A Multi–Item Production Planning Model with
Setup Times: Algorithms, Reformulations, and
Polyhedral Characterizations for a Special Case 1
Andrew J. Miller 2 , George L. Nemhauser 3 , and Martin W.P. Savelsbergh
3
August 2001
Abstract
We study a special case of a structured mixed integer programming model that
arises in production planning. For the most general case of the model, called PI,
we have earlier identified three families of facet–defining valid inequalities: (l, S)
inequalities (introduced for the uncapacitated lot–sizing problem by Barany, Van
Roy, and Wolsey), cover inequalities, and reverse cover inequalities. PI is N P–hard;
in this paper we focus on a special case, called PIC. We describe a polynomial algorithm for PIC, and we use this algorithm to derive an extended formulation of
polynomial size for PIC. Projecting from this extended formulation onto the original
space of variables, we show that (l, S) inequalities, cover inequalities, and reverse
cover inequalities suffice to solve the special case PIC by linear programming. We
also describe fast combinatorial separation algorithms for cover and reverse cover
inequalities for PIC. Finally, we discuss the relationship between our results for PIC
and a model studied previously by Goemans.
Keywords: Mixed integer programming, polyhedral combinatorics, production
planning, capacitated lot–sizing, fixed charge network flow, setup times
1
Some of the results of this paper have appeared in condensed form in “Facets, algorithms, and
polyhedral characterizations of a multi–item production planning model with setup times”, Proceedings
of the Eighth Annual IPCO conference,, pp. 318–332, by the same authors.
2
CORE, Université Catholique de Louvain, Belgium
3
Georgia Institute of Technology, School of Industrial and Systems Engineering, Atlanta, GA
30332-0205, USA
This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming. The scientific
responsibility is assumed by the authors.
This research was also supported by NSF Grant No. DMI-9700285 and by Philips Electronics North
America.
1
1
Introduction
In Miller et al. [2000a] we introduced a mixed integer programming model that occurs
as a relaxation of a number of structured MIP problems, such as capacitated production planning problems with setup times and fixed–charge network flow problems. We
originally derived this model as a single–period relaxation of the multi–item capacitated
lot–sizing problem with setup times (MCL) (Miller et al. [2000b]). We call this relaxation PI, for preceding inventory, because when it is used as a single–period submodel of
MCL, it incorporates stock or inventory entering the period in question from the period
before. PI is N P–hard.
PI can be formulated as
min
P
X
pi x i +
i=1
P
X
qi yi +
i=1
P
X
hi si
(1)
i=1
subject to
xi + si ≥ di , i = 1, ..., P,
P
X
xi +
i=1
i
P
X
ti y i ≤ c,
i=1
i
x ≤ (c − t )y i , i = 1, ..., P,
i
i
x , s ≥ 0, i = 1, ..., P,
i
y ∈ {0, 1}, i = 1, ..., P.
(2)
(3)
(4)
(5)
(6)
Perhaps the simplest way to view PI is as a single–node multi–item flow model with
two inbound arcs. The first of these arcs has capacity c, which is jointly imposed on P
items. Moreover, a fixed capacity usage ti ≥ 0 must be incurred if any amount of item
i is shipped along this arc. (Assuming that capacity is measured in time units, ti can
be considered as the setup time.) The variable xi is the amount of flow shipped along
this arc with unit cost pi , and the binary variable y i indicates whether the fixed capacity
usage for item i is incurred; the cost for setting y i = 1 is q i . The second arc can also
be used by each of the items, and it is uncapacitated. The variables si represent the
amount of flow on this second, uncapacitated arc for item i; the unit cost for shipping i
along this arc is hi . The total amount of flow shipped along these two arcs must be at
least di , i = 1, ..., P .
In Miller et al. [2000a], we studied the polyhedral structure of this model. Among
other results, we characterized the extreme points of the convex hull, and we defined
cover inequalities and reverse cover inequalities for PI, which induce facets of the convex
hull. We also showed that the (l, S) inequalities of Barany, Van Roy, and Wolsey [1984]
are facet–defining for PI.
2
In this paper we analyze a special case of PI that occurs when setup times and
demand are constant for all items, i.e., when ti = t ≥ 0, di = d > 0, i = 1, ..., P . In
defining this special case we also assume that hi > 0, q i > 0, i = 1, ..., P . For this
case, called PIC, we give polynomial–time algorithms, including algorithms based on an
extended formulation of PIC. By projecting onto the original (x, y, s) space from the
extended space of this reformulation, we show that (l, S) inequalities, cover inequalities,
and reverse cover inequalities suffice to solve the special case PIC by linear programming
(LP).
Many researchers have endeavored to characterize simple MIP models in terms of
linear inequalities. For example, in a pioneering paper on using cutting planes to solve
structured mixed integer programs, Padberg, Van Roy, and Wolsey [1985] introduced
flow cover inequalities for the single node fixed charge flow model (SNFC). They showed
that these inequalities, along with trivial facets, suffice to define the convex hull of SNFC
in the special case when arc capacities are constant. Generalizations of SNFC have been
studied by Van Roy and Wolsey [1986] and Goemans [1989], among others.
When ti = 0, i = 1, ..., P , PI resembles SNFC, but even in this case PI has a different,
more complicated structure than SNFC. Both PI and SNFC have flow variables xi , setup
variables y i , and a joint capacity constraint (3). Moreover, both have variable upper
bounds similar to (4), although in SNFC these are generally tighter than in PI. PI, in
addition, has constraints (2), which impose a demand on each of the flow variables, and
variables si , which serve as slack variables for (2) and, for example, allow demand in
excess of capacity to be satisfied. When there exists some items i with ti > 0, the
structure of PI is complicated even further.
Because of these additional structural complications, identifying a set of inequalities
that characterizes the convex hull of PIC appears to be much more difficult than identifying a set that characterizes the constant–capacity case of SNFC. Identifying such a
set for PIC remains an open question. We have, however, succeeded in characterizing a
set of inequalities that, given the conditions on the costs stated above, allows us to solve
PIC by LP.
An outline of the remainder of the paper follows. In Section 2, we state results
concerning the polyhedral structure of PIC that follow immediately from results for PI
given in Miller et al. [2000a]. Among these results are the (l, S) inequalities, cover
inequalities, and reverse cover inequalities. We will use these results in the development
of later sections.
We begin Section 3 by noting that PI remains N P–hard if either setup times or
demand, but not both, are constant. We then give a combinatorial polynomial algorithm
for PIC. In Section 4 we present an extended formulation for PIC motivated by the
polynomial algorithm described in Section 3, and we show that the LP relaxation of
this extended formulation is integral. By considering the projection of the polyhedron
defined by the constraints of this linear program onto the space of the original variables,
3
we show that the inequalities presented in Section 2 for PIC are sufficient to solve PIC
by LP in the original variable space.
