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Dynamic lead time quotation under responsive inventory and multiple customer classes

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We address the lead time quotation problem of a manufacturer serving multiple customer classes. Customers are sensitive to the quoted lead times and the manufacturer has the flexibility to keep inventory to improve responsiveness. We model the problem as a Markov decision process and characterize the optimal lead time quotation, rationing, and production policies. We then define internal and external service level measures and analyze the impact of inventory keeping decision on these measures. Before analyzing the impact, we first derive the relation between inventory level and lead time quotes. We then show that the effect of inventory level on the service levels may not follow the intuition. We also study alternative lead time quotation and production schemes. We contrast the performance of these alternative policies with that of the optimal policy. Specifically, through a numerical study, we quantify the value of controlled arrivals and customer rejection, the value of information on customer status, and the value of a far-sighted policy. Through the numerical study, we also identify the effect of revenue mix and demand mix on the inventory keeping decisions and the performance measures. Finally, we measure the impact of lead time quotation on resource pooling (i.e., inventory and capacity pooling). We show that the value of resource pooling is limited under the optimal policy, since lead time quotation is already effective in balancing the demand with capacity, and in allocating the resources among different customer classes.

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Appendix 1: Proof of Theorem 1

We first discuss the existence of an optimal policy under the average reward criteria. Establishing the existence under average reward criteria is not straightforward when state space is countable and one-step rewards are not bounded. Weber and Stidham (1987) provides the sufficient conditions for existence of an average reward optimal policy. Our Markov decision process satisfies the conditions; therefore, an average reward optimal policy exists. Also the Markov decision process under consideration is a communicating multi-chain which implies that the optimal policy has a constant gain (Puterman 1994). This implies that under the optimal policy there exists a single recurrent class and possibly a set of transient states.

In our problem, we can assume without loss of optimality that production is stopped under a sufficiently large stock level (say, \(i=-L_B<0\)), and that all customers are rejected when the cost due to the expected lateness is sufficiently high (say, \(i=L_U>0\)). This assumption can be made since the holding cost is convex increasing in stock level and lateness cost is convex increasing in number of customers in the system.

In the following we show that for \(i\in \mathbb {Z}\), \(v(i)-v(i+1)\) is increasing in i. We analyze the cases \(i<0\) and \(i\ge 0\) separately. Let \(\Delta _i=v(i)-v(i+1)\), \(\bar{\Delta _i}=v(i)-v(i-1)\), \(\Lambda =\sum _j\lambda _j\), \(\gamma =\Lambda +\mu \).

\(\underline{\hbox {Case }1\;(i<0)}\) At \(i=-L_B\) production stops, whereas for \(i>-L_B\) a decision as to whether to stop or continue production is taken. For \(i\le -L_B\), the functional equation is expressed as follows:

$$\begin{aligned} \frac{g}{\gamma }+v(i)&=-\frac{(h)i^-}{\gamma } +\sum _j\frac{\lambda _j}{\gamma }\max _{d_j}\left\{ \bar{F}_j(d_j)R_j +F_j(d_j)\Delta (i)\right\} \nonumber \\&\quad +\frac{\Lambda }{\gamma }v(i+1)+\frac{\mu }{\gamma }v(i). \nonumber \\ \Delta _i&=-\frac{g}{\Lambda }-\frac{(h)i^-}{\Lambda }+\frac{\sum _j\lambda _j\max \{R_j,\Delta _i\}}{\Lambda },\quad i\le -L_B. \end{aligned}$$

For \(-L_B<i<0\),

$$\begin{aligned} \frac{g}{\gamma }+v(i)&=-\frac{(h)i^-}{\gamma } +\sum _j\frac{\lambda _j}{\gamma }\max _{d_j}\left\{ \bar{F_j(d_j)}R_j +F_j(d_j)\Delta _i\right\} +\frac{\Lambda }{\gamma }v(i+1) \nonumber \\&\quad +\frac{\mu }{\gamma }v(i)+\frac{\mu }{\gamma }\max \{\Delta _{i-1},0\}.\end{aligned}$$
$$\begin{aligned} \Delta _i&=-\frac{g}{\Lambda }-\frac{(h)i^-}{\Lambda }\nonumber \\&\quad + \frac{\sum _j\lambda _j\max \{R_j,\Delta _i\}}{\Lambda }+ \frac{\mu }{\Lambda }\max \{\Delta _{i-1},0\},\quad -L_B<i<0. \end{aligned}$$

Note that for \(i\le -L_B\), \(\Delta _i\) and \(\Delta _{i-1}\) have almost the same expression in (7) except the term \(\frac{-(h)i^-}{\Lambda }\). Since \(\frac{-(h)i^-}{\Lambda }\) is increasing in i, \(\Delta _i>\Delta _{i-1}\) for \(i\le -L_B\). For \(i=-L_B+1\), \(\Delta _{i}\) and \(\Delta _{i-1}\) have almost the same expressions in (7) and (8) except that \(\Delta _i\) has the additional term \(\frac{\mu }{\Lambda }\max \{\Delta _{i-1},0\}\ge 0\). Thus \(\Delta _i\ge \Delta _{i-1}\). For \(i>-L_B+1\), the additional term is increasing in i which implies \(\Delta _i\ge \Delta _{i-1}\). Therefore, for \(i<0\), \(\Delta _i\) is increasing in i.

\(\underline{\hbox {Case }2\;(i\ge 0)}\) Note that \(\bar{\Delta }_i=v(i)-v(i-1)=-\Delta _{i-1}\). To show that \(\Delta _i\) is increasing in i, we show \(\bar{\Delta }_i\) is decreasing in i.

Note that in Case 1 we have shown \(\Delta _{-1}>\Delta _{-2}\), which is equivalent to \(v(-1)-v(0)>v(-2)-v(-1)\), or \(\bar{\Delta }_0<\bar{\Delta }_{-1}\). We show \(\bar{\Delta }_i<\bar{\Delta }_{i-1}\) for \(0<i\le L_U\).

For states \( 0<i\) and \(i=0\), the functional equation in (2) can be expressed, respectively, as follows:

$$\begin{aligned} \frac{g}{\gamma } + v(i)&=\sum _j\frac{\lambda _j}{\gamma } \max _{d_j}\left\{ \bar{F_j}(d_j)(R_j-\ell L_i(d_j)) +\bar{F_j}(d_j)(v(i+1)-v(i))\right\} \nonumber \\&\quad +\frac{\Lambda }{\gamma }v(i)+\frac{\mu }{\gamma }v(i-1),\quad i>0 \end{aligned}$$
$$\begin{aligned} \frac{g}{\gamma }+v(0)&=\sum _j\frac{\lambda _j}{\gamma } \max _{d_j}\left\{ \bar{F_j}(d_j)(R_j-\ell L_i(d_j)) +\bar{F}_j(d_j)(v(1)-v(0))\right\} \nonumber \\&\quad +\frac{\Lambda }{\gamma }v(0)+\frac{\mu }{\gamma }\max \{v(0),v(-1)\} \end{aligned}$$

Rearranging the terms in (10) and (11), we obtain

$$\begin{aligned} \bar{\Delta }_i&=-\frac{g}{\mu }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F_j}(d_j)(R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\right\} ,\quad i>0, \end{aligned}$$
$$\begin{aligned} \bar{\Delta _0}&= -\frac{g}{\mu }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F_j}(d_j)(R_j-\ell L_0(d_j) +\bar{\Delta }_{1})\right\} +\max \{\bar{\Delta }_0,0\} \end{aligned}$$

