A Polytope Approach to the Optimal Assembly
Problem
FRANK K. HWANG1 and URIEL G. ROTHBLUM2
1Department of Applied Mathematics, Chiaotung University, Hsinchu, 30045 Taiwan (R.O.C.) (e-mail: fhwang@math.nctu.edu.tw)
2Faculty of Industrial Engineering and Management, Technion – Israel Institute of Technology, Haifa 32000, Israel (e-mail: rothblum@ie.technion.ac.il)
(Received 17 February 2004; accepted in revised form 2 October 2005)
Abstract. The problem of assembling components into series modules to maximize the sys-tem reliability has been intensively studied in the literature. Invariably, the methods employed exploit special properties of the reliability function through standard analytical optimization techniques. We propose a geometric approach by exploiting the assembly polytope – a poly-tope generated by the potential assembly configurations. The new approach yields simpler proofs of known results, as well as new results about systems where the number of compo-nents in a module is not fixed, but subject to lower and upper bounds.
1. Introduction
Consider a monotone system with p modules where both the system and each module are in one of two states – operative or inoperative, and where the state of the system depends on the joint states of the modules. The adjective “monotone” means that the system’s performance is monotone in the performance of its modules, that is, it cannot move from the operative state to the inoperative state as the result of one of the modules moving from the inoperative state to the operative state. Each module Mi requires
components of different types that are in series structure, that is, Mi is
operative if and only if all its components are operative. The system’s state is determined by a performance function J(.) whose variables are the states of the modules, that is, when the state of module Mi for each i= 1, . . . , p
is si, the state of the system is J (s1, . . . , sp). By assigning value 1 to oper-ativeness and value 0 to inoperoper-ativeness, the function J is a Boolean func-tion, with domain {0, 1}p and range {0, 1}; system-monotonicity means that the function J is monotone. Constraints on the number of components of each type that the modules require are specified in terms of lower and upper bounds, and we refer to these constraints as the system specifications. We next consider the components of which the modules are constructed. There are t types of components indexed u=1, . . . , t, and for each u there are nu components of type u. Let ruj denote the reliability of the j th item
of type u, j= 1, . . . , nu, u = 1, . . . , t, that is, the probability that this item
is operative. Operativeness of these items are assumed to be stochastically independent. Without loss of generality, we will assume that the compo-nents of each type are indexed so that
0 < ru1 ru2 · · · runu 1 for u= 1, . . . , t; (1.1) An assembly is an assignment of the nu components of each type u
to the modules that satisfies the system specifications; formally, such an assembly consists of partitions for each u= 1, . . . , t of the set {1, . . . , nu}
into parts πu1, . . . , πup. As a module is operative if and only if all its
com-ponents are, the probability that module Mi is operative when π is applied
is given by r(π)i= t u=1 j∈πui ruj. (1.2)
and we refer to r(π )i as the reliability of Mi under assembly π . The
sys-tem reliability under assembly π , that is, the probability that the syssys-tem is
operative when π is applied, is then given by
R(π) ≡ s∈{0,1}p J (s) {i:si=0} [1− r(π)i] {i:si=1} r(π)i . (1.3)
The optimal assembly problem is to determine an assembly such that the system reliability is maximized over the set of feasible assemblies. A num-ber of authors [2, 3, 5, 6, 7, 9, 11, 13, 14, 15, 16] studied instances of this problem where the system specifications prescribe the number of compo-nents of each type that the modules require (see [10] for a review of specific results obtained in these references), Hwang and Rothblum [10] solved this problem in its generality. Their solution method first considers systems with two modules, then extends the analysis to systems with arbitrarily many modules but with one item-type required by all the modules, and finally extends the analysis to the general case. Each step requires subtle pertur-bation arguments that involve part-reliabilities and the structure function.
In this paper, we treat the assembly problem as an assignment problem, namely, every component must be assigned to a module. We then seek an optimal assembly under lower and upper bound constraints on the number of parts of each type in each module. The need for assignment may come up in various scenarios, for example, when the components are personnel each must be assigned to a job, or when the storage of components outside of the modules is expensive, or when idle components may cause degrad-ing in reliability (such as unused cars or uninhabited houses). Our solution
method uses a new polytope-approach. The new approach yields simpler proofs of existing results for the problem with prescribed specifications as well as a solution for the problem where the number of components of each type that the modules require are constrained by lower and upper bounds, a problem which has not been solved before.
In Section 2, we formalize the reliability maximization problem as an
additive assembly problem with asymmetric Schur convex objective – terms
that are defined in that section. After introducing some preliminaries about supermodular functions in Section 3, we analyze and solve the additive assembly problem with asymmetric Schur convex objective with prescribed requirements and with bounds on the requirements in Sections 4 and 5, respectively. Finally, Sections 6 contains a discussion of potential extensions of the results of the earlier sections.
2. The Additive Assembly Problem with Asymmetric Schur Convex Objective
The ith unit vector in Rp is the vector ei with eii= 1 and eit = 0 for t ∈ {1, . . . , p}\{i}. With i ranging over i = 1, . . . , p, we refer to these vectors as the standard unit vectors in Rp. Also, for i, j∈ {1, . . . , p}, let eij be the difference between the i- and the j -unit vectors in Rp.
