A Routing Model for Hybrid Hub-and-Spoke with Time Windows

Felix Arya Gunadi a*, J. Dharma Lesmono b, Kinley Aritonang c a Departement of Industrial Engineering, Parahyangan Catholic University, Indonesia Jl. Ciumbuleuit No 94, Bandung, Indonesia b Departement of Mathematics, Parahyangan Catholic University, Indonesia Jl. Ciumbuleuit No 94, Bandung, Indonesia c Departement of Industrial Engineering, Parahyangan Catholic University, Indonesia Jl. Ciumbuleuit No 94, Bandung, Indonesia *Correspondence author: felixaryagunadi@gmail.com


Introduction
Currently, competition among companies is getting tighter. In order to be competitive, distribution companies need to provide excellent service to customers. The distribution service performance can be measured with lead time. However, the reduction of lead time can result in the underutilization of a vehicle [1]. It may increase transportation costs significantly. Moreover, distribution companies in Indonesia deal with inbound and outbound distribution with separate fleets. They use a particular fleet to deal with outbound distribution, and another fleet deals with inbound distribution. It causes a backhaul as the fleet used to pick up goods at the suppliers is empty when they depart. Meanwhile, the fleet for delivering the products to the customers is also vacant as

Assumptions, Notations, and Model Formulation
The assumptions of the model comprise of 1). The constructed model only uses one hub as a consolidation center; 2). The model's performance is measured with the transportation expenses that incorporate only gas and vehicle usage costs. 3). The loading and unloading times are ignored; 4). The demand is deterministic, and 5). The sorting time/ unit is deterministic. The notations used in this model include: :  .
is a decision that arranges the route of shipping the goods to customers and pick up the goods.  is a decision that organizes the big vehicle route for transshipment in the hub-and-spoke network. One decision variable determines whether the distribution is carried via a hub or direct shipping along with the quantity. The objective function of this model is to minimize the total costs, including vehicle usage cost ( ) and transportation cost ( ). The following is the route determination model for the hybrid hub-and-spoke distribution network.

Subject to
∑ ∑ X jiv = 1 ∀v ∈ V j∈J i∈D ∑ ∑ X ijv ≤ |s| − 1 ∀s ∈ J, |s| ≥ 2, ∀v ∈ V j∈s,j≠i i∈s (15) Twh dv 1 ≤ Tl ∀d ∈ D, ∀v ∈ V (31) Twh dv 2 ≤ Tl ∀d ∈ D, ∀v ∈ V (32) Twh hv 1 ≤ Th ∀h ∈ H, ∀v ∈ V (33) The Objective function of this problem is to minimize the total distribution cost, including vehicle usage cost in equation (1) and transportation cost in equation (2). Constraint (3) ensures that each customer and supplier is served only one time. Constraint (4) ensures that every vehicle comes to and leaves the customer afterward. Constraint (5) confirms that each vehicle returns to its respective DC. Constraint (6) agrees that each vehicle departs from DC. Constraint (7) ensures that every vehicle's initial load does not exceed the capacity. Constraint (8) (9) and (10) prevents the unselected route that violates the time windows. If a specific route is not selected, the value of or is 0. Hence, M will become a considerable deduction, and the constraint will not be violated. Constraint (11) guarantees that every vehicle does not depart before the earliest departure schedule. Constraint (12) ensures that every vehicle arrives at the customer's and supplier's place at the range of the time windows. Constraints (13)- (15) are the additional constraints from the previous model. Constraint (13) ensures that picking up the goods from the suppliers is to be carried out after the vehicle finishes the shipment to the customer. Constraint (14) prevents the vehicle from stopping at another depot while shipping the goods to the customer. Constraint (15) is a sub-tour elimination constraint.
Constraints (16)- (19) are the model's novelty to ensure that the proposed lead time is achieved. Constraint (16) calculates the arrival of each vehicle at DC. Constraint (17) determines the latest arrival of the vehicle at DC. Constraint (18) estimates the earliest departure time of each vehicle at every DC, using the Tmd plus the sorting time at the depot. Constraint (19) calculates the latest arrival time of the vehicle at the hub. The latest arrival time is calculated by adding the Tmax with the processing time of the shipped goods via direct shipping. If there is a direct delivery between DCs, the number of goods that need to be processed by the hub decreases. Therefore, the time limit of the hub becomes shorter. If the Tmax is 23.00 and 60 goods shipped via direct shipping with sorting time of each good is one minute, this is 24.00 (23.00 plus sixty minutes). Constraints (20) and (21) guarantee that the goods moved from DC d to d' equal to the demand needed to be moved from DC d to d'. Constraint (22) is the capacity constraint for immediate shipping. In direct shipping, the vehicle that departs to another DC will come back with the goods from the destination to its respective DC. Also, the capacity of the vehicle needs to be considered for both directions. Constraint (23) is the capacity constraint for shipment via a hub. Constraint (24) ensures a different vehicle requirement in the shipment for both directions; it becomes the essential requirement for directions.
Constraints (25) and (26) ensure that all small and big vehicles return to their respective DC. Constraints (27) and (28) ensure that all small and big vehicles only leave DC once. Constraints (29) and (30) calculate the arrival time of each vehicle at DC. Vdv is an auxiliary variable that helps to identify the initial DC of each vehicle. Suppose the vehicle departs from the initial DC. In that case, the arrival time will be calculated using the earliest departure time from the DC plus the traveling time. Otherwise, the arrival time will be calculated using the vehicle's arrival time at the DC plus service and traveling time. Constraints (31) and (32) guarantee that every vehicle arrives at DC before the latest arrival time allowed. Constraints (33) and (34) ensure that each vehicle arrives at the hub before the latest arrival time that is permitted at the hub.