By definition of \(L_U\) rejecting all arriving customers is the optimal decision for \(i\ge L_U\). Then \(\bar{\Delta }_i=-\frac{g}{\mu }\) for \(i\ge L_U\). We use induction in the proof. For \(i\ge n+1\) suppose \(\bar{\Delta }_{i-1}\ge \bar{\Delta }_i\). For \(2\le i\le n\), let \(d_j^*(i)\) be the optimal quote in state i for customer class j. For \(i=n\), comparing \(\bar{\Delta }_{i-1}\) with \(\bar{\Delta }_i\),

$$\begin{aligned} \bar{\Delta }_{i-1}&\ge -\frac{g}{\mu }+\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*)\left( R_j-\ell L_{i-1}(d_j^*(i))+\bar{\Delta }_i\right) \\&>-\frac{g}{\mu }+\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*)\left( R_j-\ell L_{i}(d_j^*(i))+\bar{\Delta }_{i+1}\right) \\&=\bar{\Delta }_i \end{aligned}$$

The first inequality holds since \(d_j^*(i)\) is not necessarily the optimal action at state \(i-1\), and the second inequality holds since \(\bar{\Delta }_{i-1}\ge \bar{\Delta }_{i}\) for \(i\ge n+1\), and since \(L_{i-1}(d_j^*(i))<L_i(d_j^*(i))\). Finally, we show \(\bar{\Delta }_{i-1}\ge \bar{\Delta }_i\) for \(i=1\). Note that by the definition of \(\bar{\Delta }_0\) in (13), \(\bar{\Delta }_0\ge \bar{\Delta }_1\) holds, since \(\max \{\bar{\Delta }_0,0\}\ge 0\).

Analysis under \(i<0\) and \(i\ge 0\) shows that \(\Delta _i\) is increasing in i, \(i\in \mathbb {Z}\). The following are inferred from this result:

  1. (i)

    The operator used to determine the optimal production decision for a given non-negative state is \(\max \{v(i-1),v(i)\}\) (see Eq. 2). Rearranging the terms, one obtains \(\max \{0,v(i)-v(i-1)\}+v(i-1)=\max \{0,\bar{\Delta }_i\}+v(i-1)\). We have shown that \(\bar{\Delta }_i\) is decreasing in i. This implies that there exists a state s such that for \(i>s\) to produce is the optimal decision, and for \(i\le s\) not to produce is the optimal decision. This implies that optimal production policy is of control-limit type.

  2. (ii)

    Since \(\bar{\Delta }_i\) is decreasing in i (i.e., \(\Delta _i\) increasing in i), from Eq. 8 it is possible to infer that the term \(\max \{R_j,\Delta _i\}\) is equal to \(\Delta _i\) for sufficiently large values of i, and the term is equal to \(R_j\) for other values of i. Thus, for \(i<0\), there possibly exists a threshold \(K_j\), such that for \(i<K_j\), arriving customers are accepted and for \(K_j\le i\le 0\) arriving customers are rejected, when there is stock. If, on the other hand, it holds that \(R_j>\Delta _i\) for \(i\le 0\), then \(K_j\) does not exist and all arriving customers of class j are accepted whenever there is stock. In that case, there exists a threshold \(B_j\) at which customers of class j are rejected whenever the backlog in the system is \(B_j\) or more. It is possible to infer observing the Eqs. 8 and 12 that either a \(K_j\) or a \(B_j\) exists (but not both).

  3. (iii)

    Optimal quoted lead times increase with the number of customers in the system. To see why, observe that for \(i\ge 0\) for class j,

    $$\begin{aligned} d_j^*(i)&=\hbox {arg max}_{d_j}\left\{ \bar{F}_j(d_j)(R_j-\ell L_{i+1}(d_j)+\bar{\Delta }_{i+2})\right. \nonumber \\&\quad \left. +\bar{F_j}(d_j)(\bar{\Delta }_{i+1}-\bar{\Delta }_{i+2}+\ell (L_{i+1}(d_j)-L_{i}(d_j)))\right\} , \end{aligned}$$
    $$\begin{aligned} d_j^*(i+1)&=\hbox {arg max}_{d_j}\left\{ \bar{F}_j(d_j)(R_j-\ell L_{i+1}(d_j)+\bar{\Delta }_{i+2})\right\} . \end{aligned}$$

    Note that in (14) \(\bar{F}_j(d_j)(\bar{\Delta }_{i+1}-\bar{\Delta }_{i+2})+\ell (L_{i+1}(d_j)-L_{i}(d_j)))\) is decreasing in \(d_j\), since \((\bar{\Delta }_{i+1}-\bar{\Delta }_{i+2})\) is positive, and \((L_{i+1}(d_j)-L_{i}(d_j))\) is positive decreasing in \(d_j\) and \(\bar{F}_j(d_j)\) is decreasing in \(d_j\). This implies \(d_j^*(i)\le d_j^*(i+1)\), \(i\ge 0\), i.e., longer lead times are quoted to class j.

Appendix 2: Proof of Proposition 1

  1. (1)

    Consider Eq. (12). Quotes to class k in state i can be expressed as \(d_k(i)=\hbox {arg max}_{d}\{\bar{F}_k(d)(R_k-\ell L_i(d)+\bar{\Delta }_{i+1})\}\), or

    $$\begin{aligned} d_k(i)=\hbox {arg max}_{d}\left\{ \bar{F}_j(d)(R_j-\ell L_i(d)+\bar{\Delta }_{i+1})+\bar{F}_k(d)(R_k-R_j)\right\} . \end{aligned}$$

    The equivalence holds since it is assumed that \(\bar{F}_j(d)=\bar{F}_k(d)\). Since \(\bar{F}_k(d)\) is decreasing in d and \(R_k-R_j>0\) quoted lead times to class k are shorter than lead times to class j. Shorter lead times imply \(K_k\le K_j\).

  2. (2)

    If a customer class is more tolerant to lead times, i.e., \(F_k(d)<F_j(d)\), this does not imply that quoted lead times to class k are longer. However, under the further (sufficient) condition \(\bar{F}_k(d)-\bar{F}_j(d)\) is increasing in d, it is possible to show that quoted lead times to class k are longer. Optimal quote to class k in state i can be expressed as

    $$\begin{aligned} d_k(i)= & {} \hbox {arg max}_{d} \left\{ \bar{F}_j(d)(R_j-\ell L_i(d)+\bar{\Delta }_{i+1})+(\bar{F}_k(d)-\bar{F}_j(d))\right. \nonumber \\&\left. (R_k-\ell L_i(d)+\bar{\Delta }_{i+1})\right\} . \end{aligned}$$

Note that the d value defined by (15) will be in the set of values that make \(R_k-\ell L_i(d)+\bar{\Delta }_{i+1}>0\). If there is no such d value, then \(d_k(i)=d_\mathrm{max}=d_j(i)\). For states where for some \(d\in [0,d_\mathrm{max}]\), \(R_k-\ell L_i(d)+\bar{\Delta }_{i+1}>0\), lead times for class k are longer if \(\bar{F}_k(d)-\bar{F}_j(d)\) is increasing in d. The reason is that \(R_k-\ell L_i(d)+\bar{\Delta }_{i+1}\) is increasing in d which implies \((\bar{F}_k(d)-\bar{F}_j(d))(R_k-\ell L_i(d)+\bar{\Delta }_{i+1})\) is an increasing function of d. Thus lead times to class k are longer than lead times to class j.

Appendix 3: Proof of Lemma 1

We show that under a basestock level \(s\ne s^*\) quoted lead times to arriving customers are shorter, and stock rationing levels are closer to 0. We analyze \(s<s^*\) and \(s>s^*\) separately.

