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Lot-sizing polyhedra with a cardinality constraint
Operations Research Letters 11 (1992) 13-18 North-Holland
February 1992
Lot-sizing polyhedra with a cardinality constraint El Houssaine Aghezzaf and Laurence A. Wolsey CORE, Unit:ersit~;Catholique de Lout,ain, Louvain-la-Neuve, Belgium Received October 1990 Revised April 1991
Given a family of mixed 0-1 polyhedra, we suppose that the convex hull of solutions is known. Now we add a cardinality constraint such that the sum of the 0-1 variables is exactly k. W h e n is the resulting polyhedron integral, or when does the resulting linear program have an integral optimal solution for all values of k? Various equivalent conditions are given, and it is shown that uncapacitated lot-sizing polyhedra have this property when the production costs are nonincreasing over time. integral polyhedra; cardinality constraint; lot-sizing
1. Introduction Given a mixed 0-1 polyhedron P c ~p+n i.e., P is the convex hull of solutions of some set T c ~ +P× {0, 1}", we consider the family of polyhedra: for k = l . . . . ,n
(1) and we ask when the polyhedra Pk are integral for all k. In a weaker form we are given an objective vector (c, f ) and we ask when the linear program min{cx + f y ' ( x , y) ~Pk} has an optimal solution in T for all k. In the latter case we say that P has the 'integrality property' with respect to (c, f ) . Various authors have raised this question for particular families of polyhedra, in particular White and Gillenson [10] for node covers, Ward et al. [9] for locations on tree networks, and Gamble and Pulleyblank [3] for forest covers, and it also arises naturally in the algorithms for matroids and matchings, see Lawler [5] or Nemhauser and Wolsey [6]. In Section 2 we give a
very simple geometric picture of the integrality property, and state equivalent conditions that are consequences of Lagrangian duality (see Geoffrion [4]). In Section 3 we consider the mixed 0-1 polyhedron arising from the uncapacitated lot-sizing problem. Though the family of polyhedra {Pk} are not integral, we show that if the production costs are nonincreasing (after normalization so that the storage costs are zero), then optimizing over the polyhedra *ok gives optimal solutions with the set-up variables y~ . . . . , y,, E {0, 1}.
2. Conditions for integrality We consider thc following optimization problems: LG(k)
G(k) LH(k)
= min{cx + f y : ( x , y ) ~ P k ,
Y ~ {0, 1}"},
H(k)=min{cx+fy:(x, y)
and L~(A)
clg(A) = min{cx + fy -A y" yj. (x, y)
0167-6377/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
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Volume 11, Number 1
OPERATIONS RESEARCH LETTERS
The integrality property can now be formally stated as the condition H ( k ) = G ( k ) for k =
Proof. As P = cony(T), q~(A) = min{cx + f y A ~ ' 1Yj: (x, y) ~ T}. Thus by Lagrangian duality, max~{@(A) + Ak} = min{cx + fy: (x, y) conv(T), E T = l y i = k } = H ( k ) . []
l~.,,~n. Without loss of generality we assume throughout that the smallest and largest solutions in P have ~2jyj = 1 and ~2jy~ = n respectively. The following propositions tell us what happens when we slice an integral polyhedron with the cardinality constraint. P r o p o s i t i o n 1 [9]. I f xLP(k)=(xLP(k), yLP(k)) T is a vertex o f P k, xLP(k) is a convex combination o f two points (x l, yl), (x 2, y2) o f T h a v i n g Z y ) = K , , Y'.yf=K z f o r s o m e K l < k < K 2.
Proof. If the vertex is not a vertex of P, it must be at the intersection of an edge of P and the hyperplane consisting of the cardinality constraint. As P is integral, the edge joins two vertices in T. []
So L H ( k ) is the primal problem equivalent to the Lagrangian dual of L G ( k ) with the constraint En i=~YJ = k dualized, and the function H is the lower convex envelope of G. Thus we have the situation shown in Figure 1. If 1 = k~ < k 2 < . . . < k r = n are the values for which H ( k ) = G ( k ) , the slopes m i = ( G ( k i + l ) - G ( k i ) ) / ( k i + ! - k i ) of the function H are nondecreasing, and we see immediately that (i) L@(A) has an optimal solution (x, y ) ~ T with Ejyj = k i if and only if A ~ [m i_ 1, mi], i.e., A is a subgradient of the function H at k i. (ii) If k i < k < k i + l , then ki+ I - k
x L P ( k ) -- ki+l _ k i
P r o p o s i t i o n 2 [4]. (i) H(k) = maxx{cb(A) + Ak). (ii) G ( k ) = H ( k ) i f and only i f G ( k ) has a (global) subgradient at z = k.
