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Two-agent integrated scheduling of production and distribution operations with fixed departure times

  • * Corresponding author: Dongya Han

    * Corresponding author: Dongya Han 

Cheng was supported in part by The Hong Kong Polytechnic University under the Fung Yiu King-Wing Hang Bank Endowed Professorship in Business Administration. Wu was supported by the Ministry of Science and Technology of Taiwan under grant number MOST 109-2410-H-035-019

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  • We consider integrated scheduling of production and distribution operations associated with two customers (agents). Each customer has a set of orders to be processed on the single production line at a supplier on a competitive basis. The finished orders of the same customer are then packed and delivered to the customer by a third-party logistics (3PL) provider with a limited number of delivery transporters. The number of orders carried in a delivery transporter cannot exceed its delivery capacity. Each transporter incurs a fixed delivery cost regardless of the number of orders it carries, and departs from the 3PL provider to a customer at fixed times. Each customer desires to minimise a certain optimality criterion involving simultaneously the customer service level and the total delivery cost for its orders only. The customer service level for a customer is related to the times when its orders are delivered to it. The problem is to determine a joint schedule of production and distribution to minimise the objective of one customer, while keeping the objective of the other customer at or below a predefined level. Using several optimality criteria to measure the customer service level, we obtain different scenarios that depend on optimality criterion of each customer. For each scenario, we either devise an efficient solution procedure to solve it or demonstrate that such a solution procedure is impossible to exist.

    Mathematics Subject Classification: Primary: 90B35; Secondary: 68M20.

    Citation:

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  • Table 1.  Computational complexity results

    Problem Complexity
    $ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $ SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
    $ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $ PS, $ O({{s}^{2}}{{n}_{A}}n_{B}^{2}n_{\max d}^{A}n_{\max d}^{B}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 4.4
    $ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $ PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 4.5
    $ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}) $, Theorem 7.2
    $ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $ ONP, $ O({{n}_{A}}n_{B}^{2}{{(n_{\max d}^{A})}^{\bar{s}-1}}n_{\max d}^{B}{{P}^{\bar{s}-1}}\min \{{{v}^{B}},{{n}_{B}}\}\log (Q_{u}^{B}-Q_{l}^{B})) $, Theorem 7.2
    $ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
    $ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O({n_A}{n_B}{(n_{\max d}^A)^{\bar s - 1}} $ $ n_{\max d}^B{P^{\bar s - 1}}{P^B}{V^B}) $, Theorem 7.2
    $ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $ Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
    $ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $ ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
    $ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $ ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1
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    Table 2.  Overview of the problem characteristics in recent recent studies on integrate production and distribution

    Article Number of agents Delivery capacity Delivery cost Delivery mode Departure times
    Agnetis et al. [1] One Bounded Yes Non-splittable Fixed
    Hall et al. [7] One Unbounded No Non-splittable Fixed
    Han et al. [9] One Bounded Yes Non-splittable Fixed
    Kovalyov et al. [12] Two Unbounded No Non-splittable Fixed
    Leung and Chen [14] One Bounded No Non-splittable Fixed
    Li et al. [15] One Bounded No Splittable or Non-splittable Fixed
    Melo and Wolsey [18] One Bounded Yes Non-splittable Fixed
    Mor and Mosheiov [19] Two Unbounded No Non-splittable Fixed
    Seddik et al. [23] One Not involve No Not involve Fixed
    Stecke and Zhao [24] One Bounded Yes Splittable or Non-splittable Fixed
    Yin et al. [35] Two Unbounded Yes Non-splittable Fixed
    Yin et al. [37,34] Two Unbounded Yes Non-splittable No
    Our paper Multiple Bounded Yes Non-splittable Yes
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    Table 3.  Computational complexity results

    Problem Complexity
    $ (\sum w_j^AD_j^A+TC^A, \gamma^B\leq V^B) $ SNP, even if there is no capacity constraint on the delivery transporters, Theorems 5.1 and 7.2
    $ (\sum D_j^A+TC^A, f_{\max}^B+TC^B\leq V^B) $ PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d}\min v^B, n_B ) $, Theorem 4.4
    $ (f_{\max}^B+TC^B, \sum D_j^A+TC^A\leq V^A) $ PS, $ O(s^2n_An_B^2n^A_{\max d}n^B_{\max d} $ $ \min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 4.5
    $ (\sum w_j^AD_j^A+TC^A, f_{\max}^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B ) $, Theorem 7.2
    $ (f_{\max}^B+TC^B, \sum w_j^AD_j^A+TC^A\leq V^A)_{s=\overline{s}} $ ONP, $ O(n_An_B^2(n^A_{\max d})^{\overline{s}-1}n^B_{\max d}P^{\overline{s}-1}\min v^B, n_B \log(Q_u^B-Q_l^B)) $, Theorem 7.2
    $ (\sum w_j^AD_j^A+TC^A, \sum D_k^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}V^B) $, Theorem 5.6
    $ (\sum w_j^AD_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B)_{s=\overline{s}} $ ONP, $ O(n_An_B(n^A_{\max d})^{\overline{s}-1} $ $ n^B_{\max d}P^{\overline{s}-1}P^BV^B) $, Theorem 7.2
    $ (\sum D_j^A+TC^A, \sum D_k^B+TC^B\leq V^B) $ Open, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}V^B) $, Theorem 7.2
    $ (\sum D_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $ ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}P^BV^B) $, Theorem 6.5
    $ (\sum w_j^AU_j^A+TC^A, \sum w_k^BU_k^B+TC^B\leq V^B) $ ONP, $ O(s^2n_An_B(n^A_{\max d})n^B_{\max d}PV^B) $, Theorem 7.1
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