\(\underline{\hbox {Case }1.\; s<s^*}\) Let \(v_s(i)\) denote the bias and \(g_s\) the expected average gain per unit time under basestock s. Let \(\bar{\Delta }_i(s)=v_s(i)-v_s(i-1)\). First we express the Eqs. (12) and (13) under the optimal basestock, \(s^*\).

$$\begin{aligned} \bar{\Delta }_i&=-\frac{g^*}{\mu }-\frac{(h)i^-}{\mu } +\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\right\} ,\quad i>-s^*,\\ \bar{\Delta }_i&=-\frac{g^*}{\mu }-\frac{(h)i^-}{\mu } +\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\right\} +\bar{\Delta }_i,\quad i\le -s^*. \end{aligned}$$

Similarly, the following equalities are defined under the non-optimal basestock, s,

$$\begin{aligned} \bar{\Delta }_i(s)&=-\frac{g_s}{\mu }-\frac{(h)i^-}{\mu }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1}(s))\right\} ,\quad i>-s, \end{aligned}$$
$$\begin{aligned} \bar{\Delta }_i(s)&=-\frac{g_s}{\mu }-\frac{(h)i^-}{\mu }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1}(s))\right\} \nonumber \\&\quad +\bar{\Delta }_i(s),\quad i\le -s. \end{aligned}$$

In the proof, we first show that \(\bar{\Delta }_i(s)>\bar{\Delta }_i\), for \(i>-s\). As we will show, this implies quoted lead times are shorter and stock rationing levels are closer to zero. Aktaran-Kalayci and Ayhan (2009) analyze the effect of parameters on the pricing decisions in a make-to-order setting under average reward criteria. Proof follows similar lines.

\(\underline{\hbox {For }i>-s, \bar{\Delta }_i(s)>\bar{\Delta }_i}\) Consider Eq. (16). Let \(d_j^*(i)\) denote the optimal quote to class j in state i under basestock \(s^*\). Then \(\bar{\Delta }_i(s)\) can be expressed as

$$\begin{aligned} \bar{\Delta }_i(s)&=-\frac{g^*}{\mu }-\frac{(h)i^-}{\mu } +\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*(i)) (R_j-\ell L_i(d_j^*(i))+\bar{\Delta }_{i+1}) \nonumber \\&\quad +\left( \frac{g^*}{\mu }-\frac{g_s}{\mu }\right) +\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*(i)) (\bar{\Delta }_{i+1}(s)-\bar{\Delta }_{i+1}),\quad i>-s. \end{aligned}$$

In (18), note that \(-\frac{g^*}{\mu }-\frac{(h)i^-}{\mu } +\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*(i)) (R_j-\ell L_i(d_j^*(i))+\bar{\Delta }_{i+1})\) is simply \(\bar{\Delta }_i\). Furthermore, \(\frac{g^*}{\mu }-\frac{g_s}{\mu } +\sum _j\frac{\lambda _j}{\mu }\bar{F}_j(d_j^*(i))(\bar{\Delta }_{i+1}(s)-\bar{\Delta }_{i+1})\) is always positive (through induction assumption, and since \(\frac{g^*}{\mu }\ge \frac{g_s}{\mu }\)). Thus for \(i>-s\), \(\bar{\Delta }_i(s)>\bar{\Delta }_i\).

\(\underline{\bar{\Delta }_i(s)>\bar{\Delta }_i \hbox { implies shorter lead times}}\) Now we show that under s, lead times are shorter for \(i\ge -s\). Under \(s^*\), optimal quote in i is \(d_j^*(i)=\hbox {arg max}_{d}\{\bar{F}_j(d) (R_j-\ell L_i(d)+\bar{\Delta }_{i+1})\}\), whereas under s, optimal quote is \(d_j^s(i)=\hbox {arg max}_{d}\{\bar{F}_j(d) (R_j-\ell L_i(d)+\bar{\Delta }_{i+1}+\bar{F}_j(d)(\bar{\Delta }_{i+1}(s)-\bar{\Delta }_{i+1})\}\). Since for \(i\ge -s\), \(\bar{F}_j(d)(\bar{\Delta }_{i+1}(s)-\bar{\Delta }_{i+1})\) is decreasing in d, it holds that \(d_j^s(i)\le d_j^*(i)\).

\(\underline{\hbox {Case 2}.\; s>s^*}\) For \(i>-s^*\), showing that \(\bar{\Delta }_i(s)>\bar{\Delta }_i\) follows similar lines as in Case 1. And thus the conclusion of shorter lead times and lower stock rationing levels follows.

Appendix 4: Proof of Theorem 2

We show that under s, the structural results for rationing and lead time quotes hold as stated in Theorem 1: there exists a rationing level (possibly negative) for each customer and lead time quotes increase with the number of customers in the system. To show that the structure under s does not change, we equivalently show that \(\bar{\Delta }_i(s)\) is decreasing in i in the “relevant region”. Note that if \(\bar{\Delta }_i(s)\) is not decreasing in i, then lead time quotes are not necessarily monotone increasing, and there might exist several stock rationing levels for class j (where the highest of those levels would be lower than \(K_j\) under he optimal policy, as stated in Lemma 1). We analyze \(s<s^*\) and \(s>s^*\) separately.

\(\underline{\hbox {Case 1}.\;s<s^{*}}\) We show that for \(i> -s\) \(\bar{\Delta }_i(s)\) is decreasing in i. If \(s=0\), then similar analysis as in Case 2 of Theorem 1 yields \(\bar{\Delta }_i(s)\ge \bar{\Delta }_{i+1}(s)\) for \(i>0\).

If \(s>0\), then for \(i\ge 0\) similar analysis as in Case 2 of Theorem 1 yields \(\bar{\Delta }_i(s)\ge \bar{\Delta }_{i+1}(s)\). For \(-s<i<0\), due to the term \(-\frac{(h)i}{\mu }\) it is not immediate that the inequality holds. We show for \(-s<i\), \(\bar{\Delta }_i(s)>\bar{\Delta }_{i+1}(s)\) by showing that for \(-s<i\), \(\bar{\Delta }_i(s)-\bar{\Delta }_i>\bar{\Delta }_{i+1}(s)-\bar{\Delta }_{i+1}\). Since \(\bar{\Delta }_i>\bar{\Delta }_{i+1}\), this implies \(\bar{\Delta }_i(s)>\bar{\Delta }_{i+1}(s)\).

Let \(\delta _i=\bar{\Delta }_i(s)-\bar{\Delta }_i\). There exists a state \(m>0\) at which rejecting all arriving customers is the optimal decision under basestock s and \(s^*\). Let \(\delta _m=\bar{\Delta }_m(s)-\bar{\Delta }_m=-\frac{g_s}{\mu }-(-\frac{g^*}{\mu })\). For \(-s<i\) we check whether \(\delta _i>\delta _{i+1}\). In other words we check

$$\begin{aligned} \delta _i&=\left( \frac{g^*}{\mu }-\frac{g_s}{\mu }\right) - +\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1}(s))\right\} \nonumber \\&\quad -\sum _j\frac{\lambda _j}{\mu }\max _{d_j}\left\{ \bar{F}_j(d_j) (R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\right\} >\delta _{i+1}. \end{aligned}$$

Since for \(i=m\), optimal decision is to reject all arriving customers, RHS and LHS of the inequality in (19) are equal to \(\frac{g^*}{\mu }-\frac{g_s}{\mu }\). We make the proof by induction. Suppose for state \(i\ge n\), \(\delta _n\ge \delta _{n+1}\), i.e., \(\bar{\Delta }_n(s)\ge \bar{\Delta }_{n+1}(s)\). We show, \(\delta _{n-1}>\delta _n\),

$$\begin{aligned}&\delta _m+\sum _j\frac{\lambda _j}{\mu } \left( \max _{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)+\bar{\Delta }_n(s))\}\right. \nonumber \\&\quad \left. -\max _{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)+\bar{\Delta }_n)\}\right) \ge \delta _m \nonumber \\&\quad +\sum _j\frac{\lambda _j}{\mu } \left( \max _{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_{i+1}(d_j)+\bar{\Delta }_{n+1}(s))\}\right. \nonumber \\&\quad \left. -\max _{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_{i+1}(d_j)+\bar{\Delta }_{n+1})\} \right) \end{aligned}$$

