Optimization of a location-routing problem with
multiple delivery options
Rimar Lont
MASTER THESIS TOM Faculty of economics and businessUniversity of Groningen
First supervisor: dr. X. Zhu, co-assesor: prof. dr. ir. G.J.C. Gaalman r.j.lont@student.rug.nl
s2804638
June 2016
Abstract
Preface
This thesis originates from the eager to learn programming and, off course, the will to finish my Master Technology and Operations Management. The project of Alliance Healthcare was a challenging opportunity to learn this by developing and programming a heuristic for an extension of the Location-Routing Problem. This process sometimes was very frustrating and hard, but all and all I am proud of the result. Besides that, I learned the basics of programming, which is something I am delighted about.
1
Introduction
The online retail sector has become an important part of the global economy (Cata and Lee, 2006). It seems that retailers are able to successfully cope with orders when looking at their internal warehouse processes. However, successfully distributing the parcels seems to be lacking. Research has proven that 60% of the home-deliveries fail due to the absence of a customer at home (Punakivi and Tanskanen, 2002). A huge cost saving potential lies within the use of parcel boxes or pick-up lockers as an alternative for home-deliveries (Morganti, Dablanc, and Fortin, 2014). The introduction of these pick-up lockers or parcel boxes to prevend home-delivery failures is common subject in literature (Weltevreden and Rotem-Mindali, 2009; Mason-Jones, R. and Towill, 1999). A study of Punakivi and Tanskanen (2002) has showed that cost savings up to of 55-66% can be achieved by using these lockers instead of home delivery.
This thesis is motivated by a problem of Alliance Healthcare Nederland(AH). AH is a multidis-ciplinary company that produces and distributes medicines. They are active in 25 countries where they sell medicines via 13,100 own pharmacies. Overall, AH supplies more than 200,000 pharmacies through 350 distribution centres (http://www.alliance-healthcare.nl) and are seeking for ways to reduce distribution costs and increase their service level. At the moment, customers can either pick-up their medicines at a pharmacy or receive their medicines at home by means of a delivery service. Next to the home-delivery service and the option to pick-up medicines at a pharmacy, AH has introduced two other types of pick-up alternatives to reduce costs and increase the service level. These pick-up points are lockers where customers can pick up their medicines at their own convenience. The pick-up options of AH are: (1) pick-up at a local pharmacy, (2) pick-up at a regular medicine locker and (3 )pick-up at an advanced medicine locker. Since AH distributes medicines, safety is of paramount importance. New customers can therefore only pick-up their recipe through option 1 or 3 in order to receive personal user-instructions on their medicines. Customers that frequently order medicines can use all pick-up options. When regular customers are increasingly using the pick-up options, more time will be available within pharmacies to increase the service quality to customers.
where each iteration is built-up in two phases. The first phase constructs a trial solution by means of a randomized search heuristic. The second phase improves this solution by local search. The last procedure is considered as one of the most efficient and accurate heuristics for solving a LRP. Laporte and Louveaux (1989) give an overview of solving methods for the LRP which are founded before 1989 and are succeeded by Nagy and Salhi (2006) who conducted a survey of solving location-routing problems until 2006. Nagy and Salhi (2006) make a distinction between exact and heuristic methods. The difference is that the exact method can only solve small problems, while the heuristic approach can find a feasible solution in a polynomial-time for larger problems. The heuristic methods can be split up in three main solution methods according to Nagy and Salhi (2006). (1) Clustering methods that are suitable for small numbers of potential depots. (2) Iterative methods that determine the optimal location and routing by iterating between locational and routing phases. (3) Hierarchical methods that use the result of a single-depot clustering heuristic as basis.
The problem of AH can be seen as a multi-depot location-routing problem (MDLRP) (Wu, Low, and Bai, 2002) since the optimal location of lockers needs to be determined in a given set of new and regular customers, depots and pharmacies. This MDLRP (Prodhon and Prins, 2014) is a common studied subject. However, in case of AH the customer has the possibility to travel to a pick-up locker, a pharmacy or receive home-delivery. To the best of my knowledge there are no extensions of the MDLRP that include the decision if a customer travels to a pick-up point or receives home delivery. Tijhuis (2015) proposes an extension of a location-routing problem including customer pick-up. However, in this thesis there is only one type of customer and one type of pick-up locker and in addition to that there is no covering area of a locker included. Tijhuis (2015) suggests that further research should generalize this algorithm to solve a LRP including multiple pick-up options. Creating this extension is very important since, in case of AH, more patients can be served within less time and at a lower cost. Besides that, customers that pick up their medicines at the lockers do not need to come to the pharmacy, resulting in more time for the customers in the pharmacies. In addition to that, this extension might be useful for other distribution companies e.g. Bol.com and CoolBlue which are intending to introduce pick-up points.
