IN
DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS
STOCKHOLM SWEDEN 2020,
Optimization of Scheduling the Use of Vats During the Wine Fermentation Process
LINNÉA ANDERSSON
FRIDA EKNER
Abstract
In this paper the goal was to optimize the scheduling of vats for the wine fermentation process considering demand, time and quality con- straints in order to minimize the production cost. The method that was used during the project was to start of with a simple model that only took the basic parameters in to account, verify that it worked, and then add parameters until the model was completed. An integer program was formed to obtain the optimal solution to the problem and to test the pro- grams accuracy a small example that could be tested by hand was made.
It showed that the program worked as expected, furthermore the program works for larger problems, that are more based in reality. After testing the program for larger problems another conclusion that was drawn was that for instance a larger vintage period and an increased frequency of supply led to an increase in the solving time.
Sammanfattning
Syftet med den h¨ar rapporten var att optimera schemal¨aggningen av k¨arl f¨or vinfermenteringsprocessen med avseende p˚a efterfr˚agan, tid och kvalitet f¨or att minimera produktionskostnaden. Tillv¨agag˚angss¨attet under projektet var att utg˚a ifr˚an en enkel model som endast tog de grundl¨aggande parametrarna i ˚atanke, verifiera att den fungerade, och se- dan l¨agga till parametrar tills att modellen var full¨andad. Ett dynamiskt program togs fram f¨or att finna den optimala l¨osningen och f¨or att tes- ta dess tillf¨orlitlighet anv¨andes ett mindre exempel. Det visade sig att programmet fungerade som det skulle och d¨armed drogs slutsatsen att programmet ¨aven ska fungera f¨or st¨orre, mer verklighetstrogna, problem.
Efter att ha testat programmet f¨or st¨orre problem drogs ¨aven slutsat- sen att faktorer som bland annat l¨angre perioder och h¨ogre frekvens p˚a tillf¨orsel av druvor leder till st¨orre l¨osningstider.
Contents
1 Introduction 3
2 State of the art 4
3 Methodology 5
3.1 Sets . . . . 6
3.2 Parameters . . . . 7
3.3 Decision Variables . . . . 7
3.4 Model . . . . 8
3.4.1 Constraints . . . . 8
4 Results 10 4.1 Example . . . . 10
4.2 Testing the Model . . . . 12
5 Conclusion 15
1 Introduction
Wine is one of the most popular beverages in the world with a great diversity in style and quality. The production consists namely of a mix of different types of fermentation batches, alcohol concentration, flavor and many other features where the mixer has to have the right amount of each element for the fermen- tation process to occur. This leads to a distinction in quality and taste which promotes discussion and disagreement among people interested in wine, which in turn results in wine from different countries and areas varying in popularity, according to for instance perceptions of style, quality and value [10].
Nowadays most wine production is made with advanced technology methods, but there are some producers that are holding on to more traditional techniques.
Wineries often look somewhat similar to laboratories and every part of the pro- cess can be accurately controlled. In wine production there are two stages;
the growing of grapes (viticulture) and turning grapes into wine (vinification) whereas this project concerns to enhance the latter. [8]
With an industry always in motion as competition increases from different coun- tries [10] and the price of wine expecting to hit its lowest levels in five years [7], it is getting more difficult for the wine companies to make a profit. There are many ways to raise the profit margins and one of the least risky ways is to minimize the production costs [10]. As there has been proven to be a positive correlation between profits and investing in technology and digital tools [4], it is becoming more relevant for the wine companies to make these updates. One way to do this is to introduce optimization into the wine making process, and attempts at doing this have been done in the past [1][6][9]. The purpose of this project is to introduce optimization in the planning of the fermentation process.
In preparation for the fermentation process the wine companies must decide if they have to rent vats from other companies in order to meet the supply they will receive, as it is important that the grapes are quickly put into a vat or there is a risk that they will degrade and cause problems in the fermentation process [5]. The objective of this project is, through the development of an integer pro- gram, to obtain an optimal solution to minimize the total cost of fermenting the given supply considering the cost to ferment in a specific type of vat and the cost to rent a vat for a decided period of time.