+
xLP(k , -i,
k -k i
xLP(ki+l).
ki+ 1 - k i
[]
H(k)
~
February 1992
Valuesof G(k)
lope m1
~,..~
l°pe m2
[]
support o
1
2
3
4
5
k1
k2
k3
k4 Fig. 1.
14
6
k5
7
8
k6
k
V o l u m e 11, N u m b e r 1
OPERATIONS RESEARCH LETTERS
W e now restate the content of the propositions.
Theorem 3. Given an integral polyhedron P and an objective (c, f ), the following are equivalent: (i) the integrality property holds, (ii) G(k + 1) - G ( k ) is nondecreasing for k = l , . . . , n - l, (iii) whenever for some value of I~, the linear program min(cx + f y - t ~ j y j : (x, y) ~ P} has two integral optimal solutions (x 1, y l ) and (x 2, y2)
with ~ y ) = E I, ~ y / = r 2 with K I
3. Lot-sizing with nonincreasing production costs H e r e we consider the well-known uncapacitated economic lot-sizing p r o b l e m first studied by W a g n e r and Whitin [8]. Thus we consider the problem LG(k)
G(k) = min~c,x, t
Example 1 (van Hoesel [5]). Consider the uncapacitated lot-sizing p r o b l e m with the costs and d e m a n d s shown in the following table: 1
2
3
4
5
ci
8
0
5
3
1
f,
2
21
1
1
1
d,
I
1
1
1
I
Now we add the constraint that we set up k times. It is readily checked that G ( 1 ) = 42, G(2) = 31, G ( 3 ) = 31, G ( 4 ) = 30, and G ( 5 ) = 34. Thus G(3) - G(2) > G(4) - G(3) and the integrality condition does not hold. As (kj, k 2, k3, k 4 ) = (1, 2, 4, 5) the linear p r o g r a m LH(3) has an optimal solution (x, y) = ½xLP(2) + ½xLP(4) = (3, 2, l I I. I I l I~I ~_, .~, 3, 1, 3, z, ~, ~. with value H ( 3 ) = 30.5. F r o m now on we m a k e the W a g n e r - W h i t i n assumption that ct>c,+ ~ for all t. It is well known that G ( t ) = min t <~<,g(s, t), w h e r e g(s, t) = G(s - 1) + f, + Gd,,, and G(0) = 0. H e r e d , denotes Eti=sdr
Proposition 4. Under the Wagner-Whitin assumption, if G ( t ) = g(s, t), then g(s', t ' ) > g ( s , t') for all t' > t and s' < s.
+ ~f,y,,
Proof.
l
(x, y) ~ T,
g ( s ' , t') = g ( s ' , t) +cs,d,+lj,
L yj = k , j
Februuary 1992
> g( s, t) + c,dt+ l.,,
1
=g(s,t') where
T=
where
(
'
x,y)"
~xi>
Y'~difor t=l
i=1
i=1
inequality
uses
G(t) = g ( s , t)<_
[]
..... n-l,
L Xi:
L di, 0 <_x t <_Myt,
i=1
i=1
Yt ~ {0, 1} for all t}. An explicit description of P = c o n v ( T ) is given in Barany et al. [1]. T h e following example shows that the integrality p r o p e r t y does not hold for all values of c and
f.
the
g(s', t), and c,
,
Now we suppose that m i n { E t c , x , + E J t y t, has two optimal solutions (x 1, y l ) and (x 2, y2) with Y~y) = K I, Z y ~ = K2 and K2 - K l > 2. We will show that there is an alternative optimal solution w i t h Y~jYi = k for some k such that K l < k < K 2. Without loss of generality, we can replace f, by m a x ( f , , 0) in the objective function, and only consider solutions for which each period with yj = 1 is a production period. W e construct a bipartite graph G = (V~, V2, E ) as follows: Vi = {j ~ {1 . . . . . n}: yf = 1} for i = 1, 2.
(x, y ) ~ T }
15
Volume 11, Number 1
OPERATIONS RESEARCH LETTERS
ql(~)
II
February1992
t
I
V1
v2 I
I
q2 (]3)
I
I
t+2
Fig. 2.