The focus of this thesis will be to formulate an extension of the LRP including multiple pick-up options and two types of customers. The model determines the locker locations and vehicle routings which can serve as an optimization tool in AH’s investment decisions. The model is solved by a heuristic which finds the solution in two phases. The locations, number and type of lockers are determined in phase 1 whereas the route between the depot and the customers receiving home deliveries is constructed in the second phase.
2
Mathematical model and problem definition
The LRP model of Wu, Low, and Bai (2002) forms the basis for the mixed integer linear mathematical programming formulation below. In the proposed model there is an undirected graph G=(V,E,C). V is a set of nodes composed of the subsets: n Depots (D), n pharmacies (P), n possible locations for regular or advanced medicine lockers (L), a set of n regular customers (R) and a set of n new customers (N). E is a set of edges (i,j) between all the nodes consisting of dij which is the distance
between node i and node j. C consists of the costs cd
ij which are delivery costs (linear to the length
of the travelled edge), Or are the costs for opening a regular medicine locker and Oaare the costs for opening an advanced locker. Besides that, each home delivery has a fixed cost of ch. In addition the model has the following parameters: Since a pharmacies an lockers can only cover a certain range, this is specified with Rp for the range of a pharmacy, Rr the range of a regular locker and Ra the range of an advanced locker. Besides that, the pharmacies have a capacity Ip, a regular locker a capacity Ir and an advanced locker has the capacity Ia. K is a set of vehicles and Q is the capacity of each
vehicle. At least, each regular customer has a demand drj and each new customer has a demand dnj. All these characteristics are summarized in table 1.
Table 1: Model parameters
V Set of nodes Rp Range pharmacy Oa Opening costs advanced locker E Set of edges Ra Range advanced locker Or Opening costs regular locker C Set of costs Rr Range regular locker L Subset of possible locker locations K Set of vehicles Ip Capacity of a pharmacy R Subset of regular customers
Q Capacity of vehicle Ia Capacity advanced locker N Subset of new customers
D Subset of depots Ir Capacity regular locker P Subset of pharmacies
dr
j Demand regular customer dnj Demand new customer ch Fixed cost home delivery
The objective is to minimize the total distribution costs by determining the number and locations of pick-up points, the assignment of customers to pick-up or home delivery, and the route of the vehicle. The total costs consist of costs for opening a pick-up point and the distribution costs. To model this, some assumptions are made, also based on Wu, Low, and Bai (2002). The first assumption is that pick-up points are always filled since the supplier of AH is doing this. Besides that, the following assumptions are made:
- There is a given set of homogeneous vehicles which are used and the route of the vehicle always begins and ends at a depot, sub tours are not allowed.
- The cost of delivery operations is linear to the total distance travelled by the vehicle. - There is a set of pharmacies which are already opened.
- Pick-up points and pharmacies are replenished by the supplier, thus it is assumed that these are always filled.
- The distances considered are the Euclidean distances between the nodes, besides that the dis-tances are symmetric (distance i to j = j to i).
- There can only be a regular locker, advanced locker or no locker on a possible locker location. - Customers cannot pick up their medicines at a depot.
- The demand of a each customer equals 1.