2 State of the art
In this section, the aim is to explore how the problem of sorting grapes into vats is solved today, and if similar research has been done in this field. While researching this project, we were in contact with an agricultural engineer who has worked in this field for the last five years. He stated that in wine yards today, the scheduling of grapes into vats during the fermentation process is of- ten assembled more on past experience instead of a mathematical formula. It was also noticed that not much similar work has been done in this field, but a summary of relevant research has been included below.
In [9] the authors attempted to find an optimal strategy for pressing grapes in one of the largest wineries in Portugal. In contrast to previous research they used a stochastic method instead of linear programming. They came up with an optimal algorithm which they claim can be used for similar problems.
In [1], the purpose was to optimize vineyard replacement. Due to the grapevine production decaying over time, the vineyard eventually has to be replaced. One attempt at creating a strategy for this problem used binary non-linear program- ming methods, and a model that took the vineyards main dynamics like wine quality and production efficiency in to account was created.
Authors of [6] had the goal to optimize wine production in all stages, from vineyard to bottle, by developing software tools. They conclude that even wine companies with little or no research and development resources can use the tools that were produced. The article also claims that the future competitiveness of winemakers depends on their ability to use optimization potentials based on these models and simulation assisted tools.
As shown the previous research in the field significantly differs from what is attempted in this project. The current process of professional estimation has it downfalls as it becomes more difficult to create an optimal solution when the number of vats and quantity of wine increases. It can therefore be stated, that this project and a continuation of it could have a significant impact on the process and field of wine fermentation.
3 Methodology
While researching this problem we noticed that our problem had many sim- ilarities to a scheduling problem. An example of a scheduling problem from the coursebook, with an integer solution, is a hospital that wants to make a weekly night shift schedule for its nurses. Each nurse works five days in a row on the night shift and the demand for nurses on day j is dj, j = 1, 2, ..., 7. The objective is to find the minimal number of nurses the hospital needs to hire [3]. In our case, our objective is to find the minimal cost of fermenting a set amount of grapes, where the cost is divided up in the costs to run the vats that the company owns or rents, and the cost to acquire a rented vat for the vintage period (whereas the vintage period is defined as the total time the model is run).
As there has been no optimization projects on scheduling problems in the con- text we are working in, we sought out reports on scheduling problems in other fields in order to get an overview on how to create a model for a more complex scheduling problem than the example found in the coursebook. The work of [2]
was of great help for us, even though the subject matter is quite different from what we are working on. It included constraints such as putting surgeries in operation rooms that do not belong to the hospital, some operations not being able to be done in some operation rooms, the number of surgeries being less or equal to the amount of patients with that pathology and only one anesthesiol- ogist being able to in in a OR per day. These constraints were helpful as we have similar constraints such as some types of wines not being able to ferment in some types of vats, all the grapes supplied in a time period has to be put in either the company’s own vats, or a rented one, and that if a wine is fermenting in a specific vat, no more wine can be put into it until the fermentation process is complete.
In the start of the process, a simple model that only took the basic param- eters in to account was formed. Our goal was to verify that it was a working model and to have a basis in which extra complexity, such as the grapes staying in the vats for a certain time period, could be added. Confirming that our pro- totype model worked as intended, more constraints were slowly added. In the end, we have set the model to account for the supply delivered, how long the fermentation process is, the upper and lower bounds of grapes needed in a vat for the fermentation process to occur, the cost to produce a certain amount of wine, and the cost rent a specific vat. It was also defined that all supply that is delivered to the winery has to be put into vats in the same instance of time.
With this in mind, we have set up the model defined in the following sections.
3.1 Sets
The first step in creating the model was to define what will affect the process of putting the grapes into the different vats.
