2.1. Formulation of the model

Let ${\mathcal I} = (a,b) \subseteq {\mathbb R}$ . In the absence of ordering, the inventory process $X_0$ satisfies (1.1) and is a regular diffusion. Throughout the paper we assume that the functions $\mu$ and $\sigma$ are continuous on ${\mathcal I}$ , and that (1.1) is nondegenerate. The initial position of $X_0$ is taken to be $x_0$ for some $x_0 \in \mathcal{I}$ . We place the following assumptions on the underlying diffusion model.

Associated with the scale function S of Condition 1, one can define the scale measure on the Borel sets of ${\mathcal I}$ by $S[y,v] = S(v) - S(y)$ for $[y,v] \subset {\mathcal I}$ . From the modeling point of view, Condition 1(b) is reasonable since it essentially says that, in the absence of ordering, demand tends to reduce the size of the inventory. The boundary point a may be regular, exit, or natural, with a being attainable in the first two cases and unattainable in the third. In the case that a is a regular boundary, its boundary behavior must also be specified as being either reflective or sticky. The boundary point b is either natural or entrance and is unattainable from the interior in both cases. Following the approach in [Reference Helmes, Stockbridge and Zhu8, Reference Helmes, Stockbridge and Zhu9], we define the state space of possible inventory levels to be the interval $ \mathcal{E}$ which excludes any natural boundary point; it includes a when a is attainable, and b when it is entrance. Since orders typically increase the inventory level, define $ \mathcal{R} = \{(y,z) \in \mathcal{E}^2\;:\;\; y < z\}$ , the set of states cross the set of feasible actions (in a particular state), in which y denotes the pre-order inventory level and the control value z is the nominal post-order level. The actual post-order inventory level will be determined by y, z, and the realization of the slack variable of the associated order size; explained differently, the post-order inventory level is given as the realization of a transition function $Q(\!\cdot;\;y,z)$ which depends on (y, z).

Since we are using weak convergence methods for measures on $ \mathcal{E}$ and $ \mathcal{R}$ , we will need the closures of these sets as well. Define $\overline{\mathcal{E}}$ to be the closure in $\mathbb R$ of $\mathcal{E}$ ; thus when a boundary is finite and natural, it is not an element of $\mathcal{E}$ but is in $\overline{\mathcal{E}}$ . Note that $\pm \infty \notin \overline{\mathcal{E}}$ . Also set $\overline{\mathcal{R}} = \{(y,z) \in \mathcal{E}^2\;:\;\; y \leq z\}$ ; in contrast to $ \mathcal{R}$ , the set $\overline{\mathcal{R}}$ includes orders of size 0. Notice the subtle distinction between $\overline{\mathcal{E}}$ , which includes boundaries that are finite and natural, and $\overline{\mathcal{R}}$ , which does not allow either coordinate to be such a point.

The random yields are determined by the family $ \mathcal{Q} = \{Q(\!\cdot;\;y,z) \;:\; (y,z) \in \overline{\mathcal{R}}\}$ of probability measures parametrized by $(y,z)\in \overline{\mathcal{R}}$ such that (i) $Q(\!\cdot;y,z)$ is a probability measure for each $(y,z)\in \overline{\mathcal{R}}$ , and (ii) for each $E\in {\mathcal B}( \mathcal{E})$ , $(y,z) \to Q(E;\;y,z)$ is measurable. We have that Q is a transition function on $ \mathcal{E}\times \overline{\mathcal{R}}$ . The probability measure $Q(\!\cdot;\; y,z)$ is the distribution for the resulting inventory level following an order of size $z-y$ . We further impose support, continuity, and supply requirements on this family.

Condition 2(a,i) indicates that an active order of nominal size 0 will not change the inventory level. Condition 2(a,ii) implies the existence of a minimal delivery guarantee (MDG) that, with probability 1, assures the delivery of a fixed positive amount (up to the amount ordered) when a positive nominal amount is ordered. This condition is essential for showing that each admissible policy, including (s, S) policies, has a valid mathematical model for random supplies (cf. Definition 2.3 and the subsequent comments in [Reference Helmes, Stockbridge and Zhu10]). The fact that one needs to impose this kind of condition on inventory models with random supply to obtain a proper mathematical model of the controlled process has been overlooked in the literature. Condition 2(b) requires continuity of the mapping Q in the topology of weak convergence. Condition 2(c) is an assured supply commitment (ASC) that can be interpreted to be a ‘very important customer’ condition, in the sense that a customer who nominally orders to very high levels of inventory has a significant likelihood of receiving almost all of his order. This condition is used to establish the existence of an optimizer in Theorem 1 and to establish the optimality of a nominal (s, S) policy in Section 5.

We illustrate how Condition 2 may be satisfied when $b=\infty$ , a natural boundary. For fixed $0 < \Delta < 1$ , let $\widetilde{Q} \in \mathcal{P}[\Delta,1]$ be fixed. For $(y,z) \in \mathcal{R}$ , let $f_{(y,z)}\;:\; [0,1] \rightarrow \mathcal{E}$ be the linear mapping with $f_{(y,z)}(0)=y$ and $f_{(y,z)}(1)=z$ . Then the family $ \mathcal{Q}$ defined for $(y,z) \in \mathcal{R}$ by $Q(\!\cdot;\;y,z) = \widetilde{Q}f_{(y,z)}^{-1}(\!\cdot\!)$ always satisfies Condition 2. A special case of this family occurs when $\widetilde{Q}$ is the uniform distribution on $[\Delta,1]$ , resulting in a continuous-review inventory model with nearly stochastically proportional yields. A second special case having $\widetilde{Q}(\!\cdot\!)=\delta_{\{1\}}(\!\cdot\!)$ corresponds to the slack being 0 and therefore models non-deficient supply. Further examples will be examined in Section 6, for example when b is a finite natural boundary.

It will be important throughout the paper to average functions using transition functions. For a measurable function $\ell$ on $\overline{\mathcal{R}}$ and a transition function Q, we adopt the shorthand notation

(2.2) \begin{equation} \widehat \ell(y,z)\;:\!=\; \int \ell(y,v) Q(dv;\;\; y,z), \qquad (y,z) \in \overline{\mathcal{R}},\end{equation}

with the understanding that the integral exists in $\overline{\mathbb R}$ .

Turning to the cost functions, we impose the following standing assumptions throughout the paper.

The function $c_1$ is the building block for more complex cost structures of models with random supplies. For example, in the case when the decision-maker ‘pays for what he orders’, the ordering cost function is $c_1$ itself. When the cost structure is ‘you pay for what you get’, the function $\widehat{c}_1$ is used. For the remainder of the main sections, we analyze the inventory problem using $\widehat{c}_1$ , i.e. we pay for what we get; see also the following subsection.

We adapt to this inventory application the model constructed in [Reference Helmes, Stockbridge and Zhu10] for impulse-controlled processes having processes that are continuous between impulses. The model is built on an augmentation of the space $D_ \mathcal{E}[0,\infty)$ of càdlàg paths from $[0,\infty)$ to $ \mathcal{E}$ using the natural filtration $\{ \mathcal{F}_t\}$ in which X is the coordinate process and $ \mathcal{F}_t = \sigma(X(s)\;:\; 0 \leq s \leq t)$ .

We now define a nominal ordering policy. In order to do so, we need to specify the filtration of information used by the decision-maker to determine the jump-from locations and the nominal jump-to locations of a policy. Let $\{ \mathcal{F}_{t-}\}$ be given by $ \mathcal{F}_{t-} = \sigma(X(s)\;:\; 0 \leq s < t)$ for $t> 0$ , with $ \mathcal{F}_{0-}=\sigma(X(0-\!))$ being the $\sigma$ -algebra generated by the inventory level before any intervention at time 0. It is also important to specify the $\sigma$ -algebra of information available before a stopping time. Let $\eta$ be an $\{ \mathcal{F}_{t-}\}$ -stopping time. The $\sigma$ -algebra $ \mathcal{F}_{\eta-} \;:\!=\; \sigma(\{A\cap \{\eta > t\}\;:\; A\in \mathcal{F}_t, t\geq 0\})$ .

