Combinatorics -Routing Problem - Single Vehicle
The traditional traveling salesman problem involves a salesman who must make a route through some set of cities. Starting at an arbitrary city, the salesman must visit the other cities and then return to the starting city. The distance between every city pair is specified and the salesman is to visit each city once and only once. A solution is called a route and the goal is to find the minimum length route. This problem is considered by the Optimize add-in. A new add-in, the Routing add-in has a more extensive model for the routing problem.
 The routing problem addressed by the Combinatorics add-in is similar to the TSP, but here we are dealing with a vehicle that starts at a depot, visit several sites and returns to the depot. The objective is to minimize the travel cost plus a cost that depends on the delivery times to the cities. The parameters of the problem are set in the dialog below.

Features of the problem are illustrated by the example. Two additional sites are defined for the analysis. The site indexed 1 represents the depot at the start of the route, the site indexed 8 represents the depot at the end of the route, and the six specified sites are indexed 2 through 7.

Time is the essence of this problem. A distance matrix (rows 16 to 23) describes the distances between each pair of sites. Note that entry (i, j) holds the distance to site i from site j. This is different than the usual definition, but more convenient for modeling. Since the random, Euclidean and integer options were specified, the locations of the sites are randomly generated and shown in columns L and M. The first and the last sites both represent the depot and they have the same location. The Euclidean distances between the site locations are computed and the integer portions of the values are placed in the table. If the Euclidean Formula option had been chosen, Excel formulas evaluate the distances. The user can then provide non-random locations for the sites in columns L and M. If a density less than 100% is chosen cells are randomly selected to hold a numeric value with the specified probability. The remainder of the cells are made inadmissible by placing the string "***" in them. The add-in assures that at least one feasible route exists.

Since the route must start at Depot 1, all cells allowing a transition from a site other than the last site are disallowed. Similarly all cells representing a transition from the last site to other sites, except the first, are disallowed. Also the cell from Depot 1 to Depot 2 is disallowed. The cell from Depot 2 to Depot 1 completes every feasible route. The grayed cells in the table have values necessary for the add-in and should not be changed. Although the other cells are computed with formulas, their contents can be changed to more accurately reflect travel distances.

Refer to the figure below for the remainder of the discussion about the example. Distances are translated into costs by cost/distance factors (row 27), and into time by time/distance factors (row 28). The vehicle must spend time at each site, perhaps for loading and unloading, called the site time (row 29). There is also an available time (row 30) for each site that is the earliest time a site can accept the arriving vehicle. The objective in this problem is to minimize the cost associated with the route. The duration penalty (row 31) multiplies the departure time from each site to determine the cost of the route. Early and late costs may also be included as described later.

The sequence of sites visited by the vehicle is called a route. The route begins at the depot and ends at the depot. The initial default route follows the sites in numerical order as in row 10. The decision variables for the problem are in row 9. An entry specifies the next site to visit from the site associated with that index. For example, the solution shown starts at site 1 (the depot) and the next index is 2 (site 1). The route is computed from these decisions and shown in row 10. Row 12 shows the travel distance from a site to the next site in the route.

Evaluation

Immediately below the data, the sequence is evaluated. All these cells are yellow indicating that they are determined by formulas. The current sequence of site visits is in row 34. The Time Available and Site Time values are gathered from the data and presented in the order of route sequence in rows 35 and 36. The Travel Time to Next Site values in row 37 are computed by multiplying the travel distance by the Time/Distance values in the data. The Arrival Times are in row 38 and the Departure Times are in row 39. The arrival time for a site is the maximum of the time available for the site and the sum of the departure time of the previous site and the travel time. The departure time for a site is the sum of the arrival time and the site time. The arrival time for the Depot shown in column D is the time available value for the depot. The arrival time for the Depot shown in column K is the time required for the completed route.

The cost information begins in row 42. A Travel Cost in row 42 is obtained by multiplying the travel distance by the Cost/Distance data item in row 27. A Duration Cost in row 43 is computed by multiplying the departure times in row 39 by the Duration Penalty in row 31. The sum of these costs are reported in cell F5 at the top of the page. The goal is to find a sequence that minimizes this value.