• I = types of wine
The supply of grapes delivered to the fermentation vats will be of differ- ent types. A simple division is between grapes that will be turned into red wine, white wine or ros´e. Then if one gets grapes from different re- gions or vineyards it can be relevant to separate these as well in order to simplify the blending process. Different types of wines also need different conditions for the fermentation process, for example white wines generally require cooler conditions than red ones. [10]
• K = types of vat
The vats used in the fermentation process can be made of different materi- als. The most common types are stainless steel, cement and wood. These have different effects on the grapes due to their different properties and have different requirements to be able to maintain them [10]. The type of vat can also be defined by the capacity of the vat, such as all vats with a max capacity of 5 000 l being one type, max capacity of 10 000 l being another etc.
• J(k) = set of vats of type k ∈ K
This set states the number of vats we have of the different types, e.g three steel vats and two wood ones. The set is divided into two parts: the vats that we own and the vats we can acquire. They are defined as following,
– Ja(k) = the sets of vats of type k ∈ K that we already own.
– Jna(k) = the sets of vats of type k ∈ K that we can acquire during the entire planning horizon.
• T = length of planning horizon, in days.
• O(j, k, t) = set of planned occupancy (e.g. cleaning of vat) of vat j ∈ J of type k ∈ K during the time period t ∈ T
In order to maintain the vats there has to be different periods where they can not be in use for the fermentation process. How the vats are maintained and cleaned will depend on the type of vat, for example stain- less steel vats are easier to clean than wood and cement ones.[10] This set O(j, k, t) therefore contains when each vat has to be cleaned or maintained in another way.
• F (i) = set of vats where wine of type i ∈ I cannot be stored.
As earlier stated the different types of vats have different properties. As different types of wines also have different conditions in order to ferment, this can lead to some vats not being suitable for this process. This set therefore states if there is a type of vat where the fermentation process of the wine of type i ∈ I is not suitable.
3.2 Parameters
The parameters were set to constant values depending on the combination of sets they belong to.
• cj,k,t is the cost per amount of fermenting grapes e.g. cost per liter in vat j ∈ J of type k ∈ K, during time period t ∈ T .
– caj,k,t is the cost when we own the vats, j ∈ Ja,
– cnaj,k,t is the cost when we have to acquire the vats, j ∈ Jna.
• bj,k is the cost of acquiring vat j ∈ Jnaof type k ∈ K.
• Si,t is the supply of wine of type i ∈ I received during time period t ∈ T .
• Lj,k is the lower bound of volume that vat j ∈ J of type k ∈ K must be filled.
• Uj,k is the upper bound of volume that vat j ∈ J of type k ∈ K must be filled.
• Hi,j,k is the amount of time wine i ∈ I must remain in vat j ∈ J of type k ∈ K.
3.3 Decision Variables
The decision variables were used for the constraints so they could be fulfilled.
• xai,j,k,t is a variable larger than zero that represents the amount of wine of type i ∈ I, in vat j ∈ Ja of type k ∈ K, produced during time period t ∈ T .
• xnai,j,k,t is a variable larger than zero that represents the amount of wine of type i ∈ I, in vat j ∈ Jna of type k ∈ K, produced during time period t ∈ T .
• yi,j,k,ta is a binary variable where 1 means that wine of type i ∈ I is in vat j ∈ Ja of type k ∈ K, during time period t ∈ T and 0 means that there is no wine in that specific vat.
• yi,j,k,tna is a binary variable where 1 means that wine of type i ∈ I is in vat j ∈ Jna of type k ∈ K, during time period t ∈ T and 0 means that there is no wine in that specific vat.
• Bj,kis a binary variable where 1 means that the vat j ∈ Jnaof type k ∈ K has been acquired and 0 means that it has not.
3.4 Model
The goal of this project is to get an optimal solution to the problem of minimiz- ing the cost it takes to ferment all of the grapes the company gets supplied with during the entirety of the vintage period. It is defined in the following way:
min X
i∈I
X
j∈Ja
X
k∈K
X
t∈T
caj,k,txai,j,k,t+ X
j∈Jna
X
k∈K
X
i∈I
X
t∈T
cnaj,k,txnai,j,k,t+ X
j∈Jna
X
k∈K
bj,kBj,k. (1)
This minimization is in turn subject to the following constraints.