(c~,/3) ~ E if in solution 1 the p r o d u c t i o n in period a e V 1 covers period /3 e V2, or vice versa. It is readily verified that G is acyclic.
1. If q2(/3) -- ql(/3) < 0 for all /3 ~ V 1 with deg(/3) = 3, this holds for the last such node. But then Kl = ql(/3) + t for some integer t, and K2 _< q2(/3) + 2 + t (see Figure 2). Thus K2 - - K l < q2(/3) -q~(/3) + 2 < 2, which is a contradiction. Thus q2(/3) -- ql(/3) > 0 for some /3 ~ V 1 with deg(/3) = 3. As the difference can only increase by unit amounts, the claim follows. []
5. I f the graph G is connected, a) there exists a n o d e / 3 ~ VI with deg(/3) > 3; b) if there is no node /3 ~ V l with d e g ( / 3 ) > 4, there exists a n o d e / 3 ~ V] with deg(/3) = 3, and Proposition
ql(/3)
=
I{1 . . . . .
/3} n V1 I = 1{1,...,/3}
n V2 I
Proposition 6. I f (x l, yl), (x 2, y 2 ) a r e optimal solutions of min{cx + f y : (x, y ) ~ P} with E y ) = •l, ~ Y f = K2, and K2 - K l > 2, then there exists an alternaffve optimal solution of intermediate cardinality.
= q2(/3)" P r o o f . a) F r o m the definition, q l ( 1 ) = q 2 ( l ) = 1, q](n) = K1 and q2(n) = K2 with K2 >_ K 1 "{'- 2. If/31 and/32 are successive nodes of V 1, and deg(/3 l) _< 2, q2(/32 ) -- q1(/32 ) ~ q2(/31) -- ql(/31)" If deg(/3) < 2 for all / 3 ~ V 1 and /3* is the last node of V I, q2(/3") - ql(/3*) -
a) V 1 contains a node a of degree > 4. Part of the graph G is shown in Figure 3, where a, E are successive production nodes for ( x ' , y t ) and /3, y, 6 are production periods for (x 2, y2), namely the second, third, and last nodes of V 2 covered by a. Proof.
(X
V1
[3
y Fig. 3.
16
8
v2
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OPERATIONS R E S E A R C H LETTERS
V1
V2
y
periods for both solutions 1 and 2. If E j=l t2- Ll y f _ ~t2-j=tll.yjl __~>/(2 -- /¢1, the above a r g u m e n t can be applied to this interval. Otherwise we have for some interval that 1 _< ~ t%?j=,, : - ~J _ 5Zt~j=,,J.y/i_
8 Fig. 4.
Consider the solution (X 2, y2). G ( 7 - 1 ) = g(/3, y - 1) and G(+ - 1 ) = g ( y , 6 - 1). T h e r e fore by Proposition 4, g(/3, E - 1) < g ( a ,
Februuary 1992
E - 1) as a < / 3 ,
g ( y , e - I) < g ( / 3 , e -
and
I) a s / 3 _<-/.
Thus g ( y , • - 1) < g ( / 3 , • - 1) < g ( a , • - 1). H o w e v e r from solution (x 1, y t ) we have that G ( • - 1 ) = g ( a , • - 1 ) < g ( y , • - 1). T h u s g(/3, • - l ) = g ( y , • - 1). In o t h e r words g(/3, 6 - 1 ) + ct3d~,,-1 = g(Y, ~ - 1 ) + cz, da, ~_ I. As ct~ > cz, , this implies that g ( / 3 , 6 - 1 ) < g ( y , 6 - 1 ) . Now we can modify solution (x e, y 2 ) by setting yr2 = 0, x~ = dt~,~_ ~ and we have an alternative optimal solution with k = Ke - 1. b) V 1 contains no node of degree > 4. Let /3 be the first node of degree 3 with q1(/3)= q2(/3 ) as described in Proposition 5, see Figure 4. T h e cost of solution (x l, y l ) is 11 = G(/3 - 1) +ft~ +ct~d~,, ~ + H ( • ) , and the cost of solution (x , y2) is 1 2 = G ( y - 1 ) + f ~ + c r d r , ~ _ 1+H(6), where H ( w ) is the cost of an optimal solution for periods o~, w + 1. . . . . n. Now consider an alternative solution (x 3, y3) o b t a i n e d by using solution 2 for the first ( y - 1) periods, producing in y till period • - 1 and then using solution 1. T h e cost is 13 = G ( T - 1) + f ~ + c~,d~,.~_ 1 + H ( • ) . Note that this solution has Y'.yj~ = q2(/3) + 1 + K 1 - q2(/3) = t
N o w (13 + 14) - ( l l + / 2 ) = ( c v - ct~)d~ ~- B u t as c~ < c~, solutions (x 3, y3) and (x 4, y4) are alternative optima. c) W h e n G is disconnected, there exist periods t 1 and t 2 such that t I and t 2 are production
Now, by Proposition 6 and condition (iii) of T h e o r e m 3, we obtain the main result of this section.