To show that (20) holds, we introduce the following lemma:

Lemma 3

Let f(x) be a decreasing function of \(x,~x\in X=[0,\bar{x}], f(x)\ge 0\) where \(f(\bar{x})=0\). Let \(c_1,~c_2,~c_3,c_4\in \mathbb {R}\) such that \(c_1>c_2>c_4\), \(c_1>c_3>c_4\), and \(c_1-c_2>c_3-c_4>0\). Let \(L_i(x)\) be a function of \(i\in \mathbb {Z}\) and x, as defined in Eq. (1). Then for \(i\in \mathbb {Z},~x\in X\),

$$\begin{aligned}&\max _x\{f(x)(c_1-L_i(x))\}-\max _x\{f(x)(c_2-L_i(x))\}\\&\quad \ge \max _x\{f(x)(c_3-L_{i+1}(x))\}-\max _x\{f(x)(c_4-L_{i+1}(x))\}. \end{aligned}$$


We prove the lemma in three steps:

  1. (1)

    Show that for \(i\in \mathbb {Z},~x\in X\),

    $$\begin{aligned}&\max _x\{f(x)(c_1-L_i(x))\}-\max _x\{f(x)(c_2-L_i(x))\} \nonumber \\&\quad \ge \max _x\{f(x)(c_3-L_{i}(x))\}-\max _x\{f(x)(c_1-(c_3-c_4)-L_{i}(x))\}. \end{aligned}$$
  2. (2)

    Show that

    $$\begin{aligned}&\max _x\{f(x)(c_1-L_i(x))\}-\max _x\{f(x)(c_1-(c_3-c_4)-L_i(x))\} \nonumber \\&\quad \ge \max _x\{f(x)(c_3-L_{i}(x))\}-\max _x\{f(x)(c_4-L_{i}(x))\}. \end{aligned}$$
  3. (3)

    Show that

    $$\begin{aligned}&\max _x\{f(x)(c_3-L_i(x))\}-\max _x\{f(x)(c_4-L_i(x))\} \nonumber \\&\quad \ge \max _x\{f(x)(c_3-L_{i+1}(x))\}-\max _x\{f(x)(c_4-L_{i+1}(x))\}. \end{aligned}$$

    Note that \(L_i(x)\), and \(L_{i+1}(x)-L_i(x)\) are positive decreasing functions of x.

  1. (1)

    Equation (21) simply follows from \(c_1-(c_3-c_4)>c_2\), and \(\max _x\{f(x)(c_1-(c_3-c_4)-L_i(x))\}>\max _x\{f(x)(c_2-L_i(x))\}\).

  2. (2)

    Consider the function \(-f(x)L_i(x)\), which is a negative increasing function with value 0 at \(x=\bar{x}\). Let \(c_{2'}=c_1-(c_3-c_4)\) and define \(x_1, x_{2'}, x_3, x_4\) as \(x_k=\mathrm{arg\,max}_{x\in X}f(x)(c_k-L_i(x)),~k\in \{1,2',3,4\}\). Since \(c_1>c_{2'}>c_4\) and \(c_1>c_3>c_4\), it holds that \(x_1<x_{2'}<x_4\) and \(x_1<x_3<x_4\). Then two cases are possible, \(\underline{x_{2'}<x_3}\) Observe that

    $$\begin{aligned}&f(x_1)(c_1-L_i(x_1))-f(x_2)(c_{2'}-L_i(x_{2'}))\\&\qquad>f(x_{2'})(c_1-L_i(x_{2'}))\\&\qquad -f(x_{2'})(c_{2'}-L_i(x_{2'}))\\&\quad>f(x_3)(c_3-L_i(x_3))-f(x_3)(c_{4}-L_i(x_{3}))\\&\quad >f(x_3)(c_3-L_i(x_3))-f(x_4)(c_{4}-L_i(x_{4})), \end{aligned}$$

    where the first inequality is due to \(x_1\) being the maximizer of \(f(x)(c_1-L_i(x))\), the second inequality is due to \(c_1-c_{2'}=c_3-c_4\) and \(f(x_{2'})>f(x_3)\), and the last inequality is due to \(x_4\) being the maximizer of \(f(x)(c_4-L_i(x))\). This implies (22) holds. \(\underline{x_{2'}>x_3}\) Observe that

    $$\begin{aligned}&f(x_1)(c_1-L_i(x_1))-f(x_3)(c_{3}-L_i(x_{3}))\\&\quad>f(x_{3})(c_1-L_i(x_{3})) -f(x_{3})(c_{3}-L_i(x_{3}))\\&\quad>f(x_{2'})(c_{2'}-L_i(x_{2'}))-f(x_{2'})(c_{4}-L_i(x_{2'}))\\&\quad >f(x_{2'})(c_{2'}-L_i(x_{2'}))-f(x_4)(c_{4}-L_i(x_{4})), \end{aligned}$$

    where the first inequality is due to \(x_1\) being the maximizer of \(f(x)(c_1-L_i(x))\), the second inequality is due to \(c_1-c_3=c_{2'}-c_4\) and \(f(x_{2'})<f(x_3)\), and the last inequality is due to \(x_4\) being the maximizer of \(f(x)(c_4-L_i(x))\). This implies that (22) holds.

  3. (3)

    Note that \(f(x)(c_k-L_{i+1}(x))=f(x)c_k-f(x)L_i(x)-f(x)(L_{i+1}-L_i(x))\), \(k\in \{1,2,3,4\}\), and that \(L_{i+1}(x)-L_i(x)\) is a positive decreasing function of x. Let \(x_3+\) and \(x_4+\) be defined as \(x_3+=\hbox {arg max}_{x\in X}f(x)(c_3-L_{i+1}(x))\) and \(x_4+=\hbox {arg max}_{x\in X}f(x)(c_4-L_{i+1}(x))\). Since \(-f(x)(L_{i+1}(x)-L_{i}(x))\) is increasing in x, and \(c_3>c_4\), it holds that \(x_3<x_{3}+<x_4+\) and \(x_3<x_4<x_{4}+\). Then two cases are possible. \(\underline{x_4<x_3+}\) Observe that

    $$\begin{aligned}&f(x_3)(c_3-L_i(x_3))-f(x_4)(c_{4}-L_i(x_{4}))\\&\quad>f(x_{4})(c_3-L_i(x_{4}))-f(x_{4})(c_{4}-L_i(x_{4}))\\&\quad>f(x_3+)(c_3-L_{i+1}(x_3+))-f(x_3+)(c_{4}-L_{i+1}(x_{3}+))\\&\quad >f(x_3+)(c_3-L_{i+1}(x_3+))-f(x_4)(c_{4}-L_{i+1}(x_{4})), \end{aligned}$$

    where the first inequality is due to \(x_3\) being the maximizer of \(f(x)(c_3-L_i(x))\), the second inequality is due to \(f(x_{4})>f(x_3+)\), and the last inequality is due to \(x_4\) being the maximizer of \(f(x)(c_4-L_{i+1}(x))\). This implies that (23) holds. \(\underline{x_4>x_3+}\) Observe that

    $$\begin{aligned}&f(x_3)(c_3-L_i(x_3))-f(x_3+)(c_{3}-L_i(x_{3}+))\\&\quad>f(x_{3}+)(c_3-L_i(x_{3}+))-f(x_{3}+)(c_{3}-L_{i+1}(x_{3}+))\\&\quad>f(x_{4})(c_{4}-L_i(x_{4}))-f(x_{4})(c_{4}-L_{i+1}(x_{4}))\\&\quad >f(x_{4})(c_{4}-L_i(x_{4}))-f(x_4+)(c_{4}-L_{i+1}(x_{4}+)), \end{aligned}$$

    where the first inequality is due to \(x_3\) being the maximizer of \(f(x)(c_3-L_i(x))\), the second inequality is due to \(f(x_{4})(L_{i+1}(x_4)-L_i(x_4))<f(x_3+)(L_{i+1}(x_3+)-L_i(x_3+))\), and the last inequality is due to \(x_4+\) being the maximizer of \(f(x)(c_4-L_{i+1}(x))\). This implies that (23) holds.