The decision variables which are used in the model are:
yi =
(
1 if regular medicine locker is opened on location i ∈ L 0 otherwise
zi =
(
1 if advanced medicine locker is opened on location i ∈ L 0 otherwise
pRij =
(
1 if regular customer j is assigned to pick up at local pharmacy i 0 otherwise
rRij =
(
1 if regular customer j is assigned to pick up at regular medicine locker i 0 otherwise
aRij =
(
1 if regular customer j is assigned to pick up at advanced medicine locker i 0 otherwise
hRj =
(
1 if regular customer j receives home delivery from the depot 0 otherwise
pN ij =
(
1 if new customer i is assigned to pick up at local pharmacy j 0 otherwise
pNij =
(
1 if new customer i is assigned to pick up at local pharmacy j 0 otherwise
aNij =
(
1 new customer i is assigned to pick up at advanced medicine locker j 0 otherwise
hN j =
(
1 if new customer j receives home delivery from depot 0 otherwise
xijk =
(
1 if arc i to j is traversed by vehicle 0 otherwise
uj= sub tour size
pRijdij ≤ Rp ∀i ∈ R, j ∈ P (5) pNijdij ≤ Rp ∀i ∈ N j ∈ P (6) rijRdij ≤ yjRr ∀i ∈ R, j ∈ L (7) rijRdij ≤ zjRa ∀i ∈ N, j ∈ L (8) aNijdij ≤ zjRa ∀i ∈ R, j ∈ L (9) X i∈R rijRdRj ≤ yjIr ∀j ∈ L (10) X i∈R aRijdRj +X i∈N aNijdNj ≤ zjIa ∀j ∈ L (11) X i∈R∪D X k∈K xijk = X j∈P hAij ∀j ∈ R (12) X i∈N ∪D X k∈K xijk= X j∈P hNij ∀i ∈ N (13) X i∈V X j∈R∪B (dRj + dNj )xijk≤ Q ∀k ∈ K (14) X j∈V xijk− X j∈V xjik≤ 1 ∀i ∈ V, ∀k ∈ K (15) X i∈D X j∈V xijk≤ 1 ∀k ∈ K (16)
Binary constraints xijk ∈ {0, 1} ∀i ∈ V, ∀j ∈ A ∪ B, ∀k ∈ K yj ∈ {0, 1} ∀j ∈ L zj ∈ {0, 1} ∀j ∈ L pAij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V rAij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V aA ij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V hAij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V pBij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V aBij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V hBij ∈ {0, 1} ∀i ∈ D, ∀j ∈ V
The objective function (1) comprises the costs for opening a regular locker, opening an advanced locker, the costs for home delivery and the costs for travelling to customer’ home. The travelling costs are assumed to be linear to the travelled distance. The used parameters are specified in appendix A. Constraint (2) ensures that there can only be a regular medicine locker, an advanced locker or nothing on location j. Constraint (3) ensures that all regular customers and new customers pick up their medicines at a regular locker, an advanced locker, a pharmacy or receive home delivery. Constraint (4) is similar but ensures that a new customer cannot pick-up medicines at a regular locker. Constraint (5) guarantees that regular customers can only pick-up at a pharmacy if they are within the covering area of the pharmacy. Constraint (6) does the same for new customers. Constraint (7) ensures that regular customers can only pick-up at a locker if they are in the covering range of a regular locker and the locker is opened. Constraint (8) ensures the same for regular customers who are assigned to an advanced locker. Constraint (9) guarantees the same for new customers and advanced lockers. Constraint (10) ensures that the total demand of regular customers assigned to a regular customer does not exceed the capacity of the regular locker. Constraint (11) ensures the same for advanced lockers. Constraint (12) and (13) ensures that customers who are assigned to receive home delivery from the depot are admitted to the route of the vehicle. Constraint (14) guarantees that the total demand of customers who receive home delivery does not exceed the capacity of the vehicle. Constraint (15) ensures the continuity of the route of the vehicle. When a vehicle goes to a customer, it also has to leave. Constraint (16) guarantees that each vehicle at max one route. Constraint (17) is based upon Desrochers and Laporte (1991) and prevent sub-tours. This means that a route cannot only exists of customers but always has to start and end at a depot. The remaining constraints are binary constraints.
3
Proposed heuristic
for making a comparison. The pharmacies are excluded from the heuristic since all customers in the covering area of a pharmacy will be assigned to the pharmacy since the pharmacies have fixed locations and no capacity constraints. Furthermore, one depot and one vehicle are used in determining the route and the vehicle has unlimited capacity. Considering multiple depots and vehicles is more realistic but not considered within the scope of this thesis. The benefit of this assumption is that it reduces model complexity and simplifies the coding process. An overview of the two phases can be seen in figure 1. The first phase is explained in section 3.1, followed by the second phase in section 3.2. Since the routing heuristic appeared to be not very accurate (see section 4) this is improved by a repetitive nearest neighbour algorithm and a genetic algorithm.
