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Mixed Integer Linear Programming models for optimal crop selection Carlo Filippi, Renata Mansini, Elisa Stevanato PII: DOI: Reference:

S0305-0548(16)30303-3 10.1016/j.cor.2016.12.004 CAOR 4142

To appear in:

Computers and Operations Research

Received date: Revised date: Accepted date:

28 September 2015 30 November 2016 3 December 2016

Please cite this article as: Carlo Filippi, Renata Mansini, Elisa Stevanato, Mixed Integer Linear Programming models for optimal crop selection, Computers and Operations Research (2016), doi: 10.1016/j.cor.2016.12.004

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Highlights • Two MILP models for an optimal crop selection problem are given • Resource requests and timing of operations are deterministic; prices and yields are stochastic • The first model maximizes the expected profit on average data

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• The second model uses Conditional Value-at-Risk as a safety measure

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• A case-study is analyzed in detail, giving practical insights

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Mixed Integer Linear Programming models for optimal crop selection Renata Mansini†

Abstract

Elisa Stevanato∗

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Carlo Filippi∗

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In this paper, we propose the modeling of a real-case problem where a farmer has to optimize the use of his/her land by selecting the best mix of crops to cultivate. Complexity of the problem is due to the several factors that have to be considered simultaneously. These include the market prices variability of harvested products, the specific resource requests for each crop, the restrictions caused by limited machines availability, and the timing of operations required to complete each crop cultivation. We provide two different mathematical formulations for the analyzed problem. The first one represents a natural integer programming formulation looking for the cropmix that maximizes the farmer’s expected profit measured as the difference between revenues obtained by selling the harvested products and the production costs. Since the revenue of each crop depends on the price as quoted at the exchange market and the yield per hectare of harvested product, we define it as a random variable. Then, the second model uses the maximization of the Conditional Value-at-Risk (CVaR) as objective function and looks for the crop-mix that allows to maximize the average expected profit under a predefined quantile of worst realizations. To test and compare the proposed models with the cultivation choice made by the farmer, we use Italian historical data represented by monthly returns of different crops over a time period of 16 years. Computational results emphasize the advantage of using the CVaR model for a risk-averse farmer and provide interesting insights for farmers involved in similar problems. Keywords

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Agriculture; Crop selection; Mixed integer linear programming; Conditional Valueat-Risk.

Department of Economics and Management, University of Brescia, Contrada S. Chiara 50, 25122 Brescia, Italy - email:[email protected], [email protected] † Department of Information Engineering, University of Brescia, Via Branze 38, 25123 Brescia, Italy email: [email protected] ∗

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Introduction

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Optimization in agriculture is a very complex issue, typically characterized by the interaction of a large number of factors including the assignment of limited resources and information uncertainty (prices, weather). During the last years, the interest in agricultural problems and correlated issues has increased mainly because agriculture is a way to diversify economy and represents a valuable source of income and employment not only for rural countries. In this application context, the mathematical models represent valuable tools to support farmers when making decisions on the selection of crops and their allocation to fields (crop planning problem) and on the temporal successions of crops over the years (crop rotation problem). Finding a way to deal optimally with these problems may strongly affect both productivity and profitability of a farm, and has a relevant environmental impact also in terms of an efficient use of scarce natural resources. In this paper, we provide new models for a crop selection problem motivated by a real case faced by a farmer located in the North of Italy. The farmer owns an agricultural area, that he wants to cultivate with various types of crops. Each crop has its own characteristics which include, among others, the cultivation system, the specific processing time required by each of the several farming phases, the total cost of production and the return coming from the sale of the harvested quantity. Each crop requires different working phases (operations) that have to be carried out at a given time period of the year, whereas the choices related to a specific operation schedule are critically affecting the profitability of the whole system. Every operation, performed on a particular crop, has a specific cost, that can vary significantly depending on the machinery used. Finally, each crop cultivation leads to a different revenue depending on two uncertain parameters. The first one is the sale price (quotation) of the harvested product at the commodity exchange at the time it is sold. The second one is the yield per hectare mostly related to the weather in the cultivation year. In this context, a strategic farm planning can be seen as the problem to select crops while minimizing costs associated with the set of tasks required to complete crops production and maximizing the expected revenue. The problem is subject to real constraints such as the amount of resources available (hectares to cultivate, machines availability) and the correct timing of the operations involved by each crop. This work provides several contributions. Traditionally, the judgement of the farmer based on his/her experience has been the basis for planning in agriculture. Nowadays, given the development of capital intensive production systems, the use of quantitative planning methods based on the development and analysis of a mathematical model is more than desirable. Following this direction, we introduce two new models for the crop selection problem that directly incorporate the sequencing of crop operations. Both models provide as output the number of hectares to assign to each crop and the time instants at which the various operations must be performed for each crop. While very few mathematical models have been proposed in the literature for the crop planning problem, none of them explicitly models the sequencing of operations required to cultivate selected crops. This is the first important contribution of our paper. A common and simple criterion often used in practice to deal with uncertain parameters is to represent them by their expected value. Given historical data on the yield per hectare and the unit price on each crop, both parameters may be estimated using their historical mean. We use this criterion in our first model that selects the best

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crop-mix in terms of expected profit (global revenue minus costs), while satisfying the temporal constraints of the operations required by the selected crops and the resource availability constraints. An advantage of such an approach is simplicity, but a strong disadvantage is that the risk-averse attitude of the decision-maker is not taken into account. As correctly emphasized by Dury et al. [16], decisions about crop planning seldom take into account the decision maker’s behavior towards risk. The second model we propose tries to overcome this limit. More precisely, in order to perform a better economic evaluation of crops selection the model takes into account the uncertainty of information (prices, yields per hectare), and uses as objective function the maximization of a safety measure, the Conditional Value-at-Risk (CVaR), corresponding to select the crop-mix that maximizes the expected profit under a predefined percentile of worst realizations (see Mansini et al. [27], and more recently, Mansini et al. [28] for properties analysis of different risk and safety measures including CVaR). This is done by computing realizations through historical scenarios assuming they represent forecast data, while satisfying all the constraints and resource requirements for the selected crops. We decided to use CVaR, instead of a dispersion risk measure as variance, due to its nice theoretical properties that has made it very popular in financial problems and recently also in many other application contexts (see, for instance, Sarin et al. [40] for the use of CVaR as a criterion for stochastic scheduling problems). In agricultural context, CVaR has been used to quantify the risk associated with different irrigation investment strategies to address the issue of water scarcity (Paydar and Qureshi [34]); to minimize farmers’ expected losses (including insurance costs) in the selection of optimal crop insurance under climate variability and fluctuating market prices (Liu et al. [24]); to assess downside risk and investigate the impact of ENSO-based (El Niño-Southern Oscillation) climate forecasts on the optimal selection of planting schedules and financial yield-hedging strategies (like commodity future contracts) with the objective of maximizing expected revenues (Aitsahlia et al. [1]). Interested readers are also referred to Manfredo and Leuthold [25], Monjardino et al. [30], Pepelyaev and Golodnikova [35], Soltani et al. [41], Eyvindson and Cheng [18]. To the best of our knowledge, it is the first time that this safety measure is used in a crop selection model with operations management. This is the second relevant contribution of our work. Moreover, the use of CVaR as objective function can be easily adapted to other agricultural problems with a combinatorial nature such as the crop rotation problem. Finally, we test and compare the proposed models on a set of real instances based on Italian historical data. The computational results show that the models based on CVaR optimization guarantee more stable solutions and provide a definitely better choice for a risk-averse farmer than the solution based on the pure expected profit maximization or the one suggested by farmer’s experience. The results also provide interesting insights on the convenience to use the different models and indicate insights for farmers involved in similar problems. This is an interesting contribution on its own. The paper is organized as follows. In Section 2, we first analyze the state-of-the-art mathematical formulations for the crop selection problem and present the main references also on crop rotation problems. In Section 3, we provide a formal description of the analyzed problem and introduce the notation used. In Section 4, the first model based on the maximization of the expected profit is proposed. In Section 5, we briefly define the CVaR as safety measure, and then we use it as objective function of an

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innovative model formulation where we maximize the expected profit under a predefined number of worst scenarios, while guaranteeing a predefined expected return and satisfying a budget constraint. In Section 6, we describe the real case used for testing, and analyze the proposed models comparing their performance on real data. A few general insights are discussed in Section 7. Finally, in Section 8, conclusions and future developments are drawn.