For the inventory management problem with random supply, the class $ \mathcal{A}$ of admissible nominal ordering policies $(\tau,Z) =\{(\tau_{k}, Z_{k}), k\in \mathbb N \}$ is defined as follows:

1. (i) $\{\tau_k\;:\;k\in \mathbb N\}$ is a strictly increasing sequence of $\{ \mathcal{F}_{t-}\}$ -stopping times with $\tau_k \to \infty$ ;

2. (ii) for each $k \in \mathbb N$ , $Z_k \in \mathcal{E}$ is $ \mathcal{F}_{\tau_{k}-}$ -measurable with $Z_k > X(\tau_k-\!)$ ; and

3. (iii) the cost (1.3) is finite and is denoted by $J(\tau,Z)$ ; note the inclusion of the policy in the notation.

The requirement that the sequence $\{\tau_k\}$ be strictly increasing implies that at most one order can be placed at any time, while the use of $\{ \mathcal{F}_{t-}\}$ prevents the ordering decision-makers from knowing the supplied amount when an order is placed. The random variable $Z_k$ in (ii) is the nominal order-to location, so it has the value $X(\tau_k-\!)+O_k$ when $O_k$ denotes the nominal order size. The construction in [Reference Helmes, Stockbridge and Zhu10] uses the measure $Q(\!\cdot;\;X(\tau_k-\!),Z_k)$ to select the actual random supply inventory level $X(\tau_k)$ at time $\tau_k$ . Hence the corresponding random slack is $\Theta_k = Z_k - X(\tau_k)$ .

Thus, given the transition functions $ \mathcal{Q}$ and an admissible policy in the class $ \mathcal{A}$ , the associated inventory process X will be a jump-diffusion process characterized by the generator of the process $X_0$ , the jump operator determined by the decision of ordering up to a nominal level z and the transition function Q.

Looking at the infinitesimal behavior, the generator of the process X between jumps (corresponding to the diffusion $X_0$ ) is $Af = \frac{\sigma^{2}}{2} f'' + \mu f'$ , which is defined for all $f \in C^2({\mathcal I})$ ; equivalently, $Af = \frac{1}{2} \frac{d\;}{dM} \left(\frac{df}{dS}\right)$ . The effects that ordering and random yields have on the inventory process and its expected cost will be defined by the jump operator $B\;:\; C( \mathcal{E}) \rightarrow C(\overline{\mathcal{R}})$ , $Bf(y,z) \;:\!=\; f(z) - f(y)$ for $(y,z) \in \overline{\mathcal{R}}$ for an order with non-deficient supply having transition function $Q(\!\cdot;\;y,z)=\delta_{z}(\!\cdot\!)$ , and for the case of random yield by the $\widehat{\;}$ operation $\widehat{Bf}(y,z) \;:\!=\; \, \int Bf(y,v)\, Q(dv;\;y,z)$ when the order-from location is y and the action z selects a transition function $Q(\!\cdot;\;y,z)$ .

2.2. Important functions

As in [Reference Helmes, Stockbridge and Zhu9], the following two functions play a central role in our search for an optimal ordering policy. Recall that M denotes the speed measure and S represents the scale measure. Using the initial position $x_0\in \mathcal{I}$ , define the functions $g_0$ and $\zeta$ on ${\mathcal I}$ by

(2.4) \begin{align} g_{0}(x) \;:\!=\; \int_{x_0}^{x} \int_{u}^{b} 2 c_{0}(v)\, dM(v)\, dS(u) \quad \textrm{and} \quad\zeta(x) \;:\!=\; \int_{x_0}^{x} \int_{u}^{b} 2\, dM(v)\, dS(u) ,\end{align}

and extend these functions to $\overline{\mathcal{E}}$ by continuity. Observe that both $g_0$ and $\zeta$ are negative on $(a,x_0)$ and positive on $(x_0,b)$ ; also, $g_0$ may take values $\pm \infty$ at the boundaries, while $\zeta$ is $\pm \infty$ for natural boundaries. Using the second characterization of A, it immediately follows that $g_0$ and $\zeta$ , respectively, are particular solutions on $\mathcal{I}$ of

(2.5) \begin{equation} \left\{\begin{array}{l}Af = - c_0, \\[5pt] f(x_0) = 0,\end{array} \right. \quad \mbox{ and } \quad \left\{\begin{array}{l}Af = -1, \\[5pt] f(x_0) = 0.\end{array}\right.\end{equation}

Other solutions to these differential equations having value 0 at $x_0$ include summands of the form $K(S(x) - S(x_0))$ , $K \in \mathbb R$ , since the constant function and the scale function S are linearly independent solutions of the homogeneous equation $Af = 0$ . However, such additional terms grow too quickly near the boundary b, so that the transversality condition (4.3) in Proposition 6 below fails (see Remark 4.2 of [Reference Helmes, Stockbridge and Zhu9]), and therefore the definitions of $g_0$ and $\zeta$ in (2.4) exclude these terms.

To gain some intuition for the functions $g_0$ and $\zeta$ , let $y, v \in \mathcal{E}, y < v$ , and let $X_0$ satisfy (1.1) with $X_0(0) = v$ . Define $\tau_{v,y} \;:\!=\; \inf\{t \geq 0\;:\; X_0(t) = y\}$ . Then Proposition 2.6 in [Reference Helmes, Stockbridge and Zhu8] shows that

\begin{align*}\mathbb E_v\left[\int_0^{\tau_{v,y}} c_0(X_0(s))\, ds\right] = {B}g_0(y,v)\qquad \mbox{and}\qquad \mathbb E_v[\tau_{v,y}] = {B}\zeta(y,v),\end{align*}

and a simple extension establishes that if $X_0(0) \sim Q(\!\cdot;\;y,z)$ , for $(y,z) \in \mathcal{R}$ , then

\begin{align*}\int \mathbb E_v\left[\int_0^{\tau_{v,y}} c_0(X_0(s))\, ds\right]Q(dv;\;y,z) = \widehat{Bg_0}(y,z)\end{align*}

and

\begin{align*} \int \mathbb E_v[\tau_{v,y}]Q(dv;\;y,z)= \widehat{B\zeta}(y,z).\end{align*}

The proof of our basic existence result, Theorem 1, relies on the asymptotic behavior of the functions $c_0$ , $g_0$ , and $\zeta$ when the boundaries are natural. The following lemma, whose proof can be found in Lemma 2.1 of [Reference Helmes, Stockbridge and Zhu9], summarizes such asymptotic behavior.

Lemma 1. Assume Condition 1. Suppose a and b are natural boundaries, and let $c_0(a)$ and $c_0(b)$ be as in Condition 3(a). Then the following asymptotic behaviors hold:

(2.6) \begin{align} & \lim_{y\rightarrow a} \frac{Bg_0(y,v)}{B\zeta(y,v)} = c_0(a), \quad \forall v\in \mathcal{I}; \ & & \lim_{v\rightarrow b} \frac{Bg_0(y,v)}{B\zeta(y,v)} = c_0(b), \quad \forall y\in \mathcal{I}; &\end{align}

(2.7) \begin{align}& \lim_{(y,v)\rightarrow (a,a)} \frac{Bg_0(y,v)}{B\zeta(y,v)} = c_0(a); \quad & & \lim_{(y,v) \rightarrow (b,b)} \frac{Bg_0(y,v)}{B\zeta(y,v)} = c_0(b); &\end{align}

(2.8) \begin{align}& \lim_{y\rightarrow a} \frac{g_0(y)}{\zeta(y)} = c_0(a); \quad & & \lim_{v\rightarrow b} \frac{g_0(v)}{\zeta(v)} = c_0(b).&\end{align}

These behaviors imply that $\lim_{y\rightarrow a} g_0(y) = -\infty$ when $c_0(a) > 0$ and $\lim_{v\rightarrow b} g_0(v) = \infty$ when $c_0(b) > 0$ .

Another function of importance to the solution of the problem is $\widehat{c}_1$ , which we remind the reader is defined to be $\widehat{c}_1(y,z)=\int c_1(y,v) Q(dv;\;y,z)$ , where $(y,z) \in \overline{\mathcal{R}}$ . The first proposition indicates a difference between the properties of the ordering cost structure of the random supply model and the model with non-deficient deliveries.

