Title An Inverse Assignment Problem(Optimization Theory inDescrete and Continuous Mathematical Sciences)

Author(s) IWAMOTO, Seiichi; IKI, Tetsuichiro

Citation 数理解析研究所講究録 (1997), 1015: 163-175

Issue Date 1997-11

URL http://hdl.handle.net/2433/61602

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

An Inverse Assignment Problemある逆割り当て問題について

Seiichi IWAMOTO (岩本誠–)Department of Economic Engineering

Faculty of Economics, Kyushu University 27Fukuoka 812-81, Japan

Tbtsuichiro $IKI$ (伊喜哲–郎)

Department of MathematicsFaculty of Education, Miyazaki University

Miyazaki 889-21, Japan

AbstractIn this paper we focus our attention on the famous “Plane Assignment Problem”

in Beckmann’s Dynamic Programming of Economic Decisions and develop a furthertheory of the assignment problem. Formulating the problem into an optimal (main)stopping problem, we propose a new inversion of the stopping problem. By exchangeof objective function and constraint function together with replacement of optimizer$\min$ by ${\rm Max}$ , we introduce an inverse assignment problem, which is also an optimalstopping problem. We establish several inverse theorems between main and inversestopping problems. We also analyze the finite-stage (nonstopping) problems andspecip the enveloping relation to the stopping problems. Detailed numerical solutionsfor both problems are specified.

1 IntroductionIn this paper we devote ourselves exclusively to the study of the so called Plane Assign-ment Problem which has its origin in Beckmann and Laderman [1]. The Plane AassignmentProblem is one of the most typical resourse allocation $\mathrm{p}\mathrm{r}\dot{\mathrm{o}}$ blems [4]. For its simple struc-ture and elegant economic interpretation, this problem has several approaches, for instance,linear programming, integer programming, combinatorial programming, and dynamic pro-gramming (see [8]). Beckmam [2] illustrates heuristically the principle of optimality [3]through the problem.

In this paper we develop a further inverse theory of the assignment problem. We formu-late the problem into an optimal stopping problem, which we call main stopping problem.We propose an inversion of the stopping problem. By exchange of objective function andconstraint function together with replacement of optimizer $\min$ by ${\rm Max}([5],[6],[7])$ , we in-troduce an inverse assignment problem, which is called an inverse stopping problem. An

数理解析研究所講究録1015巻 1997年 163-175 163

inverse relation between main and inverse stopping problems is condensed into four inversetheorems; Weak Inverse Theorem, Strong Inverse Theorem, Strict Inverse Theorem andInverse Stopping Time Theorem. We specify detailed numerical optimal solutions for bothproblems.

In Section 2 we formulate Beckmann’s Plane Assignment Problem into an optimal stop-ping problem in a deterministic sense. We sh.ow the monotonicity of optimal value functionand derive the recursive equation.

In Section 3 we introduce its inverse problem, show the monotonicity of its optimumvalue function, and derive the recursive equation for the inverse pro.blem. The three inversetheorems are established. .

$\mathrm{s}$

In Section 4 we discuss both main and inverse nonstopping (finite-stage) problems. Wegive both problems their optimal solutions and show inverse relations. An envelopingproperty between stopping and nonstopping problems is shown for both main and inverseproblems, respectively. Further an inverse relation between both optimal stopping times isestablished.

In Section 5 we illustrate detailed $\mathrm{n}\mathrm{u}.\mathrm{m}$erical solutions both for Beckmann’s AssignmentProblem and for its inverse problem.

2 Main Stopping ProblemWe begin to consider the Beckmann’s Plane Assignment Problem [2] in the following quata-tion:

Example $[\mathrm{B}\mathrm{E}\mathrm{C}\mathrm{K}\mathrm{M}\mathrm{A}\mathrm{N}/\mathrm{L}\mathrm{A}\mathrm{D}\mathrm{E}\mathrm{R}\mathrm{M}\mathrm{A}\mathrm{N}[1](1956)]$ : Plane Assignment.As an illustration of the principle of optimality consider the problem of find-

ing the best combination of two indivisible resources to meet a given demand.Let the demand be a number of passengers and the resources to be two types

of planes

Let the cost of operating a DC 6 on a given flight be 1.4 times that ofrunning a DC 3. For any number $n$ of passengers up to $n=200\mathrm{i}.\mathrm{t}$ is desired tofind the cheapest combination of planes that will carry them.

