Synchronizing Schedules for Transportation

The formulation presented in this section assumes that the new schedule Today's manufacturing environment is highly time varying, and most of the components in the supply chain have stochastic nature of objectives and constraints due to environmental uncertainties and executional uncertainties. These uncertainties can be triggered by machine breakdowns, shortage of materials, interruption of machine operations when their performance violates quality control standards, etc. The occurrence of interruptions and the time required for assembly to resume from the interruptions are often highly stochastic in nature. These issues always lead to unexpected delays in assembly. The deterministic schedule obtained prior to the start of assembly processing is affected and becomes inappropriate. Thus, the deterministic schedule should be updated so as to minimize the disturbances due to uncertainties. The scenario of assembly process delays caused by the stochastic events is studied and a schedule repair heuristic is presented to minimize the influence of stochastic events on deliveries. 

$i \quad$ the job/order index, $i=1,2, \ldots, \mathrm{N}^{\prime}, \mathrm{N}^{\prime}$ is the total number of jobs considered at the decision instant;

$t \quad$ the delay start time;

$DU$ the delay duration;

$R_{i} \quad$ the release time of job $i$;

$P_{i} \quad$ the processing time of job $i$;

$C_{i} \quad$ the assembly completion time of job $i$

$\beta_{1 i} \quad$ per unit transportation cost of job $i$ when transported by a commercial flight;

$a_{1 i} \quad$ the per hour earliness penalty of job $i$ for assembly and it is assumed that $a_{1 i}=Q_{i}$;

$P I_{i j} \quad 1$ if job $i$ precedes job $j$ immediately, 0 otherwise;

$E F_{\text {if }} 1$ if assembly completion time of job $i$ is earlier than flight $f$ 's departure time, otherwise 0 ;

$P A_{\text {if }}$ the predetermined allocation, 1 if job $i$ is predetermined to be allocated to flight $f$ by the ILP model, 0 otherwise;

$T C_{i f}$ the transportation cost matrix which is determined by the ILP model.

The model is expressed as follows:

Min

$\sum_\limits{i=1}^{N^{\prime}}\left(\sum\limits_{f=1}^{F}\left(P A_{i f} * E F_{i f} *\left(T C_{i f}+\alpha_{1 i} * \operatorname{Max}\left(0, D_{f}-C_{i}\right)+\alpha_{i} * \operatorname{Max}\left(0, d_{i}-A_{f}\right)+\beta_{i} * \operatorname{Max}\left(0, A_{f}-d_{i}\right)\right)\right)\right)$

$+\sum\limits_{j=1}^{N^{\top}}\left(\left(1-\sum\limits_{f=1}^{F}\left(P A_{i f} * E F_{i f}\right)\right)\right.$

$\left.*\left(\beta_{1 i} * Q_{i}+\sum\limits_{f=1}\left(P A_{i f} *\left(\alpha_{i} * \operatorname{Max}\left(0, d_{i}-\left(A_{f}+C_{i}-D_{f}\right)\right)+\beta_{i} * \operatorname{Max}\left(0,\left(A_{f}+C_{i}-D_{f}\right)-d_{i}\right)\right)\right)\right)\right)$

(11)

Subject to: $\mathrm{C}_{\mathrm{i}}=\mathrm{R}_{\mathrm{i}}+\mathrm{P}_{\mathrm{i}}, \mathrm{i}=0,1, \ldots, \mathrm{N}^{\prime}, \mathrm{N}^{\prime}+1$                                   (12)

$\mathrm{R}_{0}=\mathrm{t}+\mathrm{DU}$                                                                              (13)

$R_{N+1} \geq \sum\limits_{i=1}^{N^{\prime}} P_{i}$                                                                  (14)

$\sum\limits_{j=1}^{N^{\prime}+1} P I_{i j}=1, i \neq j, \quad i=0,1, \ldots, N^{\prime}$           (15)

$\sum\limits_{j=0}^{N^{\prime}} P I_{j i}=1, i \neq j, \quad i=1, \ldots, N^{\prime}, N^{\prime}+1$ (16)

$\sum\limits_{j=1}^{N^{\prime}+1} P I_{j 0}=0$                                                                           (17)

$\sum\limits_{j=0}^{N^{\prime}} P I_{\left(N^{\prime}+1\right) j}=0$                                         (18)

$C_{i}-C_{j}-L N^{*} P I_{j i} > =P_{i}-L N \quad i, j=0,1, \ldots, N^{\prime}, N^{\prime}+1$        (19)

$E F_{i f}=1$, For $i, f$ with $C_{i} < =D_{f}$                                                                              (20)

$E F_{i f}=0$, For $i, f$ with $C_{i} > D_{f}$                                                                                 (21)

$\mathrm{PI}_{\mathrm{ij}} \in\{0,1\}, i, j=0,1, \ldots, N^{\prime}, N^{\prime}+1$                      (22)

The decision variables are R_i, PI_ij, EF_if. The objective function includes the two early and two late penalties for the orders.  Early penalties are incurred when assembly of the order is completed earlier than its transportation departure time. The late penalties are the special flight transportation cost when orders miss their predetermined flight. Since the assembly scheduling model considers synchronization with transportation, early and late penalty for assembly together with final delivery early and late penalties are taken into account in this model. The first term in the objective function is the cost of early penalties of the orders when they can catch its pre-determined flight. The early penalties consist of earliness cost before transportation, predetermined flight transportation cost, final delivery earliness/tardiness costs. The second term in the objective function is the late penalties of the orders when they miss their predetermined flights. The late penalties consist of the commercial flight transportation cost, the final delivery earliness/tardiness costs. 

Note that two dummy jobs are created in order to facilitate the representation of the immediate precedence of the jobs. They are the first and the last job which has zero quantity. Constraint (12) represents the relationship among the release time, completion time and processing time of each order. Constraint (13) sets the release time of the first job, R₀, to the assembly resume time, which is the sum of delay start time t and the delay duration DU. Constraint (14) sets the release time of the last job, R_N'+1, larger or equal to the total processing time of all the jobs. These two constraints denote that there might be inserted idle time between the release times of each two adjacent jobs. Constraint (15) and (17) ensure that all the jobs should have a precedence job except the first job. Constraint (16) and (18) ensures that all the jobs should have a successive job except the last job. Constraint (19) represents the completion time relationship between any two jobs. Constraint (20) and (21) indicate that when a job's completion time is earlier than a flight departure time, it can catch the flight. Constraint (22) indicates that PI_ij is 0-1 integer variable.