Proof. We need to show that for every $(y,z) \in {\overline{\mathcal{R}}}$ and every sequence $\{(y_n,z_n)\;:\; n\in\mathbb N\}$ in ${\overline{\mathcal{R}}}$ which converges to (y, z),

(2.9) \begin{equation} \widehat{c}_1(y,z) \le \liminf_{n\rightarrow \infty} \widehat{c}_1(y_n,z_n).\end{equation}

We may assume that the function $c_1$ is bounded; the monotone convergence theorem implies the inequality (2.9) for unbounded cost functions once it has been established for a truncated form of $c_1$ . To verify (2.9) we shall rely on the elementary but most useful Lemma 2.1 in [Reference Serfozo17]. In the sequel, we verify the hypothesis of this lemma. To this end, for the given pair (y, z) and the points $y_n, n \in \mathbb N$ , we define nonnegative continuous functions f and $f_n$ on $\mathcal{E}$ as follows. For $v \in \mathcal{E}$ , let

(2.10) \begin{equation}f(v) \;:\!=\; \left\{\begin{array}{cl}\displaystyle{c_1}(y,v), & \quad v \ge y, \rule[-15pt]{0pt}{15pt}\\[5pt] {c_1}(y,y), & \quad v \le y,\end{array} \right. \textrm{and} \quad f_n(v) \;:\!=\; \left\{\begin{array}{cl}\displaystyle{c_1}(y_n,v), & \quad v \ge y_n, \rule[-15pt]{0pt}{15pt}\\[5pt] {c_1}(y_n,y_n), & \quad v \le y_n.\end{array} \right.\end{equation}

For the remainder of this proof, we simplify notation by setting

(2.11) \begin{equation}Q(\!\cdot\!) \;:\!=\; Q(\!\cdot;\;y,z) \qquad \textrm{and}\quad Q_n(\!\cdot\!) \;:\!=\; Q(\!\cdot;\;y_n,z_n).\end{equation}

Since f is continuous, for every $t \in {\mathbb R}$ and $\epsilon > 0$ the set $\{v \in {\mathcal{E}}\;:\; f(v) > t+{\epsilon}\}$ is an open set. Moreover, $c_1$ is uniformly continuous on any compact subset in $\overline{\mathcal{R}}$ . Hence, for sufficiently large n, $v\in\{f>t+\epsilon\}$ implies $v\in\{f_n > t\}$ . By Condition 2(b), the measures $Q_n$ converge weakly to Q on $\mathcal{E}$ , and thus by the portmanteau theorem (cf. [Reference Ethier and Kurtz5, Theorem 3.3.1, p. 108]) for the first inequality below and the inclusion $\{f>t+\epsilon\}\subset \{f_n>t\}$ for n sufficiently large for the second inequality,

(2.12) \begin{equation}Q(\{ f > t+\epsilon \}) \leq \liminf_{n\rightarrow \infty} Q_n(\{ f> t+\epsilon \}) \leq \liminf_{n\rightarrow \infty} Q_n(\{f_n> t \}).\end{equation}

Since $\epsilon$ is arbitrary, the hypothesis of Lemma 2.1 in [Reference Serfozo17] is satisfied and it therefore follows that

\begin{align*}\int{f(v)Q(dv)} \le \liminf_{n\rightarrow \infty}\int{f_n(v)Q_n(dv)}.\end{align*}

By Condition 2(a) and the notation (2.11), $Q(\!\cdot\!)$ has its support in (y, z], and similarly for $Q_n(\!\cdot\!)$ . Therefore

\begin{align*}\widehat{c}_1(y,z) = \int{f(v)Q(dv)} \qquad \mbox{and} \qquad \widehat{c}_1(y_n,z_n) = \int{f_n(v)Q_n(dv)},\end{align*}

implying that (2.9) holds true.

2.3. Analysis of nominal (s, S) ordering policies

Both this paper and [Reference Helmes, Stockbridge and Zhu9] rely on characterizing the long-term average cost for (s, S) ordering policies in the cases of deficient supplies or of full supplies using a renewal reward theorem. For $(y,z) \in \mathcal{R}$ , define the nominal (y, z) ordering policy $(\tau,Z)$ so that $\tau_0=0$ and

(2.13) \begin{equation} \tau_k = \inf\{t > \tau_{k-1}\;:\; X(t-\!) \leq y\} \quad \mbox{ and } \quad Z_k = z,\quad k \geq 1,\end{equation}

in which X is the inventory level process satisfying (1.2) with this ordering policy. The above definition of $\tau_k$ must be slightly modified when $k=1$ , to $\tau_1 = \inf\{t \geq 0\;:\; X(t-\!) \leq y\}$ , to allow for the first jump to occur at time 0 when $x_0 \leq y$ . Observe that X is a delayed renewal process, since the single distribution $Q(\!\cdot;,y,z)$ is used to determine the random supply for all orders $k \geq 2$ ; it is a renewal process when $y \leq x_0$ . We note that the definition of $\tau_k$ in (2.13) needs to be more precisely stated as in Section 6 of [Reference Helmes, Stockbridge and Zhu10] because of the particular construction of the mathematical model. However, the definition in (2.13) provides the correct intuition, so we rely on this simpler statement of the intervention times.

Theorem 2.1 of [Reference Sigman and Wolff18] provides the existence and uniqueness of the stationary distribution for the process X arising from a nominal (y, z) ordering policy for any $(y,z) \in \mathcal{R}$ , and moreover, the one-dimensional distributions $\mathbb P(X(t) \in \cdot)$ converge weakly to the stationary distribution as t tends to infinity. A straightforward generalization of Proposition 3.1 of [Reference Helmes, Stockbridge and Zhu8] characterizes the density $\pi$ of the stationary distribution for X and the long-run frequency $\widehat\kappa = \frac{1}{\widehat{B\zeta}(y,z)\rule{0pt}{10pt}}$ of orders.

By renewal theory, the long-term average running cost for the nominal (y, z) ordering policy (cf. (2.13)) equals

(2.14) \begin{equation}\lim_{t\rightarrow \infty} \frac{1}{t} \int_{0}^{t} c_{0}(X(s))ds = \frac{\widehat{Bg_0}(y,z)}{\widehat{B\zeta}(y,z)} \;\;(\mbox{almost surely and in } L^1),\end{equation}

and therefore the long-term average cost $J(\tau,Z)$ of (1.3) is given by

(2.15) \begin{equation}J(\tau,Z) = \frac{\widehat{c}_{1}(y,z) + \widehat{Bg_{0}}(y,z)}{\widehat{B\zeta}(y,z)}.\end{equation}

Motivated by (2.15), define the function ${H}_0\;:\; \overline{\mathcal{R}} \rightarrow \overline{\mathbb R^+}$ by

(2.16) \begin{equation} {H}_0(y,z) \;:\!=\; \left\{\begin{array}{cl}\displaystyle\frac{\widehat{c}_1(y,z) + \widehat{Bg_0} (y,z)} {\widehat{B\zeta}(y,z)}, & \quad (y,z) \in {\mathcal R}, \rule[-15pt]{0pt}{15pt}\\[5pt] \infty, & \quad (y,y) \in \overline{\mathcal{R}}.\end{array} \right.\end{equation}

We note that $H_0$ is an adaptation of the function $F_0$ in [Reference Helmes, Stockbridge and Zhu9] to the case of random yields. Recall that $Q(\!\cdot;\;y,z)$ has its support in (y, z] and the collection is weakly convergent. Since $g_0$ and $\zeta$ are continuous, it follows that $\widehat{Bg_0}$ and $\widehat{B\zeta}$ are also continuous, as well as being nonnegative. Therefore ${H}_0$ is lower semicontinuous on $\overline{\mathcal{R}}$ by Proposition 1.

Similarly to the case of non-deficient deliveries, our goal is to minimize ${H}_0$ . Since $c_1 > 0$ , and hence $\widehat{c}_1$ is positive, ${H}_0(y,z) > 0$ for every $(y,z) \in \overline{\mathcal{R}}$ . Thus, $\inf_{(y,z)\in \overline{\mathcal{R}}} {H}_0(y,z) \;=\!:\; {H}_0^* \geq 0$ . The models with a natural boundary allow ${H}_0^* = 0$ as a limit as the appropriate coordinate approaches the boundary point, in which case it immediately follows that there is no minimizing pair $({y_0^*},{z_0^*})$ of ${H}_0$ . The imposition of Condition 4 below eliminates the possibility that $H_0^*=0$ .