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Literature review

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In this section, we provide the state-of-the-art on the mathematical formulations proposed to model the crop selection problem and the main methodologies used to tackle it. We also report some recent references for the crop rotation problems. According to Glen [19], the crop production planning problem is to determine the crops to be grown, the area to be assigned to each crop, and the resources to be used. Although the problem is integer in nature, most models available in the literature are continuous linear programming (LP) problems usually optimizing financial objectives such as profit maximization or cost minimization. For instance, Audsley et al. [5] define a crop planning policy/harvest timing using an LP model, then they obtain integer solutions by rounding. More recently, LP models have been proposed to obtain feasible agricultural production plans, see for example Haneveld and Stegeman [21] where the authors introduce a mathematical programming framework and show that crop succession requirements can be included as linear constraints in a model for agricultural production planning. Most of the models presented are deterministic, although for the correct understanding of operations in a farm business, the analysis should be done in the light of the well-known sources of uncertainty. Darby-Dowman et al. [12] introduce a twostage stochastic program with recourse for the problem of determining optimal planting plans for a vegetable crop. The first stage of the model relates to finding a planting plan which is common to all scenarios and the second stage is concerned with deriving a harvesting schedule for each scenario. The authors motivate their study by claiming that traditional linear programming models are generally unsatisfactory in dealing with the uncertainty and produce solutions that are considered to involve an unacceptable level of risk. More recently, Cherchye and Van Pyenbroeck [9] look at DEA model as a natural way for assessing profit efficiency when reliable information about true prices is lacking. Among the first mixed integer programming problems we recall: Trava et al. [42], who develop a mixed-integer programming model for short-term irrigation scheduling, using boolean variables to represent whether or not a particular field has to be irrigated on a particular day; Amir et al. [3], who propose a problem for machinery selection for hay production, where a binary variable is associated with each set of machines; and Danok et al. [11], who analyze a machinery selection and crop planning problem with a binary variable for each set of machinery under the effect of weather variability. If we exclude a few cases, most solution methods used in the past were based on simulation or on enumeration techniques. A detailed analysis of models proposed before 1986 can be found in Glen [19], who classifies the crop production models into four sections related to the determination of a crop planning policy, the methods used for planning harvesting operations, the techniques for evaluating capital investments, and the methods for

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evaluating pest and disease control strategies. The scheduling of operations, as watering and harvesting to name a few, is an important aspect of planning crop production. Nevertheless, most crop planning models ignore these complexities of scheduling activities. Van Elderen [43] compares the use of simulation and LP models in scheduling harvesting operations using historical weather data. More recently, Jiao et al. [22] propose a combination of statistical and optimization approaches to solve harvest schedule problems. The concept of cultivation planning is also suggested by Grunow et al. [20]. Their approach aims to determine the cultivation and harvesting time for each field such that the risk of undesired milling rate reductions is minimized. Almanina et al. [2] present models and algorithms used in a decision support system for water irrigation scheduling. However, harvest or water scheduling is typically the only variable being considered by these problems without any scheduling of other operations. Even though some of these models try to use the machinery as best as possible, very few of them include optimal machinery dimensionality and operation. In real agricultural contexts, farmers have to decide about field cultivation by choosing among several available crop alternatives, with the objective to develop a plan that can be sustainable over the long term while optimizing the timing of operations, cost of machinery, and resources needed in the whole production cycle (water, fuel, manpower, etc.). In the last years, several works appeared on this topic. Eto [17] analyzes the problem of alternating crops in agriculture, with the aim to induce Japanese farmers to convert traditional rice production to other crops while maximizing revenues and minimizing conversion costs. Biswas and Pal [7] show that fuzzy goal programming techniques can be efficiently used for modeling and solving land use planning problems while considering the production of several seasonal crops during a planning year. They apply it to an Indian region. Annetts and Audsley [4] consider the problem of planning a farming system under environmental conditions, and apply the developed model to a UK area. In terms of crop rotation, different problems have been addressed using linear optimization by authors such as Clarke [10], Dogliotti et al. [14], Detlefsen and Jensen [13] who take into account the fact that crop rotation influences the needs of nitrogen and the yield of the fields. In particular, the models of the latter authors consider that the amount of land to be planted with each crop is given, for each year, which enables the formulation of the crop rotation problem as a transportation problem. Boisvert and McCarl [8] consider methods for including risk in the analysis, while Nevo et al. [32] use crop yields formulas to define an integrated expert system. More recently, dos Santos et al. [15] consider, in order to meet a specified demand for each crop, the problem of determining the division of the available heterogeneous arable areas in plots to obtain an appropriate crop rotation schedule for each crop. They propose a linear formulation for this problem, in which each variable is associated with a crop rotation schedule. Their model includes a large number of variables and it is, therefore, solved by means of a column-generation approach. Finally, several contributions can be found in the multi-objective context, where different conflicting policy objectives related to farming practice are evaluated simultaneously. For example, Ragkos and Psychoudakis [37] study the possibility to achieve conflicting policy goals as acceptable incomes, reduction of agro-chemical components and water consumption. Piech and Rehman [36] and Romero and Rehman [39] em-

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ploy multi-objective programming for agricultural decisions. Other applications are presented by Berbel and Rodriguez-Ocana [6], and by Manos et al. [26]. The multiobjective nature of the problem is also analyzed in Ortuño and Vitoriano [33].

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Problem description

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We consider a general farming system where a set of crops, like for example, wheat, corn, and maize, can be grown over a limited agricultural area by using a family of tool types, like ploughs, weeding machines, and so on. A predefined number of machines per tool type is available. Each crop requires a fixed sequence of operations, like ploughing, sowing, etc. Notice that some operations may appear more than once along the sequence (e.g., watering may be required more than once during the life cycle of a plant). Every operation requires a machine of a specific type; for example, ploughing obviously requires a plough. We assume that all machines of a given type are compatible with the corresponding operation for a given crop. However, the working speed and/or the operating cost may vary inside the same subset of machines for a fixed crop and among different crops for a given tool. For instance, different ploughs may be used for corn, but a bigger one may be faster and more costly; the same plough may be used for corn and alfalfa, but it may be slower and more expensive for corn due to the deeper grooves. Tools must be mounted on a tractor machine to work and there is a limited number of tractors available. A tractor may mount any tool type and the tractors are comparable, so they can be regarded as identical machines. Every operation for every crop must be performed within a given time window. For example, soft wheat in Northern Italy should be sown within October. Once the crops to be produced are selected and a significant time horizon is fixed (e.g., one year or a season), the farmer has to assign crop operations to each time unit (e.g., half a day) over such an horizon. In a given time unit, different operations associated with different crops over different fields can be performed, provided that the required tools and tractors are available and that the time unit belongs to the time window of every active operation. The farmer owns a given total area and, for every crop-operation pair, he/she knows the number of hectares that can be worked out within a time unit using one of the compatible tools. Moreover, the cost per unit time of using any tractor-tool pair is known. Finally, historical data are available on the unit prices that can be obtained from selling the different harvested products and on the yield per hectare of every crop. The farmer’s objective is to look for an optimal selection of crops (optimal crop-mix), and an optimal assignment over time of their operations to meet time windows and resource constraints, while maximizing profit expressed as the difference between the expected revenues from selling harvested crops and the production costs. We call this problem the Crop Selection Problem (CSP).