From 1 to 38 passengerss are carried most cheaply by one DC 3; from 39 to58 passengers by one DC 6.

To decide which is the cheapest cost of trnasporting 59 passengers we denotethe minimum cost of transporting $m$ passengers by $v(m)$ and have the recursiverelation

$v(m)= \min[1.4+v(m-58), 1.0+v(m-38)]$ in particular

$v(59)= \min[1.4+v(1), 1.0+v(21)]$

164

where “$\min$” means the smaller of the two values in the brackets.Since $v(1)=v(21)=1.0$ one has $v(59)=2.0$. And so on.

Now let us formulate Beckmann’s Assignment Problem into a stopping problem andanalyze it.

Throughout the paper, we use the cost function $f$ : $\{1, 2, \ldots, 38, \ldots, 58\}arrow\{1.0,1.4\}$

defined by

$f(1)=f(2)=\cdots=f(38):=1.\mathrm{o}$ , $f(39)=\cdot*\cdot=f(58):=1.4$ (1)

Thus the a..ssignment problem is formulated into the following minimization problem :

minimize $f(x_{1})+f(x_{2})+\cdots+f(x_{t})$

MSP(200) subject to (i) $x_{1}+x_{2}+\cdots+x_{t}=200$ (2)(ii) $1\leq x_{n}\leq 58$ $1\leq n\leq t$

(iii) $1\leq t\leq 200$

The condition (iii) means when to stop assigning. Thus the deterministic variable $t$ isconsidered as a stopping time. This is the main reason why we call MSP(200) a mainstopping problem. Of course, the problem is a problem of finding not only an optimalstopping time $t$ but also an optimal assignment itself $(x_{1}, \ldots , x_{t})$ , which together yields theminimum cost.

Let $v(200)$ be the minimum value. In general, let $v(m)$ be the minimum value of $\mathrm{M}\mathrm{S}\mathrm{P}(m)$

with the right-hand side parameter $m$ in place of 200, where $m$ ranges on the set of naturalnumbers $N=\{1,2, \ldots, 200, \ldots\}$ . Let $<1.0,$ $\infty>$ be the set of discrete real numbers1.0, 1.1, ... with step-size $\mathrm{o}.\mathrm{i}$ :

$<1.0,$ $\infty>=\{1.0,1.1, \ldots, 5.2, \ldots\}$ .

Note that the set $<1.0,$ $\infty>$ contains all the possible values that the optimal valuefunction $v$ takes.

First we have the monotonicity of optimum value function $v(\cdot)$ as follows:

LEMMA 2. 1 The minimum value function $v$ : $Narrow<1.0,$ $\infty>is$ nondecreasing, andit goes to $\infty$ as so does $m$ .

Second we have the following recursive equation.

THEOREM 2. 1

$v(m)= \min[1.4+v(m-58), 1.0+v(m-38)]$ $m=59,60,$ $\ldots$ . (3)$v(1)=v(2)=\cdots=v(38)=1.0$ , $v(39)=v(40)=\cdots=v(58)=1.4$ (4)

165

Let us define the optimal pocicy $\pi^{*}:$ $Narrow\{38,58\}$ by

$\pi^{*}(m)=\{$5838 if $\{$

$1.4+v(m-58)$$1.0+v(m-38)$ attains the minimum in (3) (5)

where$\pi^{*}(1)=\cdots=\pi^{*}(38)=38$ , $\pi^{*}(39)=\cdots=\pi^{*}(58)=58$ . (6)

In the last section, Table 1 shows an optimal solution- a pair of optimal value andoptimal policy-for MPS $\{v(\cdot), \pi^{*}(\cdot)\}$ . Figure 1 illustrates that an successive applicationof optimal policy $\pi^{*}(\cdot)$ ffom the given initial state $m=200$ generates an optimal decisiontree for the given Beckmann’s problem. In summary, the optimal decision tree states thatthe cheapest cost 5.2 of transporting 200 passengers is attained by use of a combination ofone DC 3 and three DC 6.

