Routing Problems

Hub & Spoke Milk Run

VRP is to determine K vehicle delivery routes.A route is a tour that begins at the depot,

traverses a subset of the customers in a specified sequence and returns to the depot.

Each customer must be assigned to exactly one of the K vehicle routes.

The total size of deliveries for customers assigned to each vehicle must not exceed the vehicle capacity b.

The routes should be chosen to minimize total travel cost.

Vehicle Routing Problems

VRP

Depot

ObjectiveMinimize the total cost of providing the service.This includes vehicle capital costs, mileage and

personnel costs.Other objectives Public sector

School bus: Minimize the total number of students-minutes on the bus.

Dial-a-ride for handicapped or elderly: Minimize the inconvenience for all customers.

Meals-for-ME: Deliver meals at certain times of the day.

Emergency services: Minimize respond time to an incident.

Characteristicsof the ProblemsPresented as graphical networks consisting of

nodes and arcs.The problem under consideration is visualized.Depot node: the “home base” for the vehicle

or provider.Arcs describe the time, cost, or distance

required to travel from one node to another.Given an average speed or a distribution of

travel times, distance can be converted to time.(Physical barriers are ignored.)

Directed arcs / Undirected arcs

Characteristics of the ProblemsA tour: A route for the vehicleIn general, a feasible tour must satisfy the

following conditions:A tour must include all nodes.A node must be visited only once.A tour must begin and end at a depot.

A route specifies the sequence in which the nodes (or arcs) to be visited.

A schedule identifies when each node is to be visited.

Traveling Salesman Problem (TSP)The simplest case.A set of nodes is to be visited, in any order,

by a single vehicle.The travel costs between two nodes are the

same regardless of the direction traveled. (Symmetric)

No delivery time restrictions.Vehicle capacity is not considered.Objective: Minimize the total tour cost.

Multiple Traveling Salesman Problem (MTSP)An extension of the TSP.A fleet of vehicles must be routed from a single

depot.Objective: To generate a set of routes, one for each

vehicle in the fleet, that minimizes the total tour cost

A node maybe assigned to only one vehicle, but a vehicle will have more than one node assigned to it.

There are no restrictions on the size of load or number of passengers a vehicle may carry.

Vehicle Routing Problem (VRP)Restrict the capacity of the multiple vehicles Demand may vary at each node.Objective: To generate a set of routes, one

for each vehicle in the fleet, that minimizes the total tour cost

Chinese Postman Problem (CPP)The demand for the service occurs on the

arcs, rather than at the nodes.Or if demand is so high that individual

demand nodes become too numerous to specify.

Example: Street sweeping, Snow removal, Refuse collection, Postal delivery, Paper delivery.

Very difficult to solve.

Problem Classifications

Routing Problems & Scheduling ProblemsIf the customers being serviced have no

time restrictions and no precedence relationships exist. A pure routing problem

If there is a specified time for the service to take place, then a scheduling problem exists.

Solving Routing & Scheduling ProblemsConsider a product delivery problem from

a warehouse to customers. The customers have different demands,

and the vehicles have different capacities.Each vehicle is assigned a route

beginning and ending at the depot.

Solving Routing & Scheduling ProblemsFor a problem with only 10 customers,

there are 210 or 1,024 possible routings. For 50 customers, there are 250 or over 1

trillion (1012) possible routings.The problems of any size quickly become

too expensive to solve optimally. Heuristics that yield “good,” if not optimal, solutions to these problems.

TSP: The Nearest Neighbor ProcedureThe nearest neighbor procedure (NNP) builds a

tour based only on the cost or distance of traveling from the last-visited node to the closest node in the network.

The heuristic is simple, but it has the disadvantage of being rather shortsighted.

The procedure is outlined as follows:1. Start with the depot node.2. Find the node closest to the last node added to the

tour.3. Go back to step 2 until all nodes have been added.4. Connect the first and the last nodes to form a

complete tour.

Example: Nearest Neighbor

Example: Nearest Neighbor

Example: Nearest Neighbor

Example: Nearest Neighbor

Example: Nearest Neighbor

Example: Nearest NeighborThe complete tour

Example: Improved Solution

5.4 + 5.0 + 5.0 +7.8 + 3.6 +4.1 = 30.9 miles

The Nearest Neighbor ProcedureThe limitation of heuristics: they cannot

guarantee optimality. In practice, the heuristic is applied repeatedly

by assigning every node to be the depot node, resolving the problem, and then selecting the lowest-cost tour as the final solution.