Theorem 7. For the uncapacitated lot-sizing problem if the production costs {c i} are nonincreasing, then the integrality property holds. It is easy to see that the result extends to constraints of the form Y]= i Y/< k, or ~ ' = ~yj > k, and also following the proofs to constraints limiting production over the first n' < n intervals, i.e., a single constraint of the form Y~j'=1Yi = k. In [9] a related result for the K - m e d i a n p r o b l e m on a path is given for a model containing different variable costs, but no fixed costs. T h e bipartite graph used in Proposition 5 was introduced in the latter paper.
4. Open questions It is natural to ask the geometric version of the 'integrality' question: " G i v e n an integral polyhedron, when is there a family of parallel slices whose intersection with the polyhedron creates no new vertices?". It would be interesting to know of other polyhedra having this p r o p e r t y for some or all objective functions. In addition it is practically useful even if the convex hull is not known explicitly, because as we saw in Proposition 2 the integrality p r o p e r t y m e a n s that the Lagrangian dual solves the original p r o b l e m for all k. T h e special polyhedra cited in the introduction have the integrality p r o p e r t y for all objective functions and also have " a u g m e n t i n g p a t h " properties. This suggests the possibility of an algorithm based on a u g m e n t i n g an optimal lot-sizing solution with k set-ups to obtain an optimal solution with k + 1 set-ups. T h e s e p r o b l e m s also have various s u p e r m o d u l a r properties, i.e., the set function r ( S ) = min{cx + f y : (x, y) ~ T, Yi = 17
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OPERATIONS RESEARCH LETTERS
0, i ~ S}, S _c {1. . . . . n} is typically supermodular. Is this a necessary condition for the integrality property to hold? Note that typically in formulating the lot-sizing problem there are unit storage costs h t as well as unit production costs p,. Eliminating the stock costs one obtains, c t = P t + ~7=thi, or c , - ct+ 1 = pt+ht-Pt+l. Thus the Wagner-Whiting assumption holds very often in practice as this only requires i) h t > 0, and ii) Pt is either constant, or its increase P t + l - P t is less than h, for all t. What is to be done when the Wagner-Whitin does not hold? Is there a 'good' description of the convex hull of X n {(x, y): F.~= i Yj k} other than that which can be obtained from the dynamic programming recursion (Eppen and Martin [2])? Here again the results of [9] provide a 'good' characterization for the case k = 2. =
Acknowledgement We are grateful to Stan van Hoesel for the counterexample shown in Example 1, and for the important suggestions by a referee.
18
February 1992
References [1] I. Barany, T.J. Van Roy and L.A. Wolsey, "Uncapacitated lot-sizing: The convex hull solutions", Math. Programming Stud. 22, 32-43 (1984). [2] G.D. Eppen and R.K. Martin, "Solving multi-item capacitated lot-sizing problems using variable definition", Oper. Res. 35, 832-848 (1987). [3] A.B. Gamble and W.R. Pulleyblank, "Forest cover and polyhedral intersection theorems", Math. Programming 45, 45-58 (1989). [4] A.M. Geoffrion, "Lagrangean relaxation for integer programming", Math. Programming Stud. 2, 82-114 (1974). [5] S. van Hoesel, Private communication (1990). [6] E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976. [7] G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, New York, 1988. [8] H.M. Wagner and T.M. Whitin, "Dynamic version of the economic lot-size model", Management Sci. 5, 89-96 (1958). [9] J.E. Ward, R.T. Wong, P. Lemke and A. Outjit, "Properties of the tree K-median linear programming relaxation", Research Report CC-878-29, Institute for Interdisciplinary Engineering Studies, Purdue University, August 1987. [10] L.J. White and M.L. Gillerson, "An efficient algorithm for minimum k-covers in weighted graphs", Math. Programming 8, 20-42 (1975).