When the duration penalties are zero, the travel cost is minimized by minimizing the total distance traveled on the route. This is the familiar traveling salesman problem. Nonzero duration penalties penalize a delivery in proportion to the time the route departs from the site. Thus time is important in addition to distance. Preference between sites can be expressed by using higher penalties for sites of higher priority.

When all penalties are the same, the route will tend to serve sites with short durations earlier than longer ones. For example if the Cost/Distance values are set to zero, the tour will follow the shortest time rule, serving the sites with the smallest site time first. Since the objective is the sum of the distance and time effects the optimum tour will balance duration penalties with travel costs.

Optimization

We choose a solution method by clicking the Search button on the worksheet. The button calls a dialog from the Optimize add-in for selecting the solution method. The dialog shows the Exhaustive search method chosen. The 19 best solutions are to be shown in a sorted list. The best solution found appears twice in the list.

The figure below shows the worksheet with the optimum solution found with exhaustive enumeration. The cells defining the optimum route are outlined in red on the distance matrix. Of course, any other method besides exhaustive enumeration does not necessarily provide the optimum solution.

The route results are shown below. The route follows the sequence (Depot, 3, 6, 5, 4, 1, 2, Depot). The minimum total cost for the data provided is 908 (in cell F5).

When x and y coordinates are given in the data graphical display of the route is also available. Clicking the Map button creates a worksheet showing the route. Some data formats do the provide the coordinates, so the Map option will not be available.
The list below shows the best 19 solutions found by exhaustive enumeration. The optimum solution was also found by generating 10 random routes and improving each with a 2-change heuristic. Of course, such a fortuitous result cannot be guaranteed. Note that the solutions do not give the sequence directly. Rather the solution provides the next site to visit from each site. The associated sequence for for a solution can be found by placing the solution in the green area on the problem form, row 9 in the example.

Scheduled Arrival and Departure Times

To include scheduled arrivals and departures in the data, click the Early Arrival and Late Departure checkboxes. For this example, we construct the distance matrix with formulas determining the Euclidean distances.

The form for the example is shown below with data. With the Early Arrival and Late Departure options, some new data rows are provided. There is now Scheduled Arrival (row 30) and Scheduled Departure (row 31) data for each site. Early Penalty (row 33) and Late Penalty (row 34) values penalize deliveries that are earlier than the scheduled arrival time and later than the scheduled departure time.

The data below requires that the vehicle finish its route before 240 minutes or 4 hours. Sites 1, 2 and 3 have been promised that the departure times will be no later than 120 minutes. Sites 4, 5 and 6 have been promised that the vehicle will arrive no earlier than 120 minutes and that departure will be no later than 240 minutes. To encourage a completion time for the entire route of 240 minutes, we have placed 240 as the scheduled departure from the final depot site. Scheduled arrivals and departures are soft constraints. We encourage them to be satisfied by penalizing arrivals that come before the scheduled times and departures that occur after the scheduled times. A greater penalty is applied to the last depot site to give greater priority to the desired route completion time.

The data also specifies that site 2 will be available no earlier than 10 and site 3 will be available no earlier than 55. These are hard constraints.

Clicking the search button at the top left of the page, we choose Exhaustive as the search option. The solution above is the optimum solution.

The results below describe the optimum route. The optimum sequence has site 2 (index 3) as the first site on the route. The vehicle arrives at time 10, when the site is first available. This is 30 minutes before the schedule arrival resulting in a penalty of 600. The second site on the route, S3, is within the scheduled time window. The departure time for S1 is 4 minutes after the scheduled departure time resulting in a penalty of 80. The entire route is finished at 231 minutes, within the desired limit.

Clicking the Map button on the page causes a worksheet to be created that shows the solution graphically. Note that the route is different than the example without early and late times.

The single vehicle routing problem with time windows modeled here is just one of the many vehicle routing problems that could arise in practice. Many varieties have been considered in the literature including problems involving more than one vehicle and vehicles with finite capacities. These are also combinatorial problems but they involve a more complex decision structure than the simple sequencing considered on this page. The add-in is generalized for the multiple vehicle case on the next page.

Operations Research Models and Methods
Internet
by Paul A. Jensen