Nonnegative sum-symmetric matrices, optimal-score partitions, and optimal resource allocation

TL;DR

This paper investigates optimal resource allocations through the lens of nonnegative sum-symmetric matrices, revealing that such matrices can be decomposed into circuit matrices, leading to insights on optimal-score partitions.

Contribution

It introduces a novel representation of nonnegative sum-symmetric matrices as sums of circuit matrices, facilitating the analysis of optimal-score partitions and resource allocation.

Findings

Nonnegative sum-symmetric matrices can be decomposed into circuit matrices.

Optimal-score partitions correspond to certain resource allocation strategies.

The decomposition aids in understanding the structure of optimal solutions.

Abstract

The main result of the note describes certain optimal-score partitions, which can be interpreted as optimal resource allocations. This result is based on the fact that any nonnegative square matrix whose column sums are the same as the corresponding row sums can be represented as the sum of circuit matrices.

Equations

$$\forall A\in \Sigma \forall q\in G\cap [0,\nu (A)]\exists B\in \Sigma B\subseteq A\&\nu (B)=q.$$

$$\mathbf{q}=(q_{1},\dots,q_{k})\in\big{(}G\cap[0,\infty)\big{)}^{k}$$

$$q_1+\cdots +q_k=\nu (X).$$

$$\mathscr{P}_{\nu,\mathbf{q}}:=\big{\{}P=(A_{1},\dots,A_{k})\in\mathscr{P}_{k}\colon\nu(A_{i})=q_{i}\,\ \forall i\in[k]\big{\}}.$$

$$\underset{B_i}{\sup }f⩽\underset{B_j}{\inf }f\text{whenever}i<j;$$

$$s_1⩽\cdots ⩽s_k$$

$$s(P):=\sum _{i=1}^k{s}_i\mu (A_i)$$

$$F(t):=\nu\big{(}f^{-1}([0,t])\big{)}=\nu\big{(}\{x\in X\colon f(x)\leqslant t\}\big{)}$$

$$s:=\inf \{t\in [0,\infty ]:F(t)⩾q_1\}\in [0,\infty ],\text{so that}F(s-)⩽q_1⩽F(s).$$

$$\nu (C_{i,j})=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}{w}_{\pi }\mathrm{I}\left\{j=\pi (i)\in J\right\}$$

$$\nu (C_{\pi ;i,j})=w_{\pi }\mathrm{I}\left\{j=\pi (i)\in J\right\},$$

$$\nu (C_{i,j})=\sum _{\pi \in \Pi }\nu (C_{\pi ;i,j})\text{and}\mu (C_{i,j})=\sum _{\pi \in \Pi }\mu (C_{\pi ;i,j}).$$

$$r_{\pi ;i,j}:=\{\begin{array}{rlrl}& \frac{\mu (C_{\pi ;i,j})}{\nu (C_{\pi ;i,j})}=\frac{1}{\nu (C_{\pi ;i,j})}{\int }_{C_{\pi ;i,j}}fd\nu & & \text{if}\nu (C_{\pi ;i,j})\ne 0,\\ & \underset{B_j}{\sup }f& & \text{otherwise},\end{array}$$

$$\mu (C_{\pi ;i,j})=r_{\pi ;i,j}\nu (C_{\pi ;i,j});$$

$$j_1<j_2⟹r_{\pi ;i_1,j_1}⩽r_{\pi ;i_2,j_2}.$$

$$s(P)=\sum _{i\in [k]}{s}_i\mu (A_i)$$

$$=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}\sum _{i,j\in [k]}{s}_i\mu (C_{\pi ;i,j})$$

$$=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}\sum _{i,j\in [k]}{s}_i{r}_{\pi ;i,j}\nu (C_{\pi ;i,j})$$

$$=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}\sum _{i,j\in [k]}{s}_i{r}_{\pi ;i,j}{w}_{\pi }\mathrm{I}\left\{j=\pi (i)\in J\right\}$$

$$=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}{w}_{\pi }\sum _{j\in J}{s}_{{\pi }^{-1}(j)}{r}_{\pi ;{\pi }^{-1}(j),j}.$$

$$s(Q)=\sum _{i\in [k]}{s}_j\mu (B_j)$$

$$=\sum _{J\subseteq [k]}\sum _{\pi \in {\Pi }_J}{w}_{\pi }\sum _{j\in J}{s}_j{r}_{\pi ;{\pi }^{-1}(j),j},$$