Let T be an interval and g a real-valued continuously differentiable func-tion on Tp, that is, g has continuous partial derivatives with respect to all variables. The function g is called asymmetric Schur convex if for every y∈
Tp and distinct i, j∈ {1, . . . , p}, the function gij
y(γ ) ≡ g(y + γ eij) does not
decrease after an increase (here “after” means – “as γ increases”); as g is con-tinuously differentiable, this condition is equivalent to the requirement that d dγ gyij= ∂ ∂xi g(y + γ eij) − ∂ ∂xj
g(y + γ eij) does not have a sign-switch
from a positive value to a negative value as γ increases. The function g is called strictly asymmetric Schur convex if it is asymmetric Schur convex and none of the functions gyij is constant on an interval; as g is continuously
differentiable, the latter means that the derivatives of these functions cannot vanish on any interval. Asymmetric Schur convex functions were introduced in [12]; they generalize (quasi) convex and (regular) Schur convex functions and have the following important property (see [12]):
PROPOSITION 2.1. Let T be an interval and let g be an asymmetric Schur
convex function on Tp. If P is a polytope contained in Tp where the direc-tions of the edges of P are differences of pairs of standard unit vectors in Rp, then g attains a maximum at one of P’s vertices. Further, if g is strictly
asymmetric Schur convex, then every maximizer of g over P is a vertex of P .
We next show that the assembly problem described in the introduction fits within a framework of optimization problems that concern asymmet-ric Schur convex functions. For each u= 1, . . . , t and j = 1, . . . , nu, define
ρuj= ln ruj. As the indexing of ruj’s satisfies (1.1), we have
−∞ < ρu1 ρu2 · · · ρunu 0 for u= 1, . . . , t. (2.1) For each assembly π and i= 1, . . . , p, let
ρ(π)i≡ ln r(π)i= ln ⎡ ⎣t u=1 j∈πui ruj ⎤ ⎦ =t u=1 j∈πui ρuj. (2.2)
Thus, each assembly π is associated with the two p-dimensional vectors
r(π) = [r(π)1, . . . , r(π)p] and ρ(π )= [ρ(π)1, . . . , ρ(π)p)].
Given the performance function J :{0, 1}p→ {0, 1}, let f and g be the real-valued functions on (0,1]P and (−∞, 0]p, respectively, with
f (x) = s∈{0,1}p J (s) ⎡ ⎣ {i:si=0} (1 − xi) ⎤ ⎦ ⎡ ⎣ {i:si=1} xi ⎤ ⎦ for every x ∈(0,1]p, (2.3) and
g(y)= f [exp(y1), . . . , exp(yp)]
= s∈{0,1}p J (s) ⎡ ⎣ {i:si=0} (1 − eyi) ⎤ ⎦ ⎡ ⎣ {i:si=1} eyi ⎤ ⎦ for every y ∈(−∞,0]p. (2.4) Using (1.3) and (2.2), the system reliability under an assembly π is express-ible by R(π) = s∈{0,1}p J (s) ⎧ ⎨ ⎩ i:si=0 [1− r(π)i] ⎫ ⎬ ⎭ ⎧ ⎨ ⎩ i:si=1 r(π)i ⎫ ⎬ ⎭= f [r(π)] = g[ρ(π)]. (2.5) Hwang and Rothblum [10, Example 3] estabished the following result. PROPOSITION 2.2. Given any monotone performance function J :{0, 1}p→ {0, 1}, the function g defined by (2.3)–(2.4) is asymmetric Schur convex on
Proposition 2.2 implies that the problem of determining an assembly which maximizes system-reliability over a set of assemblies is an instance of the problem of maximizing an expression g[ρ(π )] over with ρ(π )i
=tu=1j∈πuiρuj for i= 1, . . . , p, with the ρujs satisfying (2.1), and with
g(.) as an asymmetric Schur convex on (−∞, 0]p. Henceforth, we consider
this more general problem to which we refer as the additive assembly
prob-lem with asymmetric Schur convex objective (in this context, the term multi-partition is used in the literature synonymously with assembly). We will
solve this problem when feasible assemblies π are those for which the size of the πuis must satisfy lower and upper bounds.
3. Preliminaries on Supermodular Functions
Given a real-valued function λ on the subsets of {1, . . . , p} with λ(Ø) = 0, we define in this section two polytopes in Rp. A permutation of {1, . . . , p} is a vector σ= (i1, . . . , ip) with {i1, . . . , ip} distinct and with {i1, . . . , ip}
= {1, . . . , p}. Each permutation σ defines a p-vector λσ with
(λσ)is= λ({i1, . . . , is}) − λ({i1, . . . , is−1}) for s = 1, . . . , p. (3.1) The permutation polytope corresponding to λ, denoted Hλ, is the convex hull of the λσ’s, with σ ranging over the set p of all permutations of
{1, . . . , p}. Also, the polytope Cλ is defined as the solution set of the linear
inequality system
i∈I
xi λ(I) for each nonempty subset I of {1, . . . , p}, (3.2)
and
p
i=1
xi= λ({1, . . . , p}). (3.3)
A real-valued function λ on the nonempty subsets of {1, . . . , p} with
λ(Ø) = 0 is called supermodular if for every pair I and J of subsets of
{1, . . . , p},
λ(I ∪ J ) + λ(I ∩ J ) λ(I) + λ(J ); (3.4)
λ is called strictly supermodular if strict inequality holds whenever the two
sets are not ordered by set inclusion, that is, I |⊂J and J |⊂I.
