Precautionary replenishment in financially-constrained inventory systems subject to credit rollover risk and supply disruption

Abstract

We consider a limited-liability firm that owns a finite single-product inventory subject to periodic-review replenishment and a corporate treasury that mediates the firm’s financial transactions related to inventory operations. The firm may elect to borrow money to purchase product via a revolving line of credit, secured by a compensating balance which determines the credit limit. The line of credit is subject to rollover risk, namely, each period the funding entity may, with some probability, refuse to roll over the line of credit. In response, the firm can search for an alternate funding entity, but in so doing it may incur search costs, primarily in the form of lost sales. The firm optimizes inventory replenishment order sizes and decides whether it should declare bankruptcy, as function of its inventory and available capital. The recent credit crunch has rendered illiquidity a critical concern for funding and operating decisions in enterprises. This paper addresses optimal inventory management in the face of liquidity shocks and supply disruptions. We show that rollover risk and supply disruption are important considerations for firms that rely on external funding. Rollover risk alone results in optimal inventory replenishment policies that differ materially from those specified by traditional supply chain models; differences manifest as state-dependent precautionary replenishment or cash hoarding. Inventory management models which fail to take rollover risk and supply disruption risk into account can prescribe suboptimal replenishment policies. Such policies would generate suboptimal profits for firms that rely on short-term financing to fund their working capital.

Appendices

A Computation of the optimal replenishment policy

In this appendix we describe how to optimize the FCI system replenishment policy by maximizing the attendant expected discounted gains (profits). Refer to Sect. 3 for background and notation.

1.1 A.1 Dynamic programming problem formulation

In this section, we discuss how to optimize the objective function of Eq. (4) over all admissible replenishment policies by computing the optimal policy

$$\begin{aligned} \pi ^{*}(s_0)\in \hbox {arg max}_{\pi \in \varPi }\{J_{\pi }(s_0)\}, \end{aligned}$$

(5)

yielding the optimal objective function value

$$\begin{aligned} J^{*}(s_0)=\max _{\pi \in \varPi }\{J_{\pi }(s_0)\}, \ s_0\in \tilde{{\mathcal {S}}}^{(f)}. \end{aligned}$$

(6)

We discretize the cash state space ${\mathcal {S}}_C$, thereby rendering $\tilde{{\mathcal {S}}}^{(f)}$ a finite state space. In our case, both the inventory size and net cash are discrete and bounded from above and below, while the credit line indicator and supply disruption indicator assume a finite number of values. Consequently, we can use vector representation for an admissible policy, $\pi \in \varPi$, and the corresponding objective function value, $J_{\pi }(s_0)$.

To solve the optimization problem of Eq. (6), we employ the value iteration method (see Bertsekas 2012) with aggregate states, followed by four lookahead steps. To this end, we use the auxiliary state process $\{ \tilde{S}_i^{(f)}:\,i\ge 0\}$, which models the state of the system just prior to applying the replenishment decision. Recalling that the only random components of the system are the Markovian credit line indicator process, the iid demand process and the iid supply disruption indicator process we conclude that the auxiliary state process $\{ \tilde{S}_i^{(f)}:\,i\ge 0\}$ is Markovian. Consequently, we can view the profit/loss, $G_i$, as a function of quantities known at the beginning of period i (the auxiliary state, $s_{i-1}\in \tilde{{\mathcal {S}}}^{(f)}$, the voluntary bankruptcy decision, $\delta _i$, and the replenishment order size, $x_i$) and the randomness pertaining to period i, namely, $\varTheta _i=(D_i,\,L_i,\,\varUpsilon _i)$. To show these dependencies, we define the discounted gains function by $g(s_{i-1},\,\delta _i,\,x_i,\,\theta _i)=\beta g_i$, where $g_i$ is the corresponding realization of $G_i$. Consequently, Eq. (6) becomes

$$\begin{aligned} J^{*}(s_0)=\max _{\pi \in \varPi }\mathbf{E }\left[ \sum _{i=1}^{+\infty }\beta ^{i-1}\ g(\tilde{S}^{(f)},\,\varDelta _i,\,X_i,\,\varTheta _i)\right] , \ s_0\in \tilde{{\mathcal {S}}}^{(f)}. \end{aligned}$$

