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A note on multi-item inventory systems with limited capacity
Volume 7, Number 2
OPERATIONS RESEARCH LETTERS
Aprii 1988
A N O T E O N M U L T I - I T E M I N V E N T O R Y S Y S T E M S W I T H L I M I T E D CAPACITY
Jose A. VENTURA and Cerry M. KLEIN Department of Industrial Engineering, 113 Electrical Engineering Building, University of Missouri-Columbia, Columbia, MO 65211, USA Received July 1987 Revised February 1988
Functional relationships between the Lagrange multiplier and system parameters for multi-item inventory systems with one restriction are identified and used to establish tight bounds on the optimal multiplier value. A recursive process which rapidly converges to the optimal multiplier is also discussed. Finally, a cor~,parative analysis of the new bounds in relation to existing bounds is presented. .~ ~. inventory/production * deterministic model • nonlinear programming • Lagrange multiplier
1. Introduction
Inventory systems models and their analysis have appeared in the literature for over the last seventy years. F.W. Harris [2] in 1915 was one of the first to present results. His lot.size formula has come to be known by a variety of names such as the Wilson formula, the Harris-Wilson formula, and the EOQ formula. Since its inception this EOQ model has undergone extensive analysis. It has been shown to be a very insensitive model to parameter errors and to be applicable to different settings. These characteristics along with the ease of use of the model have made it a widely used tool in practice. Consider an inventory system involving n items having the same assumptions as the model of Harris. Each item i has a known, uniform demand rate di (expressed in number of units pe: unit time) and instantaneous replenishme,nt. Let c~ be the holding cost of one unit of item i per unit time, and let ki be the replenishment cost of item i, which is assumed to be independent of the number of units ordered. It is also assumed that backorders are not permitted. Th, replenishment policy for item i is based on some constant reorder quantity q~ that is to be delivered in each instance upon the exhaustion of the current stock. The total variable cost per unit time of this inventory system is the sum of the holding costs
and reorder costs for each quantity qi. That is, ~., c... q:• k,d, f(q)= ,Z+ . (1) iffil q~ Clearly, f ( q ) is strictly convex. The components of q that minimize f(q) can be calculated by the Harris formula and are given by
q,=
!
, i=1,..,..
(2)
In practice however, thexe often exists a restriction on the problem. This restriction may be due to limited storage availability or a fixed budget for inventory investment. This type of restriction is incorporated in the model as a resource constraint, i.e., n
E w,q, <~ U,
(3)
i=.l
where "~ is the storage requirement per unit of item i or the cost of that item, and U is the available storage space or the fixed budget. The multi-item inventory system ~ t h limited capacity can then be formulated as follows: (P)
rain
f(q)-
+-4ffil
,
(4)
q~
n
s.t.
0167-6377/88/$3.50 © 1988, Elsevier Science Pubfisbers B.V. (North-Holland)
~,, wiq , <~U,
i'5)
iffil
if
(6) 71
Volume 7, Number 2
OPERATIONS RESEARCH LETTERS
Due to the presence of a single resource constraint and the convexity of the objective function the solution of this problem generally involves the use of a Lagrange multiplier. Holt et al. [3] presented the first of several techniques that have been developed to o b t m an approximate value to the Lagrangi~ multiplier of the constraiot. They employed a difference equation approach that was to establish an approximate linear equation for the multiplier. Hadley and Whitin [1], and Johnson and Montgomery [4] both presented marginal cost solution techniques and more classical teelm/ques based on the differentiability of the Lagrar~an function. The differentiability allows one to find the optimal multiplier by solving n + 1 nonlinear equations with n + 1 unknowns (the n decision variables and the multiplier). The optimal multiplier is determined by a line search procedure. This process has become the classical solution procedure for this problem. ~egler [5] improved on this method by establishing bounds on the optimal multiplier and by developing a primal-dual iterative scheme based on the bisection technique. In this paper a method to determine tight bounds on the optimal Lagrange multiplier for problem P is developed. This method is based on convex relationships that are shown to exist between the multiplier and the unit holding cost parameters. A comparative analysis of these bounds with the ones developed by Ziegl©r is also given as well as an efficient iterative procedure based on successive improvements of these bounds.