Lemma 3 implies that (20) holds as follows: Observe that \(\bar{\Delta }_n(s)>\bar{\Delta }_n\), \(\bar{\Delta }_{n+1}(s)>\bar{\Delta }_{n+1}\) (by Lemma 1), \(\bar{\Delta }_n(s)>\bar{\Delta }_{n+1}(s)\) (by induction assumption \(\delta _n>\delta _{n+1}\)), and \(\bar{\Delta }_{n}>\bar{\Delta }_{n+1}\) (by Theorem 1). Thus letting \(c_1=R_j+\bar{\Delta }_n(s)\), \(c_2=R_j+\bar{\Delta }_n\), \(c_3=R_j+\bar{\Delta }_{n+1}(s)\), \(c_4=R_j+\bar{\Delta }_{n+1}\), and \(f(x)=\bar{F}(x)\), Lemma 3 implies that (20) holds.

Thus the proof of for \(s<s^*\), \(\bar{\Delta }_i(s)\) is decreasing in i for \(i>-s\) is complete. We conclude that the structure of rationing and lead time quotation policy is the same with that of the optimal policy, except that the decisions taken are different. Under \(-s\), lead times are lower, and stock rationing levels are closer to zero.

\(\underline{\hbox {Case 2.}\; {s>s^*}}\) For \(i>-s^*\), showing that \(\bar{\Delta }_i(s)\) is decreasing in i follows similar lines as in Case 1. For states \(i<0\), lead time quotation corresponds to the stock rationing decision. Analysis shows that \(\bar{\Delta }_i(s)\) is not necessarily decreasing in i for states \(-s\le i\le -s^*\) (see Fig. 7). However, immediate reasoning reveals that in all states \(-s\le i <-s^*\), no rationing of stock should take place for any class of the customers even if there exists a customer class k with \(R_k=0\). Any stock over \(s^*\) is a burden which causes unnecessary stocking cost. Customer arrivals help getting rid of the unnecessary stock; thus when stock level exceeds \(s^*\) rejecting a customer cannot be the optimal action. Actually, in numerical analysis we observed that even customers with negative unit revenue might be allowed to the system in states \(-s\le i<-s^*\).

In conclusion, when \(s\ne s^*\), the rationing and lead time quotation policies have the same structure as of the optimal policy, with lower lead times and stock rationing levels close to zero.

Fig. 7
figure 7

The change in \(\bar{\Delta }_i=v(i)-v(i-1)\) as i changes for \(s>s^*\). The setting under consideration is \(\lambda _1=1.2, \lambda =2=0.4, R_1=10, R_2=3, \ell =1, h=2, \mu =1\) and \(D=\{0,d_\mathrm{max}\}\) for both customers. Optimal basestock is \(s^*=3\)

Appendix 5: Proof of Lemma 2

We show that under a given basestock s, a decrease in quoted lead times results in an increase in W and in L. For \(i<0\), “decrease in quote” implies lower stock rationing levels (i.e., stock rationing level will be closer to 0). Let policy P and P define two policies with P corresponding to the policy with shorter quotes. Let s be the basestock level implied by the policies. Let \(\underline{d}_k(i), k\in C, i\ge -s\) be the lead time quote in state i for class k under P. We first introduce the following lemma (Puterman 1994):

Lemma 4

Let \(\{x_i\}\), \(\{x_i'\}\) be real-valued non-negative sequences satisfying

$$\begin{aligned} \sum _{i=t}^{\infty } x_i\ge \sum _{i=t}^{\infty }x_i' \end{aligned}$$

for all \(t\ge 0\), with equality holding in (24) for \(t=0\).

Suppose \(v_{i+1}\ge v_i\) for \(i=0,1,\cdots \), then

$$\begin{aligned} \sum _{i=0}^{\infty }v_ix_i\ge \sum _{i=0}^{\infty }v_ix_i'. \end{aligned}$$

In the problem under consideration, let \(x_i\) stand for \(\frac{\sum _j\lambda _j\bar{F}_j(d_j(i))}{\lambda _\mathrm{eff}}\pi _i\) and \(v_i\) stand for \(E_i[wait]=(\frac{i+1}{\mu })^+, for~i\in \mathbb {Z}\). Let \(\{x_i\}\) and \(\{x_i'\}\) be the sequences defined under P and P, respectively. Note that under a given policy, \(\pi _{i+1}=\frac{\sum _j\lambda _j\bar{F}_j(d_j(i)))}{\mu }\pi _i\). For the simplicity of expression and without loss of generality, we assume \(\mu =1\). Let \(\sum _j\lambda _j\bar{F}_j(d_j(i))\) be shortly denoted as \(\underline{G_i}\) and \(G_i\) under policies P and P, respectively. The \(G_i\) is the total effective arrival rate to state i. For \(t\ge -s,\sum _{i=t}^{\infty }x_i\) would be expressed as \(\sum _{i=t}^{\infty } \frac{G_i}{\lambda _\mathrm{eff}}\pi _i\), where \(\lambda _\mathrm{eff}=\sum _{i=-s}^{\infty }G_i\pi _i\). Note that \(\sum _{i=-s}^{\infty }x_i=1\).

In the following, we first show \(\sum _{i=t}^{\infty }x_i\ge \sum _{i=t}^{\infty }x_i'\) for \(t\ge -s\):

$$\begin{aligned} \frac{\sum _{i=t}^{\infty }G_i\pi _i}{\sum _{i=-s}^{\infty }G_i\pi _i}&\ge \frac{\sum _{i=t}^{\infty }\underline{G}_i\underline{\pi }_i}{\sum _{i=-s}^{\infty }\underline{G}_i\underline{\pi }_i},\hbox {equivalently} \nonumber \\ \frac{\sum _{i=t}^{\infty }G_i\pi _i}{\sum _{i=-s}^{i=t-1}G_i\pi _i}&\ge \frac{\sum _{i=t}^{\infty }\underline{G}_i\underline{\pi }_i}{\sum _{i=-s}^{i=t-1}\underline{G}_i\underline{\pi }_i} \end{aligned}$$

Note \(G_i\pi _i=\prod _{n=-s}^iG_n\pi _i\). Thus (25) can be expressed as

$$\begin{aligned}&\dfrac{G_{-s+1}G_{-s+2}\cdots G_t(1+G_{t+1}+G_{t+1}G_{t+2}+\cdots )}{1+G_{-s+1}+G_{-s+1}G_{-s+2}+\cdots G_{-s+1}\cdots G_{t-1}}\nonumber \\&\quad \ge \dfrac{\underline{G}_{-s+1}\underline{G}_{-s+2}\cdots \underline{G}_t(1+\underline{G}_{t+1}+\underline{G}_{t+1}\underline{G}_{t+2}+\cdots )}{1+\underline{G}_{-s+1}+\underline{G}_{-s+1}\underline{G}_{-s+2}+\cdots \underline{G}_{-s+1}\cdots \underline{G}_{t-1}}. \end{aligned}$$