It is helpful to define a family $\{\mathfrak{P}(\!\cdot;\;y,z)\;:\; (y,z)\in \mathcal{R}\}$ of probability measures on $ \mathcal{E}$ as follows:

\begin{align*}\mathfrak{P}(\Gamma;\;y,z) = \int_\Gamma B\zeta(y,v) \mbox{$\frac{1}{\widehat{B\zeta}(y,z)\rule{0pt}{10pt}}$}\, Q(dv;\;y,z), \qquad \Gamma \in {\mathcal B}( \mathcal{E}).\end{align*}

Note that the value $\mathfrak{P}(\Gamma;\;y,z)$ gives the proportion of the expected cycle length $\widehat{B\zeta}(y,z)$ due to the random effect distribution $Q(\!\cdot;\;y,z)$ delivering to inventory levels $v\in \Gamma$ following the order. Also observe that $\mathfrak{P}(\!\cdot;\;y,z)$ inherits its support from $Q(\!\cdot;\;y,z)$ .

The next result shows that the infimum $F_0^*$ of the function $F_0$ in [Reference Helmes, Stockbridge and Zhu9] (see (2.17) below) is a lower bound for the value $ H_{0}^{*}$ . The function $F_0$ gives the long-term average cost of a (y, z) policy for non-deficient supply models.

Proposition 2. Assume Conditions 1–3. Define the function

(2.17) \begin{equation} F_0(y,z) \;:\!=\; \left\{\begin{array}{cl}\displaystyle\frac{c_1(y,z) + Bg_0 (y,z)} {B\zeta(y,z)}, & \quad (y,z) \in {\mathcal R}, \rule[-15pt]{0pt}{15pt}\\[5pt] \infty, & \quad { (y,z) \in \overline{\mathcal{R}} \;\textit{with}\; y=z,}\end{array} \right.\end{equation}

and let $F_0^* = \inf_{(y,z)\in \overline{\mathcal{R}}} F_0(y,z)$ . Then $ H_{0}^{*}\ge F_{0}^{*}$ .

Proof. Observe that the function ${H}_0$ defined by (2.16) can also be written as

(2.18) \begin{equation} {H}_0(y,z) \;:\!=\; \left\{\begin{array}{cl}\displaystyle\int \frac{c_1(y,v) + Bg_0 (y,v)} {\widehat{B\zeta}(y,z)}\, Q(dv;\;y,z), & \quad (y,z) \in {\mathcal R}, \rule[-15pt]{0pt}{15pt}\\[5pt] \infty, & \quad (y,z) \in \overline{\mathcal{R}} \textrm{ with } y=z. \rule{0pt}{14pt}\end{array} \right.\end{equation}

Using the factor $\frac{B\zeta(y,v)}{B\zeta(y,v)}=1$ , the expression for $H_0$ when $y < z$ yields

\begin{align*}H_0(y,z) = \int \frac{c_1(y,v) + Bg_0(y,v)}{B\zeta(y,v)}\, \mathfrak{P}(dv;\;y,z) = \int F_0(y,v)\, \mathfrak{P}(dv;\;y,z) \geq F_0^*.\end{align*}

Taking the infimum over $(y,z)\in \mathcal{R}$ therefore establishes the result.

Similarly as in [Reference Helmes, Stockbridge and Zhu9], our main optimality result depends on the existence of a minimizing pair $({y_0^*},{z_0^*})\in \mathcal{R}$ of ${H}_0$ . An important subtlety is that properties of the function ${H}_0$ on compact subsets of $ \mathcal{R}$ and close to the boundary of $ \mathcal{R}$ are not simply determined by the properties of the functions $c_1$ , $g_0$ , and $\zeta$ in these regions as they were for non-deficient supply models. In fact, the behavior of the function ${H}_0$ near the boundary depends crucially on properties of the measure-valued transition functions $Q(\!\cdot;\;y,z)$ as functions on $ \mathcal{R}$ , and in particular on the behavior of the function $\widehat{B\zeta}$ near the boundary. As a consequence, a proof of a general optimality result for an (s, S) policy for inventory models with random supply requires additional conditions. Before presenting these conditions, however, we identify an important relationship between Condition 2(c) and the family of measures $\{\mathfrak{P}(\!\cdot;\;y,z)\}$ .

Lemma 2. Let b be a natural boundary for which Condition 2(c) holds. Then for each interval $[d_1,d_2] \subset {\mathcal I}$ and for every $\check{z}$ with $d_2 < \check{z} < b$ ,

(2.19) \begin{equation} \lim_{z\to b}\, \inf_{y\in [d_1,d_2]} \mathfrak{P}((\check{z},b);\;y,z) = 1.\end{equation}

Proof. Let $[d_1,d_2] $ and $\check{z}$ be given as in the statement of the lemma. Define

\begin{align*} M \;:\!=\; \sup\{B\zeta(y,v)\;:\;\; y \in [d_1,d_2], v\in [y, \check z] \} < \infty.\end{align*}

Furthermore, let $\delta > 0$ be as in Condition 2(c). For any $\varepsilon > 0$ , choose an $N \in \mathbb N$ so that $N > \frac{2M}{\delta\,\varepsilon}$ .

Since b is a natural boundary, $\lim_{v\to b}[\zeta(v) - \zeta(y)] = \infty$ uniformly for $y\in [d_1,d_2]$ . Consequently, for the $N\in \mathbb N$ chosen above, there exists a $z_{N} < b$ (without loss of generality, we can assume that $z_{N} > \check z$ ) such that

\begin{align}\zeta(v) - \zeta(y) \ge N, \quad \textrm{ for all } v \ge z_{N} \textrm{ and }y \in [d_1,d_2].\end{align}

Now, for the chosen $z_{N}$ , Condition 2(c) says that we can find a $z_{\varepsilon} \in (z_{N}, b)$ so that

\begin{align}Q([z_{N}, z];\; y,z) \ge \frac\delta{2}, \quad \textrm{ for all }z > z_{\varepsilon} \textrm{ and } y \in [d_1,d_2].\end{align}

Then for all $y \in [d_1,d_2]$ and $z > z_{\varepsilon}$ , we have

\begin{align*} \widehat{B\zeta}(y,z) & = \int_{y}^{z_{N}} B\zeta(y,v) Q(dv;\; y,z) + \int_{z_{N}}^{z} B\zeta(y,v) Q(dv;\; y,z) \\[5pt] & \ge 0 + N Q([ z_{N}, z];\; y,z) \ge \mbox{$\frac{N \delta}{2}$}.\end{align*}

Consequently, it follows that for any $y \in [d_1,d_2]$ and $z > z_{\varepsilon}$ , we have

\begin{align*} \mathfrak{P}((\check{z},b);\;y,z) & =\int_{\check z}^{z} B\zeta(y,v) \frac{1}{\widehat{B\zeta} (y,z)} Q(dv;\;\; y,z)\\[5pt] & = \frac{\int_{y}^{z} B\zeta(y,v) Q(dv;\; y,z)- \int_{y}^{\check z} B\zeta(y,v) Q(dv;\; y,z)}{\widehat{B\zeta} (y,z)}\\[5pt] & \ge 1 - \frac{M}{\widehat{B\zeta} (y,z) } \ge 1 - \mbox{$\frac{2M}{N\delta}$} > 1-\varepsilon.\end{align*}

This establishes (2.19) and hence completes the proof.

Remark 1. Condition 2(c) is stronger than the conclusion of this lemma. To see this, assume b is a natural boundary, let Condition 1 hold, and let $\zeta$ be given by (2.4). We identify a family $\mathcal{Q}$ for which (2.19) holds but Condition 2(c) fails. We focus on the subset of $ \mathcal{R}$ for which $B\zeta > 1$ . For each such (y, z), let $\breve{y}$ satisfy $\breve{y} > y$ with $B\zeta(y,\breve{y})=\frac{1}{2}$ ; also set $m_1 \;:\!=\; \frac{1}{\sqrt{B\zeta(y,z)}}$ and $m_0\;:\!=\; 1-m_1$ . Now consider the random supply measures for (y, z) with $B\zeta(y,z) > 1$ given by

\begin{align*}Q(\!\cdot;\;y,z) = m_0 \delta_{\breve{y}}(\!\cdot\!) + m_1 \delta_{z}(\!\cdot\!).\end{align*}

Notice that

\begin{align*}\widehat{B\zeta}(y,z) = B\zeta(y,\breve{y}) m_0 + B\zeta(y,z) m_1 = \frac{m_0}{2} + \sqrt{B\zeta(y,z)},\end{align*}

so for fixed y, $\widehat{B\zeta}(y,z) \to \infty$ as $z\to b$ . This convergence then implies (2.19) holds for the fixed y, and a simple argument extends this to a uniform convergence for $y\in [d_1,d_2]$ .