April 1988
where k >~0 and/~ >t 0. The Lagrangian function can be simplified by noting that the nature of the problem ensures that the optimal solution q* is positive for each i. Therefore, p~ = 0 for each i and the KKT conditions of problem P can be stated as OL( q, ~ )
Oq,
ffi
c~
k,d~
2
q?
+ w ik
-
O,
i ffi l , . . . , n,
(s) n
(9)
w ~q~- U ~ O, i-- I
w,q,-U •o,
(lO)
i-1
<
x>.o.
(11)
Thus, if k = 0, the solution to problem P is equivalent to the unconstrained solution given by (2). For k > 0, the solution is obtained from (8) and (10). From (8), (2k,d,)
q'(>') =
'/2
c, + 2w, ,
,
if 1,...,n.
(12)
Since the optimal multiplier k* is positive, (10) n
)
implies that Y"i=P* , q , ( k * ) - U. This in turn yields an approach for finding the optimal multiplier and hence, the optimal q~'s. For the special case of ( c l / w l ) - ( c 2 / w 2 ) f f i "... - ( c , / w , ) a direct calculation for X* is possible. The solution is obtained by substituting q~(k) from (12) into Y'7..lw~q~(k)ffi U, which yields
2. Preliminaries and initial bounds Since f ( q ) is a convex [unction mid ~e ~ n straints are linear, necessary and sufficient conditions for the optimality of problem P are given by the Kurush-Kulm-Tucker (KKT)conditions. Let k be the nonnegative Lagrange multiplier for (5), and let/L~ be the multiplier for the i-th nonnegafive constraint in (6). The Lagrangian function can then be written as follows:
03) When the factors ( c j w ~ ) are not constant for all i, this technique can be used to generate initial lower and upper bounds on k*. Without loss of generality, assume that the ordering of the items is such that ( c l / w l ) <~(c2/w2) <~ . . . <~(Cn/Wn). Define
(14) =
+k
w~q~- U
- - - ~,q~, iffi,1
72
(7)
g
,-I
-
-
'
(15)
Volume 7, Number 2
OPERATIONS RESEARCH LETrERS
April 1988
Theorem 1. If the optimal solution to problem P has ~* > O, then lower and upper bounds on ?~* are given by ki <~~* <~~u.
~,,,,,,,,,,, Z.* ~ '% Proof. If (ci/wi)f(cn/wn), then ?~*=~1=~,, and ?,* satisfies the KKT conditions. If (cllwl)< ( c . / w . ) , then since ?~* > 0, equation (10) of the KKT conditions implies that
~(
2w,k,d, )1/2 (c,lw,) + 2x* = is.
h_ (c)
[
06)
~l is determined from (16) by setting (cJw~) equal to (c.lw.) for each i and solving for ?~*. Since (cJw~) and ,%* are both positive and in the denominator, an increase in (cJw~) implies a decrease in ~*. Hence, by setting (cJw~), for i ffi 1,..., n - 1, equal to ( c J w ~ ) in equation (16), ~,* decreases to ~,~. Therefore, ~,~ < ~,*. Likewise to find ~u, (cJw~) is set equal to (el~w1) for each i and a similar argument holds to show that ?,* < ~,u. []
3. Improved bounds To find tighter bounds on ~* assume the unit holding cost for item i is a variable, c. The optimal Lagrange multiplier can then be viewed as a function of c, ~,=h~(c), defined for all c >0. While h~(c) can not be expressed in closed form, its properties can be determined from its inverse function. These properties can be used to improve the initial bounds on the multiplier., Theorem 2. h~(c) is a strictly decreasing convex
.function for all c > O. Proof. For a given c > 0, let ~ be the corresponding multiplier. Then E~=lwjqj(k)= U. From this expression the inverse function of h~(c) can be found as
~
c'
cI
c" Figure I, Function ,%=/q(c),
defined for all X > max{(-cJ2wj): for all j ~ i}. For all values of ~, where g~(~,) is defined, it is easily shown that the first derivative of g~(~,) is always negative and .that the second derivative is always positive. This implies that g~(k) is a strictly decreasing convex function. Thus, its inverse, hs(c), is also a strictly decreasing convex function. ra Figure 1 graphically displays the construction of the improved bounds for 7,* using the function hi(c). In this graph the horizontal axis is c and the vertical axis is X = hi(c). The ordinate 7,i represents the intersection of the tangent line to hi(c) at E, where E ffi gx(~,l), with the vertical line c ffi cv The ordinate ~,u is the intersection of the line du'ough the points hi(c") and hl(c') with the vertical line c fficv The values c' and c" are defined as c'ffi gl(?~u) and c"ffi gl(~,,). The values Xland Xu provide the improved lower and upper bounds, respectively, and are computed as follows: R
~,,=?t, + i~hl(E)(cl-~), (~'u -- ~tl)(Ctt -- Ci)
Os) (19)
(c"-c')
2w~k~d~ i
-2w~?~, i = l , . . . , n ,
m
(17)
where 0hl(~') is the gradient of hi(c) at E. Note though, thatsince hi(c) is not known in closed form, its gradient can not be computed directly. This has been circumvented however, since the gradient of hi(c) is equal to the inverse of the 73
Volume 7, Number 2
-
OPERATIONS RESEARCH LETTERS
gradient of its inverse function g~(X). Hence, the gradient is given by abe(c)
=
April 1988
converge to the optimal Lagrange multiplier. The algorithm described below follows this recursive procedure to generate k*.