Note that showing

$$\begin{aligned}&\frac{G_{-s+1}G_{-s+2}\cdots G_t}{1+G_{-s+1}+G_{-s+1}G_{-s+2}+\cdots G_{-s+1}\cdots G_{t-1}}\nonumber \\&\quad \ge \frac{\underline{G}_{-s+1}\underline{G}_{-s+2}\cdots \underline{G}_t}{1+\underline{G}_{-s+1}+\underline{G}_{-s+1}\underline{G}_{-s+2}+\cdots \underline{G}_{-s+1}\cdots \underline{G}_{t-1}}, \end{aligned}$$

is sufficient for (26) to hold, since \(\underline{G}_i\le G_i,~ i\ge -s\). We show through recursion that the inequality in (27) holds. Assume (27) holds for \(t=r-1\),

$$\begin{aligned}&\frac{G_{-s+1}G_{-s+2}\cdots G_{r-1}}{1+G_{-s+1}+G_{-s+1}G_{-s+2}+\cdots G_{-s+1}\cdots G_{r-2}} \nonumber \\&\quad \ge \frac{\underline{G}_{-s+1}\underline{G}_{-s+2}\cdots \underline{G}_{r-1}}{1+\underline{G}_{-s+1}+\underline{G}_{-s+1}\underline{G}_{-s+2}+\cdots \underline{G}_{-s+1}\cdots \underline{G}_{r-2}}. \end{aligned}$$

If (28) holds, then (27) holds for \(t=r\), since

$$\begin{aligned} \frac{G_{r}}{\frac{1+G_{-s+1}+G_{-s+1}G_{-s+2}+\cdots G_{-s+1}\cdots G_{r-2}}{G_{-s+1}G_{-s+2}\cdots G_{r-1}}+1}\ge \frac{\underline{G}_{r}}{\frac{1+\underline{G}_{-s+1}+\underline{G}_{-s+1}\underline{G}_{-s+2}+\cdots \underline{G}_{-s+1}\cdots \underline{G}_{r-2}}{\underline{G}_{-s+1}\underline{G}_{-s+2}\cdots \underline{G}_{r-1}}+1},\nonumber \\ \end{aligned}$$

holds, due to \(G_r\ge \underline{G}_r\), and denominator of the fraction in left-hand-side (LHS) is smaller due to (28). For the initial state \(r=-s+2\), (28) holds. Thus the inequality (25) holds. Finally, since \(v_i=E_i[\hbox {wait}]=(\frac{i+1}{\mu })^+\) is increasing in i for \(i \in \mathbb {Z}\), Lemma 24 implies \(W\ge \underline{W}\).

For the comparison of the number of outstanding orders waiting to be processed, L, we show

$$\begin{aligned} \sum _{i=-s}^{\infty }i^+\underline{\pi }_i\le \sum _{i=-s}^{\infty }i^+\pi _i \end{aligned}$$

Since \(\underline{\pi }_i\) and \(\pi _i\) can be shown to follow \(\sum _{i=t}^{\infty }\underline{\pi }_i\le \sum _{i=t}^{\infty }\pi _i\) as in Lemma 24, it is trivial that (30) holds.

Appendix 6: Proof of Proposition 3

We define g and \(g^2\) as the optimal expected average gain under dynamic, and static quotation schemes, respectively. Note that the definition of the static quotation scheme implies that the stock rationing decisions are given optimally. Then for \(i<0\) it is possible to define the following functional equation under the static scheme:

$$\begin{aligned} \frac{g^2}{\gamma }+v(i)&=-\frac{(h)i^-}{\gamma }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j\in D_j}\{R_j,\Delta _i^2\} +\frac{\Lambda }{\gamma }v(i+1)+\frac{\mu }{\gamma }v(i) \nonumber \\&\quad +\frac{\mu }{\gamma }\max \{\Delta _{i-1}^2,0\}\quad i<0. \end{aligned}$$

Note that for \(i<0\) the functional equation has the same structure with that of the original dynamic quotation problem. Thus, assuming without loss of optimality that there will be a state \(-L_B\) at which stopping production is the optimal decision, for \(i<0\) it is possible to show that \(\Delta _i^2=v(i)-v(i+1)\) is increasing in i. This implies a control-limit type production policy is indeed optimal under the static and myopic quotation scheme.

Next, we show that the basestock level is higher under the static quotation scheme compared to the optimal scheme. For \(i<0\), it is possible to rewrite (7) and (8) as follows:

$$\begin{aligned} \Delta _{i}^2=-\frac{g^2}{\gamma }-\frac{(h)i^-}{\gamma }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j\in D_j}\{R_j,\Delta _i^2\},\quad i\le -L_B, \end{aligned}$$
$$\begin{aligned} \Delta _{i}^2=-\frac{g^2}{\gamma }-\frac{(h)i^-}{\gamma }+\sum _j\frac{\lambda _j}{\mu }\max _{d_j\in D_j}\{R_j,\Delta _i^2\}+\frac{\mu }{\gamma }\max \{\Delta _{i-1}^2,0\},\quad -L_B<i<0. \end{aligned}$$

The structure of the equations are the same under dynamic, zero lead time, static, and myopic schemes; however, note that \(-\frac{g^2}{\Lambda }\ge -\frac{g}{\Lambda }\) since \(g^2\le g\). This implies \(\Delta _i^2\ge \Delta _i\) for \(i\le -L_B\) and \(-L_B<i<0\). This implies the level at which the optimal decision is to stop production will be higher under non-optimal lead time quotation policies, i.e., \(s_2\ge s^*\).

Appendix 7: Proof of Proposition 4

The proof follows from the observation of the equation in (10). From the equation, the optimal quotes under dynamic quotation scheme is expressed as \(d_j(i)=\hbox {arg max}_{d_j\in D_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\}\). Since \(\bar{\Delta }_{i+1}<0\) for \(i<0\), and since \(\bar{F}_j(d_j)\) is decreasing in \(d_j\), \(\hbox {arg max}_{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)\}\le \hbox {arg max}_{d_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)+\bar{\Delta }_{i+1})\}~\forall ~j\).

Appendix 8: Proof of Proposition 5

To show the effect of the change in parameter on the optimal policy structure, we use the value function. For a related study on the effect of system parameters on the structure of the optimal policy see Cil et al. (2009) and Aktaran-Kalayci and Ayhan (2009). The value function, \(V^n(i)\), denotes the total expected reward (profit) when there are n remaining transitions, and the state is in i. We obtain \(V^n(i)\) as follows:

$$\begin{aligned} V^n(i) =-\frac{(h)i^-}{\gamma }+\frac{\sum _j\lambda _j\tau _jv^{n-1}(i)}{\gamma } +\frac{\mu }{\gamma }\tau _0v^{n-1}(i)\quad \hbox {for }i\in \mathbb {Z}, \end{aligned}$$

where \(\tau _0\) and \(\tau _j,~j\in C\) are defined on real-valued functions as

$$\begin{aligned} \tau _0v^{n}(i)&=\max \{V^{n}(i)I_{i\le 0},V^{n}(i-1)\},\\ \tau _j V^{n}(i)&=\max _{d_j\in D_j}\{\bar{F}_j(d_j)(R_j-\ell L_i(d_j)+V^{n}(i+1))+F_j(d_j)V^{n}(i)\}. \end{aligned}$$

Before the proof, note that under the optimal policy the chain has a single recurrent class plus some transient states, and thus a constant gain is obtained at all states. Puterman (1994, p. 339) states that under constant gain the bias, v, is the relative difference in total expected reward as n goes to infinity. In other words,

$$\begin{aligned} v(j)-v(k)=\lim _{n\rightarrow \infty }[V^n(j)-V^n(k)] \end{aligned}$$

This implies the structural behavior of \(v(i)-v(i+1)\) (identified in Theorem 1) also reflects the behavior of \(V^n(i+1)\) asymptotically.

In the proof, we use the structure that \(\Delta V^n(i)=V^n(i)-V^n(i+1)\) is increasing in \(i,~\forall i\in \mathbb {Z}\) as n goes to infinity.