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Basic notation

Let m be the number of possible crops, let n be the number of tool types required to perform all operations needed for the cultivation of the considered crops, and let uj be the number of tools available for tool type j, j = 1, . . . , n. A crop i is characterized by

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an ordered sequence of qi operations: j[i, 1], j[i, 2], . . . , j[i, qi ],

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where j[i, k] is the index of the tool type needed to perform the k-th operation of crop i, i = 1, . . . , m. Notice that, in general, the k-th operation is not necessarily the same for every crop, i.e., it does not necessarily requires the same tool type. Moreover, some operations may be repeated, so that we may have j[i, h] = j[i, k] for some h 6= k. Any tool must be mounted on a tractor machine to work, and we assume that w identical tractor machines are globally available. The time horizon T is divided into elementary time units (e.g., half a day), such that preemption is not allowed during a time unit. Let [si,k , fi,k ] be the time interval where the k-th operation of crop i must be executed. Time intervals are measured in time units. Without loss of generality, we assume that mini=1,..,m {si,1 } = 1, and we set T = maxi=1,..,m {fi,qi }, so that every operation must take place within the time horizon [1, T ]. Let H be the total number of hectares available for cultivation, and let ri be the expected revenue in Euros that can be yielded selling the harvested product obtained from one hectare cultivated with crop i, i = 1, ..., m. Value ri depends on both the unit price and the yield per hectare that in general are uncertain and correlated quantities. Let hi,k,` be the number of hectares that can be worked out in a time unit performing the k-th operation on crop i, using the `-th tool of type j[i, k]. Moreover, let ci,k,` be the time unit cost of using the `-th tool of type j[i, k] on the k-th operation of crop i. Usually, a farmer must also take into account costs related to fixed resources that remain constant irrespective of the revenues obtained. We refer to these fixed costs as a global amount of money F C. Parameter F C includes rent, interest on fixed capital, depreciation of building and machineries, taxes and wages of the permanent workers. In order to match operations and tool types, for all i = 1, . . . , m, and k = 1, . . . , qi , we define the following binary coefficient: 1, if j is the k-th operation for crop i; ai,j,k = 0, otherwise. For convenience, we also define I(j, t) as the subset of crop indices requiring a tool of type j that can be active at time t, i.e.,

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I(j, t) = {i = 1, . . . , m : j = j[i, k] and t ∈ [si,k , fi,k ] for some k = 1, . . . , qi }.

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In the following sections, we provide two mathematical formulations for the CSP. In the first formulation, the total expected profit is maximized and every uncertain parameter is represented by its expected value computed as the mean of historical data. In the second formulation, the objective function is formalized as the safety measure represented by CVaR and the optimal selection of crops and operations timing are done to guarantee the best expected profit (revenue minus costs) performance under a predefined number of worst case scenarios. Both models make use of the same two sets of variables, defined as follows: • a binary decision variable yi,k,`,t which is equal to 1 if machine ` is assigned to the k-th operation of crop i at time t, for all i = 1, . . . , m, k = 1, . . . , qi , ` = 1, . . . , uj[i,k] , and t ∈ [si,k , fi,k ];

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• a continuous variable zi,k,` representing the area extension, expressed in fractional number of hectares, where the k-th operation of crop i is worked out using tool `, for all i = 1, . . . , m, k = 1, . . . , qi , and ` = 1, . . . , uj[i,k] .

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An expected profit maximization model

The first model provides a time-indexed linear binary formulation of the CSP as follows:

subject to

m X n X

j=1

`=1

uj

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i=1 j=1

n X

ai,j,1 X

`=1 uj

ai,j,k−1

j=1

X `=1

qi X m X n X

zi,k,` ≤

X

i∈I(j,t) k=1

ai,j,k

k=1 j=1

uj X

fi,k X

ci,j,`

`=1

t=si,k

yi,k,`,t

(1a)

(1b)

n X

ai,j,k

j=1

uj X `=1

zi,k,` = 0

(i = 1 . . . , m; k = 2, . . . , qi )

(1c)

fi,k

X

ci,j,`

`=1

ai,j,k hi,j,`

t=si,k

fi,k X

(1d)

yi,k,`,t + F C ≤ B

yi,k,`,t

t=si,k

(i = 1 . . . , m; k = 1, . . . , qi ; ` = 1, . . . , uj[i,k] )

(1e)

ai,j,k yi,k,`,t ≤ 1

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X

qi X

zi,k−1,` −

ai,j,k

j=1

zi,1,` −

qi X n X

zi,1,` ≤ H

uj

i=1 k=1 j=1 n X

uj X

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i=1

r i

n X

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max

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m X

(j = 1, . . . , n; ` = 1, . . . , uj ; t ∈ [0, T ])

(1f)

(t ∈ [0, T ])

(1g)

(i = 1, . . . , m; k = 1, . . . , qi ; ` = 1, . . . , uj[i,k] ; t = si,k , . . . , fi,k )

(1h)

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qi m X X X i=1 j∈I(j,t) k=1

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yi,k,`,t ∈ {0, 1}

X yi,k,`,t ) ≤ w ai,j,k ( `=1

(i = 1, . . . , m; k = 1, . . . , qi ; ` = 1, . . . , uj[i,k] )

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zi,k,` ≥ 0

uj

(1i)

The objective function (1a) maximizes the sum of net profits for all the selected crops. The net profit for each crop is measured as the difference between the expected revenue and the sum of costs related to the operations required to cultivate the crop. The revenue for each crop i is obtained by multiplying the expected return ri by the total number of hectares worked out by each operation required for the crop production. Such a number is computed by counting the hectares assigned to the first operation of each crop. This number is the same for all the operations of a crop as stated in constraints (1c). The total cost is related to the operations performed during the whole time horizon T for all crops and is obtained by summing up for all crops the costs

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A CVaR model for crop selection

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related to the tools used to perform the corresponding operations in the defined time intervals. Constraint (1b) bounds the total area that can be farmed to be no more than the number of hectares H available. For a given crop, the total area devoted to its cultivation is given by the fractional number of hectares associated with its first operation which has to be the same for all its subsequent operations. Thus, as already stated, constraints (1c) compel the hectares worked out by a selected crop to be the same for every operation on that crop. For example, the number of hectares ploughed for crop i must be equal to the number of hectares seeded for the same crop and so on for the remaining operations. Constraint (1d) is the so-called budget constraint. It states that the total amount of money the farmer wants to spend in cultivating the selected crops has to be lower than or equal to a value B decided at the beginning of the time horizon considered for cultivation. Budget B will cover both proportional and fixed costs (the latter represented by parameter F C) plus a possible amount that remains not invested. Since proportional costs are minimized in the objective function, the formulation of the budget constraint as an inequality would result in setting the proportional costs to the minimum possible value under which no feasible solution exists. From this point of view, constraint (1d) is not essential for the model. Notice that an expensive, high revenue solution and a cheap, low revenue solution may be equivalent in terms of expected profit. In the computational results we will force constraint (1d) to strict equality to compare model outputs under the same level of cost. Constraints (1e) impose that a necessary quantity of resources is allocated to every operation of every crop. More precisely, the constraints impose that the strictly necessary number of time slots are assigned to every feasible triple crop-operation-tool. Constraints (1f) impose that every single tool has to be assigned to at most one operation on some crop at each time unit t. Constraints (1g) compel the number of tool-tractor pairs active at any time t to be not greater than w. Finally, constraints (1h) and (1i) are binary and non negativity conditions on the different variable sets, respectively.

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Let y and z denote the vectors of y and z variables in model CSP, respectively. Let YZ denote the feasible solutions set of problem CSP, i.e., let YZ be the set of vectors (y, z) satisfying all constraints of model CSP. The objective function of model CSP may be written as f (y, z; r), where (y, z) are the decision variables and r = (r1 , . . . , rm ) is a vector of input parameters, corresponding to the expected revenues for the different crops. Model CSP can be written succinctly as follows: max f (y, z; r) subject to (y, z) ∈ YZ

(2)

So far, we have assumed that vector r is the expected value of a random vector R = (R1 , . . . , Rm ) describing the revenue per hectare for the different crops i = 1, ..., m. In practice, we are interested to deal with the uncertainty of R in order to evaluate its value at the target time, i.e., the time at which harvested products will be sold in the market. To capture this uncertainty, we use the concept of scenarios. The revenues

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ηs (y, z) =

m X i=1

ri,s

n X j=1

ai,j,1

uj X `=1

zi,1,` −

qi X n X k=1 j=1

ai,j,k

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are discretized by defining their realizations under the specified scenarios. For us a scenario is a possible realization of crop revenues (i.e., of crop prices and yields used to compute the revenue) at the target time. Formally, a scenario is a realization of the multivariate random variable R. Let us assume that a finite number S ofP scenarios has been identified. Each scenario s has a probability ps to happen, with Ss=1 ps = 1. We also assume that for each crop i, i = 1, ..., m, its realization ris under scenario s, is known. The set of revenues of all crops {ris : i = 1, ..., m} represents PSthe scenario s. The expected revenue for crop i is computed as E(Ri ) = r¯i = s=1 ris ps . Given a feasible crop-mix (y, z) satisfying the constraints of the problem we can associate with it a random variable representing its revenue or its profit, if proportional costs required for the production of the crop-mix are also taken into account. Then, the profit of a crop-mix (y, z) under scenario s can be calculated as: uj X

ci,j,`

`=1

fi,k X

t=si,k

yi,k,`,t .

