Tactical planning of the production and distribution of fresh agricultural products under uncertainty

Agricultural Systems 112 (2012) 17–26

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Agricultural Systems journal homepage: www.elsevier.com/locate/agsy

Tactical planning of the production and distribution of fresh agricultural products under uncertainty Omar Ahumada 1, J. Rene Villalobos ⇑, A. Nicholas Mason International Logistics and Productivity Improvement Laboratory, Arizona State University, P.O. Box 878809, Tempe, AZ 85287-8809, United States

a r t i c l e

i n f o

Article history: Received 15 June 2011 Received in revised form 3 May 2012 Accepted 3 June 2012 Available online 4 July 2012 Keywords: Agricultural planning Agricultural logistics Stochastic programming Perishable products Fresh produce Tactical planning

a b s t r a c t We present a stochastic tactical planning model for the production and distribution of fresh agricultural products. The model incorporates the uncertainties encountered in the fresh produce industry when developing growing and distribution plans due to the variability of weather and demand. The main motivation for building this model is to make tools available for producers to develop robust growing plans, while allowing the flexibility to choose different levels of exposure to risk. The modeling approach selected is a two-stage stochastic program in which the decisions in a first stage are designed to meet the uncertain outcomes in a second stage. The model developed is applied to a case study of growers of fresh produce in Mexico and in a simulation of various scenarios to test the robustness of the planning decisions. The results show that significant improvements are obtained in the planning recommendations when using the proposed stochastic approach as compared to those rendered by deterministic models. For instance, for the same level of risk experimented by the producer, planning based on the proposed stochastic models rendered increases of expected profit of over 50%. At the same time when risk aversion policies were implemented, the expected losses decreased significantly over those recommended by deterministic planning models. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Growers of perishable agricultural products, such as fresh fruits and vegetables, very often face complex planning problems such as deciding how much of a particular crop to plant, the timing of planting, and harvesting. The complexity of this problem and its importance for securing the food supply chain has prompted several applications. For instance, Ahumada and Villalobos (2011) presented a deterministic tactical model for planning the production and distribution of fresh agricultural products. Given that experience indicates that some parameters used in deterministic planning models are highly dependent on weather and market conditions (Lowe and Preckel, 2004), it is necessary to develop models that capture this variability. In particular, it is necessary to capture the uncertainty on price and yield which are very important for those growers that operate under open-market conditions. For these growers, the prices of their products vary along the harvesting season due to the combined effects of supply and demand and the lack of storage opportunities because of the perishability of these crops. For this reason, models that capture these uncertainties are needed to find more robust tactical solutions that ⇑ Corresponding author. Tel.: +1 480 965 0437; fax: +1 480 965 2751. 1

E-mail address: [email protected] (J. Rene Villalobos). Present address: Mexican Ministry of Agriculture (SAGARPA), Mexico.

0308-521X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.agsy.2012.06.002

are adaptable to the situations experienced by the different types of growers and their tolerance to risk. In this paper, we develop a model that deals with the uncertainties mentioned above for the fresh produce industry. The model builds on the work introduced by Ahumada and Villalobos (2011), by adding random variables, to better reflect the variability experienced by producers. The main motivation for building this model is to develop planting and distribution plans that are robust to the uncertain effects of markets and weather. From the perspective of the growers, the model should help them achieve their goals in the fresh produce supply chain, whether these goals include maximizing the expected income of growers, also known as risk neutral approach, or at reducing the probability of experimenting a loss, which is also known as the risk-averse approach. For the development of this project, growers were involved on providing data and validating the results found. The present work follows a similar approach to the one presented in Ahumada and Villalobos (2011), which consisted of designing a model with tactical decisions such as planting, labor planning, harvesting and distribution decisions that could be applied to any fresh agricultural problem, and then demonstrating its applicability by applying this methodology to a real case study of fresh produce growers located in Mexico. The approach selected for improving the deterministic model is to use a two-stage stochastic program (Birge and Louveaux, 1997)