3.4.1 Constraints
Upper and lower bounds of the amount of wine in the vat
yi,j,k,ta Lj,k≤ xai,j,k,t≤ yi,j,k,ta Uj,k, ∀i ∈ I, ∀j ∈ Ja, ∀k ∈ K, ∀t ∈ T,
ynai,j,k,tLj,k≤ xnai,j,k,t≤ ynai,j,k,tUj,k, ∀i ∈ I, ∀j ∈ Jna, ∀k ∈ K, ∀t ∈ T. (2)
All of the grapes we receive of type i ∈ I during the time period t ∈ T must go into the vats.
X
j∈J
X
k∈K
xai,j,k,t+ xnai,j,k,t= Si,t, ∀i ∈ I, ∀t ∈ T. (3)
The variable xi,j,k,t has to be larger or equal than zero.
xai,j,k,t≥ 0, ∀i ∈ I, ∀j ∈ Ja, ∀k ∈ K, ∀t ∈ T,
xnai,j,k,t≥ 0, ∀i ∈ I, ∀j ∈ Jna, ∀k ∈ K, ∀t ∈ T. (4)
A vat that we do not own can only be used if we acquire it.
ynai,j,k≤ Bj,k, ∀i ∈ I, ∀j ∈ Jna, ∀k ∈ K, ∀t ∈ T. (5)
The vat is planned to not be in use.
yi,j,k,ta = 0, ∀i ∈ I, ∀k ∈ K, t ∈ {0, T − Hi,j,k}, if j ∈ O(j, k, p) when p ∈ {t, t + Hi,j,k}, yi,j,k,ta = 0, ∀i ∈ I, ∀k ∈ K, t ∈ {T − Hi,j,k, T }, if j ∈ O(j, k, p) when p ∈ {t, T }. (6)
yi,j,k,tna = 0, ∀i ∈ I, ∀k ∈ K, t ∈ {0, T − Hi,j,k}, if j ∈ O(j, k, p) when p ∈ {t, t + Hi,j,k}, yi,j,k,tna = 0, ∀i ∈ I, ∀k ∈ K, t ∈ {T − Hi,j,k, T }, if j ∈ O(j, k, p) when p ∈ {t, T }. (7) With these constraints we look at a vat for the time periods it would take to
ferment this grape in it. If the vat for one of these time periods is scheduled to not be in use for example maintenance we cannot put any grapes in it at these times. Therefore the y value must be zero.
The grape cannot be fermented in a specific type of vat.
yi,j,k,ta = 0, ∀i ∈ I, ∀j ∈ Ja, k ∈ F (i), ∀t ∈ T,
ynai,j,k,t= 0, ∀i ∈ I, ∀j ∈ Jna, k ∈ F (i), ∀t ∈ T. (8)
Only one type of wine can be in the vat for the duration of the fermentation process.
X
i∈I p
X
t=0
yi,j,k,ta ≤ 1, ∀j ∈ Ja, ∀k ∈ K, p ∈ {0, Hi,j,k},
X
i∈I p
X
t=p−Hi,j,k
yi,j,k,ta ≤ 1, ∀j ∈ Ja, ∀k ∈ K, p ∈ {Hi,j,k, T }.
(9)
X
i∈I p
X
t=0
ynai,j,k,t≤ 1, ∀j ∈ Jna, ∀k ∈ K, p ∈ {0, Hi,j,k},
X
i∈I p
X
t=p−Hi,j,k
ynai,j,k,t≤ 1, ∀j ∈ Jna, ∀k ∈ K, p ∈ {Hi,j,k, T }.
