Minimizing total weighted completion and batch delivery times with machine deterioration and learning effect: a case study from wax production

Abstract

This paper investigates an integrated scheduling of production and distribution activities in the supply chain where both machine deterioration and learning effects have been consequently addressed. Manufacturer aims to minimize the total weighted completion time, while a distributor focuses on reducing shipping times with batch delivery by using capacitated vehicles. The aim of this problem is to minimize the sum of weighted completion times plus total delivery times. First, a mixed integer linear programming model is proposed. Then for a special case, a branch and bound algorithm is developed with utilizing the structural features of the problem. In order to solve large-scale instances of the general problem in a short/reasonable time, a simulated annealing algorithm is provided. Computational results show that the proposed heuristic techniques have high efficiency to achieve the optimal solution, and that they are useful to solve large sizes of the problems at a short time. Finally, by providing a real-life case of wax manufacturing and its distribution system, it is shown that the application of integrated decisions can significantly reduce costs imposed on the firms.

Appendices

Appendix 1

This Proposition can be proved through exchanging the processing positions of two jobs i and j. Consider the sequence s where job j immediate precedes job i for processing at position r and $\frac{{\hat{p}_{i} }}{{w_{i} }} \le \frac{{\hat{p}_{j} }}{{w_{j} }}$. Under the agreeable ratio assumption, it is clear that $p_{i} \le p_{j}$ and $w_{i} \ge w_{j}$, Two states occur for jobs i and j:

State I Both jobs belong to a batch. We know that completion times for all jobs placed in a single batch are the same and equal to the completion time of latest job processed for that batch. In this case, the value of completion time for job i is given as below:

$$C_{i}^{s} = t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) + \hat{p}_{i} \left( {r + 1} \right)^{\beta } + \alpha \left( {r + 1} \right)^{\beta } r$$

where t is the completion time of a job scheduled before these two jobs. Like the sequence s, now, consider the sequence $s^{\prime}$ for the same batch where i is exchanged by j. So, the completion time for j which follows i for processing is given as below:

$$C_{j}^{{s^{\prime}}} = t + \hat{p}_{i} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) + \hat{p}_{j} \left( {r + 1} \right)^{\beta } + \alpha \left( {r + 1} \right)^{\beta } r$$

Now, we can calculate the value of $C_{i}^{s} - C_{j}^{{s^{\prime}}}$, since $r^{\beta } - \left( {r + 1} \right)^{\beta } > 0$:

$$\begin{aligned} C_{i}^{s} - C_{j}^{{s^{\prime}}} = \left( {\hat{p}_{j} - \hat{p}_{i} } \right)r^{\beta } + \left( {\hat{p}_{i} - \hat{p}_{j} } \right)\left( {r + 1} \right)^{\beta } \hfill \\ \, = \left( {\hat{p}_{j} - \hat{p}_{i} } \right) \times \left( {r^{\beta } - \left( {r + 1} \right)^{\beta } } \right) \ge 0 \hfill \\ \end{aligned}$$

It can be concluded that the sequence $s^{\prime}$ have priority over the sequence s. It should be noted that without considering learning effect and deterioration, the different sequences in each batch have no effect on completion time of the batch.

State II The jobs belong to different batches, i.e. the border of two consecutive batches. Suppose that in sequence s, jobs j and i belong to batch A and B, respectively. Also, it is assumed that by exchanging position of two jobs, the vehicle capacity constraint is not violated. In this case, the value of completion times for batch A and B are given as below:

$$C_{A}^{s} = t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right)$$

$$C_{B}^{s} = t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) + \hat{p}_{i} \left( {r + 1} \right)^{\beta } + \alpha \left( {r + 1} \right)^{\beta } r + t^{\prime}$$

where t is the completion time of a job scheduled before job j and $t^{\prime}$ is the sum of processing times of jobs belong to batch B and scheduled after job i. Therefore, the total weighted completion times (TWCT) of two batches A and B in sequence s is given as below:

$$TWCT(s) = W_{A} \times C_{A}^{s} + W_{B} \times C_{B}^{s} = \left( {W_{t} + w_{j} } \right) \times C_{A}^{s} + \left( {w_{i} + W_{{t^{\prime}}} } \right) \times C_{B}^{s}$$

where $W_{t}$ is the sum of weights of jobs belonging to batch A scheduled before job j and $W_{{t^{\prime}}}$ is the sum of weights of jobs belonging to batch B scheduled after job i. Like the sequence s, now, consider the sequence $s^{\prime}$ where i is exchanged by j. Therefore, the total weighted completion times (TWCT) of two batches A and B in sequence $s^{\prime}$ is given as below:

$$TWCT(s^{\prime}) = W_{A} \times C_{A}^{{s^{\prime}}} + W_{B} \times C_{B}^{{s^{\prime}}} = \left( {W_{t} + w_{i} } \right) \times C_{A}^{{s^{\prime}}} + \left( {w_{j} + W_{{t^{\prime}}} } \right) \times C_{B}^{{s^{\prime}}}$$

Then we have:

$$TWCT(s) - TWCT(s^{\prime}) = W_{t} \times \left( {C_{A}^{s} - C_{A}^{{s^{\prime}}} } \right) + W_{{t^{\prime}}} \times \left( {C_{B}^{s} - C_{B}^{{s^{\prime}}} } \right) + w_{j} \times \left( {C_{A}^{s} - C_{B}^{{s^{\prime}}} } \right) + w_{i} \times \left( {C_{B}^{s} - C_{A}^{{s^{\prime}}} } \right)$$

