Large Scale Petrol Station Replenishment Problem with Time Windows:

A Real Case Study

Pablo A. Villegas and V´ıctor M. Albornoz

Departamento de Industrias, Universidad T´ecnica Federico Santa Mar´ıa, Av. Santa Mar´ıa 6400, 7660251, Santiago, Chile

Keywords:

Petrol Station Delivery Problem, Time Windows, Trip Packing, Scheduling, Heuristic.

Abstract:

In this study we deal with a real case of the problem known as Petrol Station Replenishment Problem with

Time Windows arising in Chile. The company involved is one of the biggest actors in the country, and every

day must schedule a series of trips from their depots to their clients, delivering different kinds of fuel. This

speciﬁc case has some differences from prior formulations of this problem (e.g. trucks works in shifts with

hard time windows). Also, another challenge is the high number of orders and trucks involved in everyday

planning. To solve this problem in reasonable computing times we propose a sequential insertion heuristic.

Finally, we present results over a month of data.

1 INTRODUCTION

Oil companies have the duty to supply their clients

(gas stations and industrial clients) with different

kinds of fuel, that is why every day they are con-

fronted with the challenge of arranging a schedule

program for the next day. This work is motivated by

the challenge that one of the biggest oil companies

in Chile faces when dealing with this scheduling pro-

gram.

The problem known as Petrol Station Replenish-

ment Problem with Time Windows (Cornillier et al.,

2008a; Cornillier et al., 2012) or PSRPTW, corre-

sponds to a speciﬁc multi-compartment vehicle rout-

ing problem or MCVRP, which includes time win-

dows constraints. This problem consists in assigning

a series of products demanded by clients to available

truck compartments and schedule their delivery time.

Thus, trips must be deﬁned and assigned to trucks

considering time windows constraints.

In this case we deal with trucks that are without

ﬂow meters, thus, every truck compartment must be

delivered to no more than one client. Also, trucks

are able to make more than one trip during their work

shift.

The contribution of this work comes in two as-

pects. First, we deal with a real case which has some

slight differences over the ones previously worked,

for instance, trucks works in shifts with deﬁned

time windows. Shift starting and ending times vary

from truck to truck, different from prior formulations

(Benantar and Ouaﬁ, 2012; Cornillier et al., 2009;

Cornillier et al., 2012; Surjandari et al., 2011) that

considered either maximum worktime or closing time

windows at the depot. Here, the ﬁnal objective is to

maximize compliance with customer orders, follow-

ing a certain priority, with a limited heterogeneous

ﬂeet. Also, costs do not change based on the lapsed

time from the start of the ﬁrst trip till the end of the

last, as every trip has its own cost. Client priority

could have been addressed by forcing orders fulﬁll-

ment, however, orders may focus on a certain time

lapse making it an infeasible problem, while posing

this problem as the maximization of dispatched prod-

uct by priority enable us to get the best possible so-

lution, assuming the possibility that not every order

will be delivered in the agreed time window. Our

second contribution consists on a proposed heuris-

tic for this speciﬁc case, which main feature is get-

ting a relatively good operational solution in a short

time frame, so that people in charge of scheduling can

work over this proposal. This heuristic consist on an

adapted version of the insertion heuristic proposed by

(Solomon, 1987) for a vehicle routing or scheduling

problem with time windows.

The view from the standpoint of compliance with

customer brings a wider view than just minimizing

costs, as there is value associated with each client that

is not necessarily reﬂected by the net proﬁt of a trans-

action. This is very important in the case of fuels, as

the proﬁt per transaction is relatively low, and the gain

is in the quantity sold. Therefore, it may be important

408

Villegas, P. and Albornoz, V.

Large Scale Petrol Station Replenishment Problem with Time Windows: A Real Case Study.

DOI: 10.5220/0005823604080415

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 408-415

ISBN: 978-989-758-171-7

Copyright

c

2016 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

to give preference to a client making large quantities

of orders.

The remainder of this papers is organized as fol-

lows. In Section 2, we address some relevant litera-

ture related to previous works dealing with this prob-

lem. In section 3 we formally deﬁne the problem ad-

dressed in this work and introduce some assumptions

and characteristics that have impact in our solution

approach. In Section 4 we introduce the solution strat-

egy, including our proposed heuristic. In Section 5 we

present results obtained over a month of data on one

of the company depots. In Section 6 we provide some

ﬁnal conclusions and future research.