$$\sum _{j\in J}{s}_j{u}_j⩾\sum _{j\in J}{s}_{\sigma (j)}{u}_j$$

$$\mu (A):={\int }_{A}fd\nu =\sum _{x\in A}f(x)$$

$$\sum _{i\in [k]}\sum _{x\in A_i}f(x)s_i=\sum _{i\in [k]}{s}_i\mu (A_i)=s(P),$$

Full text

1 Nonnegative sum-symmetric matrices

A matrix is called nonnegative if all its entries are nonnegative. A square matrix $T=(t_{ij})_{i,j\in[n]}$, where $[n]:=\{1,\dots,n\}$, is called sum-symmetric if for each $i\in[n]$ the row sum $s_{i}(T):=\sum_{j\in[n]}t_{ij}$ is the same as the corresponding column sum $c_{i}(T):=\sum_{j\in[n]}t_{ji}$. A matrix $(c_{ij})_{i,j\in[n]}$ is called a circuit matrix if for some set $J\subseteq[n]$, some cyclic permutation $\pi$ of $J$, and all $i,j$ in $[n]$ we have $c_{ij}=\mathrm{I}\!\left\{j=\pi(i)\in J\right\}$, where $\mathrm{I}\!\left\{\cdot\right\}$ denotes the indicator. Clearly, any circuit matrix is sum-symmetric. A central result here is that any nonnegative sum-symmetric real matrix is a conical combination of circuit matrices; see e.g. (dantzig85, , Theorem 1) or (bapat-ragh, , Lemma 3.4.3); in dantzig85 , the sum-symmetric and circuit matrices are referred to as line-sum-symmetric and simple circuit matrices, respectively.

This result has a short and simple proof, which extends almost verbatim to the case when the entries of the matrix are from a linearly ordered Abelian group $(G,+,0,\geqslant)$, with a linear order $\geqslant$ on the set $G$ such that for any $a$ and $b$ in $G$ one has $a\geqslant b\iff a-b\geqslant 0$. Write $a>b$ to mean that $a\geqslant b\neq a$. It is shown in levi42 that an Abelian group can be linearly ordered iff it is torsion free, that is, iff [math] is its only element of finite order. It is also known (see e.g. hahn07 ; gravett ) that any linearly ordered Abelian group can be embedded into the additive group ${\mathbb{R}}^{I}$ endowed with a lexicographical order, where $I$ is a certain linearly ordered set and ${\mathbb{R}}^{I}$ is the set of all functions from $I$ to ${\mathbb{R}}$ vanishing outside a well-ordered subset of $I$. Examples of linearly ordered groups are any linearly ordered rings and, in particular, any linearly ordered fields. So, the additive groups of the ordered fields ${\mathbb{R}}$ and ${}^{*}{\mathbb{R}}$ of real and hyperreal numbers are linearly ordered groups. Any subgroup of any linearly ordered group is a linearly ordered group, with the inherited order. The direct product $G_{1}\times G_{2}\times\cdots$ of any linearly ordered groups $G_{1},G_{2},\ldots$ is a linearly ordered group with respect to the lexicographic order.

In this group context, let us also extend the notion of a circuit matrix, by defining it as a matrix $(c_{ij})_{i,j\in[n]}\in G^{n\times n}$ such that for some $c\in G$, some set $J\subseteq[n]$, some cyclic permutation $\pi$ of $J$, and all $i,j$ in $[n]$ we have $c_{ij}=c$ if $j=\pi(i)\in J$ and $c_{ij}=0$ otherwise; let us denote this circuit matrix by $C^{J,\pi,c}$. Now we can state

Theorem 1.1

Let $(G,+,0,\geqslant)$ be a linearly ordered Abelian group. Then any nonnegative sum-symmetric matrix in $G^{n\times n}$ is the sum of nonnegative circuit matrices in $G^{n\times n}$.

For readers’ convenience, let us give here

Proof

*of Theorem 1.1.  * Take any nonnegative sum-symmetric matrix $T=(t_{ij})_{i,j\in[n]}\in G^{n\times n}$. If $s_{k}(T)=0$ for some $k\in[n]$, then $c_{k}(T)=s_{k}(T)=0$, and so, all entries of the $k$th row and $k$th column of $T$ are [math]. Crossing out these row and column, we obtain a nonnegative sum-symmetric matrix in $G^{(n-1)\times(n-1)}$, and the proof can be easily completed by induction on $n$.