February 1992
Lot-sizing polyhedra with a cardinality constraint El Houssaine Aghezzaf and Laurence A. Wolsey CORE, Unit:ersit~;Catholique de Lout,ain, Louvain-la-Neuve, Belgium Received October 1990 Revised April 1991
Given a family of mixed 0-1 polyhedra, we suppose that the convex hull of solutions is known. Now we add a cardinality constraint such that the sum of the 0-1 variables is exactly k. W h e n is the resulting polyhedron integral, or when does the resulting linear program have an integral optimal solution for all values of k? Various equivalent conditions are given, and it is shown that uncapacitated lot-sizing polyhedra have this property when the production costs are nonincreasing over time. integral polyhedra; cardinality constraint; lot-sizing
1. Introduction Given a mixed 0-1 polyhedron P c ~p+n i.e., P is the convex hull of solutions of some set T c ~ +P× {0, 1}", we consider the family of polyhedra: for k = l . . . . ,n
(1) and we ask when the polyhedra Pk are integral for all k. In a weaker form we are given an objective vector (c, f ) and we ask when the linear program min{cx + f y ' ( x , y) ~Pk} has an optimal solution in T for all k. In the latter case we say that P has the 'integrality property' with respect to (c, f ) . Various authors have raised this question for particular families of polyhedra, in particular White and Gillenson [10] for node covers, Ward et al. [9] for locations on tree networks, and Gamble and Pulleyblank [3] for forest covers, and it also arises naturally in the algorithms for matroids and matchings, see Lawler [5] or Nemhauser and Wolsey [6]. In Section 2 we give a
very simple geometric picture of the integrality property, and state equivalent conditions that are consequences of Lagrangian duality (see Geoffrion [4]). In Section 3 we consider the mixed 0-1 polyhedron arising from the uncapacitated lot-sizing problem. Though the family of polyhedra {Pk} are not integral, we show that if the production costs are nonincreasing (after normalization so that the storage costs are zero), then optimizing over the polyhedra *ok gives optimal solutions with the set-up variables y~ . . . . , y,, E {0, 1}.
2. Conditions for integrality We consider thc following optimization problems: LG(k)
G(k) LH(k)
= min{cx + f y : ( x , y ) ~ P k ,
Y ~ {0, 1}"},
H(k)=min{cx+fy:(x, y)
and L~(A)
clg(A) = min{cx + fy -A y" yj. (x, y)
0167-6377/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
13
Volume 11, Number 1
OPERATIONS RESEARCH LETTERS
The integrality property can now be formally stated as the condition H ( k ) = G ( k ) for k =
Proof. As P = cony(T), q~(A) = min{cx + f y A ~ ' 1Yj: (x, y) ~ T}. Thus by Lagrangian duality, max~{@(A) + Ak} = min{cx + fy: (x, y) conv(T), E T = l y i = k } = H ( k ) . []
l~.,,~n. Without loss of generality we assume throughout that the smallest and largest solutions in P have ~2jyj = 1 and ~2jy~ = n respectively. The following propositions tell us what happens when we slice an integral polyhedron with the cardinality constraint. P r o p o s i t i o n 1 [9]. I f xLP(k)=(xLP(k), yLP(k)) T is a vertex o f P k, xLP(k) is a convex combination o f two points (x l, yl), (x 2, y2) o f T h a v i n g Z y ) = K , , Y'.yf=K z f o r s o m e K l < k < K 2.
Proof. If the vertex is not a vertex of P, it must be at the intersection of an edge of P and the hyperplane consisting of the cardinality constraint. As P is integral, the edge joins two vertices in T. []
So L H ( k ) is the primal problem equivalent to the Lagrangian dual of L G ( k ) with the constraint En i=~YJ = k dualized, and the function H is the lower convex envelope of G. Thus we have the situation shown in Figure 1. If 1 = k~ < k 2 < . . . < k r = n are the values for which H ( k ) = G ( k ) , the slopes m i = ( G ( k i + l ) - G ( k i ) ) / ( k i + ! - k i ) of the function H are nondecreasing, and we see immediately that (i) L@(A) has an optimal solution (x, y ) ~ T with Ejyj = k i if and only if A ~ [m i_ 1, mi], i.e., A is a subgradient of the function H at k i. (ii) If k i < k < k i + l , then ki+ I - k
x L P ( k ) -- ki+l _ k i
P r o p o s i t i o n 2 [4]. (i) H(k) = maxx{cb(A) + Ak). (ii) G ( k ) = H ( k ) i f and only i f G ( k ) has a (global) subgradient at z = k.