Now for $y\in [d_1,d_2]$ and (y, z) with $B\zeta(y,z) > 1$ , for any $\check{z} > d_2$ , $Q((\check{z},b);\;y,z) = $ $\frac{1}{\sqrt{B\zeta(y,z)}} \to 0$ as $z\to b$ . Hence Condition 2(c) fails.

Now, combined with Conditions 1, 2, and 3, the following set of conditions will be sufficient to guarantee the existence of a minimizer of the function $H_0$ on $\mathcal{R}$ .

We now state our main existence result, which when combined with Theorem 2 establishes the optimality of a nominal (s, S) ordering policy within the large class of admissible policies. See Section 6 for examples which illustrate these results.

Theorem 1. Assume Conditions 1–4 are satisfied. Then there exists a pair $({y_0^*},{z_0^*}) \in \mathcal R$ such that

(2.20) \begin{equation} {H}_0({y_0^*},{z_0^*}) = {H}_0^* = \inf\{{H}_0(y,z)\;:\; (y,z) \in \overline{\mathcal R}\}.\end{equation}

Proof. The proof consists of several parts, corresponding to pieces of the boundary of ${\mathcal R}$ , the type of boundary point, and the values of $c_0$ at a and b. Since much of the analysis is similar in every part, we shall only spell out the details of the case in which a and b are natural boundaries. When a is attainable or b is an entrance boundary, the boundary is included in $ \mathcal{E}$ , so the minimum of $H_0$ may be achieved using a boundary point. The proofs in these cases follow a similar line of argument.

Our method of proof is to show that $H_0$ is strictly greater than its infimum in a neighborhood of the boundary. To begin, recall that

(2.21) \begin{equation} H_0(y,z) = \int_ \mathcal{E} F_0(y,v)\, \mathfrak{P}(dv;\;y,z).\end{equation}

The challenge is that $\mathfrak{P}(\!\cdot;\;y,z)$ may place mass throughout most of the interval (y, z], so we need to be careful in developing the lower bounds of the integrand near different segments of the boundary; Figure 1 aids in visualizing this analysis. With reference to Figure 1, the bound

(2.22) \begin{equation} F_0(y,v) = \frac{c_1(y,v)+Bg_0(y,v)}{B\zeta(y,v)} > \frac{Bg_0(y,v)}{B\zeta(y,v)}\end{equation}

will be used in the regions $E_1$ , $E_2$ , $E_3$ , $E_4$ , and $E_5$ , while

(2.23) \begin{equation} F_0(y,v) = \frac{c_1(y,v)+Bg_0(y,v)}{B\zeta(y,v)} \geq \frac{c_1(y,v)}{B\zeta(y,v)} \geq \frac{k_1}{B\zeta(y,v)}\end{equation}

will be used for the region $E_6$ .

Figure 1. Neighborhoods of the boundary.

The two parts of Condition 4 can be combined to yield a single pair $(y_1,z_1)\in \mathcal{R}$ for which $c_0(a) \wedge c_0(b)> H_0(y_1,z_1)$ . Select $\varepsilon \in (0,1)$ so that

(2.24) \begin{equation} c_0(a)\wedge c_0(b) > \frac{1+\varepsilon}{1-\varepsilon} H_0(y_1,z_1) + \varepsilon \quad \mbox{and}\quad \varepsilon < \frac{k_1}{H_0(y_1,z_1)}.\end{equation}

• • By (2.7) of Lemma 1, there exists some $z_\varepsilon$ such that

  \begin{align*}\frac{Bg_0(y,v)}{B\zeta(y,v)} > H_0(y_1,z_1),\quad \forall z_\varepsilon \leq y < v < b.\end{align*}
  Define the neighborhood of (b, b) to be $E_1 = \{(y,z)\in \mathcal{R}\;:\; z_\varepsilon \leq y < z < b\}$ .

• • Again by (2.7) of Lemma 1, there exists some $y_\varepsilon$ such that

  \begin{align*}\frac{Bg_0(y,v)}{B\zeta(y,v)} > H_0(y_1,z_1), \quad \forall a < y < v \leq y_\varepsilon.\end{align*}
  Define the neighborhood of (a, a) to be $E_2 = \{(y,z)\in \mathcal{R}\;:\; a < y < z \leq y_\varepsilon\}$ .

• • Recall that $x_0$ is the initial position. Using $x_0$ as the fixed value in the two asymptotic results in (2.6) of Lemma 1, we find that there exist $\overline{y}$ and $\overline{z}$ such that for $y \leq \overline{y}$ and $v \geq \overline{z}$ , respectively,

  (2.25) \begin{equation} \frac{Bg_0(y,x_0)}{B\zeta(y,x_0)} > H_0(y_1,z_1) \quad \textrm{and}\quad \frac{Bg_0(x_0,v)}{B\zeta(x_0,v)} > H_0(y_1,z_1).\end{equation}
  For notational simplicity, we may assume $\overline{y}=y_\varepsilon$ and $\overline{z}=z_\varepsilon$ by using $y_\varepsilon\wedge \overline{y}$ and $z_\varepsilon \vee \overline{z}$ in the two previous parts as well as here. Now define
  \begin{align*}M\;:\!=\; \max_{y_\varepsilon \leq v \leq z_\varepsilon} (|g_0(v)| \vee |\zeta(v)|)\end{align*}
  and note that $M < \infty$ since $g_0$ and $\zeta$ are continuous. Using the fact that $\lim_{y\to a} \zeta(y) = -\infty$ along with (2.8) of Lemma 1, we have that there exists a $\widetilde{y} \leq y_\varepsilon$ such that for $y \leq \widetilde{y}$ ,
  \begin{align*}\frac{M}{\zeta(y)} \leq \varepsilon \quad \textrm{and}\quad \frac{g_0(y)}{\zeta(y)} > \frac{1+\varepsilon}{1-\varepsilon}H_0(y_1,z_1) + \varepsilon.\end{align*}
  Define a neighborhood of the left boundary segment between $(a,y_\varepsilon)$ and $(a,z_\varepsilon)$ to be $E_3 = \{(y,z)\in \mathcal{R}\;:\;\; y \leq \widetilde{y} \textrm{ and } y_\varepsilon \leq z \leq z_\varepsilon\}$ . Observe that for all $(y,z) \in E_3$ ,
  \begin{align*}\frac{Bg_0(y,v)}{B\zeta(y,v)} & = \frac{g_0(y) - g_0(v)}{\zeta(y)-\zeta(v)} \geq \frac{g_0(y) - M}{\zeta(y) + M} = \frac{\frac{g_0(y)}{\zeta(y)} - \frac{M}{\zeta(y)}}{1 + \frac{M}{\zeta(y)}} \\[5pt] & > \frac{\frac{1+\varepsilon}{1-\varepsilon}H_0(y_1,z_1) + \varepsilon - \varepsilon}{1+\varepsilon} = \frac{H_0(y_1,z_1)}{1-\varepsilon} > H_0(y_1,z_1).\end{align*}

• • Again, let $z_\varepsilon$ be as in the definition of $E_1$ , let $y_\varepsilon$ be from $E_2$ , and let $\widetilde{y}$ be as in $E_3$ . A key observation is that the inequalities (2.25) establish that for $a < y \leq y_\varepsilon$ and $z_\varepsilon \leq v < b$ ,

  \begin{align*}Bg_0(y,v) = Bg_0(y,x_0) + Bg_0(x_0,v) & > H_0(y_1,z_1) (B\zeta(y,x_0) + B\zeta(x_0,v)) \\[5pt] & = H_0(y_1,z_1) B\zeta(y,v),\end{align*}
  and therefore
  (2.26) \begin{equation} \frac{Bg_0(y,v)}{B\zeta(y,v)} > H_0(y_1,z_1), \quad \forall a < y \leq y_\varepsilon \textrm{ and } z_\varepsilon \leq v < b.\end{equation}
  Since $\widetilde{y} \leq y_\varepsilon$ , this inequality holds in the neighborhood of (a, b) defined by $E_4 \;:\!=\; \{(y,z)\in \mathcal{R}\;:\; a < y \leq \widetilde{y} \textrm{ and } z_\varepsilon \leq z < b\}$ .

