Cost optimization in the $(S-1, S)$ backorder inventory model with two demand classes and rationing

Abstract

Within the framework of continuous-review $(S-1, S)$ inventory systems with rationing and backorders, there are two streams of studies in the literature that involve optimization models. In the first stream, service level optimizations are studied for which exact optimization routines are provided. The second stream of studies involves cost optimization models, which relies on optimizing approximate cost models rather than the original cost model. Our main contribution in this study is to fill this research gap by providing a computationally efficient and exact optimization algorithm for determining the optimal policy parameters which minimizes the expected cost rate per unit time. One important aspect of our method is that, as the base-stock level is increased by 1 as the iteration continues, the steady-state probabilities need to be calculated only once in our optimization routine (for which the rationing level equals to zero). For the given base-stock level, the cost measures of all other policy parameters can be computed immediately through the knowledge of the probabilities computed in previous iterations. This result significantly reduces the computational complexity of the optimization routine. In the numerical study section, we show the efficiency of the proposed optimization routine under varying system parameters. We also compare the performance of our approach with the existing heuristic in the literature and show that savings up to $34.75 \%$ can be achieved.

Appendices

Appendix: A Proof of Lemma 1

$$\begin{aligned} \varphi _{h+k} (\Delta +k, k)= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta +k, k)} \\ \left( \Delta +k-r+b_{n}\right) ^{+}=h+k \end{array}}\pi _{(r,b_{n})}(\Delta +k, k), \text { by } (4) \\&\\= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta , 0)} \\ \left( \Delta +k-r+b_{n}\right) ^{+}=h+k \end{array}} \pi _{(r,b_{n})}(\Delta , 0), \text { due to Proposition } 1 \\&\\= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta , 0)} \\ \left( \Delta -r+b_{n}\right) ^{+}=h \end{array}} \pi _{(r,b_{n})}(\Delta , 0) \\&\\= & {} \varphi _{h}(\Delta , 0). \end{aligned}$$

$\square$

Appendix: B Proof of Lemma 3

$$\begin{aligned} \varphi _{h}(\Delta +k, k)= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta +k, k)} \\ \left( \Delta +k-r+b_{n}\right) ^{+}=h \end{array}}\pi _{(r,b_{n})}(\Delta +k, k) \\&\\= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta , 0)} \\ \left( \Delta +k-r+b_{n}\right) ^{+}=h \end{array}} \pi _{(r,b_{n})}(\Delta , 0), \text { due to Proposition } 1 \\&\\= & {} \sum _{\begin{array}{c} \left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta , 0)} \\ \left( \Delta -r+b_{n}\right) ^{+}=0 \\ \left( r-b_{n}-\Delta \right) ^{+}=k-h \end{array}}\pi _{(r,b_{n})}(\Delta , 0) \\&\\= & {} \psi _{k-h}(\Delta , 0). \end{aligned}$$

$\square$

Appendix: C Proof of Proposition 2

(12) follows directly from (6). The monotonicity relation in (13) has been established by Vicil and Jackson (2016) (Proposition 2).

Beginning from a regeneration point in which no orders are outstanding, let $(m,T_{m},E_{m})$ describe the $m^{th}$ event in the system: $T_{m}$ is the time of the $m^{th}$ event, and $E_{m}$ is the type of event where $E_{m}\in \{``v",``c",``n"\}$ representing events "delivery", "critical class demand", and "non-critical class demand", respectively. Let $R_{m}$ denote the number of units in resupply after the $m^{th}$ event, $B_{n,m}$ denote the number of non-critical class backorders after the $m^{th}$ event, $B_{c,m}$ denote the number of critical class backorders after the $m^{th}$ event, and $OH_{m}$ denote the physical stock after the $m^{th}$ event. Let us consider two systems with identical event sequences $\left\{ (m,T_{m},E_{m}); m=1,2,3,...\right\}$. In the first system the policy parameters are $(S,S_{c})$ and in the second system they are $(S,S_{c}^{\,\prime })$, $S_{c}^{\, \prime }>S_{c}$. In the proof of Proposition 2 in Vicil and Jackson (2016), it is shown that $R^{\, \prime }_{m}=R_{m}$, and $B^{\, \prime }_{n,m}\ge B_{n,m}$ for all m. If we consider (2) and (3), then we have $OH^{\, \prime }_{m}\ge OH_{m}$ and $B^{\, \prime }_{c,m}\le B_{c,m}$ for all m . Since these are also true for all sample paths, they also hold for the expected values. Hence, the results ${\mathbf {B}}_{n}(S,S_{c}+1) \ge {\mathbf {B}}_{n}(S,S_{c})$, ${\mathbf {B}}_{c}(S,S_{c}+1) \le {\mathbf {B}}_{n}(S,S_{c})$ and ${\mathbf {I}} (S,S_{c}+1) \ge {\mathbf {I}} (S,S_{c})$ follow immediately. $\square$

Appendix: D Proof of Theorem 1

Invariance of the non-critical class fill rate in (17) is the direct result of (6). (18) is the result of Lemma 3.2 in Vicil and Jackson (2018).