2 LITERATURE REVIEW

In this section we provide a brief overview of pub-

lished literature concerning the Petrol Station Replen-

ishment Problem (PSRP) over the last few decades.

(Brown and Graves, 1981) were the ﬁrst to pub-

lish an article dealing with the PSRP. They were faced

with a situation involving more than 80 depots and

300 tank trucks. The main features of the problem

was that all the orders corresponded to full truck-

loads, so each trip could only reach one customer.

To solve this case, they proposed an integer program-

ming model to assign commands to different trucks,

in order to minimize the amount of travel costs, es-

tablishing a penalty if the need for overtime or idle

time existed.

(Brown et al., 1987) described a decision-making

support system, where they focused on the possibility

to add more than one customer per trip to their previ-

ous work. The distribution network involved was 120

tanks and 430 tanker trucks.

(Van der Bruggen et al., 1995) in a counseling pro-

cess to a Netherlands company, proposed some sim-

ple models: assign customers to depots, determining

ﬂeet size and composition, and restructure the net-

work of depots. The problem considered ﬁve depots,

20-30 tanker trucks and about 800 customers.

(Ronen, 1995) described some of the features

present in the operating environment of this problem

and presented three models that can be applied to it:

The set partitioning model, the elastic set partitioning

model, and the set packing model.

(Taqa allah et al., 2000) addressed the problem

dealing with one tank and a homogeneous tank truck

ﬂeet. They proposed four original heuristics, three

which visited one client by lap while the fourth allows

more than one client in every trip, and stations can re-

ceive multiple visits. They showed results for cases

up to 75 clients with different evaluation horizons.

(Mal`epart et al., 2003) worked with the problem of

supply over a time horizon of several business days.

This problem stems from the notion that in some cases

the fueling stations are managed by the company, so

they can decide at their convenience which day they

will be supplied. They proposed four heuristics to

solve this problem.

(Avella et al., 2004) studied the case of one de-

pot and heterogeneous tank truck ﬂeet, where orders

were integral multiples of m

3

and truck compartments

leave either full or empty. They proposed a set parti-

tioning formulation and developeda branch-and-price

algorithm that could solve the problem up to 6 trucks

and 25 customers. They also proposed a heuristic to

select a promising set of columns.

(Ng et al., 2008) studied two distribution networks

in Hong Kong. The resulting problem corresponds

to a model of supply over several working days, as

stations inventories were managed by the company.

They proposed a model to assign trips to trucks and

stations simultaneously. The model does not guaran-

tee that the stations are not left without fuel. Between

the two areas studied, the number of stations was 48

and had a ﬂeet of 8 tank trucks.

(Cornillier et al., 2008a) worked with the problem

of one depot and unlimited heterogeneous ﬂeet. They

proposed an exact algorithm and conducted a test with

an instance of 42 stations and 8 types of vehicles.

Same year, (Cornillier et al., 2008b) treated the multi-

period case with limited heterogeneous ﬂeet, where

they proposed a multiple phase heuristic. They con-

ducted tests for this heuristic with instances up to 200

customers and a 28 days planning horizon.

(Cornillier et al., 2009) proposed two heuristics

for the case of limited heterogeneous ﬂeet and cus-

tomer time windows. To verify the performance

of these heuristics, they used instances of up to 50

clients.

(Boctor et al., 2011) addressed the trip packing

problem arising in the PSRP. They proposed a formu-

lation for the generalized version of that problem and

four heuristics to solve it.

(Surjandari et al., 2011) worked a real case with

time windows, multi-depot, split-deliveries and a lim-

ited heterogeneous ﬂeet of tank trucks. As a solution

they proposed a tabu search algorithm. In their work

they show results for that particular case, which had

two depots and a ﬂeet of 76 tank trucks.

(Benantar and Ouaﬁ, 2012) discussed the case

with time windows, one depot and a limited hetero-

geneous ﬂeet. They presented an algorithm based on

tabu search, solving instances of over 100 customers

and 20 trucks.

(Cornillier et al., 2012) proposed a heuristic for

Large Scale Petrol Station Replenishment Problem with Time Windows: A Real Case Study

409

the multi-depot problem with time windows. This

heuristic included the generation of all possible trips

visiting less than three clients and then selecting the

most promising one and solve the allocation problem.