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

in which the decisions in the first stage are made to meet the uncertain outcomes in the second stage. The use of a two-stage planning model comes natural to agricultural planning given the time elapsed from the time of planting to harvest (10–15 weeks). Most of the time, the long lead time between the time of planting and the time of harvest require that all planting decisions are made before there is a single crop harvested. These physical constraints prevent the tactical plan from being reevaluated, thus reducing the planning decisions to a single stage. At the beginning of the planting season, the farmer should decide on how much of each product will be planted without having certain information of future weather and market conditions. In accordance with the two-stage approach, the information available to the farmer at the time that tactical decisions are made is divided in two sets. The first stage set incorporates the planting constraints and the costs associated with the planting decisions, such as labor cost and availability. In the second stage the information available is the random distribution of crops’ prices and crops’ yields. Also in the second stage, there are transportation, harvesting and distribution costs. Other relevant features in the second stage include demand requirements that must be met, such as preexisting contracts, market demand, and transportation available during the harvesting period. The solution for the two-stage problem is then dependent on the first stage decisions (planting), the random realizations (Crop yield and prices) and the second stage decisions (harvesting and distribution). As mentioned before, the benefit of using stochastic programs (SPs) is that, unlike the deterministic solutions, which are based in expectations, the stochastic approach can be used to consider specific scenarios that occur according to the realizations of the different random variables explicitly considered in the model (Darby-Dowman et al., 2000). Moreover with the information provided by the solution of each scenario it is possible to implement more meaningful risk metrics that are both relevant and better understood by the growers. In this paper besides developing the methodology for a SP application for agricultural planning, we apply risk measures that can help farmers to make more robust planning decisions. One of the main benefits of the work to be presented is that it would provide the farmer with a tool to make decisions based on his/her tolerance to risk and to explore what are the expected worst case scenarios.

2. Background, related works and proposed model Planning models dealing with perishable products very often fail to incorporate realistic stochastic features present in the different echelons of the fresh produce supply chain (Ahumada and Villalobos, 2009). This may be due to the added complexity of finding solutions for the resulting models. In the few cases available in the literature that reality-based stochastic features were introduced into the models the results justified the added complexity of the model (Jones et al., 2003; Allen and Schuster, 2004). For instance, Kazaz (2004) presents an SP model for a Turkish company producing olive oil. The company has the option of leasing the olive trees to grow the olives or to buy the olives in the open market at a higher price. The planning model consists of two-stages, where the decisions at each stage depend on the stochastic distribution of demand and the uncertain yield of the olive trees. In the first stage the company determines the amount of trees to lease, and in the second stage, based on the yield and the prices of olives in the open market, the company determines the amount of olive oil to produce and olives to buy from the farmers. The objective of the model is to maximize the expected profit subject to demand and the sales price of the olive oil.

Our case is similar to that of Kazaz (2004) in the use of a twostage SP. The traditional formulation of the two-stage SP has the following structure (Birge and Louveaux, 1997):

 Max cx þ Ep Q fx; nðwÞgjAx ¼ b;

x P0g

ð1Þ

where

Q fx; nðwÞg ¼ MaxfqðwÞyjWy ¼ hðwÞ  TðwÞx; y P 0g In this notation the vector x is the first-stage decision variable and y is the second-stage decision vector with feasible sets fxjAx  b;x P 0g and fyjWy ¼ hðwÞx;y P 0g respectively. W is the matrix for the parameters of the second stage variables, h(w) are the random vectors associated with random realizations of w and T(w) is the random matrix effected by the first stage decisions variables vector (x). The objective of this problem is to maximize the revenue of the first stage cost and revenues (cx) with the expectation of the second stage solutions (q(w)y). In the case of a discrete distribution, for example, when scenarios approximate the distribution, the formulation then becomes a linear program. The deterministic equivalent for the two–stage SP has the following structure:

( Max cx þ

X ps qs yjAx ¼ b;

) xP0

ð2Þ

s

s.t.