=
-1
-4w~k,d, E w~ jr2
. . . . .
( c j + 2w)~,) 3
( 2kjdj )'/2]3
- 2w~
U - j ~- 2 w~ ci + 2wik
.
1
(20)
Theorem 3. If the optimal solution to problem P has ~* > O, then lower and upper bounds on ~* are
Algorithm Step O. Compute ~o and ~o from (14) and (15), respectively. Set ~.o _ gl(~°) and t ffi 0. Step 1. Compute _.
x,+ , = x;+ ,
+ (N',. - ?"|+1) [ g2(?'h+ x) - cl], Step 2.If (N+l-N~l)/?~t~'l¢e (e is a small
Proof. Since hi(c) is convex, it follows that (i) ~* = h~(cl) ~ h~(~) + ah~(O(cl - ~) =
+ ak~(~)(c~- ~) ffi ~.
~
(ii) h~la(c' - c") + c"] ~ a[h~(c') - hl(c")] + hz(c"), for all a ~ (0, 1). Let a ffi ( c " - c ~ ) / ( c " - c'), then
~* = h|(c~) ~ c " - ~ -
o
Note that similar bounds could be obtained by using any function h~(c), for i ffi 2,..., n, instead of h~(c). It should also be noted that for the bounds developed by Ziegler [5], the analysis was conducted on the function G(?~) = E~'. lw~q~(~). Since G(?O is a strictly decreasing convex function, it could also be used in the above ~malysis in place of hi(c). However, since the initial bounds are improved by using a tangent to the function and an inner linear approximation, it seems that the best choice for the function would be the one with the least curvature (minimum change in slope) in the search interval.
4. The algo~thm
Establishing k i as an improved bound to ~,* also yields a solution technique for finding ?,*. Since h~(~,) is strictly monotone, c" lies between cl and ~. If after obtaining ~,! and ~u, c is set equal to c , and the same process is repeated, further improvement in the bounds is obtained. As this procedure is continued, both bounds will PP
74
tolerance), set ~ = (N + 1 + Ni+ 1)/2, and terminate. Otherwise, set ~t + 1 ffi gl(~+ 1), increase t by 1, and go to Step 1.
Based on the facts that the function hi(c) is convex and that the lower bounds are computed by the Newton method, the sequence { N} converges to ~,* with a quadratic rate of convergence.
5. C o m p a r a t i v e a n a l y s i s
This section summarizes the analysis conducted to determine how the bounds defined in equations (18) and (19) compare to the improved bounds obtained by Ziegler [5] in version 2 of this second experiment. Both methods were programmed in BASlC on an IBM PC AT. In addition, the algorithm described in Section 4 was incorporated into the program to find ~,*. The value ~,* was then used in comparing the relative errors of the two bounding processes. Overall, this comparative analysis consisted of computing the optimal Lagrange multiplier, the proposed improved bounds on that multiplier, and the bounds developed by Ziegler. For each system examined, the relative errors ( ( ~ . - ~ * ) / ~ , * and (?~*-~,)/~*), and the relative range ((~,u~,l)/~,*) were compared to identify which method provided the tightest bounds for X*. One hundred randomly generated two-item inventory systems were processed using the above procedure. The R
Volume 7, Number 2
OPERATIONS RESEARCH LETTERS
April 1988
Table 1 Computational comparison Method Ziegler Eqs. (17)-(18)
Lower bound
Upper bound
Bounding interval
Best a
MItE b
ARE c
Best
MRE
ARE
Best
MRE
ARE
22 78
0.12224 0.18726
0.03779 0.02178
0 100
0.08763 0.02546
0.01666 0.00214
16 84
0.20783 0.21272
0.05445 0.02392
n,,
a Best -- Number of times each method obtains a smaller relative error. t, MRE ffi Maximum relative error. ¢ ARE - Average relative error.