Data and Experimental Procedures
The model implemented in this study consisted of three DCs, eight customers, and four suppliers. It was inspired by a real case on the automotive wholesale company in Indonesia as it owns four different suppliers and three DCs. Table 1 describes the distance between hub, depot, customers, and suppliers. Nodes 1-3 were the DCs, nodes 4-11 were the customers, and 12-15 were the suppliers. With the velocity assumption of 40 km/hour, the traveling time between the nodes was calculated. Each customer had different demand, as described in Table 2.
Each supplier had different quantities of goods that need to be taken, as described in Table 3. The demands are shown in Table 2 and Table 3 (unit measurement in box). Each customer and supplier had a specific time window. Time windows described the earliest and the latest arrival time of each customer and supplier. The vehicle was not allowed to arrive after the arrival service time; but, it was allowed to come before the earliest arrival time and wait until the initial arrival time. Table 4 was time windows and service time for each supplier and customer. The departure time for each vehicle from each DC was 8 AM. There was a distance between hub and depot. Table 5 illustrates the gap between hub and depot. It shows that nodes 1-3 were DCs, and node 4 was the hub. The unit measurement employed in this table was kilometers. There was a quantity of transit between the DCs. Table 6 describes the quantity of transit between the DCs. There was a time limit on each DC and hub. The vehicle was not allowed to arrive at DC and hub after the time limit. Table 7 shows the time limit on each DC and hub. The gas cost for the small vehicle was IDR 980 / km and IDR 1,250 for the big one. Usage cost for the small vehicles was as much as IDR 150,000 and IDR 225,000 for big vehicles.     The sensitivity analysis was performed with several parameters, such as usage cost of small and big vehicles, gas cost, and traveling time. The parameters were selected based on their effects on the model. The usage cost of small and big vehicles affected the route and the number of vehicles used. The traveling time affected the route and number of vehicles needed. The longer traveling time could increase the number of vehicles due to time window constraints. The sensitivity analysis was carried out for each parameter with change values of -20%,-10%,+10%, and +20%. One change parameter was used in one run. The case was solved using LINGO 1.0x64 bit.

Implementation
The model was implemented in a case consisting of three DCs, eight customers, and four suppliers. The case needed 9 hours and 37 minutes of running time. The result can be seen in Fig. 1. The total cost generated for the routes is IDR 4,488,905.00.

Sensitivity Analysis
The Implementation of sensitivity analysis was carried out to determine the change in the solution that occurred when the input of the parameters was changed. Sensitivity analysis was performed by changing certain parameters. Several parameters were changed. The modified parameters were vehicle usage cost for small vehicles, vehicle usage cost for big vehicles, gas cost, and traveling time. Before commencing the sensitivity analysis, the researchers operated the program several times with the change of parameters. The result of sensitivity analysis is given in Table 8.
The researchers found that the increase in the usage cost of small vehicles is followed by an increase in the total cost needed. However, the number of small and large vehicles used does not change. Meanwhile, there is also a change in the number of small and large vehicles where significant vehicle usage decreases and the number of small vehicles increases. An increase follows the increase in the usage cost of big vehicles in the total costs. The substitution of one large vehicle also follows it into two small vehicles as they have a lower cost. While the decrease in the cost of using large vehicles only reduces the total cost, the number of vehicles used is still the same.
Changes in fuel costs per km have the most significant impact as compared to other parameters. Increasing fuel cost per km by 20% raises an increase in the total costs of ISSN : 1978- In that case, the number of significant vehicles needs to be added to cut the traveling distance of another vehicle. From all the factors, the fuel cost has the highest sensitivity to the solution. The distinction of this model with the previous model is that this model generates a route that makes sure all the vehicles arrive at each DC before the time limit.