T s x þ W s y ¼ hs ; y P 0;

s ¼ 1...S

It can be observed in (2) that in the deterministic equivalent the random realizations w are approximated by the scenarios S. The previous model allows the introduction of specific scenarios of importance to the farmer. For instance, scenarios with low probability of occurrence but high impact on the potential profit, such as climate or market dislocations, could be captured into the model. This is an advantage of stochastic over deterministic models that very often are based on expected value that do not capture very well events in the tails of the distributions that could have a dramatic impact on the economic performance of the farm. For instance, some farmers may want to absolutely minimize the observation of catastrophic events at the expense of higher expected profits. In our case, the first stage decisions for the stochastic model of the grower of fresh produce are formed by the planting decisions. It is assumed, in accordance with the two-stage approach, that costs and resources in the first stage are deterministic, thus leaving the second stage or recourse variables as the only random functions. In the second-stage of the problem it is assumed that the stochastic parameters are the crops’ yield (ytjt0 ) and the market prices (ptki) for product k at shipping period t, and customer i, which are represented by the scenarios S developed in Section 4. Then the model for the first stage problem is developed in the following way:

" X X s X X max pr s Q t  k pr s ðT  Q st Þþ  Plant tjl  Cplant s

ts

ts

tjl

tl

tl

# X X X Hiretl  Chire  Opttl  Ctemp  Opltl  Clabor  tl

ð3Þ

s.t.

XX Plantpjl 6 LAl  j

all l where l 2 L; p 2 TPðj; lÞ;

and j 2 J

p

ð4Þ XXX Plantpjl  Cplantjl 6 Inv j

p

l

ð5Þ

O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

XXX Plant pjl  Waterj 6 W j

p

ð6Þ

l

Minj  Y jp 6 Plantpjl 6 Maxj  Y jp Hiretl þ Firetl ¼ Opltl  Oplt1l Hiretm l þ Opltm 1 l P Opltl Hiretl 6 Mfix;

all j; p and l

ð7Þ

all l and t < t m

ð8Þ

all l and t m 6 t

ð9Þ

Opt tl 6 Mtemp all t; l

ð10Þ

The first stage decisions include the timing and crops to plant (Plant) and the labor to hire for the whole season (Hire). The first constraint (4) states that the farmer cannot plant more than the land he/she has, the second constraint (5) makes sure that the farmer does not spend more money than the available for investments. The third constraint (6) verifies the results satisfy water restrictions, the fourth (7) constraint provides a lower bound and an upper bound on the number of hectares to plant in a single time period. The objective (3) is to maximize the overall revenue for the producers, which are obtained from the second stage expected revenues Q st minus the cost generated by the first stage; this includes a risk metric that penalizes decisions which may lead to losses and can be adjusted by the parameter lambda (k). This parameter is important for our study since it determines the penalization for target semideviation, therefore making the solution risk neutral for k equal to zero or risk adverse for higher values of k (Ruszczynski and Shapiro 2006). The definition of each of the variables and parameters used in previous and subsequent models can be consulted in Appendix A. In the second stage of the problem, given a solution from the first stage of the LP, we have that for every period t and scenario s we obtain a second stage LP as follows: " Q st ðxÞ ¼ max 