In their work they show results of instances up to 6

depots, 10 trucks and 50 customers.

(Wang et al., 2014) presented the results of a meta-

heuristic called ”guided reactive tabu search” applied

to the problem of a single depot with heterogeneous

ﬂeet. They addressed instances up to 200 orders.

(Coelho and Laporte, 2015) deﬁned and com-

pared four categories of the single period multi-

compartment vehicle routing problem. This problem

is one of the possible variants that appears when ad-

dressing the PSRP. For each of the particular cases

they proposed a mathematical model and a ”branch-

and-cut” algorithm that is applicable to all of them.

Largest instances that could be solved accurately con-

tained 50 clients in the single period case and 20 for

the multi-period version.

To the best of our knowledge there has not been

addressed a real case with instance sizes bigger than

the ones presented in this work or time windows as-

sociated with truck shifts. The challenge is to get an

acceptable solution (proposal) for planners in a rea-

sonably short time, as they will work over this so-

lution. To this effect, we followed the strategy used

by (Cornillier et al., 2012) separating routes creation,

their allocation to the various truck shifts and ﬁnally

deﬁning their schedule.

3 PROBLEM DEFINITION

The PSRPTW used in this real case can be deﬁned

as follows. Let G = ({0, n+ 1} ∪C, A) be a directed

graph, where {0, n + 1} corresponds to the depot,

C = {1, ..., n} is the set of clients, and A = {(i, j) :

i, j ∈ {0, n+ 1} ∪C, i6= j} is the set of arcs. Every arc

(i, j) has associated a travel time t

i, j

. Also, service

time s

i

at depot or station i, where i ∈ {0, n + 1} ∪C,

is known. The set of trucks associated to the depot is

denoted by K. Client i orders are composed of P

i

or-

der lines, where every line speciﬁes a different kind of

product and which type of compartment will be ﬁlled

with that order, also, every order speciﬁes an ideal

time window. Trucks must get to the client i inside

the proposed time window [a

i

, b

i

]. Furthermore, ev-

ery truck must comply with their shift time window

[α

′

k

, β

′

k

].

The real nature of the problem leads to many sit-

uations where theory differs from reality, so it is im-

portant to make a balance between which elements

of reality we will model and those we will simplify

through assumptions.

One of the most important simpliﬁcations of the

model is present in the step of route generation.

Clients are grouped by zones, so travel time is di-

rectly associated to the zone they are into. As a rule of

thumb, routes visiting more than one client shall only

visit clients inside the same zone. For this and some

other restrictions, orders may be grouped in a way so

that they will be assigned to one and just one possi-

ble route. Resulting routes will have just one kind of

truck assigned (capacity related), letting us solve the

problem for every truck type separately. Orders that

do not comply to these rules are left outside this prob-

lem, as the compliance decision is in the schedulers’

hands. As an interesting fact, volume of discarded

orders is often below 1% of total volume transported

during a work day.

Next, other important assumptions are noted for

this particular problem:

• Orders are composed of order lines, every client

may generate orders lines till they ﬁll a truck. This

orders may contain more than one kind of product,

but no more than one by compartment.

• Every client is free to generate more than one or-

der. Every one of such is treated separately as they

have their own requirements and time windows.

• Every order must be complied with just one trip.

• A route may include more than one client, but no

more than two.

• Trucks may have more than one shift during the

working day. As they are independent, they are

treated as different trucks.

• There exists relevance associated to the priority of

an order, this will be represented by constant ρ.

4 SOLUTION STRATEGY

To solve this problem in reasonably low computing

times, we have separated this problem in two phases,

addressed separately:

• Phase I: Route generation

• Phase II: Route assignment & Scheduling

We have developed a heuristic which follows the

same idea used by the sequential insertion heuristics

proposed by (Solomon, 1987). It also shows some

similarities with the construction heuristics proposed

by (Boctor et al., 2011).

In addition to parameters already deﬁned, the fol-

lowing notation will be used:

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

410

Indices:

t, l: route index

k: truck index

v: trip position index

Sets:

T: set of routes

T

k

: set of routes associated to truck k

V: set of possible trip indices.