3 Inverse Stopping ProblemIn this section, as an inverse problem, we consider the following maximization problem :

Maximize $x_{1}+x_{2}+\cdots+x_{t}$

ISP(5.2) subject to $(\mathrm{i})’f(x_{1})+f(x_{2})+\cdots+f(x_{t})\leq 5.2$ (7)(ii) $1\leq x_{n}\leq 58$ $1\leq n\leq t$

(iii) $t\geq 1$ .

This is also a stopping problem. Thus we call this problem Inverse Stopping Problem.Let $u(5.2)$ be the maximum value. In general, let $u(c)$ be the minimum value of $\mathrm{I}\mathrm{S}\mathrm{P}(c)$

with the right-hand side parameter $c$ in place of 5.2, where $c$ ranges on the set of discretereal numbers $<1.0,$ $\infty>=\{1.0,1.1.’ 1.2, \ldots\}$ . Then we have the monotonicity of optimumvalue function $u(\cdot)$ as follows:

LEMMA 3. 1 The maximum value function $u:<1.0,$ $\infty>arrow N$ is nondecreasing, andit goes to $\infty$ as so does $c$ .

We have also the following recursive equation.

THEOREM 3. 1

$u(c)={\rm Max}[58+u(c-1.4), 38+u(c-1.0)]$ $c=1.5,1.6,$ $\ldots$ . (8)$u(\mathrm{o}.1)=u(0.2)=\cdots=u(0.9)=0$ ,$u(1.0)=u(1.1)=\cdots=u(1.3)=38$ , $u(1.4)=58$ (9)

We define the optimal pocicy $\hat{\sigma}:<1.0,$ $\infty>arrow\{38,58\}$ by

$\hat{\sigma}(c)=\{$5838 if $\{$

$58+u(c-1.4)$$38+u(_{C}-1.0)$

attains the maximum in (8) (10)

166

wehere$\hat{\sigma}(1.0)=\cdots=\hat{\sigma}(1.3)=38$ , $\hat{\sigma}(1.4)=58$ . (11)

The optimal solution for IPS $\{u(\cdot),\hat{\sigma}(\cdot)\}$ is shown in Table 2 in Section 5. Figure 2illustrates that an successive application of optimal policy $\hat{\sigma}(\cdot)$ from the given initial totalcost $c=5.2$ generates an optimal decision tree for the inverse problem. The optimaldecision tree states that the maximum total number of passengers 212 for the total cost5.2 or less is also attained by use of the combination of one DC 3 and three DC 6.

Furthermore, we have the following inverse relationship between Main and Inverse Stop-ping Problems:

THEOREM 3. 2 (Weak Inverse Theorem $I$)

(i) $v(u(c))\leq c$ $c\in<1.0,$ $\infty>$ (12)(ii) $u(v(m))\geq m$ $m\in N.$ (13)

It is verified in Tables 2 and 1 that Eqs. (12),(13) hold, respectively.Let $w:Xarrow Y$ be a nondecreasing function, where $X,$ $\mathrm{Y}$ are nonempty discrete subsets

in one-dimensional Euclidean space $R^{1}$ . Then we define two kinds of its inverse functionas follows: One is the upper-semi inverse function $w^{-1}$ : $\mathrm{Y}arrow X$

$w^{-1}(y):= \min\{x\in X|w(x)\geq y\}$ . (14)

The other is the lower-semi inverse function $w_{-1}$ : $Yarrow X$

$w_{-1}(y):={\rm Max}\{x\in X|w(x)\leq y\}$ . (15)

We say that a value $y\in \mathrm{Y}$ is attainable if there exists some $x\in X$ satisfying $w(x)=y$ .Then we have the following properties.

LEMMA 3. 2

$w_{-1}(y)\geq w^{-1}(y)$ for attainable $y\in Y$ (16)$w_{-1}(y)<w^{-1}(y)$ for nonattainable $y\in Y.$ (17)

Furthermore, for any nonattainable $y\in Y$ , both $w_{-1}(y)$ and $w^{-1}(y)$ take two adjacent(neighbouring) values in $X$ .