In Section 5 we give combinatorial, polynomial–time separation algorithms for cover
and reverse cover inequalities for PIC. In Section 6, we discuss the implications of our
results on a model studied by Goemans [1989] that generalizes SNFC. Finally, we summarize our contributions and discuss potential areas of future research.
2
Polyhedral Results for PIC
The results of this section follow from results for the general case of PI stated by Miller
et al. [2000a], and are here restated for PIC for ease of reference.
Let P = {1, ..., P }. Given an instance of PIC, we denote the set of points defined by
(2)–(6) as X P IC . To avoid uninteresting cases, we assume that c > t + d.
To avoid having to consider unbounded solutions, we assume throughout that hi > 0,
i = 1, .., P . Also, we assume throughout that q i > 0, i = 1, ..., P , which ensures that in
every optimal solution, y i = 1 if and only if xi > 0. (This assumption is motivated by
the fact that in production planning setups generally do not take place unless production
occurs.) We make no assumptions on pi , i = 1, ..., P ; thus, for a given i, pi < 0 may be
true. In discussing a set of points X ⊂ {(x, y, s) : xi , si ≥ 0, 0 ≤ y i ≤ 1, i = 1, ..., P }, we
will use
Definition 1 The point (x̄, ȳ, s̄) is a dominant solution of X if there exists some cost
vector (p, q, h), such that hi > 0, q i > 0, i = 1, ..., P , for which optimizing (1) over X
yields (x̄, ȳ, s̄) as the unique optimal solution.
Proposition 1 (Characterization of the dominant solutions of conv(X P IC )) Given an
extreme point (x̄, ȳ, s̄) of conv(X P IC ), let Q = {i ∈ P : ȳ i = 1}. Partition Q into Qd
and Qr so that Qd = {i ∈ Q : x̄i = d} and Qr = {i ∈ Q : ȳ i = 1, x̄i 6= d}. Then every
extreme point (x̄, ȳ, s̄) of conv(X P IC ) has the form
0
x̄i = c −
P
x̄i = di , ȳ i = 1, s̄i = 0, i ∈ Qd
i∈Qd (t + d) − t, ȳ
x̄i = ȳ i = 0, s̄i
i0
0
0
= 1, s̄i = (d − x̄i )+ , i0 ∈ Qr
= d, i ∈ P \ Q,
(7)
where either Qr = ∅ or |Qr | = 1.
Proposition 2 The inequality
si + dy i ≥ d
is valid and facet–inducing for conv(X P IC ), i = 1, ..., P .
4
(8)
Note that (8) implies the bounds si ≥ 0, i = 1, ..., P ; therefore these bounds never induce
facets. Also note that these inequalities are (l, S) inequalities, which were introduced for
the uncapacitated lot–sizing problem by Barany, Van Roy, and Wolsey [1984]. Van Roy
and Wolsey [1987] generalized these inequalities to fixed charge network flow problems.
c
In order to define the next two sets of inequalities, we first define M = b t+d
c. To
avoid uninteresting cases, we assume throughout that c > M (t + d). Let
λ = (M + 1)(t + d) − c;
observe that λ > 0. Now define a cover of PIC to be any set S ⊆ P such that |S| ≥ M +1.
Proposition 3 (Cover Inequalities) Given a cover S of PIC, let T = P \ S, and let
(T 0 , T 00 ) be any partition of T . Then
X
si ≥ (|S| − M )λ +
i∈S
X
(d − λ)(1 − y i ) +
i∈S
λ X i
(x − (d − λ)y i )
t+d
0
(9)
i∈T
is a valid inequality for X P IC . Moreover, if d > λ, it is facet–inducing for conv(X P IC ).
Define a reverse cover of PIC to be any set S such that ∅ 6= S ⊂ P with |S| ≤ M .
Proposition 4 (Reverse Cover Inequalities) Let S 6= ∅ be a reverse cover of PIC, let
T = P \ S, and let (T 0 , T 00 ) be any partition of T . Then
X
si ≥ |S|(t + d)
i∈S
X
i∈T 0
yi −
X
t(1 − y i ) −
i∈S
X
((c − t)y i − xi )
(10)
i∈T 0
is valid for X P IC . Moreover, if T 0 6= ∅ and d > λ, it is facet–inducing for conv(X P IC ).
3
Complexity
In this section, we first note that PI is N P–hard if either demand or setup times, but
not both, are assumed to be constant. We then describe a polynomial algorithm for PIC.
Proposition 5 If either setup times or demand, but not both, are constant, then PI is
N P–hard.
For the case in which setup times are all 0 and hi = h > p = pi = 0, i = 1, ..., P , there
is a reduction from the continuous 0–1 knapsack problem (see Miller [1999], Marchand
and Wolsey [1999]). For the case in which di = 1, i = 1, ..., P , and hi > 0, p = pi = 0, i =
1, ..., P , there is a reduction from the 0–1 knapsack problem (see Miller [1999]).
5
Since PI is N P–hard, we cannot expect to find an explicit description of the convex
hull in general, or expect to solve PI by optimizing (1) over a specific set of linear
inequalities. Now we consider the case when demand and setup times are constant and
hi > 0, q i > 0, i = 1, ..., P , i.e., PIC.
Proposition 6 PIC is polynomially solvable.
Proof: We describe an O(P 2 ) algorithm for PIC. To initialize the algorithm, we sort
the items so that dp[1] + q [1] − dh[1] ≤ ... ≤ dp[P ] + q [P ] − dh[P ] . Then, for each i0 ∈ P,
we find an extreme point solution with Qr = i0 that has the minimum objective function
value among all extreme points with Qr = i0 , where Qr is defined as in Proposition 1.
0
0
0
To do this, for each i0 ∈ P, we first set y i = 1, xi = c − t, and si = 0, and
we also set y i = 0, xi = 0, si = d, i ∈ P \ i0 . Then we put i ∈ P \ i0 greedily into
Qd = {i : y i = 1, xi = d}, proceeding in the order in which we have sorted the elements,
0
decreasing xi by t + d for each element we put into Qd . We continue until doing so no
longer decreases the value of the objective function (1), or until we do not have enough
capacity left to include any more items in Qd .