V = {1, 2, ..., N}

V

k

: set of possible trip indices for truck k

Parameters:

N: maximum number of possible trips

during a shift

ρ

t

: route t’s priority associated value

α

t

: starting time of route t’s time window

β

t

: ending time of route t’s time window

λ

t

: route t’s time duration

α

k

: starting time of truck k’s time window

β

k

: ending time of truck k’s time window

Next, we describe the two phases of this problem

and how we address them.

4.1 Phase I: Route Generation

This ﬁrst phase is considered an initialization process.

It consist on the generation of the set of routes T. As

said in section 3, order aggregation in routes must fol-

low certain company rules. This allows us to group

orders in a way so that they will be assigned to one

and just one possible route. Resulting routes will have

just one kind of truck assigned, letting us solve the as-

signment problem for every truck type separately.

4.2 Phase II: Assignment & Scheduling

The main objective of the company is to maximize the

routes to be performed, weighted by their respective

priority, while complying with their time windows.

The objective function of the assignment problem can

be expressed as follows:

Maximize:

∑

t∈T

∑

v∈V

∑

k∈K

ρ

t

x

tvk

, (1)

where x

tvk

corresponds to a binary variable that takes

value 1 if route t corresponds to the v-th trip of truck

k, 0 otherwise. The complete assignment model can

be checked in the appendix section. As solving this

problem by exact methods proves to be difﬁcult with

high number of trucks and routes, we used a sequen-

tial insertion heuristic, which will assign routes to one

truck at a time until the truck can not take in another

route.

For every route in the truck, the position and start-

ing time must be deﬁned, thus leading to the Schedul-

ing problem. The next model deals with the Schedul-

ing of routes for every truck k:

Variables:

x

tv

: binary that takes value 1 if route t

corresponds to the v-th trip of truck k; 0

otherwise

d

v

: starting time of v-th trip of truck k

Minimize:

∑

v∈V

k

d

v

(2)

Subject to:

∑

v∈V

k

x

t,v

= 1, ∀t ∈ T

k

(3)

∑

t∈T

k

x

t,v

= 1, ∀v ∈ V

k

(4)

∑

t∈T

k

α

t

x

t,v

≤ d

v

≤

∑

t∈T

k

β

t

x

t,v

, (5)

∀v ∈ V

k

d

v

≥ d

v−1

+

∑

t∈T

k

λ

t

x

t,v−1

, (6)

∀v ∈ V

k

| v 6= 1

α

′

k

≤ d

1

(7)

d

n(V

k

)

≤ β

′

k

−

∑

t∈T

k

λ

t

x

t,n(V

k

)

(8)

x

t,v

∈ {0, 1} , ∀t ∈ T

k

, ∀v ∈ V

k

(9)

d

v

∈ R, ∀v ∈ V

k

(10)

Objective function (2) minimizes starting times

while privileging shorter trips before longer ones if

possible, this function corresponds to a secondary op-

timization criteria. Restrictions of type (3) ensures

that every route assigned to truck k will be performed

once. Type (4) restrictions states that every trip posi-

tion will host one route, no more, no less. Restrictions

of type (5) ensures that every route must comply their

time window if they correspond to the v-th trip. Type

(6) restrictions states that v-th trip starting time will

occur after the end of the previous one. Restriction

(7) ensures that the ﬁrst trip must comply with truck’s

starting time. Restriction (8) states that the last trip

must ﬁnish before the ending time window of truck

k. Finally, restrictions of type (9) and (10) show vari-

ables’ nature.

Our proposed sequential insertion heuristic can be

expressed as follows:

For every truck:

1. Fill a list with the set of unassigned routes.

2. Sort routes in descending order by priority.

Large Scale Petrol Station Replenishment Problem with Time Windows: A Real Case Study

411

Table 1: Main test results.