+
xLP(k , -i,
k -k i
xLP(ki+l).
ki+ 1 - k i
[]
H(k)
~
February 1992
Valuesof G(k)
lope m1
~,..~
l°pe m2
[]
support o
1
2
3
4
5
k1
k2
k3
k4 Fig. 1.
14
6
k5
7
8
k6
k
V o l u m e 11, N u m b e r 1
OPERATIONS RESEARCH LETTERS
W e now restate the content of the propositions.
Theorem 3. Given an integral polyhedron P and an objective (c, f ), the following are equivalent: (i) the integrality property holds, (ii) G(k + 1) - G ( k ) is nondecreasing for k = l , . . . , n - l, (iii) whenever for some value of I~, the linear program min(cx + f y - t ~ j y j : (x, y) ~ P} has two integral optimal solutions (x 1, y l ) and (x 2, y2)
with ~ y ) = E I, ~ y / = r 2 with K I
3. Lot-sizing with nonincreasing production costs H e r e we consider the well-known uncapacitated economic lot-sizing p r o b l e m first studied by W a g n e r and Whitin [8]. Thus we consider the problem LG(k)
G(k) = min~c,x, t
Example 1 (van Hoesel [5]). Consider the uncapacitated lot-sizing p r o b l e m with the costs and d e m a n d s shown in the following table: 1
2
3
4
5
ci
8
0
5
3
1
f,
2
21
1
1
1
d,
I
1
1
1
I
Now we add the constraint that we set up k times. It is readily checked that G ( 1 ) = 42, G(2) = 31, G ( 3 ) = 31, G ( 4 ) = 30, and G ( 5 ) = 34. Thus G(3) - G(2) > G(4) - G(3) and the integrality condition does not hold. As (kj, k 2, k3, k 4 ) = (1, 2, 4, 5) the linear p r o g r a m LH(3) has an optimal solution (x, y) = ½xLP(2) + ½xLP(4) = (3, 2, l I I. I I l I~I ~_, .~, 3, 1, 3, z, ~, ~. with value H ( 3 ) = 30.5. F r o m now on we m a k e the W a g n e r - W h i t i n assumption that ct>c,+ ~ for all t. It is well known that G ( t ) = min t <~<,g(s, t), w h e r e g(s, t) = G(s - 1) + f, + Gd,,, and G(0) = 0. H e r e d , denotes Eti=sdr
Proposition 4. Under the Wagner-Whitin assumption, if G ( t ) = g(s, t), then g(s', t ' ) > g ( s , t') for all t' > t and s' < s.
+ ~f,y,,
Proof.
l
(x, y) ~ T,
g ( s ' , t') = g ( s ' , t) +cs,d,+lj,
L yj = k , j
Februuary 1992
> g( s, t) + c,dt+ l.,,
1
=g(s,t') where
T=
where
(
'
x,y)"
~xi>
Y'~difor t=l
i=1
i=1
inequality
uses
G(t) = g ( s , t)<_
[]
..... n-l,
L Xi:
L di, 0 <_x t <_Myt,
i=1
i=1
Yt ~ {0, 1} for all t}. An explicit description of P = c o n v ( T ) is given in Barany et al. [1]. T h e following example shows that the integrality p r o p e r t y does not hold for all values of c and
f.
the
g(s', t), and c,
,
Now we suppose that m i n { E t c , x , + E J t y t, has two optimal solutions (x 1, y l ) and (x 2, y2) with Y~y) = K I, Z y ~ = K2 and K2 - K l > 2. We will show that there is an alternative optimal solution w i t h Y~jYi = k for some k such that K l < k < K 2. Without loss of generality, we can replace f, by m a x ( f , , 0) in the objective function, and only consider solutions for which each period with yj = 1 is a production period. W e construct a bipartite graph G = (V~, V2, E ) as follows: Vi = {j ~ {1 . . . . . n}: yf = 1} for i = 1, 2.
(x, y ) ~ T }
15
Volume 11, Number 1
OPERATIONS RESEARCH LETTERS
ql(~)
II
February1992
t
I
V1
v2 I
I
q2 (]3)
I
I
t+2
Fig. 2.