We then find the extreme point solution with Qr = ∅ that has the minimum objective
function value among all such extreme points, using a similar greedy procedure. Because
of Proposition 1, the solution having the minimum value among these P + 1 points must
be an optimal solution. 2
The above algorithm can be trivially modified to solve PIC if we remove from
the problem definition the assumption hi > 0, i = 1, ..., P . If we do not assume
q i > 0, i = 1, ..., P, in defining PIC, the above algorithm is no longer applicable, but
there is a polynomial algorithm based on a characterization of the extreme points (see
Miller [1999]). However, in this case we do not know how to use the result to prove the
correctness of a linear programming formulation in the original variable space, as we can
for PIC.
4
Reformulations of PIC
The main result of this section is that (l, S) inequalities, cover inequalities, and reverse
cover inequalities suffice to solve PIC by LP. To achieve this result, we first present
an extended formulation of PIC, that is, a linear reformulation of PIC with additional
variables that has integral extreme points. The projection of the extreme points of this
formulation onto the original (x, y, s) space is the set of dominant solutions of X P IC .
We then modify this formulation algebraically to obtain formulations that have the
same projection onto the (x, y, s) space. The last of these formulations incorporates
the inequalities from Section 2, in addition to the extra variables introduced for the
6
first extended formulation. By applying Farkas’ lemma, we implicitly characterize the
projection of this final formulation onto the (x, y, s) space. Moreover, we also show that
all of the dominant solutions of the polyhedron defined by (2)–(5), (8), (9), (10), and the
bounds y i ≤ 1, i = 1, ..., P , are in the projection just mentioned. From these facts we
establish an equivalence between the set of dominant solutions of (2)–(5), (8), (9), (10),
and y i ≤ 1, i = 1, ..., P , and the set of dominant solutions of X P IC . This shows that the
set of inequalities just listed suffices to solve PIC by LP.
Given an instance of PIC, let X̃ P IC be the set of dominant solutions to X P IC . We
will assume that d > λ; if this does not hold, the development is similar but simpler.
To reformulate PIC, we introduce new binary variables based on the algorithm described in the proof of Proposition 6. In defining these variables, we say that capacity
is met if (3) is tight, and we say that i is produced at demand if exactly d units of i are
produced. The variables are
½
∆m =
i
δm
ρim
i
αm
1 if m items are produced at demand
0 otherwise

 1 if m items are produced at demand
and i is produced at demand
=

0 otherwise

 1 if m items are produced at demand
and i is produced but not at demand
=

0 otherwise

 1 if m items are produced at demand
and demand for i is satisfied entirely from inventory
=

0 otherwise.
Note that δ0i = 0, i = 1, ..., P , must hold by definition, but we include these variables
in order to simplify the formulation that follows. In addition, we define the continuous
variables γ i , i = 1, ..., P to be the amount of inventory of item i that is carried in excess
of what is needed to satisfy demand.
P IC
We call the set of points defined by the following set of constraints X(∆,δ,ρ,α,γ)
:
xi = d
yi =
M
X
m=0
M
X
i
δm
+
M
X
(d − λ + (M − m)(t + d))ρim , i = 1, ..., P
(11)
m=0
i
(δm
+ ρim ), i = 1, ..., P
m=0
7
(12)
si = d(
M
X
i
αm
) + λρiM + γ i , i = 1, ..., P
(13)
m=0
M
X
∆m = 1
m=0
i
i
δm
+ ρim + αm
= ∆m , i = 1, ..., P, m
P
X
i
δm
= m∆m , m = 0, ..., M
i=1
P
X
ρim ≤ ∆m , m = 0, ..., M
i=1
i
(14)
= 0, ..., M
(15)
(16)
(17)
γ ≥ 0, i = 1, ..., P
M +1
∆ ∈ IB
P (M +1)
, δ ∈ IB
P (M +1)
, ρ ∈ IB
(18)
P (M +1)
, α ∈ IB
.
(19)
P IC
P IC
Now let X(∆,δ,ρ,α)
= {(x, y, s, ∆, δ, ρ, α) ∈ X(∆,δ,ρ,α,γ)
: γ i = 0, i = 1, ..., P }. Also, define
P roj(x,y,s) X to be the projection of a set of points X onto the space of the (x, y, s)
variables.
P IC
= X̃ P IC .
Proposition 7 P roj(x,y,s) X(∆,δ,ρ,α)
Proof: From Proposition 1 and from the variable definitions. 2
P IC
Corollary 8 The projection onto the (x, y, s) space of the dominant solutions of X(∆,δ,ρ,α,γ)
is X̃ P IC .
Proof: From Proposition 7 and from the fact that γ i = 0, i = 1, ..., P , in every dominant
P IC
solution of X(∆,δ,ρ,α,γ)
. 2
The algorithm in Proposition 6 can be seen as identifying the best solution to the
0
above formulation by searching among the O(P 2 ) choices of ρim . If it sets ρim0 = 1,
for some i0 and m0 , then clearly it would also set ∆m0 = 1, ∆m = 0, m 6= m0 . The
i = 0, i = 1, ..., P, m 6= m0 , and it would set δ i0 = 0. The values
algorithm would set δm
m0
i
0
of δm0 , i 6= i , would depend on the ordering defined in the algorithm: for the items with
i to 1, and for the other
the m smallest values of dpi + q i − dhi , the algorithm sets δm
0
i
items it sets δm0 to 0. (Note that if the capacity constraint (3) is not tight, then this
corresponds to the case ρim = 0, i = 1, ..., P, m = 0, ..., M . Also note that choosing values
i , i = 1, ..., P, m = 0, ..., M , and ∆ , m = 0, ..., M , uniquely defines the possible
for ρim , δm
m
i , i = 1, ..., P, m = 0, ..., M , and that γ i = 0, i = 1, ..., P in every optimal
value for each αm
solution.)
8
P IC
Now we consider the polyhedral structure of conv(X(∆,δ,ρ,α,γ)
). Let m be any integer
between 0 and M ; dropping the subscript m for convenience, we obtain the subformulation.
δ i + ρi + αi = 1, i = 1, ..., P,
P
X
i=1
P
X
(20)
δi = m
(21)
ρi ≤ 1
(22)
i=1
i i
δ , ρ , αi ≥ 0, i = 1, ..., P.
(23)
Note that if we fix ∆m = 1, (20)–(22) are equivalent to the constraints (15)–(17) for m.
Lemma 9 The constraint matrix associated with (20)–(23) is a network flow matrix.
P IC
P IC
Let XLP
(∆,δ,ρ,α,γ) be defined by replacing (19) in X(∆,δ,ρ,α,γ) with
i
i
δm
, ρim , αm
≥ 0, i = 1, ..., P, m = 0, ..., M ;
(24)
Note that these bounds imply 0 ≤ ∆m ≤ 1, m = 0, ..., M .
Proposition 10 The polyhedron associated with (14)–(17), (24) is integral.