(7)

Note that in view of our assumptions of finite support of the marginal distributions of demand and credit line indicator, the profit/loss in any given period is bounded. Thus, there exists $M>0$, such that the random profit/loss function satisfies $\left| g(s,\,\delta ,\,x,\,\theta )\right| \le M$ for all admissible arguments of g.

Finally, let $f(s_{i-1},\,\delta _i,\,x_i,\,\theta _i)=s_i\in \tilde{{\mathcal {S}}}^{(f)}$ be the transition function of auxiliary states and observe that for every period, the next state depends only on the previous state, controls, and random shocks, implying that it is time invariant.

1.2 A.2 Dynamic programming problem solution

In this section, we first review the general theoretical results (cf. Puterman 2014) to be used in the numerical optimization algorithm to be presented in the sequel, and then we specialize the aforementioned general theory to our model. Let ${\mathcal {J}}$ be a Banach space (complete normed linear space) of bounded real-valued functions, $J:\tilde{{\mathcal {S}}}^{(f)}\rightarrow \mathbf{R }$, equipped with the supremum norm, $||J||=\hbox {sup}_{s\in \tilde{{\mathcal {S}}}^{(f)}}\left| J(s)\right|$. Consider the following dynamic stochastic optimization problem:

$$\begin{aligned} \max _{\pi \in \varPi }\mathbf{E }\left[ \sum _{i=1}^{+\infty }\beta ^{i-1}\ g(S_{i-1},\,U_i,\,\varTheta _i)\right] , \end{aligned}$$

(8)

where $\varPi$ is the space of all admissible control policies, $\beta \in [0,\,1)$ is a discount factor, $U_i$ is an admissible control, $\varTheta _i$ is the randomness in the system, and $g(S,\,U,\,\varTheta )$ is a bounded reward function. The transition function is denoted by $f(S_{i-1},\,U_i,\,\varTheta _i)=S_i$. Define a mapping $T:{\mathcal {J}}\rightarrow {\mathcal {J}}$ by

$$\begin{aligned} T[J](s)=\max _{u\in U(s)}\mathbf{E }\left[ g(s,\,u,\,\varTheta )+\beta \ J(f(s,\,u,\,\varTheta ))\right] , \end{aligned}$$

(9)

where $J\in {\mathcal {J}}$, $s\in \tilde{{\mathcal {S}}}^{(f)}$, and U(s) is the set of admissible controls at state s. Define further the sequence

$$\begin{aligned} J_{n+1}=T[J_n], \ n\ge 0, \end{aligned}$$

(10)

for some initial $J_0\in {\mathcal {J}}$. The optimization problem solution utilizes the value iteration algorithm, which is based on the following theorem.

Theorem 1

• a. T of Eq. (9) is a contraction mapping in the Banach space ${\mathcal {J}}$ (cf. Proposition 6.2.4 of Puterman 2014), and has a unique fixed point $J^{*}$, that is, $J^{*}$ is the unique solution of the Bellman equation, $T[J^{*}]=J^{*}$.

• b. $J^{*}$ is the unique solution of the dynamic optimization problem (8), referred to as the value function (cf. Theorem 6.2.5 of Puterman 2014).

• c. The value function can be computed asymptotically as $\displaystyle J^{*}=\lim _{n\rightarrow \infty } T^n[J]\in {\mathcal {J}}$, where convergence is in the supremum norm of ${\mathcal {J}}$ (cf. Theorem 6.3.1 of Puterman 2014).