So, without loss of generality $s_{i}(T)>0$ for all $i\in[n]$, that is, for each $i\in[n]$ there is some $j\in[n]$ such that $t_{ij}>0$. Therefore, for any $i_{1}\in[n]$ we have a sequence $(i_{1},i_{2},\dots)$ in the set $[n]$ such that $t_{i_{\alpha},i_{\alpha+1}}>0$ for all natural $\alpha$. By the pigeonhole principle, there are natural $k$ and $\ell$ with the property that $k<\ell$ and $i_{k}=i_{\ell}$. Taking such $k$ and $\ell$ with the smallest value of $\ell-k$, we will have $i_{k},\dots,i_{\ell-1}$ be pairwise distinct. So, the condition $\pi(i_{\alpha})=i_{\alpha+1}$ for $\alpha=k,\dots,\ell-1$ will define a cyclic permutation $\pi$ on the set $J:=\{i_{k},\dots,i_{\ell-1}\}$. Then the matrix $\tilde{T}:=T-C^{J,\pi,t}$, where $t:=\bigwedge_{\alpha=k}^{\ell-1}t_{i_{\alpha},i_{\alpha+1}}$, will be nonnegative and sum-symmetric, and $\tilde{T}$ will have strictly fewer nonzero entries than $T$ does. Now the proof can be easily completed by induction on the number of nonzero entries of the matrix.

The case $G={\mathbb{R}}$ of Theorem 1.1 complements the famous Birkhoff–von Neumann theorem, which states that every doubly stochastic matrix is a convex combination of permutation matrices. One can similarly extend the Birkhoff–von Neumann theorem to groups:

Theorem 1.2

Let $(G,+,0,\geqslant)$ be a linearly ordered Abelian group. Then any nonnegative matrix $T\in G^{n\times n}$ with $s_{1}(T)=\cdots=s_{n}(T)=c_{1}(T)=\cdots=c_{n}(T)$ is the sum of nonnegative circuit matrices in $G^{n\times n}$ of the form $C^{J,\pi,c}$ with $J=[n]$.

For a proof of Theorem 1.2, one may take, almost verbatim (cf. the above proof of Theorem 1.1), the proof of Theorem 5.1.9 in hall67 , which is based on Ph. Hall’s theorem on distinct representatives – see e.g. Theorem 5.1.1 in hall67 ; other proofs of Ph. Hall’s theorem and its extensions can be found e.g. in ann-comb and (representant, , Section 3.3).

In the rest of the paper, we shall only need Theorem 1.1 when $G$ is ${\mathbb{R}}$ or $\mathbb{Z}$.

2 Optimal-score partitions

Let $k$ be a natural number. Let $\mu$ and $\nu$ be finite measures on a measurable space $(X,\Sigma)$ such that $\mu$ is absolutely continuous with respect to $\nu$, with a Radon–Nikodym derivative $f=\frac{d\mu}{d\nu}$. Let $\mathscr{P}_{k}$ denote the set of all partitions $P=(A_{1},\dots,A_{k})$ of $X$ such that $A_{i}\in\Sigma$ for all $i\in[k]$.

Suppose that one of the following two conditions on the group $G$, the $\sigma$-algebra $\Sigma$, and the measure $\nu$ holds:

1. (I)

$G={\mathbb{R}}$ and $\nu$ is non-atomic; 2. (II)

$G=\mathbb{Z}$, $\Sigma$ is the powerset $2^{X}$ of $X$, and $\nu$ is the counting measure (so that the set $X$ is finite).

Then

[TABLE]

Indeed, this is obvious when condition (II) holds. In the case when (I) holds, conclusion (1) follows immediately from the well-known fact that the set of all values of a non-atomic finite measure is convex; see e.g. (dudley-norv, , Proposition A.1).

Fix any $k$-tuple

[TABLE]

such that

[TABLE]

Consider

[TABLE]

In view of (1), $\mathscr{P}_{\nu,\mathbf{q}}\neq\emptyset$. Moreover, let us state

Proposition 1

There exists a partition $Q=(B_{1},\dots,B_{k})\in\mathscr{P}_{\nu,\mathbf{q}}$ such that for any $i,j$ in $[k]$

[TABLE]

recall here that $\sup\emptyset=-\infty$ and $\inf\emptyset=\infty$.