Start
Phase I heuristic:
Find locker locations
Step 1:
Input data (NbNewCustomers, NbRegularCustomers, NbLockerLocations, Depot)
Phase II heuristic:
Find route using NNA
Step 2:
Calculate total costs and elapsed time
End
Figure 1: Overview complete heuristic
3.1 Phase I: Finding optimal locker locations
The first phase of the heuristic consists of several steps. The first step is the model’s input of the data and parameters. The results of the objective function strongly depends on the parameter settings. For example, if the costs for opening a regular- or advanced locker (Or&Oa) are relatively high, the heuristic will assign all customers to home delivery. Contrary, if the costs for home delivery (CH) are
opened or not. This is done by using the following two formulations that determine the threshold, specified in percentage of the total capacity of a locker, to open a locker: (OA/CH)/IA where IA is
the capacity of an advanced locker and (OR/CH)/IR where IRis the capacity of a regular locker. For example, if the capacity of an advanced locker =IA = 100, the costs for opening an advanced locker OA= 150 and the costs for home delivery CH = 2, this means that there should be more customers
than (150/2)/100 = 75% of the total capacity in the covering area of the advanced locker to open it. After the input of the data and the parameters the heuristic starts examining each possible locker location. If the number of new customers in the covering range is greater or equal than OA/CH)/IA
Start
NbNewCustomers in covering area > x% of capacity advanced locker?
Capacity left?
RegularCustomers in covering area to fill up remaining capacity?
Open advanced locker on location j
yes
Assign RegularCustomer to advanced locker
yes
NbRegularCustomers in covering area > x% of capacity regular locker? no
yes
Do not open a locker on location j no
Step 1:
Input data (NbNewCustomers, NbRegularCustomers, NbLockerLocations, Depot) yes NewCustomers left in covering area? no Capacity left? yes yes no End no Assign RegularCustomer to regular locker j
Capacity left? yes RegularCustomers left in covering area? no
yes
no no
Open regular locker on location j
Assign NewCustomer to advanced locker j yes
3.2 Phase II: Finding route using RNN
During the second phase of the heuristic a feasible route between all customers with home delivery service is constructed. This route is constructed by using a Nearest Neighbour Algorithm (NNA). Research has proven that the NNA is a popular technique to find feasible and acceptable routes for symmetric LRP, VRP and the TSP using Euclidean distances (Gutin, Yeo, and Zverovich, 2002). The route heuristic works as follows: The route starts at the depot and searches for the nearest customer. The nearest customer will be the next point of the route. Next it searches for the unvisited customer the nearest to the current customer. This step is repeated until all customers are visited. The route is complete when the vehicle returns to the depot after visiting the last customer. The last step of the heuristic is calculating the total cost calculation, on the opened number of regular and advanced lockers, the number of people who are assigned to receive home delivery, and the total length of the route. Figure 3 visualizes the steps in the same ways as phase I.
Since the NNA appeared to be not very accurate for larger datasets (10 or more customers), it is replaced by a repetitive NNA (RNN). The RNN algorithm has proven to find more optimal routes compared to the NNA (Adewole, Agwuegbo, and Otubamawo, 2011). The RNN works as follows: Besides the depot as starting node, the algorithm is repeated by choosing all other nodes (the customers which receive home delivery and the depot) as the starting point of the algorithm. Each time a tour is generated, the distance is stored. When all tours are generated, the shortest tour is selected as the route for the vehicle.
3.3 Alternative phase II: Finding route using GA
Even though the RNN provides good results, it has the disadvantage that with the adding of one extra node the complexity increases with the order N2. Therefore a genetic algorithm (GA) is programmed to find a feasible route between the customers who receive home delivery.
Step 1:
Input data (NbClients assigned to receive home delivery from
phase I)
Start
Start from node and go to nearest neighbour
Nodes left to visit? Go to nearest neighbour
Go back to start node
Store route length
End
yes
no
Chose shortest route as final route Nodes left to chose as starting point yes no Step 1:
Input data (NbClients assigned to receive home delivery from
phase I)
Start
Create set of random tours
Select tours to be parents
Perform crossovers
End
Chose shortest route as final route Termination
criterion?
no
Yes
Evaluate set of random tours
Mutate tours
Figure 3: Repetitive Nearest Neighbour Step 1:
Input data (NbClients assigned to receive home delivery from
phase I)
Start
Start from node and go to nearest neighbour
Nodes left to visit? Go to nearest neighbour
Go back to start node
Store route length
End
yes
no
Chose shortest route as final route Nodes left to chose as starting point yes no Step 1:
Input data (NbClients assigned to receive home delivery from
phase I)
Start
Create set of random tours
Select tours to be parents
Perform crossovers
End
Chose shortest route as final route Termination
criterion?