First, we show that $C_{A}^{s} - C_{A}^{{s^{\prime}}} > 0$ and $C_{B}^{s} - C_{B}^{{s^{\prime}}} > 0$:

$$\begin{aligned} C_{A}^{s} - C_{A}^{{s^{\prime}}} &= t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) - t - \hat{p}_{i} r^{\beta } - \alpha r^{\beta } \left( {r - 1} \right) \hfill \\ &=\left( {\hat{p}_{j} - \hat{p}_{i} } \right)r^{\beta } \ge 0 \hfill \\ C_{B}^{s} - C_{B}^{{s^{\prime}}} &= t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) + \hat{p}_{i} \left( {r + 1} \right)^{\beta } + \alpha \left( {r + 1} \right)^{\beta } r + t^{\prime} \hfill \\ &\quad - t - \hat{p}_{i} r^{\beta } - \alpha r^{\beta } \left( {r - 1} \right) - \hat{p}_{j} \left( {r + 1} \right)^{\beta } - \alpha \left( {r + 1} \right)^{\beta } r - t^{\prime} \hfill \\ &= \left( {\hat{p}_{j} - \hat{p}_{i} } \right)r^{\beta } - \left( {\hat{p}_{j} - \hat{p}_{i} } \right)\left( {r + 1} \right)^{\beta } \hfill \\ &= \left( {\hat{p}_{j} - \hat{p}_{i} } \right)\left( {r^{\beta } - \left( {r + 1} \right)^{\beta } } \right) \ge 0 \hfill \\ \end{aligned}$$

So we have

$$\begin{aligned} &TWCT(s) - TWCT(s^{\prime}) \ge w_{j} \times \left( {C_{A}^{s} - C_{B}^{{s^{\prime}}} } \right) + w_{i} \times \left( {C_{B}^{s} - C_{A}^{{s^{\prime}}} } \right) \hfill \\ &\quad \, \ge w_{j} \times \left( {t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) - t - \hat{p}_{i} r^{\beta } - \alpha r^{\beta } \left( {r - 1} \right) - \hat{p}_{j} \left( {r + 1} \right)^{\beta } - \alpha \left( {r + 1} \right)^{\beta } r - t^{\prime}} \right) \hfill \\ &\quad\quad + w_{i} \times \left( {t + \hat{p}_{j} r^{\beta } + \alpha r^{\beta } \left( {r - 1} \right) + \hat{p}_{i} \left( {r + 1} \right)^{\beta } + \alpha \left( {r + 1} \right)^{\beta } r + t^{\prime} - t - \hat{p}_{i} r^{\beta } - \alpha r^{\beta } \left( {r - 1} \right)} \right) \hfill \\&\quad \Rightarrow TWCT(s) - TWCT(s^{\prime}) \ge t^{\prime} \times \left( {w_{i} - w_{j} } \right) \hfill \\&\quad \quad \, + r^{\beta } \left( {w_{j} \hat{p}_{j} - w_{j} \hat{p}_{i} + w_{i} \hat{p}_{j} - w_{i} \hat{p}_{i} } \right) \hfill \\&\quad \quad \, + \left( {r + 1} \right)^{\beta } \left( { - w_{j} \hat{p}_{j} - w_{j} \alpha r + w_{i} \alpha r + w_{i} \hat{p}_{i} } \right) \hfill \\ \end{aligned}$$

Since $w_{i} \ge w_{j}$, $p_{j} \ge p_{i}$ and $r^{\beta } > \left( {r + 1} \right)^{\beta }$:

$$\begin{aligned} TWCT(s) - TWCT(s^{\prime}) \ge \left( {r + 1} \right)^{\beta } \left( {w_{j} \hat{p}_{j} - w_{j} \hat{p}_{i} + w_{i} \hat{p}_{j} - w_{i} \hat{p}_{i} } \right) \hfill \\ \, + \left( {r + 1} \right)^{\beta } \left( { - w_{j} \hat{p}_{j} + \left( {w_{i} - w_{j} } \right)\alpha r + w_{i} \hat{p}_{i} } \right) \hfill \\ \Rightarrow TWCT(s) - TWCT(s^{\prime}) \ge \left( {r + 1} \right)^{\beta } \left( { - w_{j} \hat{p}_{i} + w_{i} \hat{p}_{j} } \right) \hfill \\ {\mathop{\longrightarrow}\limits^{{Since \, \frac{{\hat{p}_{i} }}{{w_{i} }} \le \frac{{\hat{p}_{j} }}{{w_{j} }}}}}TWCT(s) - TWCT(s^{\prime}) \ge 0 \hfill \\ \end{aligned}$$

It means that the value of total weighted completion times of Sequence $s^{\prime}$ is less than of Sequence s. Therefore, the sequence $s^{\prime}$ where jobs are sorted by WSBPT rule has priority over the latter.□

Appendix 2

As stated, 12 different settings are determined based on the initial freezing, final freezing and amount of epoch. Table 12 shows these different settings.

Table 12 All parameters settings scenarios

For each setting, the results of solving 45 random instances have been presented in Table 13.

Table 13 The results of solving 45 random instances for each setting

Table 14 shows the result of Friedman tests for each category. As is clear, the null hypothesis is rejected for all categories. In summary, for the small-size, medium-size, and large-sizes instances, the best settings are 3, 5, and 10, respectively.

Table 14 Friedman test results