Proof: Because of Lemma 9 and of (14), every fractional solution of (14)–(17), (24)
is a convex combination of M + 1 solutions to the network flow subproblems (20)–(23)
indexed by each m. Since every solution to the network flow subproblems is a convex
combination of the subproblem extreme points, every fractional solution of (14)–(17),
(24) is therefore a convex combination of extreme point solutions. 2
Corollary 11 The polyhedron associated with (14)–(18), (24) is integral.
P IC
P IC ).
Proposition 12 P roj(x,y,s) XLP
(∆,δ,ρ,α) = conv(X̃
P IC
P IC
Proof: From Proposition 10, XLP
(∆,δ,ρ,α) = conv(X(∆,δ,ρ,α) ). From Proposition 7, the
P IC
projection of conv(X(∆,δ,ρ,α)
) onto the (x, y, s) space is conv(X̃ P IC ). 2
The following corollary follows immediately from the above proposition, and from
P IC
the fact that any unique optimal solution obtained by optimizing (1) over XLP
(∆,δ,ρ,α,γ)
will have γ i = 0, i = 1, ..., P .
9
P IC
Corollary 13 Optimizing (1) over XLP
(∆,δ,ρ,α,γ) solves PIC.
Thus we have shown how to solve PIC by LP. Moreover, since the formulation defining
P IC
2
XLP
(∆,δ,ρ,α,γ) has O(P ) variables and constraints, we have shown how to solve PIC by
LP in polynomial time.
In order to use these results to help us explicitly define the set of inequalities needed
to solve PIC by LP in the original space of (x, y, s) variables, we will manipulate (11)–
(18), (24), into a different formulation which yields the same projection onto the (x, y, s)
space. We first eliminate the α variables in (11)–(18). By substituting first (15) and
then (12) into (13), we obtain
si = d − dy i + λρiM + γ i , i = 1, ..., P,
(25)
with which we can replace (13). By substituting (12) into (11), we obtain
xi = dy i +
M
X
(−λ + (M − m)(t + d))ρim , i = 1, ..., P,
(26)
m=0
which can replace (11). Now (15) is the only remaining constraint in which the α variables
i ≥ 0, i = 1, ..., P, m = 0, ..., M , we can simplify (15) to
are present. Since αm
i
δm
+ ρim ≤ ∆m , i = 1, ..., P, m = 0, ..., M,
(27)
P IC
and thus we have an equivalent formulation of XLP
(∆,δ,ρ,α,γ) that does not contain α
variables.
We continue by substituting (25) into (26) to obtain
M
−1
X
xi = d − si +
(−λ + (M − m)(t + d))ρim + γ i , i = 1, ..., P.
(28)
m=0
Note that we can replace (28) (and thus (26)) with the two inequalities
M
−1
X
xi ≥ d − si +
(−λ + (M − m)(t + d))ρim + γ i , i = 1, ..., P,
(29)
m=0
P
X
i
x ≤
i=1
P
X
i
(d − s +
i=1
M
−1
X
(−λ + (M − m)(t + d))ρim + γ i ).
(30)
m=0
By using (25) and (12) to substitute for si , i = 1, ..., P, we see that (30) is equivalent to
P
X
i=1
xi ≤
M
P X
M
X
X
i
dδm
+
(d − λ + (M − m)(t + d))ρim ).
(
i=1 m=0
m=0
10
(31)
(The inequality (31) can also be obtained directly from (11), but the above derivation
allows us to use it to replace (30).) We can also rewrite (25) and (29), respectively, as
si + dy i − d = λρiM + γ i , i = 1, ..., P,
xi + si − d ≥
M
−1
X
(32)
(−λ + (M − m)(t + d))ρim + γ i , i = 1, ..., P.
(33)
m=0
P IC
We call XLP
(∆,δ,ρ,γ) the polyhedron defined by (12), (31)–(33), (14), (27), (16)–(18),
P IC
(24). For ease of reference, we list the formulation of XLP
(∆,δ,ρ,γ) again:
M
X
i
y =
i
(δm
+ ρim ), i = 1, ..., P
(12)
m=0
P
X
xi ≤
i=1
i
P
M
M
X
X
X
i
(d
δm
+
(d − λ + (M − m)(t + d))ρim )
i=1
s + dy i − d
m=0
= λρiM
M
−1
X
xi + si − d ≥
(31)
m=0
+ γ i , i = 1, ..., P
(−λ + (M − m)(t + d))ρim + γ i , i = 1, ..., P
(32)
(33)
m=0
M
X
∆m = 1
m=0
i
δm
+
P
X
i=1
P
X
(14)
ρim ≤ ∆m , i = 1, ..., P, m = 0, ..., M
(27)
i
δm
= m∆m , m = 0, ..., M
(16)
ρim ≤ ∆m , m = 0, ..., M
(17)
i=1
γ i ≥ 0, i = 1, ..., P
(18)
i
i
δm
, ρim , αm
≥ 0, i = 1, ..., P, m = 0, ..., M.
(24)
P IC
Corollary 14 Optimizing (1) over XLP
(∆,δ,ρ,γ) solves PIC.
P IC
Now let X̄ P IC = P roj(x,y,s) XLP
(∆,δ,ρ,γ) . From Corollary 14 we immediately have
Corollary 15 The set of dominant solutions of X̄ P IC is X̃ P IC .
11
Also, from Proposition 12 and the definition of γi , i = 1, ..., P , we have
Corollary 16 X̄ P IC ⊂ conv(X P IC ).
We can obtain an implicit description of a set of inequalities which is sufficient to
solve PIC by LP in the (x, y, s) space by considering the dual polyhedron of (12), (31)–
(33), (14), (27), (16)–(18), (24). To see this, given a fixed point (x̄, ȳ, s̄), associate dual
variables ω i with constraints (12), β with constraint (31), σ i with constraints (32), π i with
i with constraints (27), τ
constraints (33), η with constraint (14), νm
m with constraints
(16), and κm with constraints (17).