Next, we specialize the discussion to the dynamic optimization problem (7) of this paper, and show that it satisfies the assumptions of Theorem 1. In our case, ${\mathcal {J}}$ is a Banach space with the supremum norm and elements of the form $J:{\mathcal {J}}\rightarrow \mathbf{R }$, while the mapping $T:{\mathcal {J}}\rightarrow {\mathcal {J}}$ becomes

$$\begin{aligned} T[J](s)=\max _{(\delta ,\,x)\in \{-1,\,1\}\times U(s)}\mathbf{E }\left[ g(s,\,\delta ,\,x,\,\varTheta )+\beta \ J(f(s,\,\delta ,\,x,\,\varTheta ))\right] , \end{aligned}$$

(11)

where $J\in {\mathcal {J}}$, $s\in \tilde{{\mathcal {S}}}^{(f)}$, $\delta$ represents a decision whether or not to declare bankruptcy, and U(s) is the space of admissible sizes of inventory replenishments at auxiliary state $s\in \tilde{{\mathcal {S}}}^{(f)}$. From Theorem 1, the Bellman equation characterizes the unique solution of the optimization problem of Eq. (7) and its corresponding value function is given by $J^{*}(s)$, $s\in \tilde{{\mathcal {S}}}^{(f)}$.

Recall that the rules for payment of dividends and state transition probabilities are time-invariant, that is, they do not depend on the time period index (decision epoch), and that all relevant information is summarized in the state $s\in \tilde{{\mathcal {S}}}^{(f)}$. Moreover, the space of admissible controls U(s) is finite by virtue of the financial and inventory constraints. Combining these facts with the boundedness of dividends over periods, discounting $(0\le \beta <1)$, and discreteness of the state space $\tilde{{\mathcal {S}}}^{(f)}$, it follows that the unique optimal policy, $\pi ^{*}(s)$, is deterministic, time-invariant, and specifies a control that depends only on the current state of the system, $s\in \tilde{{\mathcal {S}}}^{(f)}$ (cf. Theorems 6.2.5 and 6.2.10 of Puterman 2014).

To compute numerical approximations of the optimal policy, $\pi ^{*}(s)$, and value function, $J^{*}(s)$, with prescribed accuracy $\varepsilon >0$, we follow the algorithm in Puterman (2014) which is based on the results summarized in Theorem 1 (cf. Puterman 2014, Theorem 6.3.1). More precisely, a policy, $\pi ^{\varepsilon ^{*}}(s)$, is $\varepsilon$-optimal if its corresponding value function, $J^{\varepsilon ^{*}}(s)$, satisfies $J^{\varepsilon ^{*}}(s)>J^{*}(s)-\varepsilon$ for all $s\in \tilde{{\mathcal {S}}}^{(f)}$ (cf. Puterman 2014).

The algorithm for computing an $\varepsilon$-optimal policy, $\pi ^{\varepsilon ^{*}}(s)$, for the optimization problem of Eq. (6) is given by the following algorithm:

• 1. Initialization Select the precision level, $\varepsilon >0$, and an initial approximation $J_0(s)$, $s\in {\mathcal {S}}^{(f)}$. Since the choice is arbitrary, we used $J_0(s)=0$ for all $s\in \tilde{{\mathcal {S}}}^{(f)}$.

• 2. Iteration At iteration $k\ge 1$, apply the mapping T to $J_{k-1}$ recursively, yielding $J_k=T[J_{k-1}]$.

• 3. Termination If $\displaystyle \hbox {sup}_{s\in {\mathcal {S}}^{(f)}}|J_k(s)-J_{k-1}(s)|<\frac{\varepsilon (1-\beta )}{2\beta }$, then terminate the algorithm and set

  $$\begin{aligned} \pi ^{\varepsilon ^{*}}(s)=\hbox {arg max}_{(\delta ,\,x)\in \{-1,\,1\}\times U(s)}\mathbf{E }\left[ g(s,\,\delta ,\,x,\,\varTheta )+\beta \ J_k(f(s,\,\delta ,\,x,\,\varTheta ))\right] , \ s\in \tilde{{\mathcal {S}}}^{(f)}. \end{aligned}$$

  Otherwise, go to step 2.