Parts (i) and (ii) of the next result are due to Shapley [18], see also, [8]; for part (iii) see [9].
PROPOSITION 3.1. Suppose λ is supermodular on the subsets of{1, . . . , p}.
Then:
(i) Hλ= Cλ,
(ii) the vertices of Hλ=Cλ are precisely the λσ’s where σ ranges over p,
and
(iii) each direction of an edge of Hλ= Cλ is proportional to the difference of a pair of standard unit vectors in Rp.
4. Prescribed-Sizes Problem
In this section we assume that nonnegative integers {nui: u= 1, . . . , t and
i = 1, . . . , p} are given, and the system specifications require that the
num-ber of components of (each) type u in (each) module Mi must equal nui.
An assembly is called monotone if there exists a module Mi1 taking the
nui1 most reliable items of each type u , a second module Mi2 taking the
nui2 next most reliable items of each type u, and so on, until the last mod-ule Mip takes the nuip least reliable items of each type u.
Let be the set of all assemblies π with each πui having the prescribed
size nui. The assembly polytope, denoted P , is defined as the convex hull
of the ρ(π )’s, with π ranging over . Also, for each subset I of {1, . . . , p} let ρ∗(I ) = min π∈ i∈I ρ(π)i; (4.1) in particular, ρ∗(Ø) = 0 and ρ∗({1, . . . , p}) = t u=1 nu j=1 ρuj. (4.2)
For a subset I of {1, . . . , p} and u = 1, . . . , t, let nu(I ) =i∈Inui; we then
have that ρ∗(I ) = t u=1 nu(I ) j=1 ρuj. (4.3)
We observed that the nu(I )s (for all subserts I of {1, . . . , p}) and the sums
s
j=1ρujs (for s= {1, . . . , n}) can be determined, respectively, by at most 2p
andpu=1nu= n additions. Given these quantities, the nj=1u(I )ρuis are
availa-ble for each u and for each I , and (4.3) can be used to determine the ρ∗(I )’s with t−1 extra additions for each I. The total computational effort for com-puting all the ρ∗(I )’s in this way amounts to t2p+ n additions. With p fixed, the bound is linear in the number of components n.
LEMMA 4.1. ρ∗ is supermodular.
Proof. For subsets I and J of {1, . . . , p} and u = 1, . . . , t,nu(I ∪ J ) −
nu(I ) = nu(J \ I) = nu(J ) − nu(I ∩ J ), and (2.1) implies that
ρ∗(I ∪ J ) − ρ∗(I ) = t u=1 nu(I ∪J ) i=1 ρui− nu(I ) i=1 ρui = t u=1 ⎡ ⎣ nu(I ∪J ) i=nu(I )+1 ρui ⎤ ⎦ = t u=1 ⎡ ⎣ nu(I )+nu(J \I) i=nu(I )+1 ρui ⎤ ⎦ t u=1 ⎡ ⎣ nu(I ∩J )+nu(J \I) i=nu(I ∩J )+1 ρui ⎤ ⎦ = t u=1 ⎡ ⎣ nu(J ) i=nu(I ∩J )+1 ρui ⎤ ⎦ =t i=1 nu(J ) i=1 ρui− nu(I ∩J ) i=1 ρui = ρ∗(J ) − ρ∗(I ∩ J ).
Given a permutation σ= (i1, . . . , ip) of {1, . . . , p}, the assembly corre-sponding to σ , denoted πσ, is defined as the assembly with
(πσ)uis= {nu({i1, . . . , is−1}) + 1, . . . , nu({i1, . . . , is})}
for u= 1, . . . , t and s = 1, . . . , p. (4.4) LEMMA 4.2
(i) For every permutation σ of {1, . . . , p}, πσ is a monotone assembly with
ρ(πσ) = (ρ∗)σ.
(ii) The correspondence σ→πσmappingpinto monotone assemblies is onto.
Proof. (i) Consider a permutation σ = (i1, . . . , ip) of {1, . . . , p}. The monotonicity of πσ is immediate. Next, for u=1, . . . , t and s =1, . . . , p, let
nu(s) ≡ nu({i1, . . . , is}) =sq=1nuiq. It follows from (4.3) and (4.4) that for
s = 1, . . . , p, ρ∗({i1, . . . , is}) =tu=1j=1nu(s)ρuj =sq=1[ρ(πσ)]iq, and there-fore [(ρ∗)σ]is= ρ∗({i1, . . . , is}) − ρ∗({i1, . . . , is−1}) = [ρ(πσ)]is. As {i1, . . . , ip} = {1, . . . , p}, we conclude that (ρ∗)σ= ρ(πσ).
(ii) A monotone assembly π , defines a ranking on {1, . . . , p} which defines a permutation σ=(i1, . . . , ip) with is as the sth element in the rank-ing. It is straightforward to see that in this case πσ= π.
As ρ∗ is a function on the subsets of {1, . . . , p} with ρ∗(Ø) =0, the poly-topes Cρ∗ and Hρ∗ are well-defined (see Section 2).
THEOREM 4.3 (i) P= Hρ∗= Cρ∗.
(ii) The vertices of P are the vectors {ρπ: π ranging over the set of
mono-tone assemblies}.