(c~,/3) ~ E if in solution 1 the p r o d u c t i o n in period a e V 1 covers period /3 e V2, or vice versa. It is readily verified that G is acyclic.
1. If q2(/3) -- ql(/3) < 0 for all /3 ~ V 1 with deg(/3) = 3, this holds for the last such node. But then Kl = ql(/3) + t for some integer t, and K2 _< q2(/3) + 2 + t (see Figure 2). Thus K2 - - K l < q2(/3) -q~(/3) + 2 < 2, which is a contradiction. Thus q2(/3) -- ql(/3) > 0 for some /3 ~ V 1 with deg(/3) = 3. As the difference can only increase by unit amounts, the claim follows. []
5. I f the graph G is connected, a) there exists a n o d e / 3 ~ VI with deg(/3) > 3; b) if there is no node /3 ~ V l with d e g ( / 3 ) > 4, there exists a n o d e / 3 ~ V] with deg(/3) = 3, and Proposition
ql(/3)
=
I{1 . . . . .
/3} n V1 I = 1{1,...,/3}
n V2 I
Proposition 6. I f (x l, yl), (x 2, y 2 ) a r e optimal solutions of min{cx + f y : (x, y ) ~ P} with E y ) = •l, ~ Y f = K2, and K2 - K l > 2, then there exists an alternaffve optimal solution of intermediate cardinality.
= q2(/3)" P r o o f . a) F r o m the definition, q l ( 1 ) = q 2 ( l ) = 1, q](n) = K1 and q2(n) = K2 with K2 >_ K 1 "{'- 2. If/31 and/32 are successive nodes of V 1, and deg(/3 l) _< 2, q2(/32 ) -- q1(/32 ) ~ q2(/31) -- ql(/31)" If deg(/3) < 2 for all / 3 ~ V 1 and /3* is the last node of V I, q2(/3") - ql(/3*) -
a) V 1 contains a node a of degree > 4. Part of the graph G is shown in Figure 3, where a, E are successive production nodes for ( x ' , y t ) and /3, y, 6 are production periods for (x 2, y2), namely the second, third, and last nodes of V 2 covered by a. Proof.
(X
V1
[3
y Fig. 3.
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V1
V2
y
periods for both solutions 1 and 2. If E j=l t2- Ll y f _ ~t2-j=tll.yjl __~>/(2 -- /¢1, the above a r g u m e n t can be applied to this interval. Otherwise we have for some interval that 1 _< ~ t%?j=,, : - ~J _ 5Zt~j=,,J.y/i_
8 Fig. 4.
Consider the solution (X 2, y2). G ( 7 - 1 ) = g(/3, y - 1) and G(+ - 1 ) = g ( y , 6 - 1). T h e r e fore by Proposition 4, g(/3, E - 1) < g ( a ,
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E - 1) as a < / 3 ,
g ( y , e - I) < g ( / 3 , e -
and
I) a s / 3 _<-/.
Thus g ( y , • - 1) < g ( / 3 , • - 1) < g ( a , • - 1). H o w e v e r from solution (x 1, y t ) we have that G ( • - 1 ) = g ( a , • - 1 ) < g ( y , • - 1). T h u s g(/3, • - l ) = g ( y , • - 1). In o t h e r words g(/3, 6 - 1 ) + ct3d~,,-1 = g(Y, ~ - 1 ) + cz, da, ~_ I. As ct~ > cz, , this implies that g ( / 3 , 6 - 1 ) < g ( y , 6 - 1 ) . Now we can modify solution (x e, y 2 ) by setting yr2 = 0, x~ = dt~,~_ ~ and we have an alternative optimal solution with k = Ke - 1. b) V 1 contains no node of degree > 4. Let /3 be the first node of degree 3 with q1(/3)= q2(/3 ) as described in Proposition 5, see Figure 4. T h e cost of solution (x l, y l ) is 11 = G(/3 - 1) +ft~ +ct~d~,, ~ + H ( • ) , and the cost of solution (x , y2) is 1 2 = G ( y - 1 ) + f ~ + c r d r , ~ _ 1+H(6), where H ( w ) is the cost of an optimal solution for periods o~, w + 1. . . . . n. Now consider an alternative solution (x 3, y3) o b t a i n e d by using solution 2 for the first ( y - 1) periods, producing in y till period • - 1 and then using solution 1. T h e cost is 13 = G ( T - 1) + f ~ + c~,d~,.~_ 1 + H ( • ) . Note that this solution has Y'.yj~ = q2(/3) + 1 + K 1 - q2(/3) = t
N o w (13 + 14) - ( l l + / 2 ) = ( c v - ct~)d~ ~- B u t as c~ < c~, solutions (x 3, y3) and (x 4, y4) are alternative optima. c) W h e n G is disconnected, there exist periods t 1 and t 2 such that t I and t 2 are production
Now, by Proposition 6 and condition (iii) of T h e o r e m 3, we obtain the main result of this section.