(10)
As earlier stated the grapes have to ferment for a specific amount of time in the vat. As a simplification we have the constraint that if grapes have been added to the vat no more can be added until after the fermentation process is complete. We have also decided that only one type of wine can be in the vat at a time, as the blending portion of vinification is after the fermentation. [10]
With this constraint we put that the sum for the binary variable for the time period the wine is fermenting, and all types of wine, has to be equal to or less than one. This constraint works as if any of the y variables is one the rest must be zero, which results that if a vat is in use, no more grapes can be put into the vat until after its fermentation process is completed.
4 Results
During this project the model has been tested in two instances. First it has been tested in a smaller example in order to be able to confirm that the model works, then the model has been used in the problem this report is aimed to solve. In order to solve these problems we have used the programs Python 3.7 and Gurobi 9.0.1 with our own computers, they were both Intel Core i5 with 1.6 and 1.8 GHz processors, 8 GB of RAM and macOS Catalina version 10.15.4.
4.1 Example
In order to test if the model works a smaller example that can be tested by hand was created. In this case it was decided that there are two types of wine:
red and blue. There are also two different types of vats, where we own two vats of each type and we can acquire two of each type.
Upper and lower bounds of the vats, fermentation times and costs The upper and lower bounds for the vats are constant no matter the type or vat where the lower bound is 0.5 liters and the upper bound is 1 liter. The red wine ferments for two time periods and the blue wine ferments for three. The blue wine can not be in the second type of vat. We have put the cost of producing wine in our own vats to zero and the cost to acquire a vat for the entire time horizon as 10.
Occupancy and Supply
The first vat of type one is planned to not be in use during the first time period, the second vat of type one is planned to not be in use during the second time period, the first vat of type 2 is planned not to be in use during the third time period, and the second vat of type 2 is planned to not be in use during the fourth time period. This can be seen represented in Figure 1 below. The supply is the same for both types of wine each time period and it is 1 liter for the first time period, 0.5 liters for the second, and 1 liter each for the third, fourth and fifth time periods.
Figure 1: An schematic illustration to represent the scheduled occupancy, time periods, types of vats and number of vats.
Results
This results in Figure 2 where the black blocks are for when the vats are planned to not be in use, the red and darker blue are for when the wine is put directly into the vat and the orange and light blue is for when the vat is occupied for the fermentation process.
Figure 2: An schematic illustration of a smaller example to test the programs accuracy.
Explanation to figure
The idea behind is needless to say that vats 1 to 4 should be used in
preference of vats 5 to 8 since they are more affordable. The blue wine can not be put in vat 3, 4, 7, 8. Therefore the first choice is to put the red wine in 3 or 4 and the blue in 1 or 2.
Supply period 1: For the blue wine vat 1 and 2 is occupied for the three nearest periods so we have to choose 5 or 6. The red wine only ferments for two days so vat 3 will be the best choice.
Supply period 2: The blue wine was put in vat 1 since it can be there for the three nearest periods and since vat 2 is not in use. For the red wine vat 3 is busy but vat 4 is available.
Supply period 4: Vat 4 is not in use so vat 3 was chosen for the red wine. Vat 1 and 2 is occupied so vat 6 is chosen for the blue wine.
Supply period 5: Vat 1 is chosen for the blue wine and vat 2 and 4 can then be chosen for the red wine since vat 3 is not available.
With this small example it was established that it is possible to use an optimization model to create a schedule for putting grapes into vats while considering the supply of grapes, cost of renting vats, different fermentation times, schedules for e.g. cleaning vats, and incompatibilities between types of vats and types of grapes. It was therefore stated that it is possible to continue to create this type of schedule for a larger scale wine yard with numbers that are more based in reality.
4.2 Testing the Model
With the knowledge that our program finds an optimal solution for this type of problem, it is time to test the model. The tests will be comprised of running the program in larger scenarios in order to see how the program is affected when the size and complexity of the problem increases. This will be done by studying the size of the problem, when the program states that one has to rent vats, the differences between the optimal continuous solution and the optimal integer solution given by the program, and how these affect the time it takes the program to find an optimal integer solution.
Figure 3: The quantities of vats with different capacities that was used for the test.