The algorithm is guaranteed to converge in a finite number of iterations, $k^{*}$, and the corresponding value function, $J^{\varepsilon ^{*}}(s)=J_{k^{*}+1}(s)$, satisfies $\displaystyle \sup _{s\in \tilde{{\mathcal {S}}}^{(f)}}|J^{\varepsilon ^{*}}(s)-J^{*}(s)|<\frac{\varepsilon }{2}$, where $J^{*}(s)$ is the exact value function of the problem (6) (cf. Theorem 6.3.1 of Puterman 2014).

1.3 A.3 Value iteration with aggregation and look-ahead

Consider the optimization problem of Eq. (8), to be referred to as the original problem, with transition matrix $P_{\delta ,\,x}$. Since the computational complexity of the original problem grows exponentially in the state space cardinality, we mitigate the rising complexity by applying value iteration with aggregation (Bertsekas 2012). To this end, we define an associated aggregate problem with an aggregate state space,

$$\begin{aligned} {\mathcal {A}}^{(f)}={\mathcal {A}}_1^{(f)}\times {\mathcal {A}}_2^{(f)}\times \cdots \times {\mathcal {A}}_m^{(f)}\subseteq \tilde{{\mathcal {S}}}^{(f)}=\tilde{{\mathcal {S}}}_1^{(f)}\times \tilde{{\mathcal {S}}}_2^{(f)}\times \cdots \times \tilde{{\mathcal {S}}}_m^{(f)}. \end{aligned}$$

Let $\tilde{P}_{\delta ,\,x}$ be the reduced transition matrix obtained by removing all rows in $P_{\delta ,\,x}$ associated with states in $\tilde{{\mathcal {S}}}^{(f)}\backslash {\mathcal {A}}^{(f)}$. Our goal is to construct the aggregate transition matrix $W_{\delta ,\,x}=\tilde{P}_{\delta ,\,x} A$, where A is the so-called aggregation probability matrix representing transition probabilities from states $s=(s_1,\,s_2,\ldots ,\,s_m)\in \tilde{{\mathcal {S}}}^{(f)}$ to states $a=(a_1,\,a_2,\ldots ,\,a_m)\in {\mathcal {A}}^{(f)}$ to be defined next. Let ${\mathcal {N}}(s)={\mathcal {N}}_1(s_1)\times {\mathcal {N}}_2(s_2)\times \cdots \times {\mathcal {N}}_m(s_m)\subseteq {\mathcal {A}}^{(f)}$ be a neighborhood of s consisting of aggregate states only, such that for every $i\in \{1,\ldots ,\,m\}$, ${\mathcal {N}}_i(s_i)=\{{\underline{a}}_i\}\cup \{{\overline{a}}_i\}$, where ${\underline{a}}_i=\max \{a_i\in {\mathcal {A}}_i^{(f)}:\,a_i\le s_i\}$ and ${\overline{a}}_i=\min \{a_i\in {\mathcal {A}}_i^{(f)}:\,a_i\ge s_i\}$. Note that at most one subset of ${\mathcal {N}}_i(s_i)$ may be empty. We are now in a position to define the aggregation probability matrix A with elements $A(s,\,a)$ of the form

$$\begin{aligned} A(s,\,a)=\left\{ \begin{array}{ll} p_1(s_1,\,a_1)\ p_2(s_2,\,a_2)\ \ldots \ p_m(s_m,\,a_m),&{}\quad \hbox {if } a\in {\mathcal {N}}(s)\\ 0,&{}\quad \hbox {if } a\in {\mathcal {A}}^{(f)}\backslash {\mathcal {N}}(s) \end{array} \right. \end{aligned}$$