no
Yes
Evaluate set of random tours
Mutate tours
4
Computational study
To evaluate the performance of the proposed heuristic, a comparison is made between the outcome of the exact method and the outcome of the heuristic. The performance measurement is based upon the accuracy of the outcome and the speed it is solved. This is done by programming the exact model in IBM CPLEX Studio where it uses the Mixed Integer Programming Solver. The heuristic is programmed in MatLab. The computational test are performed on a HP EliteBook 8540w, with a 64-bits processor, Intel(R) Core(TM) i7 CPU and 8 GB of RAM running on windows 10. Exact solutions of larger instances (e.g. 10 possible locker locations, 24 regular customers and 12 new customers) could not be used in the comparison due to computational limitations. Therefore a good performance benchmark cannot be performed. To come up with a good benchmark of the heuristic, both phases of the heuristic are tested separately. For the first phase this is done by comparing the outcome with a facility location problem and for the second phase by comparing the outcome with a published routing benchmark dataset.
4.1 Model parameters
As explained previously, the input data such as the costs for opening an advanced locker have a significant impact on the outcome of the objective function. To make the right trade-off between opening a locker or assigning customers (regular and new) to home delivery the parameters in ap-pendix 1 are used. These are found by using the formulas presented in section 3.1 ((OA/CH)/IA & (OR/CH)/IR) and some experimenting. The possible locker locations, the location of the depot, the locations of the new customers and the regular customers are generated randomly by using the random data generator of Microsoft Office Excel 2016. The random data generator generates integer values uniformly between 0-100 which are used as X and Y coordinates. Considered distances between e.g. new clients and a locker location are Euclidean distances. These are calculated by the formulation: d(a, b) = p
(x1− x2)2+ (y1− y2)2. In addition to that, the distance from a to b is the same as b to
a. It is assumed that there is only one vehicle (K = 0) with unlimited capacity (Q = 0), one depot (D = 0) and no pharmacies. Pharmacies are left out of the model since the scope of this thesis is to find optimal locker locations for different types of customers next to the already opened pharmacies. So customers in the area of a pharmacy will always be assigned to it, since this is the cheapest op-tion. In future research the heuristic can be extended by using multiple depots, multiple vehicles with capacity constraints and the inclusion of pharmacies.
4.2 Testing complete heuristic
tried, but these lead to errors in the outcome of the objective function. To further test the performance of the heuristic, both phases are tested separately in section 4.3 and 4.4.
The comparison is made by running each problem size five times with the same parameters but with different locations for the lockers, customers and the depot. In total there are 75 sample instances tested. The entire outcome for each test can be viewed in appendix B. In table 2, L represents the number of potential locker locations, N is the number of new customers and R is the number of regular customers. Besides that, the maximum gap, the average gap and the best gap between the outcome of the exact solution and the heuristic are displayed.
From the results in table 2 it can be concluded that the heuristic performs good. It finds solutions within 5.3 % on average of the optimum. Besides that, it is very fast. Even for datasets which cannot be benchmarked by the exact solution it finds a solution within fraction of a second. In the tables in the appendix 2 it can be seen that in some cases there is a big gap between the exact solution and the heuristic. This can be clarified by the random distribution of the coordinates and the way the heuristic works. In the cases with a big gap (max 26.8 %), there were one or two customers and a locker located far away from the other points. In these cases, the heuristic will not open a locker but the exact solution will open it since it takes the distance to travel to this customers into account. The last instance, with 20 locker locations, 40 new customers and 80 regular customers is added to show the speed of the heuristic.