X

! X X s X XX s s SC tkfir þ SW shtkwir þ SDhtkdir  pricetki tki

fr

SC stkqfir CT fir 

tkqfir



hw

X

d

6h

SW shtkqwir CTW wir 

htkqwir

X

X

SDshtkqdir CTDdir

SWDshtkqwdr CTWDwdr

htkqwdr

X

# SPW shtkqfwr CTPW fwr

htkqfwr

ð11Þ

s.t. s

Har v est sphjl ¼ Plant pjl  Yieldphjl  Totalpjl

all p; h; j; l where h 2 THðj; lÞ ð12Þ

X s SPphjlf ¼ Har v est sphjl

all p; h; j; l;

where f 2 PF

ð13Þ

f

Opltl þ Opttl P

X X Plant pjl  LabPptj þ Har v est sphjl pj

 LabHphj Packhkqf ¼ Colhkq

pjl

allt; l where h ¼ t

ð14Þ

X SPphjlf ð1  Salv phjl Þ phjk

 Podphjk =Weightk

all h; f ; k where k 2 KðjÞ

The decisions in the second stage are similar to the decisions in the deterministic case; however, the SP is S times larger, since it requires running the second stage model for each one of the scenarios. With such a model, it is possible to calculate the best solution for all the potential scenarios, thus finding a planting plan that will maximize the expected income for farmers regardless of the outcome of the random variables. However the size of the problem, in particular of the second stage subproblems, might make the problem hard to solve. In the next section we present the solution methods developed for finding solutions to the current problem. 3. Solution approach for the stochastic model The solution methodology for stochastic programming is important given that the size of the model can grow very large because of the amount of scenarios computed. The size of the present problem is so large that even for small instances of the deterministic equivalent, the solver used ran out of memory. To deal with this problem decomposition methods applied to the field of SP were used. In particular, the stochastic version of Bender’s decomposition (L-shaped method) and the multi-cut version of this algorithm (Birge and Louveaux, 1997) were used. A third algorithm, a multicut for risk stochastic programs, was used to solve the model presented in Section 2. The proposed algorithm was first introduced by Takriti and Ahmed (2004) for a related risk metric. Also Kristoffersen (2005) presented a detailed algorithm for the case of minimization of the expected semideviation from the mean. We modified this algorithm to adjust it to the case of maximizing expected revenue with a penalization from target semideviation. The solution algorithm is included in Appendix B in this paper. This algorithm suffers from the same convergence problems that affect the L-shaped or Bender’s algorithm. 4. Scenario development and case study

htkqdir

SPDshtkqfdr CTPDfdr 

htkqfdr

X

19

ð15Þ

The stochastic parameters in the second stage of the model s include the price of the products (pricetki ) and the total expected s harvest from the crops (Totalpjl ); the rest of the parameters retain the same values as in the deterministic model. In (12) the planting decisions in the first stage are fixed for the second stage (Plant), the same as the case for decisions in (14) for the hired labor (Opltl and Opt tl ). The rest of the model is similar to the deterministic model presented in Ahumada and Villalobos (2011).

The proposed application incorporates transportation decisions, planting and distribution policies that enable crops of the right quality, with the right shelf life to reach the right customer in the food supply chain. Furthermore, all harvesting and transportation decisions are directly related to planting decisions made on the first stage of planning (Fig. 1). In the horticultural supply chain, crops can be contracted (sold before hand) or sold in the open or spot market. Very often the customers in the open market prefer to pick up their products free on board (FOB) at the warehouse of packers, while those who enter into contracts include the delivery in the contract; in-between there are a lot of different combinations of agreements for storage and transportation requirements as shown in Fig. 1. Another issue that needs to be considered is the final destination of the crops, since customers are becoming more specialized in their requirements. For example, in the case of tomatoes, there are different markets for different types of tomatoes such as mature green and vine-ripe tomatoes. Thus growers need to balance market demands, with their cost and operational limitations, such tradeoffs, are among the many issues that need to be included in the planning model. However, the main focus of this article is on the impact of price and yield variability on the tactical decisions made. This variability is incorporated into the model by creating a limited set of scenarios which accurately capture this uncertainty. The scenarios required were created from statistical distributions derived from the real behavior of crops yields and prices, where the distributions for each crop (e.g. tomatoes, peppers, cucumbers, etc.) were analyzed and modeled with their respective yields and prices. The scenarios created are then used directly in the SP model described in Section 2.