Let X̂ P IC be the polyhedron defined by (2)–(5), (8), (9), (10), and the bounds
i
y ≤ 1, i = 1, ..., P ; we then have
Proposition 17 Given any (x̄, ȳ, s̄) ∈ X̂ P IC , (x̄, ȳ, s̄) ∈ X̄ P IC if and only if
η+
P
X
ω i ȳ i +
i=1
P
X
σ i (s̄i + dȳ i − d) −
i=1
P
X
π i (x̄i + s̄i − d) + β
i=1
P
X
x̄i ≤ 0
(34)
i=1
for all extreme rays of the dual cone
i
(δm
)
(ρim )
(ρiM )
i
(γ )
(∆m )
i
ω i + dβ + τm − νm
≤ 0, i = 1, ..., P, m = 0, ..., M
i
ω + (d − λ + (M − m)(t + d))β − (−λ + (M − m)(d + t))π
(35)
i
i
−κm − νm
≤ 0, i = 1, ..., P, m = 0, ..., M − 1
(36)
i
(37)
i
ω + (d − λ)β + λσ − κM −
i
i
νM
≤ 0, i = 1, ..., P
i
σ − π ≤ 0, i = 1, ..., P
η = mτm − κm −
P
X
(38)
i
νm
, m = 0, ..., M
(39)
i=1
β ≥ 0; π i ≥ 0, i = 1, ..., P ; κm ≥ 0, m = 0, ..., M ;
i
νm
≥ 0, i = 1, ..., P, m = 0, ..., M.
(40)
Proof: This follows from Farkas’ Lemma. 2
Note that, in constructing the dual of (12), (31)–(33), (14), (27), (16)–(18), (24) for
a given point (x̄, ȳ, s̄), we assume that all the inequality constraints have been written
as greater than or equal to constraints in which variables appear on the left hand side
of the inequality and constant terms (including the fixed point (x̄, ȳ, s̄)) appear on the
12
right hand side. Thus, for example, we assume that the constraints (33) are expressed
as
−
M
−1
X
(−λ + (M − m)(t + d))ρim − γ i ≥ −(x̄i + s̄i − d), i = 1, ..., P,
m=0
when we associate with them the dual variables π i . Each π i will therefore have a nonpositive coefficient in the dual objective function present in (34), and it will have coefficient
−(−λ+(M −m)(t+d)) in dual rows corresponding to the variables ρim , m = 0, ..., M −1.
For a given (x̄, ȳ, s̄), if there exists an inequality that separates this point from X̄ P IC ,
we can find it in polynomial time by solving the dual linear program given above. This
allows us to define an ellipsoid algorithm for PIC that operates in the original space
of (x, y, s) variables. However, the separation oracle in this algorithm is itself a linear
program in a higher–dimensional space, and Proposition 17 does not by itself give us an
explicit description of the valid inequalities needed to solve PIC by LP in the (x, y, s)
space.
Because of Corollary 16 and Proposition 17, we know that all valid inequalities for
P
IC
X
are solutions, or rays, of (35)–(40). In particular, we can express each of (2)–(5),
(8), (9), (10), and y i ≤ 1, i = 1, ..., P , as rays of (35)–(40). For example, a given reverse
cover inequality (that is, an inequality of the form (10)) in which |S| = m̄ for some m̄
such that 0 < m̄ ≤ M , can be expressed as the extreme ray
β = 1, η = −m̄(t + d);
τm = −(t + d), κm = (m̄ − m)(t + d), m = 0, ..., m̄ − 1; τm = κm = 0, m = m̄, ..., M ;
i = (t + d), m = m̄, ..., M ; ν i = 0, m = 0, ..., m̄ − 1, i ∈ S;
π i = 1, σ i = 0, ω i = t, νm
m
i = 0, m = 0, ..., M, i ∈ T 0 ;
ω i = d − λ − (M − m̄)(t + d), π i = σ i = 0, νm
i = 0, m = 0, ..., M, i ∈ T 00 .
π i = 1, σ i = 1, ω i = −d, νm
(41)
It can be checked that the above values define an extreme ray of (35)–(40). It can also
be checked that plugging the above values into (34) yields an inequality of the form (10).
In order to explicitly describe a set of inequalities that suffices to solve PIC by LP
in the (x, y, s) space, we could attempt to characterize the extreme rays of (35)–(40).
P IC
However, this has proven to be difficult, and therefore we first alter XLP
(∆,δ,ρ,γ) slightly
in order to obtain another polyhedron with the same projection onto the (x, y, s) space,
but with a dual cone of slightly simpler structure than (35)–(40).
P IC 0
Define XLP
(∆,δ,ρ,γ) to be the polyhedron defined by (12), (31)–(33), (14), (27), (16)–
(18), (24), (2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i = 1, ..., P . Recall that (8) is
the family corresponding to the (l, S) inequalities, (9) is the family of cover inequalities,
and (10) is the family of reverse cover inequalities.
13
0
P IC
P IC .
Corollary 18 P roj(x,y,s) XLP
(∆,δ,ρ,γ) = X̄
Proof: The constraints (2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i = 1, ..., P , are
implied by the constraints (12), (31)–(33), (14), (27), (16)–(18), and (24), from Corollary
16. 2
Now replace (32) with
si + dy i − d ≥ λρiM + γ i , i = 1, ..., P,
P
P
P
X
X
X
(si + dy i − d) ≤ λ
ρiM +
γi.
i=1
i=1
(42)
(43)
i=1
00
P IC
Define XLP
(∆,δ,ρ,γ) to be the polyhedron defined by (12), (31), (43), (33), (14), (27),
(16)–(18), (24), (2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i = 1, ..., P . Note that this
P IC 0
polyhedron is obtained from XLP
(∆,δ,ρ,γ) by dropping the constraints (42).
00
P IC
P IC .
Proposition 19 P roj(x,y,s) XLP
(∆,δ,ρ,γ) = X̄
0
00
P IC
P IC
Proof: It is clear that XLP
(∆,δ,ρ,γ) ⊂ XLP (∆,δ,ρ,γ) , and so
0
00
P IC
P IC
X̄ P IC = P roj(x,y,s) XLP
(∆,δ,ρ,γ) ⊂ P roj(x,y,s) XLP (∆,δ,ρ,γ) .
We will show that
00
0
P IC
P IC
P roj(x,y,s) XLP
(∆,δ,ρ,γ) ⊂ P roj(x,y,s) XLP (∆,δ,ρ,γ) ,
¯ δ̄, ρ̄, γ̄) ∈
which will prove the desired result. To do this, choose any point (x̄, ȳ, s̄, ∆,
00
P
IC
P IC 00
ˆ
XLP (∆,δ,ρ,γ) . We will show that there exists some point (x̂, ŷ, ŝ, ∆, δ̂, ρ̂, γ̂) ∈ XLP
(∆,δ,ρ,γ)
with x̄ = x̂, ȳ = ŷ, s̄ = ŝ, that satisfies (42).