Table 2: Outcome test: complete heuristic
Exact method Heuristic Gap
L N R Time Time worst best average
2 2 4 0,16 0,24 9,4% 0,0% 2,9% 2 3 6 0,34 0,29 15,1% 0,0% 5,6% 2 4 8 1,45 0,31 11,6% 0,0% 4,4% 2 5 10 1,61 0,37 12,2% 1,8% 5,0% 3 5 10 2,56 0,35 9,3% 0,0% 5,3% 2 6 12 3,32 0,37 14,3% 0,0% 4,4% 3 6 12 7,29 0,42 18,1% 2,0% 8,0% 2 7 14 8,74 0,36 13,7% 0,0% 4,2% 3 7 14 12,70 0,35 26,8% 1,4% 8,7% 4 8 16 24,20 0,28 7,3% 2,2% 4,3% 5 8 16 65,80 0,30 13,2% 1,3% 5,4% 5 9 18 118,40 0,28 11,2% 3,7% 6,1% 5 10 20 213,40 0,30 12,3% 2,9% 6,8% 10 10 20 539,20 0,30 12,5% 0,0% 3,6% 20 40 80 x 0,35 x x x 13,4% 1,1% 5,3% 4.3 Testing phase I
same as the tests in section 4.1. So when there are 10 new customers, the number of regular customers is 20. The number of possible locker locations varies from 10 to 20. Figures 5 and 6 shown an example of a situation with 10 possible locker locations, 10 new customers and 20 regular customers. Each L is a possible locker location, R is a regular customer, N is a new customer and D is the depot. When the heuristic is finished (see figure 6), the blue dots represent the regular lockers that are opened (3 in this case). The regular customers (red dots) are assigned to the opened lockers. All the other customers (pink stars) are assigned to receive home delivery.
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 D N N N N N N N N N N R R R R R R RR R R R R R R R R R R R R R R R R R R R R R R R R R R R R R RRR L L L L L L L L L L
Figure 5: Situation before heuristic
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 D N N N N N N N N N N R R R R R R RR R R R R R R R R R R R R R R R R R R R R R R R R R R R R R RRR L L L L L L L L L L
Figure 6: Situation after heuristic
Comparing the heuristic with the exact solution can be seen as a comparison with a facility location problem (Aikens, 1985) since the decision needs to be made which type of lockers should be opened at specified locations. The outcomes of the heuristic and the exact solution appear to be exactly the same for all cases, as seen in table 2. Despite the equal objective outcomes, customers are not always assigned to lockers in the exact same way. This is due to the fact that in case a locker is opened, both methods randomly assign customers within the lockers’ covering area until its capacity is reached. The exact solution does the same, so there can be a difference. This phenomenon is only observed whit large data sets and 20 possible locker locations. The solving time is left out of table 2 as both methods solve the objective function within a fraction of a second. Therefore this time is not listed in the table.
Table 3: Outcome test: phase I
Exact method Heuristic Gap L N R Objective Objective Average
4.4 Testing phase II
Phase II covers the routing problem for customers who are assigned to receive home delivery, which is basically the same as the classical TSP. An example of how the heuristic works can be viewed in figure 7 & 8. Figure 7 represents the output of phase I and figure 8 the output after phase II. Because the exact method is limited to solve only until 40 delivery points, phase II is tested differently than phase I. to be able to assess the performance of the heuristic in the second phase, several symmetric TSP datasets, retrieved from Reinelt (1991) are used as input. Making use of these published datasets is a commonly used way of testing the performance of a routing heuristic (see for example: St¨utzle and Hoos (1999)). 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 D N N N N N N N N N N R RR R R R R R R R R R R R R R R R RR R R R R R R R R R R R R R R R R R R R R L L L L L L L L L L L L L L L L L L L L
Figure 7: Situation before heuristic
0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 D N N N N N N N N N N R RR R R R R R R R R R R R R R R R RR R R R R R R R R R R R R R R R R R R R R L L L L L L L L L L L L L L L L L L L L
Figure 8: Situation after heuristic
Table 4: Outcome: test phase II
NN RNN GA
C time gap time gap time gap
131 0,0024 26% 0,29 11% 17 5% 237 0,0035 30% 0,97 20% 42 14% 343 0,0046 56% 2,08 30% 189 20% 380 0,0052 24% 3,09 17% 202 28% 395 0,0064 30% 2,7 21% 276 28% 411 0,0067 37% 3,66 25% 152 23%
average 34% average 21% average 20%
4.5 Discussion
From table 1 in section 4.1 it can be concluded that the complete heuristic performs good on small datasets. When the size of the instance increases the heuristic still finds a good solution compared to the lower bound of the exact solution method. However, since the exact solution method can not find a solution for large instances, a good performance benchmark is missing. A suggestion for further research would be to use a supercomputer to find optimal solutions for larger instances.