Also, fix arbitrary real numbers $s_{1},\dots,s_{k}$ such that

[TABLE]

and define the “score”

[TABLE]

of any partition $P=(A_{1},\dots,A_{k})\in\mathscr{P}_{\nu,\mathbf{q}}$.

Now we can state the main result of this note:

Theorem 2.1

For any partition $Q$ as in Proposition 1 and any partition $P\in\mathscr{P}_{\nu,\mathbf{q}}$, we have $s(Q)\geqslant s(P)$; that is, any partition $Q$ as in Proposition 1 has the highest possible score among all partitions in $\mathscr{P}_{\nu,\mathbf{q}}$.

Let us now prove the above statements.

Proof

*of Proposition 1.  * This will be done by induction on $k$. The case $k=1$ is trivial. By writing $q_{1}+\dots+q_{k}=(q_{1}+\dots+q_{k-1})+q_{k}$, we reduce the consideration to the case $k=2$, so that \mathbf{q}=(q_{1},q_{2})\in\big{(}G\cap[0,\infty)\big{)}^{2} and $q_{1}+q_{2}=\nu(X)$.

Consider the (right-continuous) “distribution function” $F$ of the function $f$ with respect to the measure $\nu$, defined by the formula

[TABLE]

for $t\in(-\infty,\infty]$, and let

[TABLE]

Next, let $D:=f^{-1}(\{s\})$, and then let $D_{1}$ be any set in $\Sigma$ such that $D_{1}\subseteq D$ and $\nu(D_{1})=q_{1}-F(s-)$; such a set $D_{1}$ exists by (1), in view of the inequalities in (7) and the equality $\nu(D)=F(s)-F(s-)$. Finally, let $B_{1}:=f^{-1}([0,s))\cup D_{1}$ and $B_{2}:=X\setminus B_{1}=f^{-1}((s,\infty))\cup(D\setminus D_{1})$. Then, obviously, $(B_{1},B_{2})\in\mathscr{P}_{2}$. Also, because $D_{1}\subseteq D=f^{-1}(\{s\})$, the sets $f^{-1}([0,s))$ and $D_{1}$ are disjoint and hence \nu(B_{1})=\nu\big{(}f^{-1}([0,s))\big{)}+\nu(D_{1})=F(s-)+[q_{1}-F(s-)]=q_{1}, so that $\nu(B_{2})=\nu(X)-\nu(B_{1})=q_{2}$. Therefore, $(B_{1},B_{2})\in\mathscr{P}_{\nu,\mathbf{q}}$. Moreover, $B_{1}\subseteq f^{-1}([0,s])$ and $B_{2}\subseteq f^{-1}([s,\infty))$, so that $\sup_{B_{1}}f\leqslant s\leqslant\inf_{B_{2}}f$, and thus (4) holds, for $k=2$. This completes the proof of Proposition 1.

Proof

*of Theorem 2.1.  * Let $\Pi:=\bigcup_{J\subseteq[k]}\Pi_{J}$, where $\Pi_{J}$ stands for the set of all permutations of the set $J$. Let $\operatorname{\mathscr{T}}:=\Pi\times[k]\times[k]$.

Take any partition $Q$ as in Proposition 1 and any partition $P=(A_{1},\dots,A_{k})\break\in\mathscr{P}_{\nu,\mathbf{q}}$. Introduce $C_{i,j}:=A_{i}\cap B_{j}$ for $(i,j)\in[k]^{2}$. A crucial observation is that the matrix \big{(}\nu(C_{i,j})\big{)}_{i,j\in[k]} is nonnegative and sum-symmetric, and so, by Theorem 1.1,

[TABLE]

for all $(i,j)\in[k]^{2}$, where the $w_{\pi}$’s are some numbers in $G\cap[0,\infty)$. Therefore and in view of (1), for each $(i,j)\in[k]^{2}$ there is a partition $(C_{\pi;i,j})_{\pi\in\Pi}$ of the set $C_{i,j}$ such that for each triple $(\pi,i,j)\in\operatorname{\mathscr{T}}$ we have $C_{\pi;i,j}\in\Sigma$ and

[TABLE]

where, for any given $\pi\in\Pi$, the set $J\subseteq[k]$ is uniquely determined by the condition $\pi\in\Pi_{J}$. Hence,

[TABLE]

For each triple $(\pi,i,j)\in\operatorname{\mathscr{T}}$, let

[TABLE]

so that

[TABLE]

also, in view of the set inclusions $C_{\pi;i,j}\subseteq C_{i,j}\subseteq B_{j}$, we have $\inf_{B_{j}}\,f\leqslant r_{\pi;i,j}\leqslant\sup_{B_{j}}\,f$.