For example, if we repeat the procedure using node 6 as the depot node, the tour that results is 6 3 1 2 5 4 6 with a total length of 31.3 miles.

Clarke & Wright Savings HeuristicThe Clarke & Wright savings heuristic (C&W) is one of the

most well-known techniques for solving TSP & VRP.The heuristic begins by assuming that one vehicle travels

from the depot directly to a node and returns to the depot.

Clarke & Wright Savings HeuristicThe distance from node 2 to node 3 is 5 miles. The total distance covered by the two vehicles is 36 miles.The key to the C&W heuristic is the computation of savings. Savings is a measure of how much the trip length or cost can

be reduced by “hooking up” a pair of nodes (e.g. nodes 2 & 3) and creating the tour which can then be assigned to a single vehicle.

By linking nodes 2 & 3, we add 5 miles (the distance from node 2 to node 3), but we save 10 miles for the trip from node 2 to node 1 and 8 miles from the trip from 3 to 1.

The total tour length for the complete tour, 1 2 3 1, is 23 miles.

The savings obtained = (10+10+8+8) – (10+5+8) = 36 – 23 = 13 miles.

5 miles

Clarke & Wright Savings HeuristicSteps:1.Select any node as the depot node (node 1).2.Compute the savings, Sij for linking nodes i and

j:Sij = c1i + c1j − cij for i and j = nodes 2,3,…,n where cij = the cost of traveling from node i to node j.

3.Rank the savings from largest to smallest.4.Starting at the top of the list, form larger

subtours by linking appropriate nodes i and j.5.Stop when a complete tour is formed.

Example: C&W Savings The initial tours (with solid lines) are shown in the figure. The dashed lines show arcs that may be used but are not in use currently. The savings obtained from linking nodes 2 and 3 is (10 miles + 8 miles) -

(5 miles) = 13 miles. The savings of linking nodes 2 and 4 is (5 miles + 10 miles) - (3 miles) =

12 miles. The savings of linking nodes 4 and 3 is (5 miles + 8 miles) - (7 miles) = 6

miles. We next rank the savings for every pair of nodes not yet linked.

Rank: [2, 3], [2, 4], and [3, 4]. The first step in specifying a tour is to link the nodes with the highest

savings; nodes 2 and 3.

Example: C&W SavingsProceeding to the next highest savings, nodes 2

and 4 are linked. The complete tour is 1 4 2 3 1, which

has a total tour length of 21 miles.The total savings obtained over the “one vehicle

per node” configuration is 25 miles.

(5+5) + (10+10) + (8+8) = 46 miles 5 + 3 + 5 + 8 = 21 miles

Clarke & Wright Savings HeuristicIn general, because C&W considers cost

when constructing a tour, it yields better quality solutions than the nearest neighbor procedure.

Both the Clark & Wright savings heuristic and the nearest neighbor procedure can be easily adjusted to accommodate problems with directed arcs.

Multiple Traveling Salesman Problem (MTSP)The MTSP is a generalization of the TSP where

there are multiple vehicles and a single depot. In this problem, instead of determining a route

for a single vehicle, we wish to construct tours for all M vehicles.

Solution procedures begin by “copying” the depot node M times.

The problem is thus reduced to M single-vehicle TSPs, and it can be solved using either the nearest neighbor or Clark and Wright heuristics.

Example: MTSP

Copy Copy

The Vehicle Routing Problem (VRP)The classic VRP expands the multiple

traveling salesman problem to include different service requirements at each node and different capacities for vehicles in the fleet.

The objective of these problems is to minimize total cost or distance across all routes.

Example: VRPSuppose we have a single depot and two

buses, 1 and 2. Vehicle 1 has a capacity of 20 people and

vehicle 2 a capacity of 10. There are three nodes where travelers are to

be picked up. The number of travelers to be picked up is

shown in brackets beside each node.

Example: VRPIf the capacity of the buses and the demand at each

node are ignored, the C&W heuristic would construct a tour for each vehicle as follows:Bus 1’s tour: 1 2 3 1 (with 21 passengers)

violates the capacity constraints of bus 1

Bus 2’s tour: 1 4 1 (with 4 passengers)

Example: VRPBus 1’s tour: 1 2 2 (with 15

passengers)Bus 2’s tour: 1 4 3 1 (with 10

passengers)

Cluster First, Route Second Approach The figure presents a 12-node VRP

problem in which two vehicles must deliver cargo to 11 stations and return to the depot.