First assume γ̄ i = 0, i = 1, ..., P . From (31) and (33) we have that
P
X
i=1
s̄i ≥
P
X
d−d
P X
M
M
−1
P
X
X
X
i
(
δ̄m
+
ρ̄im ) − (d − λ)
ρ̄iM .
i=1 m=0
i=1
m=0
i=1
Substituting (12) into the above inequality yields
P
P
X
X
(s̄i + dȳ i − d) ≥ λ
ρ̄iM ,
i=1
i=1
which, with (43), implies
14
P
P
X
X
(s̄i + dȳ i − d) = λ
ρ̄iM .
i=1
(44)
i=1
Now, set
ρ̂iM =
s̄i + dȳ i − d
, i = 1, ..., P.
λ
We know that (8) is satisfied for i = 1, ..., P ; therefore ρ̂iM ≥ 0, i = 1, ..., P . Moreover,
because of (44), we know that
P
X
ρ̂iM =
i=1
P
X
ρ̄iM .
(45)
i=1
Now set
i
δ̂M
= ȳ i − ρ̂iM −
M
−1
X
i
(δ̄m
+ ρ̄im ), i = 1, ..., P.
m=0
i ≥ 0, i = 1, ..., P . Also, note that
Note that δ̂M
i + ρ̂i = δ̄ i + ρ̄i , i = 1, ..., P ,
δ̂M
M
M
M
PP i
PP i
i=1 δ̂M =
i=1 δ̄M .
(46)
(47)
We also set
i
i
δ̂m
= δ̄m
, ρ̂im = ρ̄im , i = 1, ..., P, m = 0, ..., M − 1.
Because of (45), the right hand side of (43) is the same for ρ̂ and ρ̄; therefore the
¯ δ̂, ρ̂, γ̄) satisfies (31). Because of (46), (x̄, ȳ, s̄, ∆,
¯ δ̂, ρ̂, γ̄) satisfies (12) for
point (x̄, ȳ, s̄, ∆,
i = 1, ..., P . Because of (45) and (47), the right hand side of (31) is the same for (δ̂, ρ̂)
¯ δ̂, ρ̂, γ̄) satisfies (31) as well.
and (δ̄, ρ̄); therefore (x̄, ȳ, s̄, ∆,
¯ δ̂, ρ̂, γ̄) satisfies the remaining constraints of
It can be easily checked that (x̄, ȳ, s̄, ∆,
00
P
IC
P IC 0
XLP (∆,δ,ρ,γ) . Moreover, by definition it satisfies (42), and thus (x̄, ȳ, s̄) ∈ P roj(x,y,s) XLP
(∆,δ,ρ,γ) .
i
If γ̄ > 0 for some i ∈ P, one must use an equality like (44) to define new values for
γ̂ as well as for (δ̂, ρ̂), but the approach is similar. 2
Recall that X̂ P IC is defined by (2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i =
1, ..., P . We see immediately that X̄ P IC ⊂ X̂ P IC . Moreover, we also have
15
Proposition 20 Given any (x̄, ȳ, s̄) ∈ X̂ P IC , (x̄, ȳ, s̄) ∈ X̄ P IC if and only if
η+
P
X
ω i ȳ i + σ
i=1
P
X
(s̄i + dȳ i − d) −
i=1
P
X
π i (x̄i + s̄i − d) +
P
X
i=1
βxi ≤ 0
(48)
i=1
holds for all extreme rays of the dual cone
i
(δm
)
(ρim )
i
ω i + dβ + τm − νm
≤ 0, i = 1, ..., P, m = 0, ..., M
i
ω + (d − λ + (M − m)(t + d))β − (−λ + (M − m)(t + d))π
−κm −
(ρiM )
i
(γ )
(∆m )
i
νm
(49)
i
≤ 0, i = 1, ..., P, m = 0, ..., M − 1
i
ω + (d − λ)β + λσ − κM −
i
νM
≤ 0, i = 1, ..., P
i
σ − π ≤ 0, i = 1, ..., P
η = mτm − κm −
P
X
(50)
(51)
(52)
(53)
i
νm
, m = 0, ..., M
(54)
i=1
β ≥ 0, σ ≥ 0; π i ≥ 0, i = 1, ..., P ; κm ≥ 0, m = 0, ..., M ;
i
νm
≥ 0, i = 1, ..., P, m = 0, ..., M.
(55)
00
P IC
Note that (48) and (49)–(55) are obtained from XLP
(∆,δ,ρ,γ) in the same way that
P
IC
(34) and (35)–(40) are obtained from XLP (∆,δ,ρ,γ) . The major difference in the two cones
is that we have one variable σ associated with (43) instead of σ i , i = 1, ..., P , associated
with (32).
The next lemma is necessary to prove the main results of this section.
Lemma 21 Let (x̄, ȳ, s̄) be a dominant solution of X̂ P IC . Then (48) holds for all extreme rays of the dual cone (49)–(55).
The proof of Lemma 21 proceeds by considering the rays in question by cases, which are
defined by values of β and σ. The proof is long, technical, and not intuitive, so it is not
given here. The interested reader is referred to Miller [1999].
Proposition 22 The dominant solutions of X̂ P IC and of X̄ P IC are the same.
Proof: It is obvious that X̄ P IC ⊆ X̂ P IC . Thus any dominant solution (x̄, ȳ, s̄) of X̂ P IC
is also a dominant solution of X̄ P IC if and only if (x̄, ȳ, s̄) ∈ X̄ P IC . Moreover, if all the
dominant solutions of X̂ P IC are also dominant solutions of X̄ P IC , then there exist no
other dominant solutions of X̄ P IC .
16
So, to prove Proposition 22, we need to show that if (x̄, ȳ, s̄) is a dominant solution
of X̂ P IC , then (x̄, ȳ, s̄) ∈ X̄ P IC . This holds because of Proposition 20 and Lemma 21.
2
Theorem 23 X̃ P IC is given by the dominant solutions of the polyhedron defined by
(2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i = 1, ..., P .
Proof: This follows from Corollary 15, from Proposition 22, and from the definition of
X̂ P IC . 2
Corollary 24 Optimizing (1) over (2)–(5), (8), (9), (10), and the bounds y i ≤ 1, i =
1, ..., P, solves PIC.
Thus the families of cover inequalities and reverse inequalities defined for PI in Miller
et al. [2000a], with the (l, S) inequalities, suffice to solve the special case PIC by LP.
5
Separation for Cover and Reverse Cover Inequalities for
PIC
While separation for inequalities of the forms (2)–(5), (8), is trivial, there are clearly an
exponential number of both cover and reverse cover inequalities. As a consequence of
our results in the previous section, we know that the separation problem for PIC can be
solved in polynomial time by solving an LP in a higher–dimensional space. In this section
we describe combinatorial, polynomial–time separation algorithms for inequalities of the
forms (9) and (10).