Day NR NT T (sec) T

AMPL

(sec) T

CPLEX

(sec) OF COF COF1

1 195 142 44.9 3.3 41.6 109678 83224 84849

2 207 97 42.6 2.8 39.8 80577 63613 69163

3 47 50 5.8 0.4 5.5 15647 14011 14811

4 272 142 67.6 5.2 62.4 113129 98638 103721

5 238 150 50.0 4.0 46.0 87056 66270 69610

6 294 158 81.0 6.2 74.8 117471 96616 99563

7 236 153 51.7 3.8 47.9 81992 69319 72674

8 297 156 86.8 6.1 80.6 118004 98471 99303

9 260 111 59.2 4.2 55.0 92722 74538 78578

10 62 47 7.8 0.6 7.2 15961 16013 17988

11 279 150 86.8 6.8 80.1 115868 101731 105428

12 286 154 77.2 6.0 71.2 96268.5 73765 78488

13 299 156 99.6 7.0 92.6 114833 94907 96577

14 268 154 98.8 7.0 91.8 83135 65521 66696

15 312 157 100.6 7.0 93.6 123246 108039 111279

16 261 111 53.3 3.6 49.7 88379 70378 73412

17 58 48 7.3 0.5 6.9 16342 15103 16735

18 287 141 78.8 6.0 72.9 114749.5 98706.5 110828.5

19 273 148 60.6 4.3 56.3 90570 76992 79452

20 318 152 101.5 6.9 94.5 116170 91926 101455

21 190 148 37.3 2.4 34.8 78426 61614 62914

22 255 155 58.7 4.5 54.2 121452.5 98923.5 99753.5

23 251 112 39.7 2.8 36.9 90362 71204 77964

24 58 53 8.5 0.5 8.0 19889 17900 18700

25 282 144 81.0 5.5 75.5 112043 101201 105358

26 280 154 58.3 4.2 54.1 90863.5 70729.5 75139.5

27 300 152 89.7 6.9 82.8 121682 95812 99022

28 260 153 56.2 3.4 52.8 83511.75 67608.75 72073.75

29 310 158 105.5 7.5 98.1 127477.5 105188.5 109408.5

30 282 105 58.1 4.1 54.0 88796 67359 69930

31 80 56 15.9 1.2 14.7 20518 16650 17495

3. Filter routes by truck’s time window compatibil-

ity.

4. Assign a route from the list, which complies to a

certain starting criteria, to the truck and extract it

from the list.

5. For every element in the list:

(a) Verify compatibility with current routes as-

signed. If veriﬁcation is positive, then solve the

scheduling problem including the current route.

Else, move onto the next element of the list.

(b) If scheduling problem is correctly solved, as-

sign current route to the truck, extract it from

the list and check for potential remaining

routes. Else, move onto the next element of the

list.

(c) If there are potential routes, move onto the next

element of the list. Else, move onto the next

truck.

The process ends once we have been through every

truck or all routes have been assigned.

5 RESULTS

We tested our heuristic against a month of real data.

Each day presented different number of orders and

trucks involved, so a wide variety of possibilities are

covered. The route start criteria used was: any route

that holds the highest priority on the list. As for the

main parameters used, we set the maximum number

of trips per shift to 4, and ρ(i) takes values from table

3.

Our heuristic was coded in AMPL language, using

CPLEX 12.1, in a computer with O.S. Windows 8,

Intel Core i5-4200U 1.6 - 2.3 GHz processor and 8

GB of RAM.

As shown in table 1, the objective function ob-

tained by our heuristic (OF) outperforms the ones

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

412

Table 2: Detailed Compliance Level.