cost parameters were generated from uniform distributions according to
cl e u(z0, 30),
c2 U(30, 50),
kz, k2 ~ U(IO0, 200), dl, d 2 ~ U(1000,
2000).
The two constraint parameters, wl and w 2, were set to ene, the right hand side parameter U was computed as U - a ( r h + r/2), where rh and q2 are as defined in (2), and a was uniformly distributed as
u(o.3, 0.7). The value a was used to determine the sharpness or restrictiveness of the constraint. The stopping condition for the algorithm was • ffi 10 -5. Table 1 contains a summary of the results obtained in this analysis. Of the 22 systems in which the method of Ziegler produced a tighter lower bound, the proposed method produced a smaller bot,~ding inte.,--al for 6 of ~ose systei~is.
The average size of relative errors indicated for each method shows clearly that the proposed new bounds are tighter for this selection of distributions of system parameters. However, a more extensive computational testing is required to conclude which method is better, or to determine for which choice of parameters one method is superior to the other.
References [1] G. Hadley and T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, N J, 1963. [2] F. Harris, Operations and Costs, Chicago, 1915. [3] C.C. Holt, F. Modigliani, J. Moth and H. Simon, Planning Prod,4ction, Inventories and Work Force, Prentice-Hall, Englewood Cliffs, N J, 1960. [4] L.A. Johnson and D. Montgomery, Operations Research in Production Planning, Scheduling and Inventory Control, Wiley, New York, 1974. [5] H. Ziegler, "Solving certain constrained convex optimization problems in production planning", Operations Research Letters I, 246-2521 (1982).
75
OPERATIONS RESEARCH LETTERS
Aprii 1988
A N O T E O N M U L T I - I T E M I N V E N T O R Y S Y S T E M S W I T H L I M I T E D CAPACITY
Jose A. VENTURA and Cerry M. KLEIN Department of Industrial Engineering, 113 Electrical Engineering Building, University of Missouri-Columbia, Columbia, MO 65211, USA Received July 1987 Revised February 1988
Functional relationships between the Lagrange multiplier and system parameters for multi-item inventory systems with one restriction are identified and used to establish tight bounds on the optimal multiplier value. A recursive process which rapidly converges to the optimal multiplier is also discussed. Finally, a cor~,parative analysis of the new bounds in relation to existing bounds is presented. .~ ~. inventory/production * deterministic model • nonlinear programming • Lagrange multiplier
1. Introduction
Inventory systems models and their analysis have appeared in the literature for over the last seventy years. F.W. Harris [2] in 1915 was one of the first to present results. His lot.size formula has come to be known by a variety of names such as the Wilson formula, the Harris-Wilson formula, and the EOQ formula. Since its inception this EOQ model has undergone extensive analysis. It has been shown to be a very insensitive model to parameter errors and to be applicable to different settings. These characteristics along with the ease of use of the model have made it a widely used tool in practice. Consider an inventory system involving n items having the same assumptions as the model of Harris. Each item i has a known, uniform demand rate di (expressed in number of units pe: unit time) and instantaneous replenishme,nt. Let c~ be the holding cost of one unit of item i per unit time, and let ki be the replenishment cost of item i, which is assumed to be independent of the number of units ordered. It is also assumed that backorders are not permitted. Th, replenishment policy for item i is based on some constant reorder quantity q~ that is to be delivered in each instance upon the exhaustion of the current stock. The total variable cost per unit time of this inventory system is the sum of the holding costs
and reorder costs for each quantity qi. That is, ~., c... q:• k,d, f(q)= ,Z+ . (1) iffil q~ Clearly, f ( q ) is strictly convex. The components of q that minimize f(q) can be calculated by the Harris formula and are given by
q,=
!