Moreover, we have a rather strict inverse relations as follows:

THEOREM 3. 3 (Strong Inverse Theorem $I$)

$(\mathrm{i})’$ $v_{-1}(c)=u(c)$ $c\in<1.0,$ $\infty>$ (18)$(\mathrm{i}\mathrm{i})’$ $u^{-1}(m)=v(m)$ $m\in N.$ (19)

As for Eqs. (18),(19) see Tables 2 and 1, respectively. Further, one pair of optimal valuefunction and optimal policy characterizes the other pair as follows:

167

THEOREM 3. 4 ($St\dot{\mathcal{H}}Ct$ Inverse Theorem $I$)

(iii) $\hat{\sigma}(c)=\pi^{*}(v_{-1}(.c))$ $c\in<1.0,$ $\infty>$ (20)(iv) $\pi^{*}(m)=\hat{\sigma}(u^{-1}(m))$ $m\in N$ . (21)

Table 1. Optimal Solution for MSP and Composite Solution

$\ovalbox{\tt\small REJECT}_{38\mathrm{r}}^{38_{0}581343438\mathrm{r}}201935192171715541541353813411116979627765920583381172438\mathrm{o}\mathrm{r}91450545473443838032822038711038223838825258443438\mathrm{o}_{8174}\mathrm{r}58962438540338\mathrm{o}\mathrm{r}58\mathrm{o}\mathrm{r}58\mathrm{o}\mathrm{r}58\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{r}_{5}\mathrm{o}58116285353\mathrm{r}\mathrm{r}_{5819}88581488581485815435588218888211111967339212527531620688252324844464443\mathrm{s}_{83}22211032582583434858003838\mathrm{o}\mathrm{r}38\mathrm{o}\mathrm{r}5838_{0}8_{0}8\mathrm{o}\mathrm{r}_{58}8_{0}\mathrm{r}8\mathrm{o}\mathrm{r}588\mathrm{o}_{8}\mathrm{r}_{5}5\mathrm{o}\mathrm{r}\mathrm{o}_{8}33838558\mathrm{r}_{58}888\mathrm{r}58858585888$

168

Symbolically we write

$\hat{\sigma}=\pi^{*}\mathrm{o}v_{-1}$ on $<1.0,$ $\infty>$ , $\pi^{*}=\hat{\sigma}\circ u^{-}1$ on $N$ (22)

, where $\circ$ is the composition operator between functions.As for Eqs. (20),(21) see Tables 2 and 1, respectively.

Table 2. Optimal Solution for ISP and Composite Solution

$\ovalbox{\tt\small REJECT}_{471745}^{3}522145119421745843815334133122796242219581413381081923711548116376380134397658581453838103833163232344653838_{\mathrm{o}\mathrm{r}}583815438_{\mathrm{o}\mathrm{r}}3888_{0}\mathrm{r}588_{\mathrm{o}\mathrm{r}_{58}}80_{8421}8\mathrm{o}\mathrm{r}5\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{o}\mathrm{r}_{58522}5382583882\mathrm{r}85585588838838249634848\iota_{1238\mathrm{r}}^{745}4213413282207631103844811407191543131169589238\mathrm{o}\mathrm{r}_{58}74566858238\mathrm{o}\mathrm{r}438_{0}\mathrm{r}_{58}584638\mathrm{o}\mathrm{r}58388\mathrm{o}\mathrm{r}58\mathrm{o}_{8}\mathrm{O}55\mathrm{o}\mathrm{r}35388888\mathrm{r}588885858588$

169

4 Nonstopping ProblemsIn this section we consider the nonstopping problems. Let $n$ be any given total numberof planes. Then two problems arise. One is a main problem. For any given total numberof passengers $m$ , we consider the $\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\dot{\mathrm{l}}\mathrm{e}\mathrm{m}$ of finding the minimum total cost of carrying$m$ passengers by $n$ planes. The other is its inverse problem. For any given total cost ofoperating $c$ , we consider the problem of finding the maximum total number of passengeresthat $n$ planes carry for not more than the total cost $c$ . Since all the results in the followingare proved in a similar way as in Sections 2 and 3, the proof is omitted in this section.

