## Synchronizing Schedules for Transportation

Read this article. It discusses synchronizing transportation schedules. Because the logistics segment of the cycle is a large-scale effort, the waiting and queues are magnified. How many different modes of transportation do you think are required to make a product from raw material to the customer's hand?

### 2. The 3PL Transportation Problem

The 3PL transportation problem is formulated using an Integer Linear Programming (ILP) model. As the air transportation is a typical case for the 3PL transportation, the 3PL transportation is represented by air transportation hereafter. The model allocates orders to the existing air transportation capacities with minimum costs. Synchronization is incorporated into the ILP model by including the constraint that balances the production rate of the assembly facility with the flight allocation.

The following notation is defined:

i = order index, i=1, 2, …. N;                      f, f' = flight index, f=1,2,……F;

k = destination index, k = 1,2,……L;           Af = arrival time of flight f at the destination;

Desi = order i's destination;                       Desf = flight f's destination;

LN = a large number;                                |LN| = absolute value of LN;

Qi = quantity of order i;                             di = due date of order i;

Df =departure time of flight f at the local place where the manufacturing plant is located;

NCf = transportation cost for per unit product allocated to normal capacity area of flight f; SCf = transportation cost for per unit product allocated to special capacity area of flight f;

NCapf = available normal capacity of flight f; SCapf = available special capacity of flight f;

Di = delivery earliness penalty cost (/unit/hour) of order i;

Ei = delivery tardiness penalty cost (/unit/hour) of order i;

WTi = waiting time of order i between assembly and air transportation;

PEif = per unit delivery earliness penalty cost for order i when it is transported by flight f,

$P E_{i f}=\operatorname{Max}\left(0, d_{i}-A_{f}\right)^{*} \alpha_{i}$    (1)

PLif = per unit delivery tardiness penalty cost for order i when it is transported by flight f,

$P L_{i f}=\operatorname{Max}\left(0, A_{f}-d_{i}\right)^{*} \beta_{i}$       (2)

Zif = quantity of order i allocated to flight f

Xif = the quantity of the portion of order i allocated to flight f's normal capacity area;

Yif = the quantity of the portion of order i allocated to flight f's special capacity area;

PR = the production rate of assembly manufacturing;

In case of split deliveries, an order can be split and delivered among any number of flights. The ILP model for the multi-destination air transportation problem is expressed as follows:

Minimize $\sum_{i} \sum_{f} N C_{f} X_{i f}+\sum_{i} \sum_{f} S C_{f} Y_{i f}+\sum_{i} \sum_{f} P E_{i f} Z_{i f}+\sum_{i} \sum_{f} P L_{i f} Z_{i f}$                        (3)

Subject to:

$X_{i f}+Y_{i f}=Z_{i f}$, for all $i, f$                 (4)

$L N * X_{i f} *\left|\operatorname{Des}_{i}-\operatorname{Des}_{f}\right|$, for all $i, f$  (5)

$L N * Y_{i f} *\left|\operatorname{Des}_{i}-\operatorname{Des}_{f}\right|$, for all $i, f$  (6)

$\sum\limits{i} X_{i f} \leq N \operatorname{Cap}_{f}$, for all $f$   (7)

$\sum\limits{i} Y_{i f} \leq S C a p_{f}$, for all $f$                (8)

$\sum\limits{f}\left(X_{i f}+Y_{i f}\right)=Q_{i}$, for all $i$         (9)

$\sum^{f}_\limits{f^{\prime}=1} \sum_\limits{i} \left(X_{i f^{\prime}}+Y_{i f^{\prime}}\right) \leq D_{f} P R$, for all $f$

The decision variables are: Xif, Yif, and Zif. All decision variables are non-negative integer variables. The objective is to minimize overall total cost which consists of total transportation cost for the orders allocated to the normal flight capacity, total transportation cost for orders allocated to the special flight capacity, total delivery earliness penalty cost, and total delivery tardiness penalty cost. Constraint (4) ensures that the quantity of the proportion of order i allocated into flight f consists of quantities of the proportion of order i allocated into normal capacity area of flight f and the proportion of order i allocated to special capacity area of flight f. Constraints (5) and (6) ensure that if order i and flight f have different destinations, order i cannot be allocated to flight f. Constraint (7) and (8) ensure that the normal and special capacity of flight f is not exceeded. Constraint (9) ensures that order i is completely allocated. Constraint (10) ensures that allocated orders do not exceed production capacity. It ensures that allocated quantity can be supplied by sufficient assembly capacity.

Subsequently, the equality of the above air transportation allocation problem with an unbalanced transportation problem is established. For the air transportation problem, each order can be taken as a supply point and each flight's capacity can be taken as a demand point. It is noted that the normal capacity and special capacity of each flight are considered as two demand points with different transportation costs. The unit transportation cost from a supply point SPi to a demand point DPf is the sum of the unit transportation cost and the unit delivery earliness (or tardiness) penalty cost of order SPi when transported by flight DPf . As total quantity of all orders is less than total capacity of all flights, the air transportation problem can be taken as an unbalanced transportation problem. As the transportation problem can be solved in polynomial time, this air transportation problem is also solvable using one of the commercial solvers.