Abstract
The development and application of inventory models for deteriorating items is one of the main concerns of subject matter experts. The inventory models developed in this field have focused mainly on supply chains under the assumption of constant lead time. In this study, we develop an inventory model for a main class of deteriorating items, namely perishable products, under stochastic lead time assumption. The inventory system is modeled as a continuous review system (r, Q). Demand rate per unit time is assumed to be constant over an infinite planning horizon and the shortages could be backordered completely. For modeling the deterioration process, a non-linear holding cost is considered. Taking into account the stochastic lead time as well as a non-linear holding cost makes the mathematical model more complicated. We customize the proposed model for a uniform distribution function that could be tractable to solve optimally by means of an exact approach. We then solve an example taken from the literature to demonstrate the applicability and effectiveness of the proposed model. Finally, by doing several sensitivity analyses for the key parameters of the model, some managerial insights are proposed.
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Appendix
Appendix
Proof of Lemma 1
\( {\hbox{HU}}{{\hbox{T}}_{{11}}} + {\hbox{HU}}{{\hbox{T}}_{{21}}} + {\hbox{BUT}} + {\hbox{OUT}} \) is as follows when we apply Eqs. (8), (11), (12) and (14):
In Eq. (17), it can be seen that HUT11 + HUT21 + BUT + OUT is a constant in terms of Q, indicated by C, multiplied in \( \frac{D}{Q} \). C is positive since H 11, H 21, B and O c are positive.
Consequently, \( \frac{{\partial \left( {{\text{HU}}{{{\text{T}}}_{{11}}} + {\text{HU}}{{{\text{T}}}_{{21}}} + {\text{BUT}} + {\text{OUT}}} \right)}}{{\partial Q}} = - \frac{{DC}}{{{{Q}^{2}}}} < 0 \)and \( \frac{{{{\partial }^{2}}(HU{{T}_{{11}}} + HU{{T}_{{21}}} + BUT + OUT)}}{{\partial {{Q}^{2}}}} = \frac{{DC}}{{{{Q}^{3}}}} > 0 \) which imply that\( {\hbox{ HU}}{{\hbox{T}}_{{11}}} + {\hbox{HU}}{{\hbox{T}}_{{21}}} + {\hbox{BUT}} + {\hbox{OUT}} \) is strictly decreasing and convex in terms of Q, respectively.
Proof of Lemma 2
According to Eqs. (8) and (13), HUT T2 is as follows:
So, the first differentiation of HUT T2 is derived as Eq. (19):
In Eq. (19) M 1, M 2, and N 1 to N 6 is utilized for the corresponding statements in order to simplify. Because Q ≥ Db ≥ Da then Q-Da ≥ Q-Db ≥ 0. So, we have M 1 ≥ M 2, N 1 ≥ N 2, N 3 ≥ N 4 and N 5 ≥ N 6. In other words M 1 -M 2 ≥ 0 and {N 1 − N 2} + {N 3 − N 4} + {N 5 − N 6} ≥ 0. Moreover, since Q-Da ≥ Q-Db then N 1 -N 2 ≥ M 1 -M 2. Thus {N 1 − N 2} + {N 3 − N 4} + {N 5 − N 6} ≥ M 1 − M 2. Consequently, \( \frac{{\partial {\text{HU}}{{\text{T}}_{{T2}}}}}{{\partial Q}} \geqslant 0 \) that implies HUT T2 is strictly increasing. In a similar way, it can be proven that \( \frac{{{{\partial }^2}{\text{HU}}{{\text{T}}_{{T2}}}}}{{\partial {{Q}^2}}} \geqslant 0 \).
Proof of Lemma 3
In order to prove that HUT11 + HUT21 + HUT T2 is strictly an increasing convex function in terms of r, we prove that its components are strictly increasing and convex.
According to Assumption 8 that implies r > Da, it is found that the first and second differentiation of HUT11 in terms of r is positive:
Also, HUT T2 is a linear (convex and concave) function in terms of r with a positive slope. Regarding Eq. (18), it is derived that:
Equation (22) is positive according to assumption Da ≤ Db ≤ Q.
Concerning HUT21 we have:
In Eq. (23), if and only if \( r(\gamma + 2) \leqslant \left( {\gamma + 1} \right)bD \) (condition I), then \( \frac{{\partial HU{{T}_{{21}}}}}{{\partial r}} \) is positive. Moreover, (24) is positive if and only if \( (\gamma + 2)r \leqslant \) \( \gamma bD \) (condition II). By comparing conditions I and II, it can be concluded that condition II is dominant. As a result, if and only if \( (\gamma + 2)r \leqslant \) \( \gamma bD \), then HUT11 + HUT21 + HUT T2 is a strictly increasing convex function in terms of r.
Proof of Lemma 4
Regarding Eq. (14), it can easily be found that:
Equations (25) and (26) mean that BUT is a decreasing convex function in terms of r. Moreover, since OUT is constant in terms of r, adding it to BUT does not change this result.
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Sazvar, Z., Baboli, A. & Akbari Jokar, M.R. A replenishment policy for perishable products with non-linear holding cost under stochastic supply lead time. Int J Adv Manuf Technol 64, 1087–1098 (2013). https://doi.org/10.1007/s00170-012-4042-2
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DOI: https://doi.org/10.1007/s00170-012-4042-2