(3)

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To evaluate uncertainty on profit, and the risk associated with this uncertainty we may consider the variance as the most classical statistical quantity used to measure the dispersion of a random variable around its mean. Variance equally penalizes positive and negative deviations from the average value, whereas it is clear that for a maximization problem positive deviations are better than negative ones. Moreover, variance involves a quadratic expression. Nowadays, convex Quadratic Programming (QP) models are not hard to solve. Nevertheless, Linear Programming (LP) models remain much more attractive from a computational point of view, especially if integer or binary variables are required in the problem formulation. In the literature, alternatives to variance, have been proposed to measure the dispersion of a random variable. One of these measures is the Mean Absolute Deviation (MAD) that considers, with respect to the variance, absolute values instead of squared values. When the returns are discretized, as we do, the MAD as well as other risk measures have the advantage to be LP computable (see Konno and Yamazaki [23] and Mansini et al. [27]). As an alternative to risk measures, one can use safety measures to be maximized. The idea behind a safety measure is that we are not interested in the deviations of a random variable from its expected value but we try instead to protect the investor (in our case the farmer) from the worst scenarios in terms of expected profits (revenues). One of the most appealing safety measure is the maximization of the worst realization, which is LP computable (see Young [44]). A natural generalization of worst realization is the statistical concept of quantile. Let us denote as ξβ (y, z) the value of the β-quantile, that is the value of the profit (revenue) defined as ξβ (y, z) = inf {ξ : F(y,z) (ξ) ≥ β}

for 0 < β ≤ 1,

(4)

where F(y,z) (·) is the cumulative distribution function (cdf) of profits. In the finance literature, and in particular in portfolio optimization, this quantile is usually called Value-at-Risk or simply VaR measure. Actually, for a given portfolio, its VaR depicts the worst (maximum) loss within a given confidence interval. With a change of sign, losses can be seen as negative revenues, and we can define the quantile

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ξβ (y, z) as in (4). Since the cdf can be discontinuous, the VaR measure is, generally, not an LP computable measure. The corresponding portfolio optimization model can be formulated as a MILP problem (see, for a detailed analysis, Mansini et al. [29]). More recently, risk measures based on averaged quantiles have been introduced in different ways by several authors. The tail mean or worst conditional expectation, defined as the mean return of the portfolio taken over a given tolerance level (percentage) 0 < β ≤ 1 of the worst scenarios probability is usually called Tail VaR, Average VaR or Conditional VaR. Actually, the name CVaR, after Rockafellar and Uryasev [38], is now the most commonly used. We refer to the survey by Mansini et al. [28] and the recent book by the same authors [29] for a detailed analysis of CVaR and other risk measures that can be modelled as Linear Programming problems. Let η1 , η2 , . . . , ηS be the profit realizations associated with a vector (y, z) ∈ Y Z. An approximation of the CVaR associated with (y, z) is the average value of the worst (lowest) realizations with total probability at least β. This average can be computed by the following LP [29]: P max η − β1 Ss=1 ps ds (5) subject to ds ≥ η − ηs (s = 1, . . . , S) ds ≥ 0 (s = 1, . . . , S)

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At optimality in the above problem, the free variable η is the β-quantile of the considered realizations, ds = 0 for all scenarios s with ηs ≥ η, and ds = η−ηs for all scenarios s with ηs < η. Thus, the optimal objective function value is the average of the realizations not exceeding the β-quantile. In other words, the optimal objective function value is the CVaR computed over the realizations. We can embed model (5) in our CSP, obtaining the following MILP formulation: (CSP(β)) S 1X ps ds β

(6a)

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ri

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6

A real case study

We apply our models to the real-case of a single farm located in Veneto region, North of Italy. In the following, we first describe the data collection and the instances generation, then the setting of model parameters. Finally, we present and comment the obtained results.

6.1

Input data and parameters setting

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Most of the data necessary to our models have been obtained through interviews to the entrepreneur whose farm is taken as our case study. These data include the land available to grow crops, the sequence of operations required for each crop, the corresponding time windows, the availability of tools and tractors, their operating costs and the working speeds. In particular, in the analyzed farm, the time horizon T is equal to 145 time slots each lasting half a day, whereas the total cultivable area is equal to 140 hectares of which about 100 hectares of land are usually cultivated each year, while the remaining part is left unused for crop rotation. For convenience, we assume in our experiments a total area H equal to 100 hectares. With regard to the resources available for cultivation, the farmer owns w = 4 tractors and two tools for each operation type (hence two plows, two harrows, and so forth). After several years of experience, the entrepreneur has decided to cultivate his land according to the following fixed composition of crops: 40% of the area is cultivated with corn, 30 % with wheat, 10% with soy, and the remaining 20% with barley. According to his past experience, this composition implies a lower risk of loss upon the sale of the harvested crops. We take this crop-mix as reference solution, labeling it as the Farmer’s solution. In particular, in order to correctly evaluate the maximum expected profit associated with this solution, we solve the CSP model by substituting constraint (1b) with the following ones where Hi , i = 1, ..., m, are the number of hectares the farmer has decided to assign to each crop: n X

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Moreover, the described four crops are the ones we take into account in our experiments (i.e., m = 4). The total number of operations is equal to 9, corresponding to the following activities: ploughing, harrowing, sowing, weeding, hoeing, fertilizing, watering, phyto-sanitary treatment, and harvesting. Table 1 provides the sequence of operations required by each crop, along with the first (s) and the last (f ) time slots in the corresponding time window. For each crop-operation pair, in Table 2 we report the number of hectares that can be worked out during a time slot (h) and the corresponding cost in Euro (c). Notice that the reported figures are valid for both tools available for every operation in our case study. As previously mentioned, there is substantial uncertainty about the actual revenues that can be obtained from a given crop mix. This uncertainty has two main sources, namely the yield that can be obtained from each hectare grown with a given crop (measured in tons per hectare), and the price at which the different crops can be sold (measured in Euros per ton). In order to obtain a proper input for our models, we considered historical data over sixteen years, from 2000 to 2015.

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Barley Operation s f Ploughing 15 28 Harrowing 29 45 Fertilizing 50 55 Sowing 56 66 Fertilizing 67 75 Weeding 76 85 Watering 86 90 Treatment 91 95 Harvesting 96 105

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Soy Operation s Ploughing 1 Harrowing 40 Sowing 71 Weeding 81 Hoeing 96 Watering 106 Treatment 111 Watering 117 Harvesting 136

Table 1: Sequence of operations and time windows for each crop.

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Corn Wheat Operation s f Operation s f Ploughing 1 20 Ploughing 1 15 Harrowing 40 60 Harrowing 16 25 Sowing 61 65 Fertilizing 26 30 Weeding 66 70 Sowing 31 40 Hoeing 71 80 Fertilizing 41 46 Watering 81 90 Weeding 51 60 Watering 96 105 Fertilizing 61 65 Treatment 106 110 Treatment 76 80 Harvesting 116 130 Harvesting 116 130

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Table 2: Number of hectares worked out per time unit (h) and cost in Euros per time unit (c), for each crop and each operation.