The capacities of vehicles 1 and 2 are 45 and 35 tons, respectively.

Cargo demand is bracketed at each node, and distances, in miles, are shown on the arcs.

The 12 nodes have been clustered initially into two groups, one for each vehicle. Nodes 2 through 6 are assigned to

vehicle 1. Nodes 7 through 12 to vehicle 2. Node 1 is the depot node.

Cluster First, Route Second Approach In practice, clustering takes into account

physical barriers such as rivers, mountains, or interstate highways, as well as geographic areas such as towns and cities that form a natural cluster.

Capacity restrictions are also taken into account when developing the clusters.

From the initial clustering, vehicle 1 carries 40 tons and vehicle 2 carries 34 tons.

Using the C&W heuristic, tours are constructed as follows:Vehicle 1’s: 1 2 3 4 5 6 1 (A total tour length is 330 miles)Vehicle 2’s: 1 7 8 9 10 11 12 1

(A total tour length is 410 miles)

Cluster First, Route Second Approach The next phase is to determine whether a

node or nodes can be switched from the longest tour (tour 2) to tour 1 such that the capacity of vehicle 1 is not exceeded and the sum of the two tour lengths is reduced.

This step is referred to as tour improvement.

The nodes in tour 2 that are closest to tour 1 are nodes 7 and 8.

Node 8 has a demand of 6 tons and cannot be switched to tour 1 without exceeding vehicle 1’s capacity.

Node 7 has a demand of 3 tons and is eligible to switch.

Evaluate where the node 7 should be inserted into tour 1 and whether it will reduce the distance traveled. The minimum cost of insertion technique.

Cluster First, Route Second ApproachThe minimum cost of insertion is calculated in

the same way as the Clark and Wright heuristic.

If all distances are symmetrical, then the cost of insertion, Iij, can be calculated as follows:

Iij = ci,k + cj,k − cij for all i and j, i ≠ j

where cij = the cost of traveling from node i to node j.

Nodes i and j are already in the tour.Node k is the node we are trying to insert.

Cluster First, Route Second Approach Node 7 is a candidate for insertion because

it is near tour 1. Evaluate all alternatives :

Node 7 could be inserted between nodes 6 and 1

Or it could be inserted between nodes 5 and 6.

The cost of inserting node 7 between nodes 1 and 6 is (30 + 50 - 50) = 30 miles.

The cost of inserting node 7 between nodes 5 and 6 is (60 + 30 - 90) = 0 miles.

The lowest cost is found by inserting node 7 between nodes 5 and 6, resulting in a completed tour for vehicle 1 of 1-2-3-4-5-7-6-1.

The total length of tour 1 is now 330 miles, and the length of tour 2 is 400 miles.

The distance traveled by the two vehicles has decreased from 740 to 730 miles.

30 miles60 miles

60 miles

130 miles

Cluster First, Route Second Approach

Initial solution:Tour 1: 330 miles ; Tour 2: 410 milesTotal distance traveled: 740 miles

Revised solution:Tour 1: 330 miles ; Tour 2: 400 milesTotal distance traveled: 730 miles

Scheduling Service VehiclesScheduling problems are characterized by

delivery-time restrictions. The starting and ending times for a service may be

specified in advance. “Time windows” A one-sided time window either specifies that a

service precede a given time or follow a given time. (e.g., most newspapers attempt to have papers delivered before 7:00 A.M.)

Two-sided time window: In the Meals-for-ME program, meals had to be

delivered between 11:30 A.M. and 1:00 P.M.Furniture delivery is usually scheduled after 9:00

A.M. and before 4:30 P.M.

A VRP with Time Windows An arc may join node i to node j if the

start time of task j is greater than the end time of task i and the start time of task j must include a user-specified period of time (“deadhead time”) longer than the end time of task i.

In this example, the deadhead time is 45 minutes.

This is the nonproductive time required for the vehicle to travel from one task location to another or return to the depot empty.

The paths are not restricted in length.

Each vehicle must start and end at the depot.

A VRP with Time WindowsThe VRP with time windows (VRPTW) may be formulated

as a special type of network problem called a minimal-cost-flow problem.

Alternatively, a heuristic approach such as “the concurrent scheduler approach” may be used.