Day NR NT T (sec) C CC CC1 CCCL CCCL1 CCT

1 195 142 44.9 97.9% 77.4% 79.0% 90.3% 92.8% 86.2%

2 207 97 42.6 89.9% 81.2% 86.0% 91.3% 98.1% 87.9%

3 47 50 5.8 87.2% 83.0% 85.1% 93.6% 95.7% 89.4%

4 272 142 67.6 93.4% 83.5% 87.1% 94.1% 98.2% 88.6%

5 238 150 50.0 97.1% 81.9% 84.9% 93.3% 96.6% 88.2%

6 294 158 81.0 98.6% 85.0% 87.8% 94.9% 97.6% 90.1%

7 236 153 51.7 99.2% 87.3% 91.5% 92.4% 96.6% 94.9%

8 297 156 86.8 98.0% 84.2% 84.8% 96.3% 97.3% 87.5%

9 260 111 59.2 98.1% 85.0% 88.1% 94.2% 98.8% 89.2%

10 62 47 7.8 72.6% 82.3% 87.1% 93.5% 98.4% 88.7%

11 279 150 86.8 93.9% 83.5% 86.4% 96.4% 99.3% 87.1%

12 286 154 77.2 99.3% 83.9% 87.8% 93.0% 96.9% 90.6%

13 299 156 99.6 97.3% 85.3% 87.0% 96.3% 98.3% 88.6%

14 268 154 98.8 99.3% 83.6% 84.3% 96.3% 97.4% 86.6%

15 312 157 100.6 98.4% 86.5% 88.1% 96.2% 98.1% 89.4%

16 261 111 53.3 98.1% 82.8% 85.4% 94.6% 97.7% 87.4%

17 58 48 7.3 77.6% 81.0% 86.2% 93.1% 98.3% 86.2%

18 287 141 78.8 94.8% 81.9% 89.2% 88.9% 98.3% 89.9%

19 273 148 60.6 96.7% 81.7% 87.9% 90.1% 96.3% 91.6%

20 318 152 101.5 93.7% 83.3% 87.4% 92.1% 96.2% 91.2%

21 190 148 37.3 100.0% 83.7% 85.8% 95.3% 97.4% 88.4%

22 255 155 58.7 99.2% 85.5% 86.7% 96.5% 98.0% 88.6%

23 251 112 39.7 96.8% 85.7% 90.4% 92.8% 99.2% 91.2%

24 58 53 8.5 96.6% 89.7% 91.4% 94.8% 96.6% 94.8%

25 282 144 81.0 90.8% 81.9% 86.5% 92.2% 98.2% 87.2%

26 280 154 58.3 99.6% 82.1% 86.8% 91.4% 97.1% 88.9%

27 300 152 89.7 99.0% 84.7% 87.0% 93.7% 96.0% 91.0%

28 260 153 56.2 99.2% 83.8% 87.3% 93.1% 96.9% 90.0%

29 310 158 105.5 97.7% 81.0% 84.2% 94.8% 98.1% 85.8%

30 282 105 58.1 94.7% 78.4% 81.2% 92.9% 96.5% 84.4%

31 80 56 15.9 92.5% 85.0% 90.0% 91.3% 96.3% 92.5%

Table 3: ρ values.

Type A Client Type B Client

Type A Order 125 25

Type B Order 5 1

from the company (COF) in 30 out of 31 cases. There

is a second acceptance criteria which states that any

order may have plus/less one hour of error. Consider-

ing this criteria, our heuristic gets a higher objective

function than the ones from the company (COF1) in

29 out of 31 cases. Looking deeper within these two

cases, it is important to state that few of the routes

were made by a truck with higher capacity, action that

needs further conﬁrmation and is not included in the

set obtained in Phase I. Overall, our heuristic is able

to ﬁnd good solutions despite being simple and fast.

The average increase in the objective function respect

COF is of 22%.

The lowest computing times (T) are reported on

days 3, 10, 17, 24 and 31, which corresponds to

Sundays, where there is little movement compared to

other working days. Time required to solve this prob-

lem on these days ranges from 6 to 16 seconds. In-

terestingly, the only two days where our heuristic’s

resulting objective function was lower than the com-

pany’s one are present in this group. There is no clear

difference between other days of the week.

Excluding Sundays, the number of routes (NR)

ranges from 190 to 318, the number of trucks (NT)

ranges from 97 to 158 and solution time ranges from

37 to 106 seconds. An increase in any of them, routes

and trucks, have a negative impact in computing time,

although it is expected that a higher number of routes

encourages the company to try and get enough truck

shifts to perform them, so they often grow together.

Breaking down total computing time in time used

by CPLEX solver (T

CPLEX

) and the rest of the phases

(T

AMPL

), it is possible to see that most of the time

is used to solve the single truck scheduling problem

with tentative sets of routes assigned. Any improve-

Large Scale Petrol Station Replenishment Problem with Time Windows: A Real Case Study

413

ment to the selection of routes to be included or to the

model itself is expected to have a huge impact on total

computing time.

Table 2 shows compliance levels. We can see that

our heuristic is capable of obtaining a proposal with a

really high number of trips while fulﬁlling all restric-

tions proposed, even outperforming the compliance

of the company (CC), even considering the error ac-

ceptance criteria (CC1) in most of the cases. We can

see that the compliance of the company with clients’

time windows is really high (CCCL), their main prob-

lem resides in being able to manage compliance with

trucks’ time windows (CCT). If we go by the criteria

of an acceptable error of plus/less one hour, the com-

pany compliance with the clients (CCCL1) is higher

than our heuristic in 18 out of the 31 days.