4.1 Main ProblemsFor any $n\in N$ , let us define the following two discrete intervals;

$N_{n}$ $:=$ $\{n, n+1, \ldots, 58n\}$ (23)$C_{n}$ $:=$ $\{1.0n, 1.0n+0.1, \ldots , 1.4n+0.5\}$ . (24)

Then the interval $N_{n}$ contains all the possible totaI numbers of passengers $n$ planes cancarry. The interval $C_{n}$ does all the possible total costs for which or less $n$ planes can carry.

Given two positive integers $n,$ $m$ satisfing $m\in N_{n}$ , we consider the problem.of dividing$m$ into $n$ possible natural numbers between 1 and 58 and minimizing the summed valuemeasured through the cost function $f$ :

minimize $f(x_{1})+f(x_{2})+\cdots+f(x_{n})$

$\mathrm{N}\mathrm{M}\mathrm{P}(m;n)$ subject to (i) $x_{1}+x_{2}+\cdots+x_{n}=m$ (25)

(ii) $1\leq x_{i}\leq 58$ $1\leq i\leq n$ .

Let $v_{n}(m)$ be the minimum value. Then we have the following double-monotone propertyand recursive equation:

LEMMA 4. 1 (i) The minimum value function $v_{n}$ : $N_{n}arrow C_{n}$ is nondecreasing :

$v_{n}(m)\leq v_{n}(m+1)$ $m,$ $m+1\in N_{n}$ . (26)

(ii) The sequence of minimum functions $\{v_{n}\}_{n\geq 1}$ is nondecreasing:

$v_{n}(m)\leq v_{n+1}(m)$ $m\in N_{n}\cap N_{n+1}$ . (27)

THEOREM 4. 1

$v_{1}(m)=f(m)$ $m\in N_{1}$ (28)

$v_{n+1}(m)= \min_{x\cdot*}.[f(X)+v_{n}(m-X)]$ $m\in N_{n+1}$ , $n\geq 1$ (29)

where $x:*means$ that the minimization is taken for all $x$ satisfying

$1\leq x\leq 58$ , $m-x\in N_{n}$ . (30)

In particular, when {m--38, m–58} $\subset N_{n}$ , Eq. (29) reduces

$v_{n+1}(m)= \min[1.0+v_{n}(m-38), 1.4+v_{n}(m-58)]$ . (31)

170

Let $\pi_{n+1}^{*}(m)\subset\{1,2, \ldots, 58\}$ be the set of all minimizers for (29), where

$\pi_{1}^{*}(1)=\cdots=\pi_{1}^{*}(38)=38$ , $\pi_{1}^{*}(39)=\cdots=\pi_{1}^{*}(58)=58$ . (32)

Then the $\mathrm{P}^{\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}- \mathrm{t}}\mathrm{O}^{-\mathrm{S}\mathrm{e}\mathrm{t}}$ valued function $\pi_{n}^{*}$ : $N_{n}arrow\{1,2, \ldots, 58\}$ is called n-th optimal deci-sion function. We call the sequence of optimal decision functions $\pi^{*}=\{\pi_{1}^{*}, \pi_{2}^{*}, \ldots , \pi_{n}^{*}, \ldots\}$

an optimal policy for Nonstopping Main Problem $\mathrm{N}\mathrm{M}\mathrm{P}(m;n)$ .Further we have the following relation $\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{w}\mathrm{e}\vee \mathrm{e}\mathrm{n}$ Stopping Problem $\mathrm{a}\mathrm{n}\dot{\mathrm{d}}\mathrm{N}\mathrm{Q}\mathrm{n}\mathrm{S}\mathrm{t}_{0}\mathrm{p}\mathrm{p}\dot{\mathrm{i}}\mathrm{n}\mathrm{g}‘ \mathrm{P}\mathrm{r}\mathrm{o}\mathrm{b}-$

$\mathrm{l}\mathrm{e}\mathrm{m}$ :

THEOREM 4. 2 (Main Envelope Theorem)

$v(m)= \min_{n|m\in Nn}v_{n}(m)$ $m\in N$ . (33)

Let $t^{*}(m)$ be the first positive integer $n$ such that $v(m)=v_{n}(m)$ . Then $t^{*}$ is the o..ptimalstopping time for MSP:

$v(m)=v_{t^{*}}(m)$ $m\in N$ . (34)

As for Eqs. (33),(34), see Table 3.