(12)

where for every $1\le i\le m$,

$$\begin{aligned} p_i(s_i,\,a_i)=\left\{ \begin{array}{ll} \displaystyle \frac{s_i-{\underline{a}}_i}{{\overline{a}}_i-{\underline{a}}_i},&{}\quad \hbox {if } \min \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\}<s_i<\max \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\}\hbox { and } a_i={\overline{a}}_i\\ \displaystyle 1-\frac{s_i-{\underline{a}}_i}{{\overline{a}}_i-{\underline{a}}_i},&{}\quad \hbox {if }\min \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\}<s_i<\max \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\} \hbox { and }a_i={\underline{a}}_i\\ 1, &{}\quad \hbox {if } s_i\ge \max \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\} \hbox { or } s_i\le \min \{a_j:\,a_j\in {\mathcal {A}}_i^{(f)}\} \end{array} \right. \end{aligned}$$

(13)

The solution to the original optimization problem via its aggregate problem counterpart proceeds as follows:

• 1. Apply the value iteration algorithm described in Sect. 1.2 to the aggregate problem using the Bellman equation,

  $$\begin{aligned} J^{*}_{{\mathcal {A}}}\left( a\right) =\max _{(\delta ,\,x)\in \{-1,\,1\}\times U(a)}\{\tilde{P}_{\delta ,\,x}\left( a\right) \ g_{\delta ,\,x}\left( a\right) +\beta \ W_{\delta ,\,x}\left( a\right) \ J^{*}_{{\mathcal {A}}}\}, \ a \in {\mathcal {A}}^{(f)}, \end{aligned}$$

  where $g_{\delta ,\,x}\left( a\right)$ is the vector of possible discounted gains over a period given a control pair $(\delta ,\,x)$ and initial aggregate state $a\in {\mathcal {A}}^{(f)}$; $J^{*}_{{\mathcal {A}}}$ is the vector representing the value function of the aggregate problem with elements $J^{*}_{{\mathcal {A}}}\left( a\right)$, $a \in {\mathcal {A}}^{(f)}$; $\tilde{P}_{\delta ,\,x}\left( a\right)$ and $W_{\delta ,\,x}\left( a\right)$ are the rows of the corresponding matrices, $\tilde{P}_{\delta ,\,x}$ and $W_{\delta ,\,x}$, respectively, which contain the transition probabilities from state $a\in {\mathcal {A}}^{(f)}$.

• 2. Use the aggregation probabilities of Eq. (12) to obtain an estimate $J^{*}_{\mathcal {S}}$ of the value function for the original problem, where

  $$\begin{aligned} J^{*}_{{\mathcal {S}}}=A\ J^{*}_{{\mathcal {A}}} \end{aligned}$$

• 3. Compute a sequence of value functions, $\{J^{*}_l:\,l\ge 0\}$, where $J^{*}_0=J^{*}_{\mathcal {S}}$, and for $l>0$, apply recursively 4 successive lookahead iterations of the form

  $$\begin{aligned} J^{*}_{l+1}\left( s\right) =\max _{(\delta ,\,x)\in \{-1,\,1\}\times U(s)}\{P_{\delta ,\,x}\left( s\right) \ g_{\delta ,\,x}\left( s\right) +\beta \ P_{\delta ,\,x}\left( s\right) \ J^{*}_l\}, \ s \in \tilde{{\mathcal {S}}}^{(f)}, \end{aligned}$$

  where $g_{\delta ,\,x}\left( s\right)$ is the vector of possible discounted gains over a period given a control pair $(\delta ,\,x)$ and initial aggregate state $s\in \tilde{{\mathcal {S}}}^{(f)}$; $J^{*}_{l}$ is the vector representing the l-step lookahead estimate of the value function of the original problem with elements $J^{*}_{l}\left( s\right)$, $s \in \tilde{{\mathcal {S}}}^{(f)}$; $P_{\delta ,\,x}\left( s\right)$ is the row of $P_{\delta ,\,x}$ which contains the transition probabilities from state $s\in \tilde{{\mathcal {S}}}^{(f)}$. For each iteration, the corresponding optimal policy is computed by