Steps:1.Order all tasks by starting times. Assign the first task to

vehicle 1.2.For the remaining number of tasks, do the following. If it

is feasible to assign the next task to an existing vehicle, assign it to the vehicle that has the minimum deadhead time to that task. Otherwise, create a new vehicle and assign the task to the new vehicle.

The Concurrent Scheduler Approach

Initially, vehicle 1 is assigned to task 1.

Because task 2 begins before vehicle 1 is available, a second vehicle is assigned to this task.

Vehicle 2 finishes task 2 in time to take care of task 3 also.

In the meantime, vehicle 1 completes task 1 and is available for task 4.

A third vehicle is not required until task 5, when vehicles 1 and 2 are busy with tasks 4 and 3, respectively.

Continuing in a similar fashion, the schedule for vehicle 1 is 1-4-7-10-12, for

vehicle 2 the schedule is 2-3-6-9, and for vehicle 3 the schedule is 5-8-11.

Assume that the deadhead time is 15 minutes.

Nearest NeighborAdd nearest customer to end of the route.

depot depot depot

1 2 3

Nearest Neighbor

depot depot depot

4 5 6

Nearest InsertionInsert customer closest to the route in the

best sequence.

depot depot depot

1 2 3

Nearest Insertion

depot depot depot

4 5 6

Savings Method(Clarke & Wright, 1964)

depot

1 2 3

4

Remove Add

depot

1 2 3

4

Savings = S12 = C10+C02 -C12

Savings Method

depot depot depot

depot depot depot

S12

1 2

3

4 5

6

1 2

3

45

6

S13

S14 S15 S16

1 2

3

4

6

5

1 24

5

6

3

1 24

6

3

5

1 24

6

3

5

Small savings

Savings Method (Clarke & Wright, 1964)Form route by linking customers according to

savings.

depot depot

1 2

3

4 5

6

1 2

3

45

6

S35:link 3&5 S34:link 3&4 (keep 3-5)

0-3-5-0 0-4-3-5-0

Savings Method

depot depot depot

depot depot depot

S24

1 2

3

4 5

6

1 2

3

45

6

S25

S26 S34 S35

1 2

3

4

6

5

1 24

5

6

3

1 24

6

3

5

1 24

6

3

5

Large savings

S23

Large savings

Savings Method

depot depot depot

depot

S45

1 2

3

4 5

6

1 2

3

45

6

S46

S56

1 2

3

4

6

5

1 24

5

6

3

Large savingsS36

In general, with n customers there are n(n-1)/2 savings to calculate.

Order savings from largest to smallest.S35, S34, S45 , S36, S56, S23, S46, S24, S25, S12, S26, S13, etc.

Savings Method

depot depot depot

1 2

3

4 5

6

1 2

3

45

6

1 2

3

4

6

S45: skip S36: skip S56:link 5&6

5

Savings Method

depot depot depot

1 2

3

4

6

5

S23: skip S46: skip

1 2

3

4

6

5

1 2

3

4

6

S24:link 2&4

Savings Method

depot depot

1 2

3

4 5

6

1 2

3

45

6

S25: skip S12: link 1&2

Final route: 0-1-2-4-3-5-6-0

The Sweep Method (Gillett and Miller, 1974)SimpleSolves problems rapidly, without requiring

enormous amounts of computer memory.The level of computational error is considered

acceptable where results must be obtained in a short time.

Disadvantage: The routing process is two-staged, with stop being

assigned to vehicles first. Then, the stop sequence on the routes is determined.

Timing issues such as total time spent on route and time windows allowance are not well handled.

The Sweep Method1. Locate all stops including the depot on a map or grid.2. Extend a straight line from the depot in any

direction. Rotate the line clockwise or counterclockwise, until it intersects a stop.

The inserted stop will be included on the route, if the vehicle capacity is not exceeded.

Use the largest vehicle first. Continue with the sweep until all points have been

assigned to routes.3. Within each route, sequence the stops to minimize

distance. (The tear-drop method or algorithms for TSP.)

Example: P.K. Smith Trucking CompanyThe P.K. Smith Trucking Company uses vans

to pick up merchandise from customers to a depot point.

The vans can haul 10,000 units.To complete a route typically requires the

entire day.The company wants to determine: How many routes (trucks) are needed?Which stops should be on the routes and in

which sequence?