(1) Effect of \(R_k\). We show the impact of \(R_k\) on the structure of the optimal policy. Let \(R_k^+\) denote the increased revenue for class k, \(R_k^+>R_k\). Let \(V^n_+(i)\) denote the value function under \(R_k^+\). Our aim is to show that under \(R_k^+\), quotes to class k are shorter while quotes to classes \(j\ne k\) are longer, compared to those under \(R_k\).

To show the effect of \(R_k\) on the optimal policy, we first show that under \(R_k^+\) the following hold:

  1. (i)

    \(V^n(i)-V^n(i+1) < V^n(i)-V^n(i+1)\), i.e., \(\Delta V^n(i) <\Delta V_+^n(i),\)

  2. (ii)

    \(\Delta V^n(i)-R_k>\Delta V_+^n(i)-R_k^+, ~\forall i.\)

We use induction in the proof. Initial step of induction assumes that \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\).

(i) Show \(\Delta V^n(i)<\Delta V_+^n(i)\).

We first show that under each operator \(\tau _j\) and \(\tau _0\), \(\Delta \tau _jV^{n-1}(i)<\Delta \tau _jV_+^{n-1}(i),~\forall ~j\in {0}\cup C\).

\(\underline{\hbox {For }\tau _j,j\in C}\) Let \(d_i, d_{i+1}, d_i^+,d_{i+1}^+\) be maximizers of \(\tau _j V^{n-1}(i), \tau _j V^{n-1}(i+1),\tau _j V_+^{n-1}(i),\tau _j V_+^{n-1}(i+1)\), respectively. Let \(\varepsilon = R_k^+-R_k\). We check whether

$$\begin{aligned}&\tau _j V^{n-1}(i) - \tau _j V^{n-1}(i+1)< \tau _j V_+^{n-1}(i) -\tau _j V_+^{n-1}(i+1), \nonumber \\&\bar{F}_j(d_i)(R-\ell L_i(d_i)+V^{n-1}(i+1)) +F_j(d_i)V^{n-1}(i) \nonumber \\&\quad -\bar{F}_j(d_{i+1})(R-\ell L_{i+1}(d_{i+1})+V^{n-1}(i+2)) \nonumber \\&-F_j(d_{i+1})V^{n-1}(i+1)< \bar{F}_j(d_i^+)(R+\varepsilon I_{k}-\ell L_i(d_i^+)+V_+^{n-1}(i+1)) \nonumber \\&\quad +F_j(d_i^+)V_+^{n-1}(i) -\bar{F}_j(d_{i+1}^+)(R+\varepsilon I_k-\ell L_{i+1}(d_{i+1}^+)+V_+^{n-1}(i+2)) \nonumber \\ {}&\quad - F_j(d_{i+1}^+)V_+^{n-1}(i+1). \end{aligned}$$

In (33) \(\varepsilon I_k\) takes value \(\varepsilon \in \mathbb {R}^+\) if the operator under consideration is \(\tau _k\) and 0 otherwise. The analysis is made under \(j\ne k\) and \(j=k\) separately.

\(\underline{\hbox {For }j\ne k}\) If in (33) \(d_{i+1}\) is replaced with \(d_{i+1}^+\) and \(d_i^+\) with \(d_i\):

$$\begin{aligned}&\bar{F}_j(d_i)V^{n-1}(i+1)+F_j(d_i)V^{n-1}(i) \nonumber \\ {}&\quad - \bar{F}_j(d_{i+1}^+)V^{n-1}(i+2)-F_j(d_{i+1}^+)V^{n-1}(i+1)\nonumber \\ {}&\qquad< \bar{F}_j(d_i)V_+^{n-1}(i+1)+F_j(d_i)V_+^{n-1}(i) \nonumber \\ {}&\quad \qquad - \bar{F}_j(d_{i+1}^+)V_+^{n-1}(i+2)-F_j(d_{i+1}^+)V_+^{n-1}(i+1),\nonumber \\&F_j(d_i)\Delta V^{n-1}(i)+\bar{F}_j(d_{i+1}^+)\Delta V^{n-1}(i+1)\nonumber \\ {}&\quad < F_j(d_i)\Delta V_+^{n-1}(i)+\bar{F}_j(d_{i+1}^+)\Delta V_+^{n-1}(i+1). \end{aligned}$$

Under the induction assumption of \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\), the inequality in (34) and thus in (33) are satisfied.

\(\underline{For j=k}\) Note that when \(j=k\), \(\tau _jV_+^{n-1}(i)\) is defined with \(R_k^+\) and let \(d_i^+\) be the corresponding maximizer of \(\tau _jV_+^{n-1}(i)\).

\(\underline{\hbox {When }d_{i+1}^+<d_i}\) In (33) replace \(d_{i+1}\) with \(d_i\) and \(d_i^+\) with \(d_{i+1}^+\). Then,

$$\begin{aligned}&\bar{F}_j(d_i)V^{n-1}(i+1)+F_j(d_i)V^{n-1}(i)\\&\qquad -\bar{F}_j(d_i)V^{n-1}(i+2)-F_j(d_i)V^{n-1}(i+1)\\&\qquad +\bar{F}_j(d_i)(\ell (L_{i+1}(d_i)-L_i(d_i)))\\&\quad <\bar{F}_j(d_{i+1}^+)V_+^{n-1}(i+1)+F_j(d_{i+1}^+)V_+^{n-1}(i)\\&\qquad -\bar{F}_j(d_{i+1}^+)V_+^{n-1}(i+2)-F_j(d_{i+1}^+)V_+^{n-1}(i+1)\\&\qquad +\bar{F}_j(d_{i+1}^+)(\ell (L_{i+1} (d_{i+1}^+)-L_i(d_{i+1}^+))),\\ \end{aligned}$$
$$\begin{aligned}&F_j(d_i)\Delta V^{n-1}(i)+\bar{F}_j(d_i)\Delta V^{n-1}(i+1) \nonumber \\ {}&\qquad +\bar{F}_j(d_{i})(\ell (L_{i+1}(d_{i})- L_i(d_{i})))\nonumber \\ {}&\quad < F_j(d_{i+1}^+)\Delta V_+^{n-1}(i) +\bar{F}_j(d_{i+1}^+)\Delta V_+^{n-1}(i+1) \nonumber \\ {}&\qquad +\bar{F}_j(d_{i+1}^+) (\ell (L_{i+1}(d_{i+1}^+)-L_i(d_{i+1}^+))). \end{aligned}$$

Note \(d_{i+1}^+<d_i\) implies \(F_j(d_{i+1}^+)<F_j(d_i)\). Since \(L_{i+1}(d)-L_i(d)>0\) and decreasing in d, and \(\Delta V^{n-1}(i)\) is increasing in i, inequality (35) and thus, (33) hold.

\(\underline{\hbox {When }d_{i+1}^+>d_i}\) In (33) replace \(d_{i+1}\) with \(d_{i+1}^+\) and \(d_i^+\) with \(d_i\). Then as in (34),

$$\begin{aligned} \begin{aligned}&F_j(d_i)\Delta V^{n- 1}(i)+\bar{F}_j(d_{i+1}^+)\Delta V^{n-1}(i+1)\\ {}&\quad <F_j(d_{i})\Delta V_+^{n-1}(i) +\bar{F}_j(d_{i+1}^+)\Delta V_+^{n-1}(i+1)\\ {}&\qquad +[\bar{F}_j(d_i)- \bar{F}_j(d_{i+1}^+)]\varepsilon . \end{aligned} \end{aligned}$$

The inequality holds since \(\bar{F}_j(d_i)-\bar{F}_j(d_{i+1}^+)>0\).

\(\underline{\hbox {For }\tau _0}\) For \(i> 0\), \(\tau _0V^{n-1}(i)=V^{n-1}(i-1)\) and thus \(\Delta \tau _0V^{n-1}(i)<\Delta \tau _0V_+^{n-1}(i)\) immediately holds since \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\) by assumption.