(iii) Each direction of an edge of P is proportional to the difference of a
pair of standard unit vectors in Rp.
Proof. From (4.1), for every assembly π and subset I of {1, . . . , p}, ρ∗(I )
i∈Iρ(π)i and, using (4.2), ρ∗({1, . . . , p}) =tu=1
nu j=1ρuj= t u=1 p i=1 j∈πiρuj= p
i=1ρ(π)i. So, ρ(π )∈ Cρ∗. It follows that the convex hulls of
the ρ(π )’s, namely P , is contained in Cρ∗. Next, Lemma 4.2 implies that
P = conv {ρ(π); π is an assembly } ⊇ conv {ρ(π); π
is a monotone assembly }
= conv {(ρ∗)σ; σ is a permutation of {1, . . . , p}} = Hρ∗.
Next, Proposition 3.1 and the supermodularity of ρ∗ (Lemma 4.1) imply that Hρ∗= Cρ∗ and that the vertices of this polytope are the (ρ∗)σs with σ ranging over the permutations of {1, . . . , p}. So, P ⊆ Cρ∗= Hρ∗⊆ P , imply-ing that P = Cρ∗ = Hρ∗; further, Lemma 4.2 implies that the vertices of this polytope are the ρ(π )s with π ranging over the monotone assemblies. Finally, part (iii) follows directly from the third part of Proposition 3.1 and the above results.
The linear inequality representation of P through Cρ∗ has p variables and 2p constraints that use the ρ∗(I )’s. We recall the paragraph following (4.3) that shows how to determine the ρ∗(I )’s with t2p+ n additions. THEOREM 4.4. There exists a monotone optimal assembly.
Proof. Proposition 2.1, the asymmetric Schur convexity of g and part (iii)
of Theorem 4.3 assure that g attains a maximum over P at a vertex of that polytope; Theorem 4.3 also assures that such a vertex has the representa-tion ρ(π∗) for some monotone assembly π∗. Now, let π be an arbitrary assembly; then ρ(π )∈ P and R(π) = g[ρ(π)] g[ρ(π∗)] = R(π∗).
Theorem 4.4 implies that the assembly problem with prescribed sizes can be solved by evaluating the monotone assemblies; by Lemma 4.2, these are precisely the p! πσ’s, determined by the permutations over {1, . . . , p}. In
order to evaluate R(πσ) = g[ρ(πσ)] for a permutation σ , one has to
eval-uate g(.) at ρ(πσ) = (ρ∗)σ (the equality following from Lemma 4.2). Now,
again, see the paragraph following (4.3) for a discussion of the computa-tion of all ρ∗(I )s using at most t2p+ n additons.
5. The Bounded-Sizes Problem
In the current section we consider the optimal assembly problem where lower and upper bounds are prescribed on the number of components of each type to be assigned to each module. Specifically, throughout this sec-tion, integers Lui and Uui for u= 1, . . . , t and i = 1, . . . , p are given with
Lui Uui and pi=1Lui nupi=1Uui for u= 1, . . . , t. We let L,U be the
set of assemblies π with Lui |πui| Uui for u= 1, . . . , t and i = 1, . . . , p,
and the assembly polytope PL,U be the convex hull of the vectors ρ(π ) with π ranging over L,U.
In this section we show that an asymmetric Schur convex function g(.) over PL,U attains a maximum at a vertex of PL,U and that vertices corre-spond to monotone assemblies; further, we provide a linear inequality rep-resentation of the polytope PL,U.
The next result asserts optimality of monotone assemblies and assures a representation of vertices of PL,U via monotone assemblies. It follows from the results of Section 4.
THEOREM 5.1
(i) There exists a monotone assembly which is optimal over L,U.
(ii) Each vertex of PL,U has a representation ρ(π) with π as a monotone assembly in L,U.
Proof. As there are finitely many assemblies, there exists an optimal
assembly. Let π∗ be an optimal assembly. By considering the problem with prescribed number |(π∗)ui| of components of each type u for each
mod-ule Mi, we have from Theorem 4.4 that this problem has a monotone
opti-mal assembly, say π. It follows ρ(π∗) ρ(π); the optimality of π∗ for the bounded-sizes problem then implies that ρ(π) = ρ(π∗), assuring that π is also optimal for that problem.
A vertex v of PL,U has a representation ρ(¯π) with ¯π ∈L,U. Let P be the polytope corresponding to the assembly problem where the number of compo-nents of each type u for each module Mi is| ¯πui|. As P is the convex hull of a
set that is smaller than the one defining PL,U and as v∈ P, v is a vertex of P as well. By Theorem 4.3, v has a representation as ρ(π) where πis a monotone assembly having|πui | = | ¯πui| for each u and i, in particular, π∈L,U.
The conclusions and proof of Theorem 5.1 extend to sets of assemblies with arbitrary constraints on the number of components of each type in each module (not just constraints that are determined through lower and
upper bounds). In the remainder of this section we explore the extra struc-ture available in bounded-sizes problems to obtain a refined analysis.