Therefore, in view of inequalities (4), we now arrive at the second important point in this proof: that for all triples $(\pi,i_{1},j_{1})$ and $(\pi,i_{2},j_{2})$ in $\operatorname{\mathscr{T}}$ we have the implication

[TABLE]

By (6), (9), (10), and (8),

[TABLE]

Similarly to this, we have

[TABLE]

with the only difference that $s_{i}$ in $\sum_{i,j\in[k]}s_{i}\mu(C_{i,j})$ and in the two subsequent expressions in multi-line display (2) is now replaced by $s_{j}$.

So, to compete the proof of Theorem 2.1, it suffices to show that

[TABLE]

for any permutation $\sigma\in\Pi_{J}$, where $u_{j}:=r_{\pi;\pi^{-1}(j),j}$. Since any permutation can be obtained from the identity permutation by finitely many inversions, it is enough to verify (12) in the case when the cardinality of $J$ is $1$ or $2$, so that $J=\{j,m\}$ for some $j,m$ in $[k]$. Then (12) can be rewritten as $s_{j}u_{j}+s_{m}u_{m}\geqslant s_{m}u_{j}+s_{j}u_{m}$ or, equivalently, as $(s_{j}-s_{m})(u_{j}-u_{m})\geqslant 0$, which is true – because, by (5) and (11), $s_{j}$ and $u_{j}=r_{\pi;\pi^{-1}(j),j}$ are each nondecreasing in $j\in J$. This concludes the proof of Theorem 2.1.

3 Optimal resource allocation

Theorem 2.1, appropriately interpreted, provides a solution to an optimal resource allocation (ORA) problem. For simplicity, let us state here this problem and its solution for the “discrete” setting, corresponding to alternative (II) on page I. The ORA problem is as follows.

• •

Each member $x$ of a finite set $X$ is to be subjected to exactly one of $k$ treatments, labeled by $1,\dots,k$, with potencies $s_{1},\dots,s_{k}$ and available in quantities $q_{1},\dots,q_{k}$, respectively.

• •

In accordance with condition (5), we assume that the potencies $s_{1},\dots,s_{k}$ are real numbers such that $s_{1}\leqslant\dots\leqslant s_{k}$; that is, the $k$ treatments are enumerated according to their potencies, from the lowest to the highest. Potencies are allowed to take negative values, corresponding to negative treatment effects.

• •

In this “discrete” setting, the available quantities $q_{1},\dots,q_{k}$ of treatments $1,\dots,k$ are nonnegative integers such that the total of the quantities $q_{1},\dots,q_{k}$ equals the number $\nu(X)$ of the members of the set $X$.

• •

For each member $x$ of the set $X$, the effect of any treatment $i\in[k]$ is proportional to the potency $s_{i}$ of the treatment, with a proportionality coefficient $f(x)\in[0,\infty)$, so that the just mentioned effect is $f(x)s_{i}$. It is then natural to refer to $f(x)$ as the responsiveness of member $x$ to treatment.

• •

For each $i\in[k]$, let $A_{i}$ denote the set of all members $x$ of the set $X$ assigned to treatment $i$, so that $P:=(A_{1},\dots,A_{k})$ is a partition of $X$. This partition represents a treatment allocation. In accordance with what has been said, we only consider “feasible” treatment allocations, that is, the ones satisfying the conditions $\nu(A_{i})=q_{i}$ for all $i\in[k]$; cf. (3) (recall that here $\nu$ stands for the counting measure). Letting now

[TABLE]

for any set $A\subseteq X$, we see that the overall effect of a treatment allocation $P=(A_{1},\dots,A_{k})$ will then be

[TABLE]

in accordance with (6).

Now Theorem 2.1 tells us that the overall effect $s(P)$ of a treatment allocation $P=(A_{1},\dots,A_{k})$ will be the largest possible if members of the set $X$ with higher responsiveness are assigned to higher-potency treatments. More specifically, for the optimal treatment allocation, $q_{k}$ members $x$ of the set $X$ with the highest values of responsiveness $f(x)$ are selected to constitute the set $A_{k}$ and thus to receive treatment $k$, of the highest-potency, $s_{k}$; then $q_{k-1}$ members of the remaining set $X\setminus A_{k}$ with the highest values of responsiveness are selected to constitute the set $A_{k-1}$ and thus to receive treatment $k-1$, of the second highest-potency, $s_{k-1}$; etc.