, i=1,..,..
(2)
In practice however, thexe often exists a restriction on the problem. This restriction may be due to limited storage availability or a fixed budget for inventory investment. This type of restriction is incorporated in the model as a resource constraint, i.e., n
E w,q, <~ U,
(3)
i=.l
where "~ is the storage requirement per unit of item i or the cost of that item, and U is the available storage space or the fixed budget. The multi-item inventory system ~ t h limited capacity can then be formulated as follows: (P)
rain
f(q)-
+-4ffil
,
(4)
q~
n
s.t.
0167-6377/88/$3.50 © 1988, Elsevier Science Pubfisbers B.V. (North-Holland)
~,, wiq , <~U,
i'5)
iffil
if
(6) 71
Volume 7, Number 2
OPERATIONS RESEARCH LETTERS
Due to the presence of a single resource constraint and the convexity of the objective function the solution of this problem generally involves the use of a Lagrange multiplier. Holt et al. [3] presented the first of several techniques that have been developed to o b t m an approximate value to the Lagrangi~ multiplier of the constraiot. They employed a difference equation approach that was to establish an approximate linear equation for the multiplier. Hadley and Whitin [1], and Johnson and Montgomery [4] both presented marginal cost solution techniques and more classical teelm/ques based on the differentiability of the Lagrar~an function. The differentiability allows one to find the optimal multiplier by solving n + 1 nonlinear equations with n + 1 unknowns (the n decision variables and the multiplier). The optimal multiplier is determined by a line search procedure. This process has become the classical solution procedure for this problem. ~egler [5] improved on this method by establishing bounds on the optimal multiplier and by developing a primal-dual iterative scheme based on the bisection technique. In this paper a method to determine tight bounds on the optimal Lagrange multiplier for problem P is developed. This method is based on convex relationships that are shown to exist between the multiplier and the unit holding cost parameters. A comparative analysis of these bounds with the ones developed by Ziegl©r is also given as well as an efficient iterative procedure based on successive improvements of these bounds.
April 1988
where k >~0 and/~ >t 0. The Lagrangian function can be simplified by noting that the nature of the problem ensures that the optimal solution q* is positive for each i. Therefore, p~ = 0 for each i and the KKT conditions of problem P can be stated as OL( q, ~ )
Oq,
ffi
c~
k,d~
2
q?
+ w ik
-
O,
i ffi l , . . . , n,
(s) n
(9)
w ~q~- U ~ O, i-- I
w,q,-U •o,
(lO)
i-1
<
x>.o.
(11)
Thus, if k = 0, the solution to problem P is equivalent to the unconstrained solution given by (2). For k > 0, the solution is obtained from (8) and (10). From (8), (2k,d,)
q'(>') =
'/2
c, + 2w, ,
,
if 1,...,n.
(12)
Since the optimal multiplier k* is positive, (10) n
)
implies that Y"i=P* , q , ( k * ) - U. This in turn yields an approach for finding the optimal multiplier and hence, the optimal q~'s. For the special case of ( c l / w l ) - ( c 2 / w 2 ) f f i "... - ( c , / w , ) a direct calculation for X* is possible. The solution is obtained by substituting q~(k) from (12) into Y'7..lw~q~(k)ffi U, which yields
2. Preliminaries and initial bounds Since f ( q ) is a convex [unction mid ~e ~ n straints are linear, necessary and sufficient conditions for the optimality of problem P are given by the Kurush-Kulm-Tucker (KKT)conditions. Let k be the nonnegative Lagrange multiplier for (5), and let/L~ be the multiplier for the i-th nonnegafive constraint in (6). The Lagrangian function can then be written as follows:
03) When the factors ( c j w ~ ) are not constant for all i, this technique can be used to generate initial lower and upper bounds on k*. Without loss of generality, assume that the ordering of the items is such that ( c l / w l ) <~(c2/w2) <~ . . . <~(Cn/Wn). Define
(14) =
+k
w~q~- U
- - - ~,q~, iffi,1
72
(7)
g
,-I
-
-
'
(15)
Volume 7, Number 2
OPERATIONS RESEARCH LETrERS
April 1988
Theorem 1. If the optimal solution to problem P has ~* > O, then lower and upper bounds on ?~* are given by ki <~~* <~~u.