4.2 Inverse ProblemsWe consider the inverse problem of $\mathrm{N}\mathrm{M}\mathrm{P}(m;n)$ as follows:

Maximize $x_{1}+x_{2}+\cdots+x_{n}$

NIP $(c;n)$ subject $.\mathrm{t}\mathrm{o}$ $(\mathrm{i})’f(x_{1})+f(x_{2})+\cdots+f(x_{n})\leq c$ (35)(ii) $1\leq x_{i}\leq 58$ $1\leq i\leq n$

where $c\in C_{n}$ , $n\geq 1$ . Let $u_{n}(c)$ be the maximum value. Then the maximum valuefunctions enjoy the following double-monotone property and recursive equation:

LEMMA 4. 2 (i) The maximum value function $u_{n}$ : $C_{n}arrow N_{n}i\mathit{8}$ nondecreasing:

$u_{n}(c)\leq u_{n}(c+\mathrm{o}.1)$ $c,$ $c+\mathrm{o}.1\in C_{n}$ . (36)

(ii) The sequence of maximum functions $\{u_{n}\}_{n\geq 1}$ is nonincreasing:

$u_{n}(c)\leq u_{n+1}(c)$ $c\in C_{n}\cap C_{n}+1$ . (37)

THEOREM 4. 3

$u_{1}(1.0)=u_{1}(1.1)=\cdots=u_{1}(1.3)=38$ , $u_{1}(1.4)=\cdots=u_{1}(1.9)=58$ (38)$u_{n+1}(c)={\rm Max}.[xx\cdot**+un(c-f(X))]$ $c\in C_{n+1}$ (39)

where $x:**denoteS$ that the maximization is taken for all $x$ satisfying

$1\leq x\leq 58$ , $c-f(_{X})\in C_{n}$ . (40)

In $parti_{C}ular$, when {c-1.0, c–1.4} $\subset C_{n}$ , Eq. (39) reduces

$u_{n+1}(c)={\rm Max}[38+u_{n}(c-1.\mathrm{o}), 58+u_{n}(c-1.4)]$ . (41)

171

Let $\hat{\sigma}_{n+1}(c)\subset\{1,2, \ldots, 58\}$ be the set of all maximizers for (39), where

$\hat{\sigma}_{1}(1.\mathrm{o})=\cdots=\hat{\sigma}_{1}(1.3)=38$ , $\hat{\sigma}_{1}(1.4)=\cdots=\hat{\sigma}_{1}(1.9)=58$ . (42)

Then the $\mathrm{P}^{\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}- \mathrm{t}}\mathrm{O}^{-\mathrm{S}\mathrm{e}\mathrm{t}}$ valued function $\hat{\sigma}_{n}$ : $C_{n}arrow\{1,2, \ldots , 58\}$ is an n-th optimal decisionfunction. Thus the sequence of $\mathrm{o}\mathrm{p}$timal- decision functions $\hat{\sigma}=\{\hat{\sigma}_{1},\hat{\sigma}_{2}, \ldots,.\hat{\sigma_{n}:.}.’\ldots\}\mathrm{i}‘..\mathrm{s}.\mathrm{a}\mathrm{n}$

optimal policy for Nonstopping Inverse Problem NIP$(c;n)$ .We have the following enveloping relation. $.\wedge$ . $\cdot$

. . $\cdot$

.l

THEOREM 4. 4 (Inverse Envelope Theorem)

$u(c)={\rm Max} u_{n}(Cn|c\in cn)$ $c\in<1.0,$ $\infty>$ . (43)

Let $\hat{t}(c)$ be the first positive integer $n$ such that $u(c)=u_{n}(c)$ . Then $\hat{t}$ is the optimal stoppingtime for ISP:

$u(c)=u(\hat{t}c)$ $c\in<1.\mathrm{o},$ $\infty>$ . (44)

As for Eqs. (43),(44), see Table 4. Furthermore, we have three corresponding inversetheorems for Nonstopping Problems.