While this solution to this ORA problem appears to agree with intuition, we saw that it takes some effort to prove it rigorously, by using the decomposition of nonnegative sum-symmetric matrices provided by Theorem 1.1.

Let us now provide a few possible specific interpretations of the general ORA setting described above:

The set $X$ may be a human population to be vaccinated against a certain disease. Here, the treatments $1,\dots,k$ correspond to $k$ kinds of a vaccine, with potencies $s_{1},\dots,s_{k}$. The total quantity of the available vaccine, $q_{1}+\dots+q_{k}$ units, is the same as the population size, so that each member of the population be able to receive exactly one unit of the vaccine. For each individual $x$ in the population, $f(x)$ is the individual’s responsiveness to vaccination. The goal here is to maximize the overall vaccination effect $s(P)$. 2. 2.

Here $X$ is the set of workers of a certain specialty in an industrial company. Now the treatments $1,\dots,k$ correspond to $k$ kinds of equipment, with efficiencies $s_{1},\dots,s_{k}$. The total quantity of the equipment units, $q_{1}+\dots+q_{k}$, is the same as the size of the set $X$ of workers, and each worker will be assigned to exactly one unit of the available equipment. For each worker $x$, $f(x)$ is the worker’s individual productivity coefficient. The goal here is to maximize the overall production $s(P)$. 3. 3.

Now $X$ is a set of agricultural plots. The treatments $1,\dots,k$ correspond to $k$ grades of a fertilizer, with efficiencies $s_{1},\dots,s_{k}$. The total quantity of the fertilizer units, $q_{1}+\dots+q_{k}$, is the same as the the number of plots, and each plot will receive exactly one unit of a fertilizer. For each plot $x$, $f(x)$ is the plot’s responsiveness to fertilization. The goal here is to maximize the overall response $s(P)$ to the fertilization. 4. 4.

This is a “non-atomic” modification of the latter “discrete” scenario. Here $X$ is the set of points on an agricultural field, and the measure $\nu$ of a (measurable) part $A$ of $X$ is $c|A|$, where $c$ is a positive real number and $|A|$ is the area of $A$. The treatments $1,\dots,k$ again correspond to $k$ grades of a fertilizer, with efficiencies $s_{1},\dots,s_{k}$. The field $X$ is partitioned into parts $A_{1},\dots,A_{k}$ so that the part $A_{j}$ receive the $j$th grade of the fertilizer, for each $j=1,\dots,k$. The corresponding quantities $q_{1},\dots,q_{k}$ of the $k$ grades of the fertilizer may now take any nonnegative real values such that the total quantity of the fertilizer, $q_{1}+\dots+q_{k}$, equals $\nu(X)=c|X|$ so that the entire field be covered by the fertilizer with the uniform density $c$ per unit area. For each point $x$ on the field, $f(x)$ is the corresponding local responsiveness to fertilization. The goal here is, again, to maximize the overall response $s(P)$ to the fertilization.

In all these specific scenarios, the maximum overall effect occurs when higher levels of responsiveness are coupled with higher potencies, as specified in the general conclusion.

A search for articles containing the phrase “optimal resource allocation” in Google Scholar reveals about 35400 results. Optimal resource allocation (ORA) problems arise in a great variety of fields and a great variety of settings. A very small sample representing such problems includes ORA studies in biology govern-wolde , computing shahab-etal , economics arrow , electrical engineering seong-etal , health care richter-etal , information theory li-goldsmith , operations research azaiez-bier , risk analysis bier-etal , and transportation dafermos-sparrow .

Kantorovich was apparently the first to consider ORA problems systematically; see e.g. (kantorovich, , Section “Linear programming”) and (koopmans, , page 240). Methods used in the work by Kantorovich and his great many followers are analytical, based on separation of convex sets, with the feasible solutions being points in a finite- or infinite-dimensional linear space.

On the other hand, the main tool used in the present paper is the decomposition of nonnegative sum-symmetric matrices into nonnegative circuit matrices, provided by Theorem 1.1, whose proof is rather combinatorial, and the feasible solutions in our setting are partitions, rather than points in linear spaces over ${\mathbb{R}}$. It is hoped that the simple and rather general resource allocation model considered here, as well as the corresponding results, will be of use in a variety of specific applications.