~,,,,,,,,,,, Z.* ~ '% Proof. If (ci/wi)f(cn/wn), then ?~*=~1=~,, and ?,* satisfies the KKT conditions. If (cllwl)< ( c . / w . ) , then since ?~* > 0, equation (10) of the KKT conditions implies that
~(
2w,k,d, )1/2 (c,lw,) + 2x* = is.
h_ (c)
[
06)
~l is determined from (16) by setting (cJw~) equal to (c.lw.) for each i and solving for ?~*. Since (cJw~) and ,%* are both positive and in the denominator, an increase in (cJw~) implies a decrease in ~*. Hence, by setting (cJw~), for i ffi 1,..., n - 1, equal to ( c J w ~ ) in equation (16), ~,* decreases to ~,~. Therefore, ~,~ < ~,*. Likewise to find ~u, (cJw~) is set equal to (el~w1) for each i and a similar argument holds to show that ?,* < ~,u. []
3. Improved bounds To find tighter bounds on ~* assume the unit holding cost for item i is a variable, c. The optimal Lagrange multiplier can then be viewed as a function of c, ~,=h~(c), defined for all c >0. While h~(c) can not be expressed in closed form, its properties can be determined from its inverse function. These properties can be used to improve the initial bounds on the multiplier., Theorem 2. h~(c) is a strictly decreasing convex
.function for all c > O. Proof. For a given c > 0, let ~ be the corresponding multiplier. Then E~=lwjqj(k)= U. From this expression the inverse function of h~(c) can be found as
~
c'
cI
c" Figure I, Function ,%=/q(c),
defined for all X > max{(-cJ2wj): for all j ~ i}. For all values of ~, where g~(~,) is defined, it is easily shown that the first derivative of g~(~,) is always negative and .that the second derivative is always positive. This implies that g~(k) is a strictly decreasing convex function. Thus, its inverse, hs(c), is also a strictly decreasing convex function. ra Figure 1 graphically displays the construction of the improved bounds for 7,* using the function hi(c). In this graph the horizontal axis is c and the vertical axis is X = hi(c). The ordinate 7,i represents the intersection of the tangent line to hi(c) at E, where E ffi gx(~,l), with the vertical line c ffi cv The ordinate ~,u is the intersection of the line du'ough the points hi(c") and hl(c') with the vertical line c fficv The values c' and c" are defined as c'ffi gl(?~u) and c"ffi gl(~,,). The values Xland Xu provide the improved lower and upper bounds, respectively, and are computed as follows: R
~,,=?t, + i~hl(E)(cl-~), (~'u -- ~tl)(Ctt -- Ci)
Os) (19)
(c"-c')
2w~k~d~ i
-2w~?~, i = l , . . . , n ,
m
(17)
where 0hl(~') is the gradient of hi(c) at E. Note though, thatsince hi(c) is not known in closed form, its gradient can not be computed directly. This has been circumvented however, since the gradient of hi(c) is equal to the inverse of the 73
Volume 7, Number 2
-
OPERATIONS RESEARCH LETTERS
gradient of its inverse function g~(X). Hence, the gradient is given by abe(c)
=
April 1988
converge to the optimal Lagrange multiplier. The algorithm described below follows this recursive procedure to generate k*.
=
-1
-4w~k,d, E w~ jr2
. . . . .
( c j + 2w)~,) 3
( 2kjdj )'/2]3
- 2w~
U - j ~- 2 w~ ci + 2wik
.
1
(20)
Theorem 3. If the optimal solution to problem P has ~* > O, then lower and upper bounds on ~* are
Algorithm Step O. Compute ~o and ~o from (14) and (15), respectively. Set ~.o _ gl(~°) and t ffi 0. Step 1. Compute _.
x,+ , = x;+ ,
+ (N',. - ?"|+1) [ g2(?'h+ x) - cl], Step 2.If (N+l-N~l)/?~t~'l¢e (e is a small
Proof. Since hi(c) is convex, it follows that (i) ~* = h~(cl) ~ h~(~) + ah~(O(cl - ~) =
+ ak~(~)(c~- ~) ffi ~.