Example: P.K. Smith Trucking Company

The Sweep MethodThe “sweep” method has the potential to

giving very good solutions when:

1. Each stop volume is a small fraction of the vehicle capacity,

2. All vehicles are the same size,3. There are no time restrictions on routes.

Typical Complications in Real VRPs1. Travel costs can be asymmetric so that cij cji in

general.2. The characteristics of the vehicles can introduce a

variety of constraints beyond the simple vehicle capacity constraints. Heterogeneous fleet (each with bk)

(Vehicles have different capacity limitations to both weight and volume.)

Multiple capacity constraints (weight & volume) Different products in different compartments Customer -Vehicle compatibility Multiple-trip VRP Pickup & Delivery VRP (Precedence constraints imposed)

Typical Complications in Real VRPs3. The total time duration in a route may be constrained.

Department of Transportation (DOT) safety restriction: Max. drive duration = 10 hrs. before 8 hrs. break. Max. work duration = 15 hrs. before 8 hrs. break. Max. trip duration = 144 hrs.

4. The time of delivery to customer i may be constrained to fall with in a designated ‘time window’. (Customers allow pickup and/or delivery only at certain times of the day.)

5. Drivers may be allowed to take short rests or lunch breaks at certain times of the day.

Typical Complications in Real VRPs6. There may be multiple depots with each vehicle in the

fleet assigned to a particular depot.7. Precedence constraints can impose a partial ordering on

the customer delivery sequence.8. In fleet planning problems, the number of vehicle used is

a variable rather than a constant. (Fixed charge included)9. Delivery to some customers may be optional provided we

incur a specified penalty cost for not delivering (or hiring an outside carrier at a known cost).

10.Period VRPs arise in the distributing firms that are interested in developing a set of daily routes for some T-day period so that each customer receives delivery at a designated frequency.

Typical Complications in Real VRPs11. Inventory routing problems (IRP) arise in the distribution

of liquid products (e.g., gases or gasoline). Each customer has an inventory of product, and the distributor must determine the timing and amount of deliveries so that the customer does not run out of product.

12. The objective function in real problems can be complex, including terms dependent on the distance traveled, the number of vehicle used, the time duration of routes, and penalties.Many costs involved: Fixed charge; Variable costs per loaded mile & per empty mile; Waiting time; Loading & unloading cost; Penalties; etc.

VRP with Time WindowsHard time windows:

The deliveries must be made within specified time windows.

Soft time windows: The deliveries may be made outside the specified time windows with penalties.

Mixed Pickup & Delivery VRP

Can pickups and deliveries be made on same trip?Can they be interspersed?

Depot

Pickup

Delivery

Mixed P&D VRP Interspersed

Depot

Pickup

Delivery

Depot

Not Interspersed

Depot

Separate routes

Interspersed Routes

depot

For clockwise trip:

Load at depot

Stop 1: Deliver A

Stop 2: Pickup B

Stop 3: Deliver C

Stop4: Deliver D

etc.

A

B

ED

C

F

G

HI

J

K

L

ACDFIJK

CDFIJK

BCDFIJK

BDFIJK Delivering C requires moving B

BFIJK Delivering D requires moving B

Pickup

Delivery

VRP with BackhaulsShould a vehicle return empty (deadhead) or find a

backhaul?How far out of the way should it look for a backhaul?Compare profit from deadheading and carrying

backhaul.

D

Depot

AB

CA

B

C

D

Empty Return

Backhaul

Pickup

Delivery

Cognitive Approach for VRPsWhen points are spatially related, good

solutions can be found by using the pattern recognition capabilities of the human mind.Good stop sequences are formed when the

paths of the route do not cross.The route shape will usually bulge, or form a

teardrop shape, where possible.An analyst may be able to quickly sketch

out a route plan that might require a computer many hours to find.

Poor vs. Good Routing

ExampleA company uses route salespersons to sell beer and other

beverages from a truck owned by the local distributor. The salesperson is paid on a commission and, like the

distributor, does not wish to spend more time or travel any greater distance than necessary to cover the accounts on a daily basis.

Pins on a map are used to locate the current accounts for a particular salesperson.

An example of this type of information for 20 rural accounts has been transferred to the grid-overlay map shown as Figure 7-7(a).

The coordinate relate to distance. The truck is to start at the depot and return to the depot,

visit all accounts and travel the least distance possible.

ExampleTry the cognitive approach. Compare your solution with the one

generates by ROUTESEQ module in LOGWARE, as shown in Figure 7-7(b).

The total tour distance (cost) is 37.59 coordinate units.

This is a good solution, but not necessarily an optimal one.