In average our heuristic got a 95.1% compliance

level, which correspond to an increase of 14% with

respect of the current compliance level of the com-

pany over that month.

The CCT column in table 2 shows that truck com-

pliance is fairly lower when comparing it to client

compliance or our heuristic’s result. This means that

truck compliance is one of the key points that can be

drastically improved in practice.

6 CONCLUSIONS AND FUTURE

RESEARCH

From the results obtained, we can say it is possible

to include trucks shifts’ time windows in the schedul-

ing problem faced by the company, as it is possible to

reach at least the same compliance level the company

already has without considering it, for most of cases.

The end result from our heuristic correspond to a

proposal which schedulers usually request at the start

of their day or along it, specially when big changes

to the current schedule plan is required, so it must be

able to present a good starting plan in a short span

of time. As stated in results section, the heuristic is

able to handle the biggest instances of this problem in

at most 2 minutes, which is a fairly good time span

considering the compliance yielded.

Future research will be focused on the impact of

including more real life aspects to the two phases

mentioned in 3, and test different aspects of the

heuristic so it can solve them, such as: the efﬁciency

of using a parallel insertion heuristic for this case; the

impact on time solution of the secondary optimization

criteria; and lastly, test different route starting criteria,

as they might boost the heuristic performance.

ACKNOWLEDGEMENTS

This research was partially supported by Direccion

General de Investigacion y Postgrado (DGIP) from

Universidad Tecnica Federico Santa Maria, Grant

USM 28.15.20. Pablo Villegas also wishes to ac-

knowledge the Graduate Scholarship also from DGIP.

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APPENDIX

Assignment Model

Variables:

x

tvk

: binary that takes value 1 if route t

corresponds to the v-th trip of truck k; 0

otherwise

d

tvk

: starting time of route t if it corresponds

to the v-th trip of truck k

Maximize:

∑

t∈T

∑

v∈V

∑

k∈K

ρ

t

x

tvk

(11)

Subject to:

∑

v∈V

∑

k∈K

x

tvk

≤ 1, ∀t ∈ T (12)

∑

t∈T

k

x

tvk

≤ 1, ∀v ∈ V, ∀k ∈ K (13)

∑

t∈T

k

x

t,v−1,k

≥

∑

t∈T

k

x

t,v,k

, (14)

∀v ∈ V | v 6= 1, ∀k ∈ K

α

t

x

tvk

≤ d

tvk

≤ β

t

x

tvk

, ∀t ∈ T, (15)

∀v ∈ V, ∀k ∈ K

α

′

k

x

tvk

≤ d

tvk

≤

β

′

k

− λ

t

x

tvk

, (16)

∀t ∈ T, ∀v ∈ V, ∀k ∈ K

d

tvk

≥

∑

l∈T

k

,l6=t

d

l,v−1,k

+ λ

l

x

l,v−1,k

,

(17)

∀t ∈ T, ∀v ∈ V | v 6= 1,

∀k ∈ K

x

t,v,k

∈ {0, 1} , d

t,v,k

∈ R, (18)

∀t ∈ T, ∀v ∈ V, ∀k ∈ K

Objective function (11) maximizes priority as-

signed. Restrictions of type (12) ensures that routes

can be performed at most once. Type (13) restrictions

ensures that at most one route can be performed as

v-th trip of truck k. Restrictions of type (14) ensures

that in truck k, the v-th trip exists just if trip v-1 also

exists. Type (15) restrictions states that if route t is

the v-th trip of truck k, its departing time must be

inside the route’s time window. Restrictions of type

(16) states that if route t corresponds to the v-th trip

of truck k, its departing and arriving time must be in-

side truck’s time window. Type (17) restrictions states

that starting time of route t in truck k must be higher

than the arrival time of the previous trip. Finally, type

(18) restrictions show variables’ nature.

ρ

t

constant is calculated as follows:

ρ

t

=

∑

i∈C

t

ρ

i

∗ Q

i

(19)

Where C

t

is the set of orders involved in route t, ρ

i

is the priority modiﬁer of order i, and ﬁnally Q

i

is the

quantity of product to be delivered in order i.

Large Scale Petrol Station Replenishment Problem with Time Windows: A Real Case Study

415