Proposition 25 (Separation for Cover Inequalities) Given a point (x̄, ȳ, s̄), there exists
a violated inequality of the form (9) if and only if the optimal solution to the IP
max
subject to
P
X
i=1
i
P
λ X i
((d − λ)(1 − ȳ ) + λ − s̄ )χ +
(x̄ − (d − λ)ȳ i )ξ i
t+d
i
i
(56)
i=1
χ + ξ i ≤ 1, i = 1, ..., P
P
X
i
χi ≥ M + 1
(57)
(58)
i=1
χi , ξ i ∈ {0, 1}, i = 1, ..., P
(59)
is strictly greater than M λ. Moreover, if the optimal solution is strictly greater than M λ,
then a most violated inequality of the form (9) is obtained by taking S = {i ∈ P : χi = 1},
T 0 = {i ∈ P : ξ i = 1}, and T 00 = {i ∈ P : χi = ξ i = 0}.
17
The IP (56)–(59) can be solved by finding a maximum weight perfect matching on
a complete bipartite graph G. In G the vertex set V is partitioned into (A, B), with
|A| = |B| = P . The nodes in A represent items in P, while the nodes in B are used to
represent positions in S, T 0 or T 00 .
We partition B further into B 0 and B 00 such that |B 0 | = M + 1. For each i ∈ A and
j ∈ B 0 , we let the edge weight wij be
wij = (d − λ)(1 − ȳ i ) + λ − s̄i ;
that is, wij is the value added to the objective function (56) by setting χi = 1. For each
i ∈ A and j ∈ B 00 , we let the edge weight wij be
λ
(x̄i − (d − λ)ȳ i , 0};
t+d
that is, wij is the value added to (56) by setting to 1 the choice between χi or ξ i that
increases the value of (56) the most, or by setting χi = ξ i = 0 if both (d−λ)(1− ȳ i )+λ−s̄i
λ
and t+d
(x̄i − (d − λ)ȳ i ) are nonpositive.
A maximum weight perfect matching in G yields the optimal solution to (56)–(59).
The most violated cover inequality is then defined as prescribed in Proposition 25.
It is well known that finding a maximum weight matching in a bipartite graph can
be accomplished by using a maximum flow algorithm (see e.g. Cook et al. [1998]). For
a general graph with unit capacities, it is possible to solve the maximum flow problem
3
in O(E 2 ), where E is the number of edges (see e.g. Even and Tarjan [1975]). Since G
has P 2 edges, we have
wij = max{(d − λ)(1 − ȳ i ) + λ − s̄i ,
Proposition 26 The separation problem for cover inequalities for PIC can be solved in
O(P 3 ) time.
Separation for reverse cover inequalities can be accomplished in a manner similar to
cover inequality separation.
Proposition 27 (Separation for Reverse Cover Inequalities) Given a fractional solution
P IC , there is a violated inequality of the form (10) if and only if, for some
(x̄, ȳ, s̄) of XLP
m, 0 ≤ m ≤ M , the value of the optimal solution to the IP
max
subject to
P
P
X
X
(tȳ i − s̄i )χi +
((m(t + d) − c + t)ȳ i + x̄i )ξ i
i=1
i
χ + ξi ≤ 1
P
X
(60)
i=1
(61)
χi = m
(62)
i=1
i i
χ , ξ ∈ {0, 1},
(63)
18
is strictly greater than tm. Moreover, if the optimal solution is strictly greater than tm
for some such m, then a most violated inequality of the form (10) with |S| = m is defined
by putting i into S if χi = 1, putting i into T 0 if ξ i = 1, and putting i into T 00 otherwise.
Thus the separation problem for all reverse cover inequalities amounts to solving M
IP’s of the form (60)–(63), which are identical except for the value of m. Each of these
IP’s can be solved by finding a maximum weight perfect matching in a complete bipartite
graph similar to the graph described in the discussion on cover inequality separation.
Therefore, for each m = 1, ..., M , the separation problem for reverse cover inequalities
in which |S| = m can be solved in O(P 3 ) time. This fact yields
Proposition 28 The separation problem for reverse cover inequalities for PIC can be
solved in O(P 4 ) time.
Given the separation algorithms that we have described in this section, we can now
define an ellipsoid algorithm for PIC that runs entirely in the space of (x, y, s) variables.
The separation algorithms presented here can be applied to problems for which PIC is
a substructure. Moreover, while these separation algorithms are not directly applicable to
the general case of PI, we can use the results of this section to help to develop separation
heuristics for PI, and for problems for which PI provides a relaxation. For example, the
analysis in this section contributed to the development of the separation heuristics for
MCL discussed in Miller et al. [2000b].
6
Implications for Goemans’ Model
Goemans [1989] studied a model composed of a mixed constraint with variable upper
bounds. This model, which we call MVUB, can be written as
min
P
X
p̂i xi +
i=1
subject to
P
X
P
X
qiyi
(64)
ti y i ≤ c
(65)
i=1
xi +
i=1
P
X
i=1
i i
0 ≤ xi ≤ d y , i = 1, ..., P
i
y ∈ {0, 1}, i = 1, ..., P.
(66)
(67)
Note that if ti = 0, i = 1, ..., P , then MVUB becomes the model SNFC studied by
Padberg, Van Roy, and Wolsey [1985]. Thus, MVUB generalizes SNFC.
19
Goemans also considered the special case of MVUB in which ti = t ≥ 0, di = d >
0, i = 1, ..., P ; we call this case MVUBC. We define X M V U BC to be the set of feasible
solution to (65)–(67) for MVUBC. For MVUBC as for PIC, we assume that c > t + d to
avoid simple cases.
Recalling our assumption that in PIC hi > 0, q i > 0, i = 1, ..., P , if we further impose
pi > 0, i = 1, ..., P , then we know that xi + si = d, i = 1, ..., P , in every optimal solution.
We can then eliminate the variables si and set p̂i = pi − hi , i = 1, ..., P . This allows us to
write PIC as a special case of MVUBC in which the fixed costs q i are strictly positive.
(Note that in general PIC cannot be expressed as a special case of MVUBC, since, for
each i, the possibility that pi < 0 means that there are dominant solutions in which
xi > d.)
Goemans [1989] derived a class of generalized cover inequalities for MVUBC by extending results of Van Roy and Wolsey [1986] for a variant of SNFC. To define this class,
let a minimal cover of MVUBC be a set S ∗ ⊆ P such that t + d > λ = |S ∗ |(t + d) − c > 0.
Proposition 29 (Goemans [1989]) Let S ∗ be a minimal cover of MVUBC, and let L be
any subset of P \ S ∗ . Then
X
xi +
i∈S ∗ ∪L
X
ty i +
i∈S ∗ ∪L
X
(t + d − λ)(1 − y i ) ≤ c +
i∈S ∗
X
(t + d − λ)y i
(68)
i∈L
is a valid inequality for X M V U BC .