Parallel to our development in Section 4, define the real-valued function on subsets of {1, . . . , p} where for subset I,
ρL,U ∗ (I ) = min i∈I ρ(π)i: π∈ L,U ; (5.1)
in particular, ρ∗L,U(Ø) = 0 and
ρL,U ∗ ({1, . . . , p}) = t u=1 nu j=1 ρuj. (5.2)
For each subset I of {1, . . . , p}, let
n− u(I ) = min i∈1 Uui, nu− i∈Ic Lui , for u = 1, . . . , t. (5.3) where Ic ≡ {1, . . . , p}\I. The next lemma provides a representation of
ρL,U
∗ (.) that resembles (4.3) (which applied to the fixed sizes case). LEMMA 5.2
(i) For each permutation σ=(i1, . . . , ip) of {1, . . . , p}, there exists a
mono-tone assembly πσ∈L,U with (ρ∗L,U)σ= ρ(π) and
(πσ)uis= {n−u(i1, . . . , is−i}) + 1, . . . , n−u({i1, . . . , is})}
for u = 1, . . . , t and s = 1, . . . , p. (5.4) (ii) For each subset I of {1, . . . , p},
ρL,U ∗ (I ) = t u=1 n− u(I ) j=1 ρuj. (5.5)
Proof. (i) For each assembly π ∈L,U and u= 1, . . . , t,i∈I|πui|
i∈IUui and i∈I|πui| =pi=1|πui| −i∈Ic|πui| n −i∈IcLui, implying
that i∈I|πui| n−u(I ); hence, (2.1) (which includes the assertion that the
ρuj’s are nonpositive) implies that i∈Ij∈πuiρuj n
− u(I ) j=1 ρuj. It follows that i∈I [ρ(π )]i= i∈I t u=1 j∈πui ρuj= t u=1 i∈I j∈πui ρuj t u=1 n− u(I ) j=1 ρuj,
and therefore ρL,U ∗ (I ) t u=1 n− u(I ) j=1 ρuj. (5.6)
Let σ= (i1, . . . , ip) be a permutation of {1, . . . , p}. For u = 1, . . . , t and
i = 1, . . . , p define n−
ui by setting for each s= 1, . . . , p, n−uis≡ n−u({i1, . . . , is}) − n−
u({i1, . . . , is−1}). Fix u ∈ {1, . . . , p}. We will show that for u = 1, . . . , t and i=1, . . . , p, Luin−uiUui; this will be established by showing that for
u =1, . . . , t and s =1, . . . , p, Luisn−uisUuis. So, fix u and s and set Is−1≡ {i1, . . . , is−1} and Is= {i1, . . . , is}. We consider two cases, one of which has
two subcases. Case I: n−u(Is−1) = i∈Is−1Uui n − i∈Ic s−1Lui. In this case, if n − u(Is)
=i∈IsUui, then n−uis= n−u(Is) − n−u(Is−1) = Uuis Luis, assuring that Luis
n− uis Uuis. Alternatively, if n − u(Is) = n −i∈Ic s Lui i∈IsUui, then n − uis = n− u(Is) − n−u(Is−i) i∈IsUui − i∈Is−1Uui = Uuis and n−uis = n−u(Is) − n− u(Is−1) (n −i∈Ic s Lui) − (n − i∈Ic s−1Lui) = Luis. Case II: nu−(Is−1) = n − i∈Ic s−1Lui
i∈Is−1Uui. In this case,
i∈IsUui =i∈Is−1Uui+ Uuis (n − c i∈Is−1Lui) + Luis = n − i∈Ic s Lui, assuring that n− u(Is) = n −i∈Ic s Lui and n − uiS = n−u(Is) − n−u(Is−1) = (n − i∈Ic s Lui) − (n − i∈Ic s−1Lui) = Luis Uuis, assuring that Luis n − uis Uuis.
Consider the assembly problem with prescribed sizes n−ui for u= 1, . . . , t and i= 1, . . . , p and the monotone assembly πσ corresponding to σ as
defined by (4.4), that is, πσ is determined by (5.7). In particular, |(πσ)ui|
= n−
ui for each u and i, and the above paragraphs assure that πσ∈L,U.
Next, for s= 1, . . . , p, ∪i∈Is(πσ)ui= {1, . . . , n−u(Is)} and
i∈Is [ρ(πσ)]i= i∈Is t u=1 j∈(πσ)ui ρuj= t u=1 i∈Is j∈(πσ)ui ρuj = t u=1 n− u(Is) j=1 ρuj ρ∗L,U(Is), (5.7)
the last inequality following form (5.6). As the definition of ρ∗L,U in (5.1) assures that ρ∗L,U(Is) i∈Is[ρ(πσ)]i, we conclude that equality
holds throughout (5.7). Thus, for s = 1, . . . , p,i∈I
s[ρ(πσ)]i = ρ
L,U
∗ (Is)
=i∈Is[(ρ∗L,U)σ]i. It follows that for s= 1, . . . , p, ρ(πσ)is= [(ρL,U∗ )σ]is; as {i1, . . . , ip} = {1, . . . , p}, we conclude that ρ(πσ) = (ρ∗L,U)σ.
(ii) Let I⊆ {1, . . . , p}, say I = {i1, . . . , is}. One can construct a
respect to the notation of the proof of (i), we have that I= IS and the
established equalities in (5.7) prove (5.5).