  $$\begin{aligned} \pi _l^{\varepsilon ^{*}}(s)=\hbox {arg max}_{(\delta ,\,x)\in \{-1,\,1\}\times U(s)}\{P_{\delta ,\,x}\left( s\right) \ g_{\delta ,\,x}\left( s\right) +\beta \ P_{\delta ,\,x}\left( s\right) \ J^{*}_l\}, \ s \in \tilde{{\mathcal {S}}}^{(f)}. \end{aligned}$$

The numerical implementation of the above algorithm performs the required operations in an efficient manner, both in terms of computation time and memory requirements. More specifically, it exploits the sparseness of all matrices involved and computes dynamically only the required elements for each operation.

To further reduce computational complexity, a 2-stage algorithm is employed. In the first stage, the aggregate value iteration algorithm is used with the prescribed accuracy parameter $\varepsilon =0.25$ and with the proviso that no policy change occurs for a series of at least 3 consecutive iterations; experience shows that this rule is sufficient for policy convergence in our case. Once the policy is deemed to have converged by this rule, 4 lookahead iterations are performed, as described above. Finally, in the second stage, to further increase the precision of the value function estimate, we carry out policy iterations with the accuracy parameter set to $\varepsilon =0.05$, which results in good approximations in the problems to be considered.

B Parametrization

In this appendix, we list the parameter values used in the numerical study of Sect. 4 of the main paper. The demand distribution is assumed uniform with support $\{2,\,4,\ldots ,\,50\}$ and time-invariant across periods. The credit line indicator process, $\{L_i:\,i\ge 1\}$, is Markovian over the state space $\{0,\,1\}$, where state 0 means that the firm has access to a credit line and state 1 means that the firm does not have access to a credit line (recall that the size of the credit line is endogenously determined). The transition probabilities of $\{L_i:\,i\ge 1\}$ are $p_{0,\,0}=0.5$, $p_{0,\,1}=0.5$, $p_{1,\,0}=0.25$, and $p_{1,\,1}=0.75$, where the asymmetry in the transition probabilities reflects the value of having an ongoing relationship with the funding entity. Throughout this appendix we focus on an underlying FCI system with initial states endowed with a line of credit (that is, Pr$\{L_1=1\}=1$), and we set the basis for establishing the initial credit limit to $G_0=2.5$. The remaining parameter values are listed below:

• Borrowing interest rate, charged by the revolving line of credit: $r_b=2.5\%$.

• Earned interest rate, paid on the firm’s cash on hand: $r_e=0.0\%$.

• Compensating balance: $b_c=3.0$.

• Sensitivity of credit to profits: $\zeta =0.5$.

• Leverage-related parameter: ${\overline{l}}=5.0$.

• Discount factor: $\beta =0.975$.

• Unit price of sold product: $\psi =1.00$.

• Unit cost of ordered product: $\gamma =0.5$.

• Fixed cost: $K=2.5$.

• Inventory holding cost: $h=0.25$.

• Backordering penalty: $g=0.375$.

• Lost-sales penalty: $z=0.5$.

• Forced-sale unit price: $\varphi = 0.375$.

• Dividend threshold: ${\overline{C}}=20$.

• Maximal inventory level (capacity): ${\overline{I}}=50$.

• Maximal backorder level: ${\overline{B}}=10$.

• Supply disruption probability: $p_d=0$.

C Glossary of notation

This Appendix contains a glossary of key notation used in this paper.

• $\beta$: discount factor corresponding to the required rate of return.

• $\gamma$: unit cost of ordered product.

• $\varphi$: forced sale price per unit inventory.

• $\psi$: unit price of sold product.

• $\zeta$: sensitivity of the credit limit to profits.

• $b_c$: compensating balance.

• ${\overline{B}}$: maximal backorder level beyond which demand is lost.