Theorem 7. For the uncapacitated lot-sizing problem if the production costs {c i} are nonincreasing, then the integrality property holds. It is easy to see that the result extends to constraints of the form Y]= i Y/< k, or ~ ' = ~yj > k, and also following the proofs to constraints limiting production over the first n' < n intervals, i.e., a single constraint of the form Y~j'=1Yi = k. In [9] a related result for the K - m e d i a n p r o b l e m on a path is given for a model containing different variable costs, but no fixed costs. T h e bipartite graph used in Proposition 5 was introduced in the latter paper.
4. Open questions It is natural to ask the geometric version of the 'integrality' question: " G i v e n an integral polyhedron, when is there a family of parallel slices whose intersection with the polyhedron creates no new vertices?". It would be interesting to know of other polyhedra having this p r o p e r t y for some or all objective functions. In addition it is practically useful even if the convex hull is not known explicitly, because as we saw in Proposition 2 the integrality p r o p e r t y m e a n s that the Lagrangian dual solves the original p r o b l e m for all k. T h e special polyhedra cited in the introduction have the integrality p r o p e r t y for all objective functions and also have " a u g m e n t i n g p a t h " properties. This suggests the possibility of an algorithm based on a u g m e n t i n g an optimal lot-sizing solution with k set-ups to obtain an optimal solution with k + 1 set-ups. T h e s e p r o b l e m s also have various s u p e r m o d u l a r properties, i.e., the set function r ( S ) = min{cx + f y : (x, y) ~ T, Yi = 17
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0, i ~ S}, S _c {1. . . . . n} is typically supermodular. Is this a necessary condition for the integrality property to hold? Note that typically in formulating the lot-sizing problem there are unit storage costs h t as well as unit production costs p,. Eliminating the stock costs one obtains, c t = P t + ~7=thi, or c , - ct+ 1 = pt+ht-Pt+l. Thus the Wagner-Whiting assumption holds very often in practice as this only requires i) h t > 0, and ii) Pt is either constant, or its increase P t + l - P t is less than h, for all t. What is to be done when the Wagner-Whitin does not hold? Is there a 'good' description of the convex hull of X n {(x, y): F.~= i Yj k} other than that which can be obtained from the dynamic programming recursion (Eppen and Martin [2])? Here again the results of [9] provide a 'good' characterization for the case k = 2. =
Acknowledgement We are grateful to Stan van Hoesel for the counterexample shown in Example 1, and for the important suggestions by a referee.
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References [1] I. Barany, T.J. Van Roy and L.A. Wolsey, "Uncapacitated lot-sizing: The convex hull solutions", Math. Programming Stud. 22, 32-43 (1984). [2] G.D. Eppen and R.K. Martin, "Solving multi-item capacitated lot-sizing problems using variable definition", Oper. Res. 35, 832-848 (1987). [3] A.B. Gamble and W.R. Pulleyblank, "Forest cover and polyhedral intersection theorems", Math. Programming 45, 45-58 (1989). [4] A.M. Geoffrion, "Lagrangean relaxation for integer programming", Math. Programming Stud. 2, 82-114 (1974). [5] S. van Hoesel, Private communication (1990). [6] E.L. Lawler, Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York, 1976. [7] G.L. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, Wiley, New York, 1988. [8] H.M. Wagner and T.M. Whitin, "Dynamic version of the economic lot-size model", Management Sci. 5, 89-96 (1958). [9] J.E. Ward, R.T. Wong, P. Lemke and A. Outjit, "Properties of the tree K-median linear programming relaxation", Research Report CC-878-29, Institute for Interdisciplinary Engineering Studies, Purdue University, August 1987. [10] L.J. White and M.L. Gillerson, "An efficient algorithm for minimum k-covers in weighted graphs", Math. Programming 8, 20-42 (1975).