Part (i) of Lemma 5.2 generalizes part (i) of Lemma 4.2 from the pre-scribed-sizes case to the bounded-sizes case. But, part (ii) of Lemma 4.2 does not have a corresponding generalization as the number of monotone assemblies is generally larger than the number of permutations of{1, . . . , p} which equals p!. In fact, for each (feasible) collection{nui: LuinuiUui, u =
1, . . . , t and i= 1, . . . , p} there exists a monotone assembly for each permu-tation σ of{1, . . . , p}; with L,U, as the set of such feasible collections, the number of monotone assemblies is||p!. When ||1, this number is larger than the number p!. It is noted that the effort of determining the vector asso-ciated with each monotone assembly is tn, so the total effort in computing all vectors associated with monotone assemblies is tn||p!.
COROLLARY 5.3. Suppose I1, . . . , Ik are subsets of {1, . . . , p} with Ø ⊆
I1⊂ · · · ⊂ Ik⊆ {1, . . . , p}. Then there exists an assembly π ∈L,U satisfying
ρL,U
∗ (It) =i∈It[ρ(π )]i.
Proof. We first assume that I1 = Ø and Ik = {1, . . . , p}. The list I1, . . . , Ik can be extended by adding sets to one having p− 1 elements, that is, we can assume that k= p − 1. Let I0= Ø and Ip= {1, . . . , p}. We observe that
in this case σ≡(I1\I0, . . . , Ip\Ip−1) is a permutation. (here we identify set containing a single element with that element). Let π∈L,U be a mono-tone assembly satisfying ρ(π )=(ρ∗L,U)σ (whose existence was established in
Lemma 5.1). Now, suppose σ= (i1, . . . , ip). For each t = 1, . . . , p then It= {i1, . . . , it} and i∈It [ρ(π )]i= t s=1 [ρ(π )]is= t s=1 [(ρ∗L,U)σ]is= ρ L,U ∗ ({i1, . . . , it}) = ρ∗L,U(It). The extension to the cases where I1 =Ø and/or Ik ={1, . . . , p} is straightfor-ward.
LEMMA 5.4. ρ∗L,U is supermodular.
Proof. Let I and J be subsets of{1, . . . , p} which are not ordered by set
inclusion. Then Ø⊆I ∩J ⊂I ∪J ⊆{1, . . . , p} and Corollary 5.3 implies that there exists an assembly π in L,U satisfying ρ∗L,U(I ∩ J ) =i∈I∩J[ρ(π )]i
and ρ∗L,U(I ∪ J ) =i∈I∪J[ρ(π )]i. Consider the assembly problem with
pre-scribed sizes nui≡ |πui| for u = 1, . . . , t and i = 1, . . . , p and let ρ∗ be given by (4.1). For each S⊆ {1, . . . , p}, ρ∗L,U(S) ρ∗(S) i∈S[ρ(π )]i. Thus, the
and ρ∗L,U(I ∪ J ) = ρ∗(I ∪ J ) =i∈I∪J[ρ(π )]i. From the supermodularity of
ρ∗ (Lemma 4.1), then
ρL,U
∗ (I ∪ J ) + ρ∗L,U(I ∩ J ) = ρ∗(I ∪ J ) + ρ∗(I ∩ J ) ρ∗(I ) +ρ∗(J ) ρ∗L,U(I ) + ρ∗L,U(J ), proving that ρ∗L,U is supermodular.
Recall that the set of permutations of {1, . . . , p} is denoted p. Given
σ in p, the monotone assemblies in L,U that is constructed in Lemma 5.2 (determined by (5.4) ) will be denoted πσ.
THEOREM 5.5
(i) PL,U= Hρ∗L,U= Cρ∗L,U.
(ii) The vertices of PL,U are precisely the ρ(πσ)s where σ ranges over p.
(iii) Each direction of an edge of PL,U is proportional to the difference of a pair of standard unit vectors in RP.
Proof. (i) From (5.1), for every assembly π ∈ L,U and subset I of {1, . . . , p},i∈Iρ(π)i ρ∗L,U(I ), and from (5.2), pi=1ρ(π)i =
t u=1 p i=1 j∈πiρuj= t u=1 nu
j=1ρuj=CρL,U∗ ({1, . . . , p}), proving that ρ(π)∈
CρL,U
∗ . It follows that the convex hulls of the ρ(π )s, namely PL,U, is contained in Cρ∗L,U, that is, PL,U⊆ Cρ∗L,U. Next, the supermodularity of ρ∗L,U (Lemma 5.4) and Proposition 3.1 imply that Hρ∗L,U=Cρ∗L,U. Finally, Lemma 5.2 implies that
PL,U= conv {ρ(π); π ∈ L,U} ⊇ conv{ρ(π); π
is a monotone assembly in L.U} ⊇ conv (ρL,U ∗ )σ; σ ∈ p = HρL,U ∗ ,
So, PL,U⊆ Cρ∗L,U= Hρ∗U,L⊆ PL,U, implying that PL,U= Cρ∗L,U= Hρ∗U,L.
(ii) The supermodularity of ρ∗L,U (Lemma 5.4), part (ii) of Lemma 3.1 and the established part (i) show that the vertices of PL,U= HρL,U∗ = Cρ∗L,U are the (ρ∗L,U)σ with σ ranging over p, and part (i) of Lemma 5.2 shows
that these are precisely the ρ(πσ)s.
(iii) Part (iii) of Lemma 3.1, the supermodularity of ρ∗L,U (Lemma 5.4) and (the established) part (i), immediately imply that the directions of the edges of PL,U= Hρ∗L,U= CρL,U∗ are proportional to differences of unit vectors.