Ploughing Harrowing Sowing Weeding Hoeing Fertilizing Watering Treatment Harvesting

Corn h c 8.0 50.00 16.0 15.00 16.0 10.00 16.0 10.00 6.4 37.00 – – 3.6 250.00 32.0 12.50 20.0 28.00

Wheat h c 8.0 50.00 16.0 15.00 16.0 10.00 32.0 5.00 – – 40.0 4.00 – – 32.0 10.00 20.0 28.00

Soy h 8.0 16.0 16.0 16.0 6.4 – 3.6 32.0 20.0

c 40.00 15.00 10.00 10.00 25.00 – 100.00 10.00 20.00

Barley h c 8.0 30.00 16.0 12.00 16.0 6.00 25.0 6.00 – – 30.0 4.00 10.8 50.00 32.0 10.00 20.0 20.00

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For each considered year, we compute an average yield for each crop dividing the total Italian production, in tons, by the total cultivated area, in hectares. These data have been collected by Eurostat (statistical office of the European union) and are publicly available at http://ec.europa.eu/eurostat/web/agriculture/data/database. Computed average yields are shown in Table 3. Since such figures represent averages on the whole Italian territory, they may hide significant variability through a risk pooling phenomenon. We then decide to perturb such figures assuming they are the average of a Beta distribution. More precisely, we assume that the actual yield in a given rural district behaves like a Beta distribution with parameters α = 5 and β = 2, see Figure 1. We use τ = 1, . . . , 16 to index the considered years, from 2000 to 2015. For each entry yieldτ,i of Table 3, with τ = 1, . . . , 16 and i = 1, . . . , 4, we generate independently five numbers from the Beta distribution b1 , . . . , b5 and we obtain five independent perturbations of the given yield:

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where 5/7 = α/(α + β) is the expected value of the Beta distribution. The Beta distribution represents a very versatile way to represent outcomes such as yields. In agricultural contexts, its use has been commonly associated with crop yield. We refer, for instance, to Nelson and Preckel [31], where the Beta distribution is proposed as a parametric model of the probability distribution of the agricultural output. In the paper, the authors mention different reasons for which crop yield may be distributed as a Beta random variable. Our values for α and β parameters are consistent with the ones provided in [31] and directly estimated from real data. For the same years 2000-2015, monthly prices, expressed in Euro per ton, are available for each considered crop; these data have been kindly supplied by ISMEA (Institute of Services for the Agricultural and Food Market) and are publicly available at https://sites.google.com/a/unibs.it/or-brescia/instances. In Figure 2, we plot the price series during the considered time period.

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Table 3: Average yields in Italy, in tons per hectare. Barley 3.667 3.380 3.470 3.293 3.806 4.419 3.896 3.552 3.781 3.448 3.463 3.402 2.759 3.390 4.007 4.282

Soy 3.571 3.785 4.529 2.556 3.454 3.638 3.096 3.142 3.276 3.482 3.622 3.643 3.820 3.680 3.647 3.913

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Wheat 4.730 4.463 4.809 4.354 5.314 5.450 5.477 4.913 5.354 5.182 5.158 5.326 5.888 5.290 5.295 5.412

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Corn 9.528 9.516 9.491 7.483 9.496 9.369 8.688 9.316 9.878 8.959 9.286 9.803 8.078 8.699 10.633 9.725

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Corn Wheat Soy Barley

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Figure 2: Price series of the different crops from January 2000 to December 2015.

priceuτ,i · yieldvτ,i

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where priceuτ,i is the price of crop i recorded in month u of year τ . Notice that each yield value is multiplied by a price associated with the corresponding year. In this way, possible correlations between yields and prices are preserved. We denote by rs = (r1,s , . . . , r4,s ), with s = 1, . . . , 960, the obtained revenue realizations, or scenarios. These are used as input in our CSP(β) models. The vector of expected revenues r obtained from the scenarios as historical means over all the crops is equal to r = [ 1669.10, 1073.56, 1146.51, 608.23 ] (Corn) (Wheat) (Soy) (Barley)

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6.2

Experimental analysis

We implemented and solved the models by using IBM ILOG CPLEX Optimization Studio 12.6 (64 bit version) with default settings on a PC Intel Xeon ES-1650, 3.50GHz, 16 GB RAM processor. In a first set of experiments we have imposed the constraint on the budget as a strict equality to allow a direct comparison among models that optimize different objective

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functions. In this way, all the models are forced to provide the best possible solution given a predefined amount of proportional production costs. For these experiments, we have considered three different values for the budget equal to B = e68000, e70000 and e72000, respectively and a fixed cost F C = e60000. In Section 6.5, we complete the analysis by showing additional results obtained by assuming that the farmer does not consider a budget restriction (constraint (1d) is maintained as an inequality and the budget value is set to a very large value), and µ0 is set equal to 0.5 for models CSP(0.05) and CSP(0.01). The number w of available tractors has been set to 4, as in the real case, and then halved to 2 to see if feasible solutions can still be found with reduced resources. Thus, we have 3 × 2 = 6 different instances, over which we test the CSP model and the CVaR model, CSP(β), where we use the two values of β equal to 0.05 and 0.01. Given 960 scenarios, model CSP(0.05) maximizes the expected profit associated with the worst 0.05 × 960 = 48 scenarios, whereas model CSP(0.01) maximizes the expected profit associated with the worst d9.6e = 10 scenarios. Moreover, while model CSP finds a solution that maximizes the expected profit, models CSP(β) optimize a safety measure while forcing the expected revenue to be at least a certain fraction µ0 of the invested capital. In the following experiments, we have considered five values for the rate µ0 , which are equal to 0.5, 0.6, 0.7, 0.8, and 0.9, respectively.

Computational results: the case with four tractors

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Tables 4–6 show the details of the results obtained by the different models in all the instances when the number of available tractors, as in the real case, is equal to four (w = 4) and constraint (1d) is considered as a strict equality. All monetary values are rounded to the closest integer. In Table 4, the first column reports the name of the model solved, whereas the second and third columns indicate the value of the budget B, and the required rate of revenue µ0 (if requested by the model), respectively. The remaining four columns provide statistics. In particular, “Avg”, “Min”, “Max” and “StdDev” indicate the average, the minimum, the maximum and the standard deviation value of the profit provided by the model over the 960 scenarios. Columns CVaR(0.05) and CVaR(0.01) provide the CVaR values of the solutions obtained by the different models. Note that Avg is the optimal objective function value for the CSP model, whereas for models CSP(β) such a value is obtained by dividing the total revenue of the solution by the number of scenarios minus the costs, that are independent from the scenarios. Moreover, CVaR(0.05) and CVaR(0.01) are the optimal objective function values for models CSP(0.05) and CSP(0.01), respectively. So, for instance, the value CVaR(0.05) is computed by taking the average profit out of the worst 48 scenarios for model CSP and CSP(0.01), respectively. Finally, columns “Time” and “Gap” show the computational time required to solve the instance (in seconds), and the percentage gap (if any) of the incumbent integer solution value provided by CPLEX (possibly the optimal solution) from the best upper bound value found in the search tree. We set a threshold time equal to 3600 s for the solution of each model. Whenever CPLEX is able to find the optimal solution within such a time limit, then Gap(%) is equal to zero and the time is strictly lower than 3600 s.

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Table 5 provides the difference in percentage between the performance of CSP(β) models with respect to CSP solution values. The percentage deviation, for all the statistics described in Table 5, is computed as StatCSP (β) − StatCSP StatCSP

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where StatCSP(β) and StatCSP are the values of the considered statistics for models CSP(β) and CSP, respectively. We have already explained that a direct comparison of the models is possible since we assume they have the same level of investment. So, with a budget of e68000, all models provide the best possible solution obtainable with an amount of proportional production cost exactly equal to e8000. Looking at the solutions, it is evident that CSP provides higher average expected profits at the cost of a higher standard deviation value over the scenarios. On the contrary, CSP(β) models are more conservative by generating lower average expected profit, but guaranteeing lower deviations and better performance in terms of minimum profit, thus better protecting from worst scenarios. Comparing CSP(0.05) with CSP(0.01) one can notice that the more risk-averse CSP(0.01) provides lower values of average profit and standard deviation. Moreover, such values tend to increase when the value of the required rate of return µ0 grows, and the statistics of the model are very close to the ones of CSP(0.05). This is obvious since the constraint on expected revenue forces the farmer to become more aggressive by asking for higher returns and thus more risky solutions. Interestingly enough, the optimal solution for model CSP(0.05) does not change when µ0 moves from 50% to 90% for each level of budget. For µ0 = 0.9 and a budget equal to e68000, the required expected revenue in constraint (7) has to be at least equal to 1.9 × 68000 = 129200 Euros. Such value grows to e133000 and e136800 for a budget equal to e70000 and e72000, respectively. When solving the model for µ0 = 0.5 and the three different budgets, one can notice that the optimal solutions have an average revenue equal to e132107, e142184 and e150498, respectively. These values are already higher than the value of the required return for µ0 = 0.9. This explains why the solution does not change when increasing the value of µ0 . It is clear that the CSP model finds a solution that does not protect the farmer from worst case scenarios. For instance, for a budget equal to e68000 the CVaR(0.01) value for model CSP is 20.96% worse than the one of model CSP(0.01) and its value for CVaR(0.05) is 0.33% worse than the optimal solution value of model CSP(0.05) (see Table 5). With e70000, the budget is higher and thus the average profit obtainable by all models increases along with its standard deviation. Similar considerations apply when moving from a budget of e70000 to a budget of e72000. The only exception is the average profit for model CSP(0.01) that decreases when the budget increases. Model CSP(0.01) is more conservative with respect to model CSP(0.05). This explains why given a budget the solution changes when increasing the value of required return µ0 . The average revenue obtained by model CSP(0.01) with a budget equal to e68000 and µ0 = 0.5 is e119400, thus larger than the budget B incremented by 60% and 70%, but lower than the budget incremented by 80%. This explains why the constraint on expected revenue becomes effective starting with µ0 = 0.8. Thus, given any value of proportional production cost investment, a required rate of return of 50%, 60% or 70% does not force model CSP(0.01) to hold a position that is more risky than the one it

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Table 4: Computational results for w = 4: absolute values.