~
(ii) h~la(c' - c") + c"] ~ a[h~(c') - hl(c")] + hz(c"), for all a ~ (0, 1). Let a ffi ( c " - c ~ ) / ( c " - c'), then
~* = h|(c~) ~ c " - ~ -
o
Note that similar bounds could be obtained by using any function h~(c), for i ffi 2,..., n, instead of h~(c). It should also be noted that for the bounds developed by Ziegler [5], the analysis was conducted on the function G(?~) = E~'. lw~q~(~). Since G(?O is a strictly decreasing convex function, it could also be used in the above ~malysis in place of hi(c). However, since the initial bounds are improved by using a tangent to the function and an inner linear approximation, it seems that the best choice for the function would be the one with the least curvature (minimum change in slope) in the search interval.
4. The algo~thm
Establishing k i as an improved bound to ~,* also yields a solution technique for finding ?,*. Since h~(~,) is strictly monotone, c" lies between cl and ~. If after obtaining ~,! and ~u, c is set equal to c , and the same process is repeated, further improvement in the bounds is obtained. As this procedure is continued, both bounds will PP
74
tolerance), set ~ = (N + 1 + Ni+ 1)/2, and terminate. Otherwise, set ~t + 1 ffi gl(~+ 1), increase t by 1, and go to Step 1.
Based on the facts that the function hi(c) is convex and that the lower bounds are computed by the Newton method, the sequence { N} converges to ~,* with a quadratic rate of convergence.
5. C o m p a r a t i v e a n a l y s i s
This section summarizes the analysis conducted to determine how the bounds defined in equations (18) and (19) compare to the improved bounds obtained by Ziegler [5] in version 2 of this second experiment. Both methods were programmed in BASlC on an IBM PC AT. In addition, the algorithm described in Section 4 was incorporated into the program to find ~,*. The value ~,* was then used in comparing the relative errors of the two bounding processes. Overall, this comparative analysis consisted of computing the optimal Lagrange multiplier, the proposed improved bounds on that multiplier, and the bounds developed by Ziegler. For each system examined, the relative errors ( ( ~ . - ~ * ) / ~ , * and (?~*-~,)/~*), and the relative range ((~,u~,l)/~,*) were compared to identify which method provided the tightest bounds for X*. One hundred randomly generated two-item inventory systems were processed using the above procedure. The R
Volume 7, Number 2
OPERATIONS RESEARCH LETTERS
April 1988
Table 1 Computational comparison Method Ziegler Eqs. (17)-(18)
Lower bound
Upper bound
Bounding interval
Best a
MItE b
ARE c
Best
MRE
ARE
Best
MRE
ARE
22 78
0.12224 0.18726
0.03779 0.02178
0 100
0.08763 0.02546
0.01666 0.00214
16 84
0.20783 0.21272
0.05445 0.02392
n,,
a Best -- Number of times each method obtains a smaller relative error. t, MRE ffi Maximum relative error. ¢ ARE - Average relative error.
cost parameters were generated from uniform distributions according to
cl e u(z0, 30),
c2 U(30, 50),
kz, k2 ~ U(IO0, 200), dl, d 2 ~ U(1000,
2000).
The two constraint parameters, wl and w 2, were set to ene, the right hand side parameter U was computed as U - a ( r h + r/2), where rh and q2 are as defined in (2), and a was uniformly distributed as
u(o.3, 0.7). The value a was used to determine the sharpness or restrictiveness of the constraint. The stopping condition for the algorithm was • ffi 10 -5. Table 1 contains a summary of the results obtained in this analysis. Of the 22 systems in which the method of Ziegler produced a tighter lower bound, the proposed method produced a smaller bot,~ding inte.,--al for 6 of ~ose systei~is.
The average size of relative errors indicated for each method shows clearly that the proposed new bounds are tighter for this selection of distributions of system parameters. However, a more extensive computational testing is required to conclude which method is better, or to determine for which choice of parameters one method is superior to the other.
References [1] G. Hadley and T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, N J, 1963. [2] F. Harris, Operations and Costs, Chicago, 1915. [3] C.C. Holt, F. Modigliani, J. Moth and H. Simon, Planning Prod,4ction, Inventories and Work Force, Prentice-Hall, Englewood Cliffs, N J, 1960. [4] L.A. Johnson and D. Montgomery, Operations Research in Production Planning, Scheduling and Inventory Control, Wiley, New York, 1974. [5] H. Ziegler, "Solving certain constrained convex optimization problems in production planning", Operations Research Letters I, 246-2521 (1982).
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