The equality PL.U= Cρ∗L,U in part (i) of Theorem 5.5 provides a linear inequality representation for PL.U in terms of the ρ∗L,U(I )s, using p vari-ables and 2p constraints. Now, the method for determining the ρ∗(I )s in Sec-tion 4 (following Theorem 4.3) can be modified for computing the ρ∗L,U(I )s, with (5.5) replacing (4.3). As (5.5) uses the n−u(I )s instead of using the nus
in (4.3), the only needed modification is the computation of the n−u(I )s (instead of the nu(I )s). But, these are available from (5.3) – their
compu-tation requires the compucompu-tation of all (nu−i∈ILui)s and (i∈IUui)s and
executing a minimization of two terms for each I . So, the n−u(I )s can be determined with at most 3t2p arithmetic operations instead of the t2p oper-ations needed to compute the nu(I )s. The resulting effort to compute the
ρL,U
∗ (I )s is then bounded by tu=1nu+ 4t2p arithmetic operations. With p
fixed, the bound is linear in the number of componentstu=1nu.
Part (i) of Theorem 5.5 and part (i) of Lemma 5.2 show that each vertex of PL,U has the representation ρ(πσ) for some permutation σ of {1, . . . , p};
in particular, the number of vertices of PL,U is bounded by the number of permutations over {1, . . . , p}, namely by p!. Once all the n−u(I )s and
(ρL,U
∗ )(I )s are determined, the πσs are readily available and the
compu-tation of each ρ(πσ) =(ρ∗L,U)σ from (3.1) with λ=ρ∗L,U, requires t
subtrac-tions. This procedure will generate a set of p! (monotone) assemblies along with the associated vectors such that these vectors are precisely the verti-ces of PL,U; the additional computational effort, beyond the computation of the n−u(I )s and the (ρ∗L,U)(I )s, consists of p!t arithmetic operations.
As the πσs are monotone assemblies, part (ii) of Theorem 5.5 implies that
each vertex of PL,U has a representation as the vector that is associated with some monotone assembly. But, the next example demonstrates that the equality of the set of vertices of the assembly polytope and the set of vectors associated with monotone assemblies does not extend from the prescribed-sizes problem (part (ii) of Theorem 4.3) to the bounded-prescribed-sizes problem.
Example 1 Suppose t=1, p =2, n1=4, ρ1i=−1 for i =1, . . . , 4. Consider the set of assemblies π with 1|π11|3 and 1|π12|3. The assemblies in this set are the (ordered) partitions of 1, 2, 3, 4 into two nonempty parts. Consider the monotone assemblies π0= ({1, 2}, {3, 4}), π1= ({1, 2, 3}, {4}, ) and π2= ({1}, {2, 3, 4}). We then have that ρ(π0) = −(2, 2), ρ(π1) = −(3, 1) and ρ(π2) = −(1, 3) and ρ(π0) =12ρ(π1) +12ρ(π2), assuring that ρ(π0) is not a vertex of the corresponding assembly polytope.
We next use Theorem 5.5 to solve the optimal assembly problem. COROLLARY 5.6 There exists a permutation σ such that πσ is a monotone
Proof. Proposition 2.1, the asymmetric Schur convexity of g and part (iv)
of Theorem 5.5 assure that g attains a maximum over PL.U at a vertex of that polytope. Part (ii) of Theorem 5.5 assures that such a vertex has a representation ρ(πσ). Now, let π be an arbitrary assembly; then ρ(π) ∈
PL,U and R(π )= g[ρ(π)] g[ρ(π
σ)] = R(πσ). Part (ii) of Theorem 5.5 also
assures that π is monotone.
The two paragraphs following Theorem 5.5 showed how the set of vertices of
PL,U can be generated with computational effort that is linear in n (the size of
the partitioned set), while proportional to p!. By Theorem 5.6, the evaluation of R(.) for these assemblies and the selection of the best one yields an optimal solution to the optimal assembly problem with bounded part-sizes.
6. Discussion and Extensions
The equality PL,U = Cρ∗L,U, established in Theorem 5.5, provides a repre-sentation of PL,U as the feasible set of a system of 2P linear inequalities with p variables; it follows that when p is small, linear functions can be efficiently optimized over PL,U. This conclusion together with part (iii) of Theorem 5.5 allow one to apply the vertex enumeration method that was developed in [17] for solving (the newly defined) Convex Combinatorial Optimization Problems. Specifically, [17] describes an algorithm that will efficiently enumerate vertices of a polytope P under two assumptions: (i) the efficient solvability of linear programs over P , and (ii) the availability of a (short) list of vectors that contains directions of all of Ps edges. The-orem 5.5 establishes these properties for the assembly polytope. We have already seen that linear functions can be efficiently optimized over PL,U and part (iii) of Theorem 5.5 identifies p2 vectors which cover the direc-tions of all the edges of PL,U. The algorithm that is described in [17] can then enumerate the vertices of PL,U by solving at most O[p(p2)] linear pro-grams over PL,U. Unfortunately, the complexity bound of this method is not better than that of enumerating the πσs.