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

127228 124107 111400 124107 111400 124107 111400 124107 114400 124107 121200

272245 248779 200584 248779 200584 248779 200584 248779 205505 248779 232237

43835 48085 55956 48085 55956 48085 55956 48085 55510 48085 50968

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

133803 132184 109400 132184 109400 132184 109400 132184 116000 132184 123000

293752 282605 198584 282605 198584 282605 198584 282605 212459 282605 234338

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

138498 138498 107400 138498 107400 138498 110400 138498 117600 138498 124800

306748 306748 196584 306748 196584 306748 201139 306748 221582 306748 244359

Profit Max Min

CVaR(0.05)

CVaR(0.01)

Time (s.)

Gap (%)

41126 38942 34455 38942 34455 38942 34455 38942 35908 38942 37752

64220 64433 60585 64433 60585 64433 60585 64433 62085 64433 64008

47722 52011 57727 52011 57727 52011 57727 52011 57498 52011 54560

6.12 3600.00 10.76 3600.00 9.45 3600.00 10.94 3600.00 11.45 3600.00 3600.00

0.00 0.32 0.00 0.32 0.00 0.32 0.00 0.32 0.00 0.32 3.63

38444 40442 53956 40442 53956 40442 53956 40442 52269 40442 49502

44862 43647 34455 43647 34455 43647 34455 43647 36912 43647 39292

63846 63967 58585 63967 58585 63967 58585 63967 61056 63967 62109

42151 44170 55727 44170 55727 44170 55727 44170 55127 44170 53289

7.80 2720.69 10.67 64.85 11.26 1168.74 11.03 24.84 12.36 56.77 10.83

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

34476 34476 51956 34476 51956 34476 51602 34476 48846 34476 46000

48511 48511 34455 48511 34455 48511 36031 48511 37993 48511 41031

62786 62786 56585 62786 56585 62786 58140 62786 59646 62786 60460

38036 38036 53727 38036 53727 38036 53488 38036 52420 38036 49793

281.69 3600.00 10.30 3600.00 11.62 3220.20 11.15 3600.00 11.03 3600.00 9.52

0.00 0.08 0.00 0.10 0.00 0.00 0.00 0.08 0.00 0.06 0.00

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would accept given the imposed budget. This is not true any more for larger required returns. This explains the results reported in Table 5. In terms of computational complexity, we can notice that CPLEX is able to find the optimal solution for models CSP and CSP(0.01) in a few seconds. Conversely, model CSP(0.05) reaches the time limit of one hour in ten instances. It terminates with a percentage gap lower than 0.32% but in one case, where it hits 3.63%. In the remaining instances, the model finds the optimal solution in a few seconds with the exception of two instances with a budget of e70000, which require more than 19 and 45 minutes, respectively. In Table 6, we present the solutions obtained by the models expressed in terms of percentage of the total area devoted to the cultivation of each crop. Corn and wheat are the most common crops cultivated in North Italy. Many farmers decide to only grow these crops, because they are the most profitable ones according to their experience. This is indeed what is done by model CSP. On the contrary models CSP(0.05) and CSP(0.01) suggest to diversify the risk by investing also in barley and soy. While this diversification suggestion is only mildly followed by the model CSP(0.05) that keeps

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Table 5: Computational results for w = 4: percentage deviations CSP(β) vs CSP. Budget

µ0

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

-2.45 -12.44 -2.45 -12.44 -2.45 -12.44 -2.45 -10.08 -2.45 -4.74

-8.62 -26.32 -8.62 -26.32 -8.62 -26.32 -8.62 -24.51 -8.62 -14.70

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

-1.21 -18.24 -1.21 -18.24 -1.21 -18.24 -1.21 -13.31 -1.21 -8.07

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

0.00 -22.45 0.00 -22.45 0.00 -20.29 0.00 -15.09 0.00 -9.89

Profit Max Min

CVaR(0.05)

CVaR(0.01)

9.70 27.65 9.70 27.65 9.70 27.65 9.70 26.64 9.70 16.27

-5.31 -16.22 -5.31 -16.22 -5.31 -16.22 -5.31 -12.69 -5.31 -8.20

0.33 -5.66 0.33 -5.66 0.33 -5.66 0.33 -3.32 0.33 -0.33

8.99 20.96 8.99 20.96 8.99 20.96 8.99 20.48 8.99 14.33

-3.79 -32.40 -3.79 -32.40 -3.79 -32.40 -3.79 -27.67 -3.79 -20.23

5.20 40.35 5.20 40.35 5.20 40.35 5.20 35.96 5.20 28.76

-2.71 -23.20 -2.71 -23.20 -2.71 -23.20 -2.71 -17.72 -2.71 -12.41

0.19 -8.24 0.19 -8.24 0.19 -8.24 0.19 -4.37 0.19 -2.72

4.79 32.21 4.79 32.21 4.79 32.21 4.79 30.78 4.79 26.42

0.00 -35.91 0.00 -35.91 0.00 -34.43 0.00 -27.76 0.00 -20.34

0.00 50.70 0.00 50.70 0.00 49.67 0.00 41.68 0.00 33.42

0.00 -28.97 0.00 -28.97 0.00 -25.73 0.00 -21.68 0.00 -15.42

0.00 -9.88 0.00 -9.88 0.00 -7.40 0.00 -5.00 0.00 -3.71

0.00 41.25 0.00 41.25 0.00 40.62 0.00 37.82 0.00 30.91

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Table 6: Crop-mix composition: case with w = 4. µ0

Corn

Wheat

Soy

Barley

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

46.80 39.60 22.80 39.60 22.80 39.60 22.80 39.60 22.23 39.60 33.59

53.20 44.40 32.26 44.40 32.26 44.40 32.26 44.40 40.00 44.40 41.21

0.00 16.00 36.00 16.00 36.00 16.00 36.00 16.00 36.00 16.00 25.20

0.00 0.00 8.94 0.00 8.94 0.00 8.94 0.00 1.77 0.00 0.00

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

61.20 57.60 22.80 57.60 22.80 57.60 22.80 57.60 26.90 57.60 38.65

38.80 35.20 32.26 35.20 32.26 35.20 32.26 35.20 37.10 35.20 25.35

0.00 7.20 36.00 7.20 36.00 7.20 36.00 7.20 36.00 7.20 36.00

0.00 0.00 8.94 0.00 8.94 0.00 8.94 0.00 0.00 0.00 0.00

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

72.00 72.00 22.80 72.00 22.80 72.00 21.54 72.00 32.94 72.00 45.03

24.40 24.40 32.26 24.40 32.26 24.40 41.59 24.40 31.06 24.40 18.97

3.60 3.60 36.00 3.60 36.00 3.60 36.00 3.60 36.00 3.60 36.00

0.00 0.00 8.94 0.00 8.94 0.00 0.88 0.00 0.00 0.00 0.00

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excluding barley and grows soy for a small percentage of the area, it becomes stronger for model CSP(0.01) where the percentage of soy is constantly equal to 36% whereas the percentage of barley oscillates between 8.94% and 1.77%.