For \(i\le 0\), \(\tau _0V^{n-1}(i)=\max \{V^{n-1}(i),V^{n-1}(i-1)\}\). Let \(u_i,~u_i^+\in \{0,1\}\) be the indicator variables associated with \(\tau _0V^{n-1}(i)\) and \(\tau _0V_+^{n-1}(i)\). We check whether

$$\begin{aligned} \Delta \tau _0 V^{n-1}(i)<&\Delta \tau _0V_+^{n-1}(i),\nonumber \\ u_iV^{n-1}(i)+(1-u_i)V^{n-1}(i-1) \nonumber \\ -u_{i+1}V^{n-1}(i+1)-(1- u_{i+1})v_i <&u_i^+V_+^{n-1}(i)+(1-u_i^+)V_+^{n-1}(i-1)\nonumber \\ {}&-u_{i+1}^+V^{n- 1}(i+1)-(1-u_{i+1}^+)V_+^{n-1}(i). \end{aligned}$$

Replacing \(u_{i+1}\) with \(u_{i+1}^+\) and \(u_i^+\) with \(u_i\) gives

$$\begin{aligned} (1-u_i)\Delta V^{n-1}(i-1)+u_{i+1}^+\Delta V^{n-1}(i)<(1-u_i)\Delta V_+^{n- 1}(i-1)+u_{i+1}^+\Delta V_+^{n-1}(i). \end{aligned}$$

Since \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\), inequality (37) and thus (36) holds.

That \(\Delta \tau _jV^{n-1} (i)<\Delta \tau _jV_+^{n-1}(i),~j\in \{0\}\cup C\) is sufficient for \(\Delta V^n(i)<\Delta V_+^n(i)\).

(ii) Show \(\Delta V^n(i)-R_k>\Delta V_+^n(i)-R_k^+\).

We show that under each operator \(\tau _j\), \(\tau _0\) \(\Delta \tau _j V^{n-1}(i)-R_k>\Delta \tau _j V_+^{n-1}(i)-R_k^+\) and \(\Delta \tau _0 V^{n-1}(i)-R_k>\Delta \tau _0 V_+^{n-1}(i)-R_k^+\).

For \(\tau _j,j\in C\), let \(d_i\) and \(d_i^+\) be maximizers of \(\tau _j V^{n-1}(i)\) and \(\tau _j V_+^{n-1}(i)\). We check whether

$$\begin{aligned}&\tau _j V^{n-1}(i) - \tau _j V^{n-1}(i+1)< \tau _j V_+^{n-1}(i) -\tau _j V_+^{n-1}(i+1), \nonumber \\ {}&\bar{F}_j(d_i)(R-\ell L_i(d_i)+V^{n-1}(i+1)) +F_j(d_i)V^{n-1}(i) \nonumber \\ {}&\quad - \bar{F}_j(d_{i+1})(R-\ell L_{i+1}(d_{i+1})+V^{n-1}(i+2)) \nonumber \\ {}&\quad -F_j(d_{i+1})V^{n- 1}(i+1) -R_k < \bar{F}_j(d_i^+)(R+\varepsilon I_{k}-\ell L_i(d_i^+)+V_+^{n-1}(i+1)) \nonumber \\ {}&\quad +F_j(d_i^+)V_+^{n-1}(i) -\bar{F}_j(d_{i+1}^+)(R+\varepsilon I_k-\ell L_{i+1}(d_{i+1}^+)+V_+^{n-1}(i+2)) \nonumber \\ {}&\quad - F_j(d_{i+1}^+)V_+^{n-1}(i+1)- R_k^+. \end{aligned}$$

\(\underline{\hbox {For }j\ne k}\) We know that \(d_i<d_{i+1}<d_{i+1}^+\). Similar to inequality (34) the following inequality is obtained:


Observe that \(1-F_j(d_i)-\bar{F}_j(d_{i+1}^+)>0\), so inequality (39) and thus (38) hold. \(\underline{\mathrm{For} j=k.}\) This case is analyzed separately for \(d_{i+1}^+>d_i\) and \(d_i<d_{i+1}^+\).

\(\underline{\hbox {When }d_{i+1}^+<d_i}\) In (38) replace \(d_{i+1}\) with \(d_{i+1}^+\) and \(d_i^+\) with \(d_i\), then as in (34) we obtain


By the induction assumption the inequality holds and thus (38) holds.

\(\underline{\hbox {When }d_{i+1}^+>d_i}\) Replace \(d_{i+1}\) with \(d_i\) and \(d_i^+\) with \(d_{i+1}^+\) in (38). Similar to (40) the following is obtained:


Since \(d_{i+1}^+>d_i\), \(\bar{F}_j(d_i)>\bar{F}_j(d_{i+1}^+)\). Induction assumption implies \(\Delta V^{n-1}(i)-R_k>\Delta V^{n-1}(i)-R_k^+\) and thus in (41) the sum of first two terms in the left-hand side (LHS) is greater than the sum of first two terms on the right-hand side (RHS). Furthermore \(L_{i+1}(d)-L_i(d)>0\) and decreasing in d and thus the inequality (41) holds. Therefore (38) holds.

For the analysis of \(\tau _0\), let \(u_i, u_{i+1}\in \{0,1\}\) be indicator variables associated with \(\tau _0 V^{n-1}(i)\) and \(\tau _0 V_+^{n-1}(i)\), respectively. We check

$$\begin{aligned} \begin{aligned} \Delta \tau _0 V^{n-1}(i)-R_k~>~\tau _0 V^{n-1}(i)-R_k^+. \end{aligned} \end{aligned}$$

Replacing \(u_{i+1}\) with \(u_i\) on LHS of (42) and \(u_i^+\) with \(u_{i+1}^+\) on RHS yields

the inequality and thus (42) holds.

(2) Effect of \(\lambda _k\). We show the impact of \(\lambda _k\) on the structure of the optimal policy. Let \(\lambda _k^+\) denote the increased arrival rate for class k. We show that under \(\lambda _k^+\), \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\). This condition is sufficient to conclude that under \(\lambda _k^+\), for all classes basestock and rationing levels increase, and the lead time quotes are longer. For \(\tau _j,~j\in \{0\}\cup C\), it is shown that \(\Delta \tau _jV^{n-1}(i)<\Delta \tau _j V_+^{n-1}(i)\).

\(\underline{\hbox {For }\tau _j,~j\in C}\) Let \(d_i\), \(d_i^+\), \(d_{i+1}\), \(d_{i+1}^+\) be maximizers of \(\tau V^{n-1}(i)\), \(\tau V_+^{n-1}(i), \tau V^{n-1}(i+1)\), and \(\tau V_+^{n-1}(i+1)\), respectively. \(\underline{\hbox {For }\tau _0.}\) Let \(u_i\), \(u_i^+\), \(u_{i+1}\), \(u_{i+1}^+\) be maximizers of \(\tau _0 V^{n-1}(i)\), \(\tau _0 V_+^{n-1}(i), \tau _0 V^{n-1}(i+1)\), and \(\tau _0 V_+^{n-1}(i+1)\), respectively.

Similar analysis as in (33), (34), (36) and (37) would imply that \(\Delta V^{n-1}(i)<\Delta V_+^{n-1}(i)\) holds.

From (8), this implies as \(\lambda _k\) increases, the basestock \(s^*\), and the rationing levels increase. From (12), a decrease in \(\bar{\Delta }_{i+1}\) leads to an increase in the optimal quote for state i. Thus, as \(\lambda _k\) increase so do the optimal quotes.

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Savasaneril, S., Sayin, E. Dynamic lead time quotation under responsive inventory and multiple customer classes. OR Spectrum 39, 95–135 (2017).

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