Theorem 5.6 and 4.4 provide conditions for the existence of optimal assembly that have some restricted structure. We next discuss conditions under which every optimal assembly has that structure. We say that a per-formance function is coherent if no module i is irrelevant in the sense that J (.) is independent of the ith variable. Following [12, example 3] we observe that when J (.) is coherent the function g(.) defined by (2.3) is strictly asymmetric Schur convex on (−∞, 0)p.1 It then follows from 1It was observed in [11] that coherence suffices for g to be strictly assymmetric Schur convex, but the statement therein ignores the fact that the conclusion applies only to the restriction of g to (−∞, 0)p.
Proposition 2.1 that every maximizer of g(.) over PL,U is a vertex of PL,U, implying that every optimal assembly π has ρ(π ) as a vertex if PL,U. We further observe that the (rather complicated) analysis of [1] can be used to show that when all the inequalities of (2.1) hold strictly, each vertex of
PL,U corresponds to a unique assembly. It then follows from the results of
Section 5 that every optimal assembly is monotone, has the form πσ and
has its associated vector as a vertex of PL,U.
We finally describe extensions of our results to assembly problems in which the objective function does not necessarily express system-reliability. Consider an assembly problem in which ρuis are associated with the
par-titioned elements, but without having the interpretation of element-reliabil-ity. Also, the objective function R(.) is expressible as g[ρ(π )] with ρ(π )is
defined by the right-hand side of (2.2), without R(.) having an interpre-tation of system-reliability. We observe that the results and methods of Section 5 (and 4) extend to such assembly problems when g(.) is any asym-metric Schur convex function on PL,U and the ρuis are arbitrary
num-bers that satisfy (2.1). In particular, the nonpositivity of the ρuis is needed
for the representation of ρ∗L,U through (5.3) (and its instance (4.3) that applies to the prescribed-sizes case). Still, these formulae – (5.3) and (4.3) – do have simple counterparts when nonnegativity replaces the nonpositivity, that is, (2.1) is replaced by
−∞ < ρu1< ρu2< · · · < ρunu< 0 for u = 1, . . . , t. (6.1) Consequently, one can derive counterparts of the results of Section 5, in particular, PL,U=CρL,U∗ , (providing a representation of PL,U through a sys-tem of linear inequalities), the vertices of PL,U are the vectors associated with simple monotone assemblies and directions of edges of PL,U are pro-portional to differences of standard unit vectors. When g is strictly asym-metric Schur convex, and the inequalities in (2.1) or (6.1) are strict, we get that every optimal assembly has the special structure.
References
1. Barnes, E.R., Hoffman, A.J. and Rothblum, U.G. (1992), On optimal partitions having disjoint convex and conic hulls, Mathematical Programming 54, 69–86.
2. Baxter, L.A. and Harche, F. (1992), Note: on the greedy algorithm for optimal assembly,
Naval Research Logistics 39, 833–837.
3. Derman, C., Lieberman, G.J. and Ross, S.M. (1972), On optimal assembly of systems,
Naval Research Logistics Quarterly 19, 564–574.
4. Du, D.Z. (1987), When is a monotonic grouping optimal? In: Osaki, S. and Cao, J. (eds.), Reliability Theory and Application, World Scientific, New Jersey, pp. 66–76. 5. Du, D.Z. and Hwang, F.K. (1990), Optimal assembly of an s-stage k-out-of-n system,
6. El-Neweihi, E., Proschan, F. and Sethuraman, J. (1986), Optimal allocation of com-ponents in parallel-series and series-parallel systems, Journal of Applied Probability 23, 770–777.
7. El-Neweihi, E., Proschan, F. and Sethuraman, J. (1987), Optimal allocation assembly of systems using Schur functions and majorization, Naval Research Logistics Quarterly 34, 705–712.
8. Fujishige, S. (1991), Submodular functions and optimization, Annals of Discrete
Mathe-matics 47, North Holland, Amsterdam.
9. Hwang, F.K., Lee, J.S. and Rothblum, U.G. (2004), Permutation polytopes correspond-ing to strongly supermodular functions, Discrete Applied Mathematics, 142, 87–97. 10. Hwang, F.K. and Rothblum, U.G. (1994), Optimality of monotone assemblies for
coher-ent systems composed of series modules, Operations Research 42, 709–720.
11. Hwang F.K. and Rothblum, U.G. (1995), Some comments on the optimal assembly problem, Naval Research Logistics 42, 757–771.
12. Hwang, F.K. and Rothblum, U.G. (1996), Directional-quasi-convexity, asymmetric Schur-convexity and optimality of consecutive partitions, Mathematics of Operations
Research 21, 540–554.
13. Hwang, F.K. and Rothblum, U.G. (2004), Partitions: Optimality and Clustering, World Scientific, forthcoming.
14. Hwang, F.K., Sun, J. and Yao, E.Y. (1985), Optimal set-partitioning, SIAM Journal on
Algebraic and Discrete Methods 6, 163–170.
15. Hwang, F.K., Wang, Y.M. and Lee, J.S. (2002), Sortability of multi-partitions, Journal
of Global Optimization 24, 463–472.
16. Malon, D.M. (1990), When is greedy module assembly optimal, Naval Research
Logis-tics Quarterly 37, 847–854.
17. Onn, S. and Rothblum, U.G. (2004), Convex combinatiorial optimization, Discrete and
Computational Geometry 32, 549–566.
18. Shapley, L.S. (1971), Cores of convex games, International Journal of Game Theory 1, 11–26.
