Sony/Sequencing
Therefore, g,(z)
the probability
density function,
gl(z), will be
- 6(1/a’
+ l/a)z2 + Bzla,
O
4z3/a2
=
picking operations
a
2 - 22,
Letting E 1(TB) denote the expected travel between time under the first condition, obtained. Ei(TB)
= - $a3
+ ia’
the expected
E,(TB) = ECI x2
II,
-
Xl
Yl
the third condition.
J%CW = ECmW I x2
Therefore,
lY2 -
I +
travel between
time is
*(b - a + 1).
Lastly, we consider
=
the following is
+ 3,
Next, under the second condition,
=
87
and travel time models
-
x1
I, a - yl) +
For y, < y,, y2 -
al,
i(b - a) - &a” + 3a’ + 4. the expected
E(TB) = E1(TB)(a/b)2 = (4a5 -
travel time between + E2(TB)[(b
storage
and retrieval
point, E(TB), will be
- a)/b12 + E3(TB) [2a(b - a)]/b2,
15a4 + 20a3 + lob3 + lob2 - 5a4b + 20a3b - 30a2b)/(30b2).
Let E(DC) denote
the expected
travel time for a complete
dual command
cycle. Then,
E(DC) = 2 E(SC) + E(TB). Hence, E(DC) = (4a5 - 15a4 + 20a3 + 40b3 + 40b2 - 5a4b + 30a2b2 - 60ab2)/(30b2). Note that the results of the expected travel times for single and dual commands system are the same as those obtained by Bozer and White [13] for a = b.
in the MOB
4. Conclusions In the MOB system, the operator works on the platform of the S/R truck and handles a number of retrieval (or storage) requests between successive returns to the I/O point. The system efficiency is greatly affected by how these requests are sequenced. The sequencing problem can be formulated as the TSP problem whose optimal solution can be hardly found for large problems within the computational time limitation. Thus, this paper presents the combined hull heuristic procedure which consists of three steps. Through computer simulation, we show that the procedure generates solutions which are better than those by existing algorithms. The computational experiences
88
H. Hwang, .I.- Y. Song/Sequencing
picking operations
and travel time models
indicate that the procedure can be easily implemented on microcomputer for the real time use. Additionally, with the application of probabilitic analysis the expected travel time models of the S/R truck are developed for single and dual commands under randomized storage assignment policy. References Cl1 Apple, J.M., 1972. Material Handling Systems Design. Wiley, New York. Tompkins, J.A. and White, J.A., 1984. Facilities Planning, Wiley, New York. A Guide to the Theory of NP-completeness. F:; Carey, M.R. and Johnson, D.S., 1979. Computers and Intractability: W.H. Freeman and Co. c41 Bellmore, M. and Nemhauser, G.L., 1968. The traveling salesman problem: A survey. Oper. Res., 16: 538. CSI Parker, R.G. and Rardin, R.L., 1983. The traveling salesman problem: An update of research. Nav. Res. Logist. Q,, 30: 69. [61 Golden, B., Bodin. L., Doyle, T. and Stewart, W., 1980. Approximate traveling salesman algorithms. Oper. Res., 28: 694. M. and Ratliff, H.D., 1988. Sequencing picking operations in a man-aboard order picking system. c71 Goetschalckx, Material Flow, 4: 255. PI Bozer, Y.A., Schorn, E.C. and Sharp, G.P., 1990. Geometric approaches to solve the Chebyshev traveling salesman problem. HE Trans., 22: 238. c91 Allison, D.S. and Noga, M.T., 1984. The L, traveling salesman problem. Inf. Proc. Letters, 18: 195. Cl01 Barachet, L.L., 1957. Graphic solution of the traveling salesman problem. Oper. Res., 5: 841. Cl11 Akl, S.G. and Toussaint, G.T., 1978. A fast convex hull algorithm. Inf. Proc. Letters, 7: 219. Cl21 Lin, S., 1965. Computer solutions of the traveling salesman problem. Bell Syst. Tech. J., 44: 2245. systems. IIE Trans., 16: 329. Cl31 Bozer, Y.A. and White, J.A., 1984. Travel-time models for automated storage/retrieval