For this test it was decided to only use one type of wine with the fermentation time of 15 days and no limits to where the wine can be put. This means that none of the vats will have to be maintained during the period of the tests, and all wine can be put into all vats. The types of vats and the amount of each
type is visible in Figure 3. The cost of running a vat whether the company owns it or not is put as 0, while the cost to rent a vat is put as 10. The tests were also done at the vintage periods of 30, 50 and 70 days, where 70 days is the most similar to the actual vintage period received from the wine company our advisor and his colleague in Chile have been in contact with.
It was also decides to limit the running time of the programs to one hour in order to raise the amount of tests that could be done, and compare what the program can do within that time limit.
While running the code it was visible that this problem could be solved with- out renting any vats until a daily supply between 175 000 and 200 000 liters was reached. A daily supply of 200 000 liters was also a point in which it was difficult for the program to solve the problem, as seen in Figure 4, as the time decreased with a jump of 25 000 liters in both directions. The fact that it takes longer time to find a solution at this point can be caused by a large increase of nodes needing to be explored in order to reach the optimal solution. This may in turn be due to the best bound result of 9,0259 being very close to the optimal integer solution of 10.
Figure 4: Depicts relevant results for the solutions for each supply.
A break between the supply dates, in other words that supply was delivered every second, third, fourth, and fifth day, resulted in a larger quantity of supply being able to be delivered without having to rent any vats. This can be explained by considering the quantity of grapes received during a week, where a daily delivery will give twice the amount of grapes compared to receiving grapes every second day. The time to reach a solution for these situations were also all under 10 seconds, which can be caused by the program having simple
Figure 5: Different node times for different supplies.
It was also visible that an increase in the length of the vintage period led to an increase in the time it took to find the optimal solution, as shown in Figure 5.
This increase is caused by the size of the problem increasing, as increasing the vintage period with 20 days adds an additional 42140 rows and 22080 columns to the problem, which results in close to 11 000 new integer and continuous variables for the problem to solve. This is represented by an increased amount of rows, columns, continuous, and integer variables of roughly 65-66% between the vintage periods of 30 and 50 days, and 39-40% between the vintage periods of 50 and 70 days.
While looking at the vintage period of 70 days, it was noticeable that the pro- gram struggled once it reached a daily supply of 200 000 liters, as there was only one tested point after this that took less than an hour to solve. Unfortunately due to the current pandemic we were unable to receive any data concerning the expected supply of grapes and costs of renting vats during a vintage period from the wine company that our advisor and his colleague in Chile have been working with. Therefore we can not make any comparisons between our program and the real life situation and are unable to draw any conclusions of how helpful this model would be in real life, or if a continuation of this project would have to use techniques to shorten the run time of the program.
5 Conclusion
The objective of this project was to minimize the cost of fermenting a set amount of grapes during a set vintage period. This problem is often solved by the use of past experience at the wine companies, but this project aims to solve it by the creation of an integer program. This is of interest as there has not been much similar work done with optimization in the wine industry, and this project is a new application of integer programming in the field of wine fermentation.
Once an integer program was created, it was tested by using a small example that could be verified by hand. This example proved that the program could find an optimal solution for this type of problem and a conclusion that was drawn is therefore that the program works for larger problems as well. While testing this program for larger problems it could be concluded that a larger vintage period, an increased frequency of supply, an increase of the amount of supply, led to an increase in the solving time. There was a breaking point between 175 000 and 200 000 liters per day where the model stated that one had to start renting vats, and there was a peak at the daily supply of 200 000 liters where the program had difficulties finding an optimal solution. With that said, perfectly good and feasible solutions were found which can be read and implemented for all of the problems that were tested. Even though as stated the size of the problem does play a large role in how the results turned out, these problems can be solved, or at least minimized by the use of a high power computer.
Unfortunately due to the COVID-19 pandemic, we were unable to receive infor- mation of the expected supply of grapes and costs of renting vats for a vintage period, and we are therefore unable to conclude how much help this model is in practice. With that said, further examination considering the missing data and including the uncertainties of supply that exists in practice is encouraged.
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