Example

When Points are NOT spatially related.The spatial relationships become distorted

for practical reasons.(Travel barrier, one-way streets, traffic congestion)

Exact distances or times should be specified between stop pairs.

Cognitive procedures are less applicable.Use one of the mathematical procedures

suggested in the researches to find the stop sequences on a route.

Solution procedures tend to give approximates answers.

Principles for Good Routing & SchedulingGood solutions to the VRP problems can be

found by applying “principles for good routing and scheduling” or some “logical heuristic procedures”.

Consider the routing & scheduling problem where trucks are to start at a central depot, visit multiple stops to make deliveries, and return to the depot in the same day.

Principles for Good Routing & Scheduling1. Form truck routes around clusters of stops

that are nearest to each other in order to minimize the inter-stop travel between them.

Principles for Good Routing & Scheduling2. Stops on different days should be arranged to

produce tight clusters. The stops should be segmented into separate

routing & scheduling problems for each day of the week.

The daily segments for which routes & schedules are to be developed should avoid overlapping stop clusters.

Minimize the number of trucks needed to serve all stops.

Minimize truck travel time & distance during the week.

Daily Segments

Principles for Good Routing & Scheduling3. Build routes beginning with the farthest stop from

the depot and then working back toward the depot. Identify the farthest stop. Select the volume from the tightest cluster of stops

around that stop to fill out the assigned truck capacity.

After the stop volumes have been assigned to the vehicle, select another vehicle and identify the farthest stop from the depot among the remaining stops not yet routed.

Proceed until all stops have been assigned to the vehicles.

Principles for Good Routing & Scheduling4. The sequence of stops on a truck route

should form a teardrop pattern. No routes paths cross. The route appears to have a teardrop shape. However, time window restrictions and the

forcing of stop pickups after deliveries may cause route paths to cross.

Principles for Good Routing & Scheduling5. The most efficient routes are built using the

largest vehicles available. Ideally, using a vehicle large enough to handle

all stops in one route will minimize total distance or time traveled to serve the stops.

Therefore, the largest vehicles should be allocated first, providing that good utilizations for them can be utilized.

Principles for Good Routing & Scheduling6. Pickups should be mixed into delivery routes rather

than assigned to the end of routes. To minimize the amount of path crossing that can

occur when such stops are served after all deliveries are made.

Depend on: The vehicle configuration The size of the pickup volumes The degree to which they may block access to the

delivery merchandise inside the vehicle The compatibility of merchandises (Cross-contamination

among food, raw materials, and chemicals)

Principles for Good Routing & Scheduling7. A stop that is isolated from a route

cluster (especially that with low volume) is a good candidate for an alternate means of delivery.

Using small trucks to handle such stops may prove to be more economical, depending on the isolation of particular stops and volumes.

Using a for-hire transportation service would be a good alternative.

Principles for Good Routing & Scheduling8. Narrow stop time window restrictions

should be avoided. Time window restrictions on stops, where they

are narrow, can force stop sequencing away from ideal patterns.

Any stop(s) forced to be served in a less-than-desired routing pattern should have its time window limits renegotiated and hopefully widened.

Example: The Case Casket CompanyThe company manufactures and distributes a complete

line of burial caskets to funeral homes.Typically, a funeral director will order caskets to

replenish his or her inventory or to meet the particular needs of a family.

Orders are usually in small quantities, frequently no more than one casket at a time.

The company’s warehouse operates two specially racked trucks that each haul a maximum of 18 caskets.

Delivery are made 5 days a week.Develop a routing & scheduling plan for the given

week for the territory in Figure 7-11.

1. Segment the territory into 5 daily customer clusters based on 5 delivery days per week.Cluster customers starting with the farthest customer

and then adding customers by progressively moving toward the warehouse.

Construct 4 customer groups for outlying stops for the first four days of the week, and one group for the fifth day that serves stops close to the warehouse.

Balancing the workload for each of the five groups is desired to avoid the need for more than 2 trucks.

Since Detroit is a dominant market point, we divide it over 2 days.

Sweeping in both directions to capture approximately equal stop volume in each of the four stop clusters.

2. Load the truck and design routing patterns.Each day is considered separately.Starting with the farthest stop, cluster enough

stops in close proximity to each other until the capacity of a truck is filled.

Then, select the farthest stop from the remaining ones and repeat the process for loading the next truck.

For the stops assigned to a truck, the stops are sequenced so that the paths within a route do not cross, and they bulge outward. Cluster-First, Route-Second