• • Yet again, let $z_\varepsilon$ be as in the definition of $E_1$ and $\widetilde{y}$ be from $E_3$ . Now set $M_1 = \max_{\widetilde{y}\leq v\leq z_\varepsilon} (g_0(v)\vee \zeta(v))$ , noting that $M_1 \geq M$ since $[\widetilde{y},z_\varepsilon] \supset [y_\varepsilon,z_\varepsilon]$ . Since b is a natural boundary, $\lim_{v\to b} \zeta(v) = \infty$ and the asymptotic relation in (2.8) of Lemma 1 holds. Thus there exists some $\check{z} \geq z_\varepsilon$ such that for $v \geq \check{z}$ ,

  \begin{align*}\frac{M_1}{\zeta(v)} \leq \varepsilon \quad \textrm{and}\quad \frac{g_0(v)}{\zeta(v)} > \frac{1+\varepsilon}{1-\varepsilon}H_0(y_1,z_1) + \varepsilon, \end{align*}
  and hence
  (2.27) \begin{align} \nonumber\frac{Bg_0(y,v)}{B\zeta(y,v)} = \frac{g_0(z) - g_0(y)}{\zeta(z)-\zeta(y)} & \geq \frac{g_0(v) - M_1}{\zeta(v) + M_1} = \frac{\frac{g_0(z)}{\zeta(z)} - \frac{M_1}{\zeta(z)}}{1 + \frac{M_1}{\zeta(z)}} \\[5pt] & > \frac{\frac{1+\varepsilon}{1-\varepsilon}H_0(y_1,z_1) + \varepsilon-\varepsilon}{1+\varepsilon} = \frac{H_0(y_1,z_1)}{1-\varepsilon}.\end{align}
  Using this $\check{z}$ in (2.19) of Lemma 2, we have that there is some $\widetilde{z} > \check{z}$ such that for $z > \widetilde{z}$ ,
  (2.28) \begin{equation} \inf_{y\in [y_\varepsilon,z_\varepsilon]} \mathfrak{P}((\check{z},b);\;y,z) > 1-\varepsilon.\end{equation}
  Define a neighborhood of the top boundary segment between $(y_\varepsilon,b)$ and $(z_\varepsilon,b)$ by $E_5 = \{(y,z)\in \mathcal{R}\;:\; \widetilde{y} \leq y \leq z_\varepsilon \textrm{ and } z \geq \widetilde{z}\}$ .

• • Let $y_\varepsilon$ , $z_\varepsilon$ , $E_1$ , and $E_2$ be as in the previous steps. From the first two analyses we know that for all $(y,z)\in E_1\cup E_2$ , $H_0(y,z) \geq H_0(y_1,z_1)$ . We therefore only need to consider a neighborhood of the diagonal segment having $y\in [y_\varepsilon,z_\varepsilon]$ . Pick $\check{y}$ with $a < \check{y} < y_\varepsilon$ to allow a slight overlap with the region $E_2$ .

  Since $\zeta$ is continuous, it is uniformly continuous on the interval $[\check{y},z_\varepsilon]$ . Let $\delta$ be such that $\check{y} \leq y \leq z_\varepsilon$ and $y \leq z \leq y+\delta$ implies $B\zeta(y,z) < \varepsilon$ . Define the neighborhood of the cropped diagonal to be $E_6 = \{(y,z)\in \mathcal{R}\;:\; \check{y} \leq y \leq z_\varepsilon, y < z \leq y+\delta \}$ . Recall from (2.24) that $\varepsilon < \frac{k_1}{H_0(y_1,z_1)}$ ; it therefore follows from (2.23) that for all $(y,z)\in E_6$ and $y <v \leq z$ ,

  \begin{align*}F_0(y,v) > \frac{k_1}{B\zeta(y,v)} > \frac{k_1}{\varepsilon} > H_0(y_1,z_1).\end{align*}

Returning to (2.21), observe that the integration is with respect to the second variable v, so it is integration over the vertical line segment from the point (y, y) on the diagonal to (y, z). In particular, for $(y,z) \in E_1\cup E_2 \cup E_3 \cup E_4 \cup E_6$ , supp $(\mathfrak{P}(\!\cdot;\;y,z))$ is contained in this union.

Now in the regions $E_1$ to $E_4$ , combine (2.22) with the fact that $\frac{Bg_0(y,v)}{B\zeta(y,v)} > H_0(y_1,z_1)$ to see that $F_0(y,v) > H_0(y_1,z_1)$ . Similarly for the region $E_6$ , use the relation $F_0(y,v) > H_0(y_1,z_1)$ to obtain the same result. It now follows from (2.21) and the fact that $\mathfrak{P}(\!\cdot;\;y,z)$ is a probability measure that on the regions $E_1$ , $E_2$ , $E_3$ , $E_4$ , and $E_6$ , we have $H_0(y,z) > H_0(y_1,z_1)$ , and hence the infimum does not occur in these regions or in the limit at the outer boundaries.

More care must be taken in the region $E_5$ , since for $(y,z) \in E_5$ , supp $(\mathfrak{P}(\!\cdot;\;y,z))$ may not be contained in $\cup_{i=1}^6 E_i$ where $F_0(y,v)$ is larger than $H_0(y,v)$ . Using (2.21), (2.22), (2.27), and (2.28), for $(y,z) \in E_5$ ,

\begin{align*}H_0(y,z) = \int_ \mathcal{E} F_0(y,v)\, \mathfrak{P}(dv;\;y,z) & \geq \int_{(\check{z},b)} F_0(y,v)\, \mathfrak{P}(dv;\;y,z) \\[5pt] & > \frac{H_0(y_1,z_1)}{1-\varepsilon}\, \mathfrak{P}((\check{z},b);\;y,z) > H_0(y_1,z_1).\end{align*}

It thus follows that the infimum $H_0^*$ is not achieved or approached in $\cup_{i=1}^6 E_i$ . Therefore $H_0^*$ is achieved at some $(y_0^*,z_0^*)\in \overline{(\cup_{i=1}^6 E_i)^c}\subsetneq \mathcal{R}$ since $H_0$ is lower semicontinuous on this compact region.

6.1. Drifted Brownian motion inventory models

The first inventory problem concerns the classical fundamental process of a drifted Brownian motion $X_0$ satisfying the stochastic differential equation

(6.1) \begin{equation} dX_0(t) = - \mu\, dt + \sigma\, dW(t), \qquad X_0(0)=x_0,\end{equation}

where $\mu, \sigma > 0$ and W is a standard Brownian motion, under the cost structure

\begin{align*} c_0(x) = \begin{cases}-c_b\, x, & x < 0, \\[5pt] c_h\, x, & x\geq 0,\end{cases} \ \quad \mbox{and} \ c_1(y,z) = k_1 + k_2(z-y), \ -\infty < y \leq z < \infty,\end{align*}

with $c_b, c_h, k_1, k_2 > 0$ .

A modification of the problem has reflection at 0, so that no back-ordering is allowed, with the cost structure

\begin{align*}c_0(x) = k_3 x + k_4 e^{-x}, \ \ x\geq 0, \quad \mbox{ and } \ \ c_1(y,z) = k_1 + k_2 \sqrt{z-y}, \ \ 0 \leq y \leq z < \infty,\end{align*}

again with $k_1, k_2, k_3, k_4 > 0$ .

As mentioned previously, Condition 1 is the same in both papers, and Condition 2.2 of [Reference Helmes, Stockbridge and Zhu9] is the same as Condition 3 in this paper. Further, Condition 2.3 of the previous paper is more restrictive than Condition 4 here. Thus, Conditions 1, 3, and 4 are satisfied by both of these models, as established in [Reference Helmes, Stockbridge and Zhu9]. Consequently, for any family $ \mathcal{Q}$ satisfying Condition 2, the conditions of Theorem 1 are satisfied and there exists an optimizing pair $({y_0^*},{z_0^*}) \in \mathcal{R}$ of $H_0$ .