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Tables 7–9 refer to the results obtained with only two tractors instead of four. The headers of the columns have already been explained. Most of the comments introduced for w = 4 are still valid. Notice that the model CSP(0.01) becomes harder to solve, and the time limit of 3600 s is reached more frequently than for w = 4. However, the optimality gap is negligible being always lower than 0.34% for model CSP(0.05) and lower than 1.26% for model CSP(0.01). In terms of crop-mix composition, nothing changes for models CSP and CSP(0.05) when halving the number of tractors up to a budget equal to e72000. For this budget, the CSP model increments wheat by 3.60% through the elimination of soy, whereas CSP(0.05) decreases corn by 3.60% in favor of an increment of soy by 2.80% and wheat by 0.80%. The changes for model CSP(0.01) are more substantial. The availability of

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µ0

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

127228 124107 110340 124107 110340 124107 110340 124107 114400 124107 121200

272245 248779 203609 248779 203609 248779 203609 248779 210242 248779 232237

43835 48085 54831 48085 54831 48085 54831 48085 54335 48085 50968

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

133803 132184 108340 132184 108340 132184 109000 132184 116000 132184 123000

293752 282605 201609 282605 201609 282605 202664 282605 218110 282605 237299

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

138235 136558 106340 136558 106340 136558 110400 136558 117600 136558 124800

309383 298822 199609 298822 199609 298822 206242 298822 225979 298822 245715

Profit Max Min

CVaR(0.05)

CVaR(0.01)

Time (s.)

Gap (%)

41126 38942 33910 38942 33910 38942 33910 38942 36190 38942 37752

64220 64433 60641 64433 60641 64433 60641 64433 62205 64433 64008

47722 52011 56545 52011 56545 52011 56545 52011 56197 52011 54560

7.86 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 3600.00 25.37

0.00 0.33 1.09 0.32 1.09 0.33 1.09 0.34 1.17 0.34 0.00

38444 40442 52831 40442 52831 40442 52750 40442 50912 40442 48145

44862 43647 33910 43647 33910 43647 34256 43647 36857 43647 39087

63846 63967 58641 63967 58641 63967 58945 63967 61329 63967 62563

42151 44170 54545 44171 54545 44171 54519 44171 53827 44170 52030

5.63 3600.00 3600.00 3600.00 3600.00 2691.59 3600.00 619.95 3600.00 2305.52 10.90

0.00 0.06 1.13 0.07 1.13 0.00 1.18 0.00 1.23 0.00 0.00

33902 35771 50831 35771 50831 35771 50335 35771 47489 35771 44642

48482 47240 33910 47240 33910 47240 36190 47240 37858 47240 40755

62660 62680 56641 62680 56641 62680 58205 62680 60047 62680 61075

37477 39367 52545 39367 52545 39367 52197 39367 51117 39367 48448

5.84 13.04 3600.00 12.14 3600.00 11.58 3600.00 11.62 11.31 12.82 10.94

0.00 0.00 1.17 0.00 1.17 0.00 1.26 0.00 0.00 0.00 0.00

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Table 7: Computational results for w = 2: absolute values.

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Table 8: Computational results for w = 2: percentage deviations CSP(β) vs CSP. Budget

µ0

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

-2.45 -13.27 -2.45 -13.27 -2.45 -13.27 -2.45 -10.08 -2.45 -4.74

-8.62 -25.21 -8.62 -25.21 -8.62 -25.21 -8.62 -22.77 -8.62 -14.70

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

-1.21 -19.03 -1.21 -19.03 -1.21 -18.54 -1.21 -13.31 -1.21 -8.07

CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

-1.21 -23.07 -1.21 -23.07 -1.21 -20.14 -1.21 -14.93 -1.21 -9.72

Profit Max Min

CVaR(0.05)

CVaR(0.01)

9.70 25.08 9.70 25.08 9.70 25.08 9.70 23.95 9.70 16.27

-5.31 -17.54 -5.31 -17.54 -5.31 -17.54 -5.31 -12.00 -5.31 -8.20

0.33 -5.57 0.33 -5.57 0.33 -5.57 0.33 -3.14 0.33 -0.33

8.99 18.49 8.99 18.49 8.99 18.49 8.99 17.76 8.99 14.33

-3.79 -31.37 -3.79 -31.37 -3.79 -31.01 -3.79 -25.75 -3.79 -19.22

5.20 37.42 5.20 37.42 5.20 37.21 5.20 32.43 5.20 25.23

-2.71 -24.41 -2.71 -24.41 -2.71 -23.64 -2.71 -17.84 -2.71 -12.87

0.19 -8.15 0.19 -8.15 0.19 -7.68 0.19 -3.94 0.19 -2.01

4.79 29.40 4.79 29.40 4.79 29.34 4.79 27.70 4.79 23.44

-3.41 -35.48 -3.41 -35.48 -3.41 -33.34 -3.41 -26.96 -3.41 -20.58

5.51 49.94 5.51 49.94 5.51 48.48 5.51 40.08 5.51 31.68

-2.56 -30.06 -2.56 -30.06 -2.56 -25.35 -2.56 -21.91 -2.56 -15.94

0.03 -9.61 0.03 -9.61 0.03 -7.11 0.03 -4.17 0.03 -2.53

5.05 40.22 5.05 40.22 5.05 39.29 5.05 36.41 5.05 29.29

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Table 9: Crop-mix composition: case with w = 2. µ0

Corn

Wheat

Soy

Barley

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

68000 68000 68000 68000 68000 68000 68000 68000 68000 68000 68000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

46.80 39.60 23.32 39.60 23.32 39.60 23.32 39.60 21.73 39.60 33.59

53.20 44.40 37.12 44.40 37.12 44.40 37.12 44.40 49.47 44.40 41.21

0.00 16.00 28.80 16.00 28.80 16.00 28.80 16.00 28.80 16.00 25.20

0.00 0.00 10.76 0.00 10.76 0.00 10.76 0.00 0.00 0.00 0.00

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

70000 70000 70000 70000 70000 70000 70000 70000 70000 70000 70000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

61.20 57.60 23.32 57.60 23.32 57.60 23.06 57.60 27.78 57.60 39.53

38.80 35.20 37.12 35.20 37.12 35.20 39.13 35.20 43.42 35.20 31.67

0.00 7.20 28.80 7.20 28.80 7.20 28.80 7.20 28.80 7.20 28.80

0.00 0.00 10.76 0.00 10.76 0.00 9.01 0.00 0.00 0.00 0.00

CSP CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01) CSP(0.05) CSP(0.01)

72000 72000 72000 72000 72000 72000 72000 72000 72000 72000 72000

– 0.5 0.5 0.6 0.6 0.7 0.7 0.8 0.8 0.9 0.9

72.00 68.40 23.32 68.40 23.32 68.40 21.73 68.40 33.82 68.40 45.91

28.00 25.20 37.12 25.20 37.12 25.20 49.47 25.20 37.38 25.20 25.29

0.00 6.40 28.80 6.40 28.80 6.40 28.80 6.40 28.80 6.40 28.80

0.00 0.00 10.76 0.00 10.76 0.00 0.00 0.00 0.00 0.00 0.00

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a lower number of tractors impacts the profits by lowering them due to a more costly combination of machine types to satisfy operation constraints. Thus, to guarantee the required revenue, the crop-mix reduces soy in favor of more profitable crops such as corn.

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6.5 Analysis of the solutions for the case with no budget constraint In this section, we study the solutions obtained by the models for the case in which no budget restriction is imposed (B is set to a large non binding value). Each model freely decides the amount of proportional cost that optimizes the expected profit (model CSP) or the expected profit under the predefined percentile of worst realizations (models CSP(β)). These results are compared with the farmer’s solution. The farmer decides about his crop-mix by taking into account several issues including the evaluation of the regional incentives, the availability of the resources, the production costs, the trends in market prices and, of course, his personal experience.

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Table 10: Computational results: CSP(β) and CSP vs Farmer, four and two tractors, no budget. w

µ0

Farmer CSP CSP(0.05) CSP(0.01) Farmer CSP CSP(0.05) CSP(0.01)

4 4 4 4 2 2 2 2

– – 0.5 0.5 – – 0.5 0.5

Avg 114511 139054 120913 113013 114511 138684 120913 112628

Profit Max Min 235713 289832 233197 201468 235713 309832 233197 203226

48559 37149 51692 57580 48559 34350 51692 57321

StdDev 35286 49989 37496 34671 35286 48482 37496 34358

CVaR(0.05)

CVaR(0.01)

Cost (prop.)