• ${\overline{C}}$: dividend threshold characterizing the dividend policy.

• g: backordering penalty per unit inventory backordered per unit time.

• h: inventory holding cost per unit inventory per unit time.

• K: fixed cost of supporting the firm’s operating activity.

• ${\overline{l}}$: leverage-related parameter.

• $r_b$: borrowing interest rate charged by the revolving line of credit.

• $r_e$: earned interest rate paid to the firm on its cash on hand.

• $s_B$: bankruptcy state.

• ${\overline{I}}$: maximal inventory level the firm is allowed to hold.

• $p_d$: probability of a supply chain disruption.

Processes

• $\{\varDelta _i:\,i\ge 1\}$: voluntary bankruptcy decision process.

• $\{\varUpsilon _i:\,i\ge 0\}$: supply disruption indicator process.

• $\{C_i^{(d)}:\,i\ge 1\}$: intermediate cash process (after costs of replenishment, fixed costs, revenues, backorder penalties, holding costs (if any), interest earned/owed).

• $\{C_i^{(s)}:\,i\ge 1\}$: intermediate cash process (after proceeds from forced sale of inventory, if any).

• $\{C_i:\,i\ge 0\}$: net cash process (after paying out dividends, if any).

• $\{D_i:\,i\ge 1\}$: demand-size process.

• $\{I_i^{(d)}:\,i\ge 1\}$: intermediate inventory process (after replenishment and demand).

• $\{I_i:\,i\ge 0\}$: inventory process.

• $\{G_i:\,i\ge 1\}$: period gains process.

• $\{L_i:\,i\ge 1\}$: credit line indicator process.

• $\{Q_i:\,i\ge 1\}$: credit limit multiplier process.

• $\{R_i:\,i\ge 1\}$: sales revenue process.

• $\{V_i:\,i\ge 1\}$: dividend process.

• $\{X_i:\,i\ge 1\}$: inventory order process.

• $\{X_i^{(r)}:\,i\ge 1\}$: received inventory process.

• $\{Z_i:\,i\ge 1\}$: disrupted inventory order process.

State spaces

• ${\mathcal {A}}:$ action state space with elements $a=(\delta ,\,x)$.

• ${\mathcal {S}}_C$: state space of the cash process.

• ${\mathcal {S}}_D$: state space of the demand process.

• ${\mathcal {S}}_I$: state space of the inventory process.

• ${\mathcal {S}}_L$: state space of the credit line indicator process.

• ${\mathcal {S}}_Q$: state space of the credit limit multiplier process.

• ${\mathcal {S}}_G$: state space of the period gains process.

• ${\mathcal {S}}_Z$: state space of the disrupted order inventory process.

• ${\mathcal {S}}=({\mathcal {S}}_I,\,{\mathcal {S}}_C,\,{\mathcal {S}}_D,\,{\mathcal {S}}_L,\,{\mathcal {S}}_Q)\cup \{s_B\}$: full state space.

• ${\mathcal {S}}^{(f)}=({\mathcal {S}}_I,\,{\mathcal {S}}_C,\,{\mathcal {S}}_L,\,{\mathcal {S}}_G)\cup \{s_B\}$: auxiliary state space of the FCI model with rollover risk.

• $\tilde{{\mathcal {S}}}^{(f)}=({\mathcal {S}}_I,\,{\mathcal {S}}_C,\,{\mathcal {S}}_L,\,{\mathcal {S}}_G,\,{\mathcal {S}}_Z)\cup \{s_B\}$: auxiliary state space of the FCI model with rollover risk and supply disruptions.

Functions

• $\pi ^{*}_B(s)$: optimal benchmark policy.

• $\pi ^{*}_R(s)$: optimal rollover policy.

• $J^{*}_B(i,\,c)$: profits generated by the optimal benchmark policy.

• $J^{*}_R(i,\,c)$: profits generated by the optimal rollover policy.