Turning to Theorem 2 to establish the optimality of the $({y_0^*},{z_0^*})$ policy, we note that Condition 5 of this paper differs from Condition 5.1 of [Reference Helmes, Stockbridge and Zhu9] only in the use of $U_0 = g_0 - H_0^* \zeta$ in place of $G_0 = g_0 - F_0^* \zeta$ . The verification of Condition 5.1 of the previous paper does not rely on $F_0^*$ . Thus the same argument using $U_0$ in place of $G_0$ demonstrates that Condition 5 holds for both problems involving the drifted Brownian motion model. Theorem 2 therefore establishes that the $({y_0^*},{z_0^*})$ ordering policy is optimal.

6.2. Geometric Brownian motion storage models

In the second model examined in [Reference Helmes, Stockbridge and Zhu9], we take the fundamental dynamics to be a geometric Brownian motion process satisfying the stochastic differential equation

\begin{align*}dX_0(t) = -\mu X_0(t)\, dt + \sigma X_0(t)\, dW(t), \qquad X_0(t) = x_0 \in (0,\infty),\end{align*}

where $\mu, \sigma > 0$ . Two cost structures are analyzed:

\begin{align*}\begin{aligned}c_0(x)& = k_3 x + k_4 x^\beta \ \quad \mbox{ for } 0 < x < \infty, \\[5pt] c_1(y,z) & = k_1 + k_2\sqrt{z-y}\ \quad \mbox{ for } 0 < y \leq z < \infty, \end{aligned}\end{align*}

and

\begin{eqnarray*}c_0(x)\; &=& \;\left\{\begin{array}{cl}k_3 (1-x) & \qquad \qquad \qquad \qquad \mbox{for } 0 < x < 1, \\[5pt] k_4(x-1) & \qquad \qquad \qquad \qquad \mbox{for } 1 \leq x < \infty,\end{array}\right. \\[5pt] c_1(y,z)\; &=& \; k_1 + \mbox{$\frac{1}{2}$}\left(y^{-\frac{1}{2}} - z^{-\frac{1}{2}}\right) + \mbox{$\frac{1}{2}$} (z-y) \quad \; \; \mbox{for } 0 < y \leq z < \infty,\end{eqnarray*}

with the parameters $k_1, k_2, k_3, k_4 > 0$ and $\beta < 0$ .

For the geometric Brownian motion model, Conditions 1, 3, and 4 are shown to be satisfied in [Reference Helmes, Stockbridge and Zhu9]. Thus, for any family $ \mathcal{Q}$ satisfying Condition 2, Theorem 1 establishes the existence of an optimizing pair $({y_0^*},{z_0^*}) \in \mathcal{R}$ for $H_0$ . Furthermore, as in the case of the drifted Brownian motion model, Condition 5 follows from the same analysis as in the proof of Theorem 6.4 of [Reference Helmes, Stockbridge and Zhu9], with $U_0$ replacing $G_0$ . Therefore Theorem 2 shows that the $({y_0^*},{z_0^*})$ ordering policy is optimal for deficient supply models.

6.3. Logistic storage model

Our third example is a logistic inventory model in a random environment with a special family of random supplies. The process is an adaptation to an inventory set-up of a population model analyzed in [Reference Lungu and Øksendal15] in the context of a particular harvesting study.

For this model, the inventory level of a product (in the absence of orders) satisfies the stochastic differential equation

(6.2) \begin{equation} dX_0(t) = - \mu X_0(t) (k-X_0(t))\, dt + \sigma X_0(t) (k-X_0(t))\, dW(t), \quad X_0(0) = x_0,\end{equation}

where k, $\mu$ and $\sigma$ are positive constants. Set $\beta \;:\!=\; - \frac{2 \mu }{k \sigma^2}$ and require $\beta < -1$ . The process $X_0$ evolves on the bounded state space ${\mathcal I} = (0,k)$ . With reference to Chapter 15 of [Reference Karlin and Taylor13], straightforward calculations verify that this model satisfies Condition 1. In particular, both endpoints are natural, 0 is attracting, and k is non-attracting; see also [Reference Helland7]. In comparison with geometric Brownian motion, both boundary points are finite. We identify the scale function and speed measure in (6.3) and (6.4) for a particular scaling of the logistic model.

A common yield structure when there are deficient supplies is provided by the uniform distribution on (y, z), which represents proportional yields. When ${\mathcal I}$ is unbounded above, this family of uniform distributions on (y, z) for $y,z \in {\mathcal I}$ is easily seen to satisfy Condition 2(c), since the mass escapes to $\infty$ as $z\to\infty$ . Unfortunately this condition is no longer true for a uniform distribution with y fixed, and $z \to k$ for this example since the right boundary is a finite value. Thus, we adopt the family of ‘z-skewed uniform distributions’ as a surrogate, resulting in a model with nearly proportional yields.

To be precise, choose a large integer j, and for each $(y, z) \in {\mathcal R}$ let $Q(\!\cdot \,;\;y,z)$ be the uniform distribution on the interval having left endpoint $(1-(z/k)^j) y + (z/k)^j z$ and right endpoint z. In this choice, the left endpoint is a convex combination of y and z with a weight factor $(z/k)^j$ that more heavily favors z as z approaches the upper boundary k. Clearly, this family of distributions satisfies the ASC condition as well as the MDG condition in Condition 2(a,ii). Furthermore, depending on the choice of j, the measure $Q(\!\cdot;\; y,z)$ is a ‘reasonable’ approximation to the uniform distribution on (y, z) when z is not too close to k. Therefore this family of random effects distributions results in a model having nearly proportional yields. Finally, we take $Q(\!\cdot\,,y,y) = \delta_y(\!\cdot\!)$ so that Condition 2(a,i) holds and it is easy to verify the weak convergence of the measures in Condition 2(b).

For this example, we choose the bounded holding cost function $c_0(x) \;:\!=\; k_0 (x-{\bar x})^2$ for $0 < x < k$ , where $k_0$ is a positive constant and the number ${\bar x} \in (0,k)$ characterizes a ‘preferred’ inventory level. Furthermore, we choose the order cost function $c_1(y,z)$ in (6.1). Again, straightforward analysis verifies (2.3) and hence Condition 3 is satisfied.

Scaling the inventory process by the factor k and adjusting the parameters appropriately, we can set $k = 1$ without loss of generality. The scale function S and the speed measure M associated with $X_0$ can be determined as follows. Let $C_1 = ({x_0/(1-x_0)})^{\beta}$ , $C_2 = 1/(\sigma^2C_1) = ((1-x_0)/x_0)^{\beta}/\sigma ^2$ , and let ${_2F_1}$ denote the (Gaussian) hypergeometric function. Let $\tilde S(x) = C_1 x^{(1-\beta)}/(1-\beta)\, {_2F_1}(1-\beta,-\beta;\;2-\beta;\;x)$ . Then

(6.3) \begin{equation} S(x) = C_1 \int_{x_0}^{x} {((1-u)/u)}^{\beta} \, du = \, \,{\tilde S(x)} - {\tilde S(x_0)},\qquad 0 < x < 1,\end{equation}

while $M[a,b] = \int_{a}^{b}m(v) dv$ for any $[a, b] \subset (0,1)$ , where the speed density m is given by

(6.4) \begin{equation} m(v) = C_2 (1-v)^{-(\beta+2)}{v}^{\beta - 2}, \qquad 0 < v < 1.\end{equation}

For later reference, we note that $S'(x) = C_1 (\frac{1-x}{x})^\beta$ for $x\in (0,1)$ .