In section 4.1 it is observed that some instance have a large gap in the objective outcome. This can be clarified by the operation of phase I of the heuristic. The decision whether to open a locker or not depends on the number of new and regular customers in the covering area of a locker. It neglects its location with respect to the other customers, lockers, and the depot. The exact solution method does take this into account. Moreover, the used parameters are not very realistic. However, they are chosen in this way to make the right trade-off between opening a locker versus deliver the medicines at home. Using the low parameter values (e.g. the advanced locker capacity Ia= 5) is also beneficial for the solving time of the exact method.
In section 4.3 it is observed that when the first phase of the heuristic is replicated multiple times for the same data set, it might assign different customers to a locker in case the number of customers within the covering area exceeds the total capacity of the locker (applies to both types of lockers and customers). This phenomenon affects the outcome objective function of the heuristic but only occurs at larger instances. This can be explained by the way the heuristic works. The heuristic randomly selects customers in the covering area to assign to the locker. Further research should adapt it such that it selects the customers in a way that is beneficial for the route of the vehicle.
From table 4 in section 4.4 it can be concluded that the GA doesn’t find much better solutions compared to the to the RNN except for some instances where the total distance is up to 10% shorter. This result can be clarified by the fact that the coordinates of the new and regular customers, the locker locations and the depot are uniformly distributed and the distance between them are considered as symmetric Euclidean distances. This is beneficial for the RNN. However, when actual road distances would be taken into account by using GPS data, the RNN is expected to find worse solutions than the GA. Therefore my suggestion for further research would be to benchmark the heuristic with the GA as the second phase with more challenging datasets.
one depot, unlimited vehicle capacity and a static locker capacity should be carefully discussed when used in investment decisions.
5
Conclusion
In this thesis an attempt has been made to design an investment decision tool which can lower the costs and improve the quality of the distribution process of AH. The goal was to find the optimal locations for pick-up lockers where customers can pick-up their medicines at their own convenience or deliver the medicines at their homes. This is done by formulating an extension of the location-routing problem where two types of customers can travel to pick-up lockers is proposed. Since this is a NP-hard problem a heuristic is designed. The heuristic consist of two phases, in the first phase the locations of the lockers is determined and the decision is made which customers are assigned to a locker and which customers receive home delivery. The second phase consist of creating the route between the customers which are assigned to receive home delivery using a RNN. The heuristic seems to work good. It is extensively tested on small instances to come up with a good benchmark. The heuristic finds the solution within 5.3 % of the optimum and some times it even finds the optimal solution, in addition to that it finds a solution within a second. However, for larger instances the heuristic could not be compared with the exact solution, since the exact solver cannot find the solution for larger instances. The lack of a published benchmark set to test the performance of the heuristic is the biggest limitation of this research. Therefore, no conclusion can be made about the performance of the heuristic for larger instances. Further research should at least provide a proper benchmark for both heuristics, by for example programming the mathematical model in C++ and use a supercomputer to find the optimal solution for larger instances.
The performance of my heuristic and the heuristic of Tijhuis (2015) is almost the same. However, my heuristic in fact solves a FLP and a single depot TSP in a smart way, where Tijhuis (2015) heuristic solves a single depot TSP including pick-up. Future research could improve my heuristic by e.g. using a GRASP heuristic, which has proven to be one of the most efficient and accurate heuristics for solving a LRP.