$$\begin{aligned} {\mathbf {B}}_{n}(\Delta +k,k)= & {} \sum _{b_n=1}^{\infty } b_n \sum _{\left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta +k,k)}}\pi _{(r,b_{n})}(\Delta +k,k), \,\mathrm {by}\, (8) \\= & {} \sum _{b_n=1}^{\infty } b_n \sum _{\left( r,b_{n}\right) \in {\mathbb {F}}_{(\Delta ,0)}}\pi _{(r,b_{n})}(\Delta ,0), \,\text {due to Proposition}\, 1 \\= & {} {\mathbf {B}}_{n}(\Delta ,0). \\ {\mathbf {B}}_{c}(\Delta +k, k)= & {} \sum _{u=1}^{\infty } u \, \psi _{u}(\Delta +k, k), \text {by } (9) \\= & {} \sum _{u=1}^{\infty } u \, \psi _{u+k}(\Delta , 0), \text { by Lemma}\, 2 \\= & {} \sum _{u=k+1}^{\infty }(u-k) \, \psi _{u}(\Delta , 0), \text {by change of variables.}\\ {\mathbf {I}}(\Delta +k,k)= & {} \sum _{h=1}^{\Delta +k}h \, \varphi _{h}(\Delta +k,k), \text { by } (10) \\= & {} \sum _{h=k+1}^{\Delta +k}h \, \varphi _{h}(\Delta +k,k) \, + \, \sum _{h=1}^{k}h \, \varphi _{h}(\Delta +k,k) \\= & {} \sum _{h=1}^{\Delta } (k+h) \, \varphi _{h+k}(\Delta +k,k) \, + \, \sum _{h=1}^{k}h \, \varphi _{h}(\Delta +k,k), \text {by change of variables} \\= & {} \sum _{h=1}^{\Delta }(k+h) \, \varphi _{h}(\Delta ,0) \, + \, \sum _{h=1}^{k}h \, \varphi _{h}(\Delta +k,k), \text { by Lemma}\, 1 \\= & {} \sum _{h=1}^{\Delta }(k+h) \, \varphi _{h}(\Delta ,0) \, + \, \sum _{h=1}^{k}h \, \psi _{k-h}(\Delta ,0), \text { by Lemma}\, 3 \\= & {} \sum _{h=1}^{\Delta }(k+h) \, \varphi _{h}(\Delta ,0) \, + \, \sum _{u=1}^{k} u \, \psi _{k-u}(\Delta ,0), \text { by change of variables}. \end{aligned}$$

$\square$

Appendix: E Proof of Lemma 4

Due to Proposition 2, $\rho _{n}\lambda _{n}(1-\beta _{n}(S,S_{c}))$ is monotonically increasing in $S_{c}$, and ${\mathbf {B}}_{n}(S,S_{c})$ and ${\mathbf {I}}(S,S_{c})$ are nondecreasing in $S_{c}$. $\square$

Appendix: F Proof of Proposition 4

The following relations hold for any $(S,S_{c})$ pair:

$$\begin{aligned} C(S,S_{c})= & {} \rho _{n}\lambda _{n}(1-\beta _{n}(S,S_{c}))+\rho _{c}\lambda _{c}(1-\beta _{c}(S,S_{c})) + \omega _{n} \; {\mathbf {B}}_{n}(S,S_{c}) \nonumber \\&+ \omega _{c} \; {\mathbf {B}}_{c}(S,S_{c}) + h \; {\mathbf {I}}(S,S_{c}) \nonumber \\\ge & {} \omega _{n} \; {\mathbf {B}}_{n}(S,S_{c}) + \omega _{c} \; {\mathbf {B}}_{c}(S,S_{c}) + h \; {\mathbf {I}}(S,S_{c}) \nonumber \\= & {} \left( \omega _{n} + h \right) \; {\mathbf {B}}_{n}(S,S_{c}) + \left( \omega _{c} + h \right) \; {\mathbf {B}}_{c}(S,S_{c}) + h \; \Big \{ {\mathbf {I}}(S,S_{c}) - {\mathbf {B}}_{n}(S,S_{c}) - {\mathbf {B}}_{c}(S,S_{c}) \Big \} \nonumber \\\ge & {} h \; \Big \{ {\mathbf {I}}(S,S_{c}) - {\mathbf {B}}_{n}(S,S_{c}) - {\mathbf {B}}_{c}(S,S_{c}) \Big \} \end{aligned}$$

(F.1)

$$\begin{aligned}= & {} h \; \left( S-{\mathbf {R}}(S,S_{c}) \right) \end{aligned}$$

(F.2)

$$\begin{aligned}= & {} h \; \left( S-\lambda T \right) . \end{aligned}$$

(F.3)

(F.2) follows from (F.1) due to (1), which also holds for the expected values. (F.3) follows from (F.2) due to (11). Furthermore, for given S the relation $C(S,S_{c})\ge h \left( S-\lambda T \right)$ holds for all $S_{c}$ because RHS of (F.3) is independent of $S_{c}$. Hence, $C^{*}(S)\ge h \left( S-\lambda T \right)$ should hold by definition. $\square$