Now turning to Condition 4, since each boundary is natural, we need to check that there is some $(y,z)\in {\mathbb R}$ for which $H_0(y,z)=(\widehat{Bc_1}(y,z)+\widehat{Bg_0}(y,z))/\widehat{B\zeta}(y,z)$ is smaller than the holding cost rates at the boundaries. To this end, note that the expressions for $\zeta$ and $g_0$ simplify considerably when we set $x_0 = {\bar x} = 1/2$ , so for this example we make this selection. These functions are then

(6.5) \begin{align} \zeta(x) & = -\,\frac{2 \left(1-2 x+2 \beta \ln(2-2 x)+\beta (1+\beta ) \ln\left(\frac{x}{1-x}\right)\right)}{\sigma ^2 \beta \left(-1+\beta^2\right)},\end{align}

(6.6) \begin{align}g_0(x) &= \frac{k_0 \left((-1+2 x) \left(-1+2 \beta ^2\right)-2 \beta \ln(2-2 x)-\beta (1+\beta ) \ln\left(\frac{x}{1-x}\right)\right)}{2\sigma ^2 \beta \left(-1+\beta ^2\right)}\ .\end{align}

The functions $\widehat{Bc_1}$ , $\widehat{B\zeta}$ , and $\widehat{Bg_0}$ are then obtained by integrating the functions given above with respect to the measures Q. Usually this integration is best accomplished using software packages such as Maple or Mathematica, since the formulas become messy. Then, by elementary but rather lengthy calculations, one verifies Condition 4.

For more general parameters in this model, (6.5) and (6.6) become more involved and even analytically intractable for some families of random effects measures. An alternative approach to verifying Condition 4 is to simply optimize $H_0$ and then compare the optimal value $H_0^*$ with the cost rates $c_0(0) = k_0/4 = c_0(1)$ . An optimizing pair $(y^*,z^*)$ in the interior would then satisfy Condition 4 for this model when $H_0^* < k_0/4$ . Minimizing $H_0$ is a two-dimensional optimization problem. Since Condition 4 only requires the existence of a pair $(y,z)\in \mathcal{R}$ , other alternatives for verifying this condition would be (i) to fix one of the variables or a relation between the variables, perform a one-dimensional optimization, and compare this value of $H_0$ against $k_0/4$ , or (ii) to compare the values of $H_0$ from a random search of $\mathcal{R}$ . Each of these approaches is numerical, rather than analytic.

Finally, to see that an (s, S) policy is optimal, we need to verify parts (a,ii) and (b,ii) of Condition 5. To this end, recall that $U_0(x) = g_0(x) - H_{0}^{*} \zeta(x)$ and observe that $c_{0}(x) - H_{0}^{*}$ is uniformly bounded on the unit interval. With $\zeta$ and $g_0$ given in (2.4) and using the expressions for the scale density (6.3) and the speed density (6.4), we have

(6.7) \begin{align} \nonumber |\sigma x (1 - x) U'_{\!\!0}(x)| & \le |\sigma x (1 - x) S'(x)| \int_{x}^{1} |c_0(v) - H_{0}^{*}| dM(v) \\[5pt] & \le K x^{1-\beta}(1-x)^{1+\beta}\int_{x}^{1} v^{\beta-2} (1-v)^{-\beta-2} dv,\end{align}

where K is a positive constant independent of x or $x_{0}$ . To see that the left-hand side of (6.7) is uniformly bounded on [0, 1] which, in turn, implies (a,ii) and (b,ii) of Condition 5, it is clearly sufficient to find bounds in some neighborhoods of the two endpoints. The simple idea is to verify the following: (i) for x close to 1, the integral on the right-hand side of the inequalities decreases at the same rate as the factor $(1-x)^{1+\beta}$ increases; (ii) when x is close to zero, the integral increases at a rate no faster than the rate at which the factor $x^{1-\beta}$ decreases.

(i) For $x\in (\frac12, 1)$ the integral in (6.7) is dominated by

\begin{align*}\int_{x}^{1} v^{\beta-2} (1-v)^{-\beta-2} dv \quad \le \quad 2^{2-\beta} \int_{x}^{1}(1-v)^{-\beta-2} dv \quad \le \quad \mbox{$\frac{2^{2-\beta}}{-\beta-1}$} (1-x)^{-\beta-1},\end{align*}

and hence the left-hand side of (6.7) is bounded by some $K_1$ for $x\in (\frac12,1)$ . (ii) Similarly, for $x\in (0, \frac12)$ , a dominating function for the integral in (6.7) is determined as follows:

\begin{eqnarray*}\int_{x}^{1} v^{\beta-2} (1-v)^{-\beta-2} dv\; &=& \;\int_{x}^{1/2} v^{\beta-2} (1-v)^{-\beta-2} dv + \int_{1/2}^{1} v^{\beta-2} (1-v)^{-\beta-2} dv \\[5pt]\; &\leq&\;(2^{2+\beta}\vee 1) \int_{x}^{1/2} v^{\beta-2} dv + K_2 \\[5pt] \; &=& \;\mbox{$\frac{2^{2+\beta}\vee 1}{1-\beta}$}\, x^{\beta-1} + K_3,\end{eqnarray*}

where $K_2$ is the value of the integral over $\big[\frac12,1\big]$ and $K_3$ then adjusts this value by the contribution of the first integral at the boundary $1/2$ . Thus, taking into account the factor $x^{1-\beta}$ on the right-hand side of (6.7), we see that the left-hand side of (6.7) is bounded for $x\in (0,1/2)$ .

Using both estimates in (6.7), together with the fact that $\lim_{x\to 0} \sigma x (1-x) U'_{\!\!0}(x)$ exists and is finite, we have thus shown that $|\sigma x (1-x) U'_{\!\!0}(x)|$ is uniformly bounded on [0, 1]. Since the denominators are bounded below by 1, Condition 5 holds.

In summary, the model satisfies Conditions 1, 2, 3, and 4. Therefore Theorem 1 establishes the existence of an optimizing pair $({y_0^*},{z_0^*}) \in \mathcal{R}$ of $H_0$ . Furthermore, since Condition 5 holds, Theorem 2 shows that the $({y_0^*},{z_0^*})$ ordering policy is optimal for this particular logistic inventory model.

Finally, we numerically illustrate the effect of using the optimization results in this paper for a particular set of parameters. For comparison purposes, three models based on the logistic dynamics in (6.2) are examined. Model 1 assumes no noise by setting $\sigma=0$ so that the dynamics are deterministic, and it uses the non-deficient supply measures $Q(\!\cdot;\;y,z) = \delta_{\{z\}}(\!\cdot\!)$ for all $(y,z) \in \mathcal{R}$ . Model 2 has $\sigma=1/10$ , resulting in random fluctuations in the inventory level, but also uses $Q(\!\cdot;\;y,z) = \delta_{\{z\}}(\!\cdot\!)$ for all $(y,z) \in \mathcal{R}$ , so that the amount ordered is the amount delivered. Model 3 takes $\sigma=1/10$ and uses the nearly proportional yield transition functions Q defined earlier in this subsection, with $j=10$ . The other parameters in this illustration are $k=1$ , $\mu=1/20$ , $k_0=100$ , $k_1=9$ , $k_2=4$ , and $x_0 = {\bar x} = 1/2$ .

Table 1 illustrates the impact a random environment and/or random supplies have on the optimal characteristics of the logistic inventory model. Specifically, the following characteristics of the optimal solutions have been computed:

• • the order ‘from’ level ${y_0^*}$ and the deterministic order ‘to’ or nominal order ‘to’ level ${z_0^*}$

• • the ‘mean supply’, a deterministic quantity in Models 1 and 2

• • the optimal expected long-run average ‘Cost’

• • the ‘mean cycle length’ (the cycle length is again deterministic for Model 1)

Observe that the optimal value of $H_0^* = 1.33092 = H_0(0.384973,0.6575) < 25 = k_0/4$ , so Condition 4 is satisfied.

Table 1. Comparison of three logistic inventory models.

From a management point of view, the following observations are important. The nearly proportional yield model’s having random fluctuations in inventory results in cost increases of 42% and 33% over Models 1 and 2, respectively. Also, the uncertainty of the environment and the fluctuating deliveries typically shorten the mean cycle length, even though the nominal order interval increases in length as randomness is added to the process and to the delivered amounts. Thus, ordering tends to occur more frequently for the stochastic models.

Additional insights into the characteristics of the optimal nominal policy and optimal inventory process can be obtained by more extensive sensitivity analysis. For instance, for modifications of this example, various statistics of the aforementioned quantities, such as the mean cycle time, can be computed or derived from simulation studies.

As indicated earlier, uniqueness of the optimal policy is not analytically guaranteed. However, one may obtain contour plots of $H_0$ numerically and thereby determine the uniqueness of the optimal policy for this particular model and for more general stochastic differential equations and Q distributions.