The class of generalized cover inequalities defined by Goemans for MVUB is more general
than the one we give for MVUBC above; however, this definition is sufficient for our
discussion.
Goemans posed the question of whether the generalized cover inequalities suffice
to describe conv(X M V U BC ). This question is still unresolved, but our results for PIC
imply new results for MVUBC that support the conjecture that the generalized cover
inequalities suffice to describe conv(X M V U BC ).
Note that since Proposition 29 requires S ∗ to be a minimal cover, |S ∗ | = M + 1 must
be true. Using this fact and simple algebra, we can rewrite (68) as
X
i∈S ∗ ∪L
xi −
X
(d − λ)y i ≤ c − (M + 1)(t + d − λ) = −λ + (M + 1)λ = M λ.
(69)
i∈S ∗ ∪L
Turning our attention to PIC, we state
Lemma 30 If pi > 0, hi > 0, i = 1, ..., P in an instance of PIC, then in any optimal
solution
1. xi + si = di , i = 1, ..., P ;
20
2. xi ≤ dy i , i = 1, ..., P .
Proof: If xi + si > d for some i, we can find a feasible solution with a better value of
(1) by lowering either xi or si . Thus 1 holds. Statement 2 follows from substituting
si = d − xi in (8). 2
Note that since by assumption t + d < c, then the inequality (3) becomes redundant
when pi > 0, i = 1, ..., P . Thus, in this case we can transform PIC into an instance of
MVUBC by eliminating the variables si . We will use this fact in some of the proofs in
this section.
Lemma 31 Given an instance of PIC in which pi > 0, hi > 0, i = 1, ..., P , it suffices to
consider only the set of cover inequalities (9) in which T 0 = ∅.
Proof: Replacing si with d − xi in (9) yields
P
i∈S (d
P
− xi ) ≥ (|S| − M )λ +
Mλ ≥
P
i∈S
xi
−
i∈S (d
− λ)(1 − y i ) +
⇐⇒
P
i∈S (d
− λ)y i +
λ
t+d
P
λ
t+d
i
i∈T 0 (x
P
i∈T 0 (x
i
− (d − λ)y i )
− (d − λ)y i ).
(70)
Let I be any inequality of the form (70) in which T 0 6= ∅, and let SI and TI0 be the
particular choice of sets S and T 0 defining this inequality. Then I is a linear combination
of two other inequalities in which T 0 = ∅. The first of these inequalities is defined by
taking S = SI ; the second is defined by taking S = SI ∪ TI0 . 2
Lemma 32 Given an instance of PIC in which pi > 0, hi > 0, i = 1, ..., P , the set of
reverse cover inequalities (10) is implied by (66), the inequalities y i ≤ 1, i = 1, ..., P , and
the set of cover inequalities (9) in which T 0 = ∅.
Proof: Replacing si with d − xi in (10) yields
P
i∈S (d
− xi ) ≥ |S|(t + d)
|S|(t + d) ≥
P
i∈S∪T 0
P
i∈T 0
xi
+
yi −
P
P
i∈S
⇐⇒
i∈S
ty i −
t(1 − y i ) −
P
i∈T 0 (c
P
i∈T 0 ((c
− t)y i − xi )
− t − |S|(t + d))y i .
(71)
Let I be any inequality of the form (71), and let SI and TI0 be the particular choice of
sets S and T 0 defining I.
If |S| < M , then I is a linear combination the inequalities xi ≤ d, i ∈ SI , y i ≤ 1, i ∈
SI , and xi ≤ dy i , i ∈ TI0 (since c − t − |SI |(t + d) > d).
21
If |S| = M , then I can be rewritten
M (t + d) ≥
X
i∈SI ∪TI0
X
xi +
i∈SI
ty i −
X
(d − λ)y i
i∈TI0
Then I is a linear combination of the inequality (70) defined by S = SI ∪ TI0 and T 0 = ∅
(multiplied by 1), and of the inequalities y i ≤ 1, i ∈ SI (each multiplied by d + t − λ). 2
Proposition 33 Given an instance of PIC in which pi > 0, hi > 0, q i > 0, i = 1, ..., P ,
(65), (66), the inequalities y i ≤ 1, i = 1, ..., P , and the set of cover inequalities (9) in
which T 0 = ∅ suffice to solve PIC by LP.
This follows from Lemmas 30, 31, and 32 and from Theorem 22, and it immediately
yields
Proposition 34 Given an instance of MVUBC in which q i > 0, i = 1, ..., P , the set
of generalized cover inequalities (68), along with trivial facets (65), (66), and y i ≤ 1,
i = 1, ..., P , suffice to solve MVUBC by LP.
Proof: This follows from Proposition 33, and from observing that choosing S = S ∗ ∪ L
and T 0 = ∅ in (70) yields an inequality that is equivalent to (69), after substituting
si = d − xi . 2
Thus, the class of generalized cover inequalities suffices to solve MVUBC by LP. While
this does not imply that this class suffices to describe the convex hull of X M V U BC , it
does provide evidence in support of this conjecture.
7
Conclusions and Future Directions
In this paper we have studied PIC, which is a special case of the more general model
PI. We have given three nontrivial classes of facet–inducing valid inequalities for PIC:
the (l, S) inequalities, cover inequalities, and reverse cover inequalities. Barany, Van
Roy, and Wolsey [1984] introduced the (l, S) inequalities for the uncapacitated lot–sizing
problem, while the other two families were originally defined for PI.
We then gave polynomial–time algorithms for PIC. One result of the developments
necessary to define these algorithms is that we are able to show that the families of inequalities mentioned above enable us to solve PIC in polynomial time by linear programming in the original space of variables. That is, the inequalities that we have identified
for PI in Miller et al. [2000a], are sufficient to solve PI by LP in a polynomially solvable
case.
An interesting question that remains open is a characterization of the convex hull
of X P IC in terms of linear inequalities. Not only would such a description enable us to
22
solve problems with the same constraint set but more general cost conditions, it could
also provide intuition into the structure of the convex hull of closely related problems.
The same is true for related models, such as the model MVUBC discussed in Section 6.
Given the results that we have established for PI and PIC, we expect that the application of the general forms of (9) and (10) would be computationally effective for many
problems for which PI provides a relaxation. This has been confirmed for multi–item
capacitated lot–sizing problems (Miller et al. [2000b]).
8
Acknowledgements
We would like to thank Laurence Wolsey and two anonymous referees for comments that
helped considerably to improve the presentation of the results in this paper.
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