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Appendix A: Model parameters
Table 5: Model parameters Parameters used in the model
Symbol Description Value
Oa Opening costs for advanced locker 5
Or Opening costs for regular locker 4
Ch Costs for home delivery 2
Ia Capacity of advanced locker 5
Appendix B: Test instances complete heuristic
Table 6:
2 lockers, 2 new customers, 4 regular customers instance heuristic time exact time difference
1 262 0,25 262 0,2 0% 2 244 0,24 235 0,16 4% 3 267 0,25 267 0,12 0% 4 317 0,218 312 0,14 2% 5 255 0,23 231 0,19 9% Max 9% Min 0% Average 3%
Average solving time H 0,2376 Average solving time E 0,162
Table 7:
2 lockers, 3 new customers, 6 regular customers Instance Heuristic time exact time Difference
1 290 0,29 290 0,41 0% 2 274 0,28 259 0,62 5% 3 260 0,31 260 0,3 0% 4 324 0,29 275 0,12 15% 5 349 0,28 324 0,25 7% Max 15% Min 0% Average 6%
Table 8:
2 lockers, 4 new customers, 8 regular customers Instance Heuristic time exact time Difference
1 398 0,35 352 1,37 12% 2 335 0,3 326 1,41 3% 3 294 0,28 294 1,52 0% 4 434 0,32 401 1,54 8% 5 351 0,32 351 1,41 0% Max 12% Min 0% Average 4%
Average solving time H 0,314 Average solving time E 1,45
Table 9:
2 lockers, 5 new customers, 10 regular customers Instance Heuristic time exact time Difference
1 437 0,38 412 1,51 6% 2 456 0,4 446 1,53 2% 3 382 0,41 371 1,56 3% 4 458 0,33 402 1,81 12% 5 391 0,34 384 1,62 2% Max 12% Min 2% Average 5%
Average solving time H 0,372 Average solving time E 1,606
Table 10:
3 lockers, 5 new customers, 10 regular customers Instance Heuristic time exact time Difference
1 357 0,34 346 2,4 3% 2 345 0,38 313 2,5 9% 3 384 0,36 363 2,6 5% 4 425 0,35 388 2,8 9% 5 364 0,34 364 2,5 0% Max 9% Min 0% Average 5%
Table 11:
2 lockers, 6 new customers, 12 regular customers Instance Heuristic time exact time Difference
1 379 0,36 379 4,12 0% 2 364 0,38 360 4,2 1% 3 388 0,38 378 2,5 3% 4 463 0,37 397 3,8 14% 5 359 0,38 344 2 4% Max 14% Min 0% Average 4%
Average solving time H 0,374 Average solving time E 3,324
Table 12:
3 lockers, 6 new customers, 12 regular customers Instance Heuristic time exact time Difference
1 394 0,42 350 8 11% 2 458 0,54 375 10 18% 3 416 0,39 400 7,2 4% 4 481 0,36 457 6,23 5% 5 499 0,37 489 5 2% Max 18% Min 2% Average 8%
Average solving time H 0,416 Average solving time E 7,286
Table 13:
2 lockers, 7 new customers, 14 regular customers Instance Heuristic time exact time Difference
1 416 0,4 416 9 0% 2 553 0,41 477 10,2 14% 3 476 0,38 470 8,2 1% 4 476 0,31 472 9 1% 5 462 0,29 438 7,3 5% Max 14% Min 0% Average 4%
Table 14:
3 lockers, 7 new customers, 14 regular customers Instance Heuristic time exact time Difference
1 549 0,4 402 10,5 27% 2 553 0,41 501 12 9% 3 501 0,38 491 15 2% 4 485 0,31 478 12 1% 5 478 0,29 460 14 4% Max 27% Min 1% Average 9%
Average solving time H 0,358 Average solving time E 12,7
Table 15: My caption
4 lockers, 8 new customers, 16 regular customers Instance Heuristic time exact time Difference
1 499 0,31 483 22 3% 2 496 0,28 465 21 6% 3 480 0,29 468 23 3% 4 540 0,26 528 25 2% 5 504 0,27 467 30 7% Max 7% Min 2% Average 4%
Average solving time H 0,282 Average solving time E 24,2
Table 16: My caption
5 lockers, 8 new customers, 16 regular customers Instance Heuristic time exact time Difference
1 492 0,35 477 60 3% 2 462 0,28 456 67 1% 3 526 0,29 516 75 2% 4 474 0,31 437 72 8% 5 524 0,29 455 55 13% Max 13% Min 1% Average 5%
Table 17:
5 lockers, 9 new customers, 18 regular customers Instance Heuristic time exact time Difference
1 492 0,34 437 110 11% 2 512 0,29 480 115 6% 3 515 0,28 496 119 4% 4 508 0,25 482 130 5% 5 576 0,26 551 118 4% Max 11% Min 4% Average 6%
Average solving time H 0,284 Average solving time E 118,4
Table 18:
5 lockers, 10 new customers, 20 regular customers Instance Heuristic time exact time Difference
1 460 0,31 421 200 8% 2 560 0,29 540 205 4% 3 608 0,26 533 196 12% 4 558 0,32 542 250 3% 5 565 0,31 528 216 7% Max 12% Min 3% Average 7%
Average solving time H 0,298 Average solving time E 213,4
Table 19:
10 lockers, 10 new customers, 20 regular customers Instance Heuristic time exact time Difference
1 461 0,29 460 500 0% 2 572 0,31 573 525 0% 3 601 0,32 578 601 4% 4 547 0,3 537 575 2% 5 585 0,29 512 495 12% Max 12% Min 0% Average 4%