Time (s.)

Gap (%)

60823 60466 64660 62315 60823 63108 64660 62640

51095 40629 55068 59148 51095 37921 55068 58762

8090 13223 7076 6513 8090 11551 7076 6203

5.38 5.07 3600.00 3600.00 4.65 4.77 3600.00 3600.00

0.00 0.00 0.17 0.09 0.00 0.00 0.19 0.27

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Farmer’s crop-mix includes all the four crops with a predominance for wheat, as shown in Figure 3 where the crop-mix obtained by the models are also shown (case with four tractors). It confirms that our farmer is on average wiser than other farmers in North Italy who concentrate on only one, maximum two crops (mainly corn and wheat). To obtain this solution the farmer invests in proportional costs an amount slightly larger than e8000. As expected, the solution provided by the CSP model selects the two crops with highest average revenue, i.e., corn and soy. Solutions obtained with models CSP(0.05) and CSP(0.01) are more balanced. In particular, the solution suggested by CSP(0.01), the most conservative model, selects all the crops as in the farmer’s solution. In Table 10, we report a comparison of statistics on the obtained solutions. The crop-mix selected by the farmer allows an average profit of e114511 and requires a proportional cost investment equal to e8090. The solution proposed by model CSP improves considerably the expected profit, but requires a much higher investment in terms of costs (e13223). Conversely, the solutions proposed by models CSP(0.05) and CSP(0.01) reduce the expected profit with respect to CSP but also imply definitely smaller costs. Interestingly enough, with respect to the farmer’s solution, model CSP(0.05) provides a better value in terms of expected profit but with a definitely lower investment in proportional costs. Similar considerations apply to model CSP(0.01). This confirms that models are valid tools when making decisions, leading to solutions with the best trade-off profit/costs. We then turn to variability measures. We see that the range of results is much larger for the solutions suggested by CSP, and significantly smaller for the solutions suggested by the CVaR models. In particular, the worst possible result (minimum profit) for model CSP(0.05) is 6.45% higher than in the farmer’s case, and is 18.58% higher for model CSP(0.01). This may be relevant, since if we take into account fixed costs, then increasing the minimum profit might reduce the risk of working at a loss. In terms of standard deviation of the profits under the different scenarios, we see that the solution suggested by model CSP has a standard deviation that is almost one and a half the standard deviation of the farmer’s solution. Conversely, the standard deviation associated with CVaR models (CSP(0.05) and CSP(0.01)) is significantly lower. The second part of Table 10 reports the values and the statistics for the case with only two tractors and no budget constraint. In Figure 4, we look at the cumulative frequency of profits. Here we see that

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Figure 3: Comparison of crop-mix selected by models and the farmer.

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the distributions associated with the farmer’s solution, CSP(0.05) and CSP(0.01) are quite similar, whereas the distribution associated with CSP stochastically dominates the other for large profits. However, the situation is much different when risky, low profits are considered. Figure 5 zooms in the left part of the cumulative distributions. It shows that, for small profit values, the solutions suggested by models CSP(0.05) and CSP(0.01) dominate both the solution suggested by model CSP and the farmer’s solution. This means that the solutions identified by the models incorporating the CVaR measure protect against risky situations. In Figure 6, we show a Gantt chart of the operations for the three crop solutions found by model CSP(0.05) when four tractors are taken into account. We see that the activities are rather sparse. Moreover, at most three tractors are used simultaneously. This fact suggests that the available resources could be better employed. In fact, as shown in Table 10, the model is able to implement the same crop allocation with two tractors instead of the available four. Furthermore, CSP(0.05) is able to find a solution even if 140 hectares are cultivated using two tractors, though with different proportions of corn, wheat, and soy. This suggests that the proposed models may be easily used to implement a sensitivity analysis on the resources available for production. In summary, the CSP model, that does not take into account the uncertainty in prices and yields, produces a very aggressive strategy, with a very good expected profit but with the possibility of bad outcomes. On the other hand, the models addressing CVaR as a safety measure produce solutions with an expected result similar to the farmer’s solution but with less variability in the outcome and in particular with an improvement in the worst possible results. In particular, the solution suggested by model CSP(0.05) is competitive with the farmer’s solution in terms of expected economic result and better than the farmer’s solution in terms of variability of the result. It is evident that solutions with higher expected profits are also more expensive to

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1,000

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600 400 200

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Cumulative frequency

800

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Farmer CSP CSP(0.05) CSP(0.01)

PT

20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 Profit (e×1000)

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Figure 4: Cumulative frequency distributions of profits, case with w = 4 and no budget constraint.

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Farmer CSP CSP(0.05) CSP(0.01)

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30 20

0

40

45

50 55 60 Profit (e×1000)

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65

70

75

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Figure 5: Detail of the right part of the cumulative frequency distributions, case with w = 4 and no budget constraint.

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Figure 6: The Gantt chart of the solution returned by model CSP(0.05), where a double vertical line corresponds to a few idle time slots.

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obtain, and have a lower rate of return in terms of profits over costs. The solutions identified by the CSP(β) models are significantly cheaper than the farmer’s solution. As a consequence, the rates of return of the solutions by models CSP(0.05) and CSP(0.01) are higher than the rate of return of the farmer’s solution.

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Insights for practitioners

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The conducted analysis allows us to draw some general guidelines that can be used by farmers and practitioners when involved with the crop selection problem. We try to make a list of them although we are aware that it is far from being exhaustive: 1. Although experience is extremely important when making investment decisions, it is not enough when dealing with complex real problems. In such a case, mathematical models represent a valuable and essential tool to support farmers. The use of models allows farmers to easily investigate different options under uncertain conditions.

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2. Models proposed are extremely different and the choice of the right one should take into account the risk aversion of the farmer. One can notice that CSP is an aggressive model, whereas CSP(β) models guarantee a stronger protection against worst cases. Moreover, models based on CVaR offer a gamut of possible alternatives according to the selected values for the parameter β and the required rate of return µ0 . In our experiments, the decision maker or the farmer himself could decide among the more risk-averse model CSP(0.01) up to model CSP(0.05) passing through CSP(0.01) with increasing required rate of return µ0 making it closer and closer to CSP(0.05).

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3. Farmers consider risk only in an intuitive way. This frequently leads to large losses due to irrational use of resources (purchase of tractors and machines that are not needed). We have shown that feasible solutions exist by halving the number of tractors available in the real case under analysis. This suggests that the proposed models can be easily used to implement a sensitivity analysis on the resources used for the crops production. In this sense, the appropriate model may help different entrepreneurs to share efficiently their production resources in order to reduce the total amount of costs. This could be particularly valuable in a low-margin industry like agriculture.

Conclusions

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We have proposed two mixed-integer linear programming models for the crop selection problem. Both models prescribe the amount of land to devote to each possible crop and provide the timings of all operations to be performed within the specified time intervals and the tools to be used. The first model (CSP) suggests an aggressive strategy, where the expected total profit is maximized. The second type of model (CSP(β)) takes into account the variability in prices and yields by incorporating the Conditional Value-atRisk as a safety measure. On the data obtained from a real case, we compare the solutions suggested by the different models with the solution implemented by the farmer according to his experi-

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ence. On the one hand, model CSP returns a solution with a significant improvement in the expected profit but with a higher variability in the results with respect to the farmer’s solution. On the other hand, models CSP(β), for both β = 0.05 and β = 0.01, return solutions with a significant reduction in the variability of the results and in particular a significant increment of the worst outcomes with respect to the farmer’s solution. These results validate model CSP(β) as an appropriate tool for crop allocation decisions and point to possible developments for the present work. First, model CSP(β) may be used to study the effects of different resource configurations and the advantages of coordination among different farms in order to share tools and tractors without resorting to third party suppliers. Second, the model could be enriched by incorporating explicitly decisions about other scarce resources. Third, the CVaR measure could be extended to other agricultural problems such as the crop rotation problem.

Acknowledgements

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The authors wish to thank the two anonymous referees and the Area Editor for their valuable comments, that improved the quality of the paper.

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