THEOREM 4. 5 (Weak Inverse Theorem II) For $n\geq 1$

(i) $v_{n}(u_{n}(c))\leq c$ $c\in C_{n}$ (45)(ii) $u_{n}(v_{n}(m))\geq m$ $m\in N_{n}$ . (46)

THEOREM 4. 6 (Strong Inverse Theorem II) For $n\geq 1$

$(\mathrm{i})’$ $(v_{n})_{-1}(c)=un(c)$ $c\in C_{n}$ (47)$(\mathrm{i}\mathrm{i})’$

$(.u_{n}. )^{-1}(m)$

.

$=$

.$v_{n}(m)-.\cdot$ $m\in.N_{n}....\cdot$

$\backslash (4\dot{8})\backslash \sim$

THEOREM 4. 7 (Strict Inverse Theorem II) For $n\geq 1$

$(\mathrm{i}\mathrm{i}\mathrm{i})’$ $\hat{\sigma}_{n}(c)=\pi^{*}n((v_{n})-1(c))$ $c\in C_{n}$ (49)$(\mathrm{i}\mathrm{v})’$ $\pi_{n}^{*}(m)=\hat{\sigma}n((u_{n})-1(m))$ $m\in N_{n}$ . (50)

Symbolically we have

$\hat{\sigma}_{n}=\pi_{n}^{*}\circ(v_{n})_{-1}$ on $C_{n}$ , $\pi_{n}^{*}=\hat{\sigma}_{n}\circ(u_{n})^{-1}$ on $N_{n}$ . (51)

Further both optimal stopping times are characterized in the following inverse sense:

THEOREM 4. 8 (Inverse Stopping Time Theorem)

$\hat{t}=t^{*}\circ v_{-1}$ on $<1.0,$ $\infty>$ , $t^{*}=\hat{t}\circ u-1$ on N. , $:$. . (52)

Finally we we $\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{i}\mathfrak{g}_{r}$ Tables 4 and 5, which illustrate optimal value functions, optimalpolicies and optimal stopping times for the main and inverse nonstopping problems, re-spectively. Further the forementioned relations are also shown in tables. The specificationverifies that all the results in both Inverse Theorems and Envelope Theorems are valid.

172

Table.3. Envelope Property and Optimal $\mathrm{s}_{\mathrm{t}\mathrm{O}\mathrm{p}\mathrm{P}}\mathrm{i}\mathrm{n}\mathrm{g}$ Time for MPS

$\ovalbox{\tt\small REJECT}_{4}^{201}11428283011115282831173134131523815151171724217174423914143811919245859624276279724310_{8384}21110_{8}10154300592027246253543425484843525101014142030_{2}401343301\mathrm{o}\mathrm{o}1028488322428210200203040222283042803040_{0}104304030_{24}33330402334343344\mathrm{o}_{8443}\mathrm{o}_{8}\mathrm{o}_{84}\mathrm{o}_{2}4444040440_{83}4440_{24}440344444845002020_{2}200343438311212223334$

173

Table 4. Envelope Property and Optimal Stopping Time for IPS

$\ovalbox{\tt\small REJECT}_{1}^{-7}525114819250474442411541541524391541533713433301122272242076762114110341317401541548413319837693212212419174174117151119219249696962585813835741741161611628581219248384174154136116114268119162626116217134344143443421921515237275232324333311$

174

References[1] M.J. Beckmann and J. Laderman, A bound on the use.of inefficient indivisible units,

Naval Research Logistics Quartely 3(1956), 245-252

[2] M.J. Beckmann, Dynamic Programming of Economic Decisions, Springer, NY, 1968.

[3] R.E. Bellman, Dynamic Programming, Princeton Univ. Press, NJ, 1957.

[4] T. Ibaraki and N. Katoh, Resource Allocation Problems: Algorithmic Approaches, MITPress, MA, 1988.

[5] S. Iwamoto, Inverse theorem in dynamic programming I, J. Math. Anal. Appl.58(1977), 113-134.

[6] S. Iwamoto, Inverse theorem in dynamic programming II, J. Math. Anal. Appl.58(1977), 249-279.

[7] S. Iwamoto, Inverse theorem in dynamic programming III, J. Math. Anal. Appl.58(1977), 439-448.

[8] S. Iwamoto, Theory of Dynamic Programs (in Japanese), Kyushu Univ. Press,Fukuoka, 1987.

175