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

Fig. 1. Supply chain decisions of horticultural crops.

We present next a general description of the methodology to estimate the distribution of prices for each product. Later we present the steps followed to determine the distribution of yield for the crops. 4.1. Finding distribution for prices of products

XðtÞ ¼ l  SðtÞ þ e

ð16Þ

The estimated mean and seasonal components for the historical data are presented in Ahumada (2008). This model was applied directly to the prices of peppers; however, the historical prices of tomatoes had to be adjusted because there is a minimum price (floor price) that importers of tomatoes must pay to Mexican

Fig. 3. Weekly prices for tomatoes in years 1999–2006.

suppliers. To correct this issue, the median was used instead of the mean to calculate the center moving average for each period of the cycle. With these new center points we calculated the seasonal components, the error and the expected value of the cycle for tomatoes (Ahumada, 2008). Finally, after adjusting for seasonal behavior, autoregressive components were identified. These were modeled using a lag one autoregressive model (AR) with a constant term to fit for the data (17).

XðtÞ ¼ C þ aXðt  1Þ þ e

ð17Þ

Once the weekly price distributions and seasonality factors were captured, the scenarios for the prices of the crops were generated. These scenarios can be created using the expected values of prices for the year (Expected mean), together with the AR and seasonal multiplicative models. An example of the results obtained by this procedure can be observed in Figs. 4 and 5. These graphs show the average 2.5th percentile and the 97.5th percentiles of the prices for tomatoes and peppers scenarios

30

Price per Box in US$

In the particular case of prices for tomatoes and peppers, the data used are the weekly historical prices of peppers and tomatoes for the 1999–2006 seasons (USDA, 2007a) and 25 previous years (USDA, 2006b) after adjusting for inflation. Fig. 2 presents the histogram of the historical monthly prices for the case of tomatoes in the month of January. As it can be observed in Fig. 2, the distribution of the prices is skewed to the right and has a lower bound, suggesting a lognormal distribution. This distribution was confirmed by a v2 goodness of fit test with 35° of freedom, yielding a p-value of 0.038; the lognormal distribution was confirmed for all months. A similar approach was used for the weekly price distribution of peppers. The next step is to estimate the seasonal patterns followed by prices. As it can be observed in Fig. 3, there are some clear patterns in the data, with high prices at the beginning of the season followed by lower prices in weeks 6–10, which correspond to the peak of the harvesting season. To better estimate the seasonal behavior, a multiplicative seasonal model without a linear trend was used, as presented (16). Where the S(t) is the seasonal component, l is the mean of the cycle (year) and e is the error term (Farnum and Stanton, 1989).

25 20 15 10 5 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Week Floor Price Fig. 2. Histogram for historical monthly data of tomatoes (25 lbs box).

Average

Percentile 97.5

Fig. 4. Range of prices sampled for tomatoes.

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

Price per Box in US$

30

Table 1 Parameters estimated for yield distributions.

25 20

Distribution

Min

Max

a

b

Beta Normal

37.3

64.0

0.46

0.50

Mean

Std. dev.

49.5

11.9

15 10 5 0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Week Percentile 2.5

Average

Percentile 97.5

With both parametric distributions for yields selected, the beta for the case of tomatoes and the normal with a linear component for the case of peppers, we can fit the county level data to these distributions. The estimated parameters for the beta and normal distributions respectively, are presented in Table 1 below. Finally, these distributions can be used to develop random samples from the estimated yield.

Fig. 5. Range of prices sampled for peppers.

4.3. Correlation between prices and yields respectively. The main difference between the two graphs is the floor price for tomatoes, which does not apply to peppers, and it was estimated that the 2.5th percentile serves as the minimum expected. 4.2. Finding distribution for Yields Once the price distributions are defined, the next step is the estimation of the crop’s yields distribution. For estimating the crop’s yield we split the estimation of weekly data into two components: One that determines the total amount obtained for the season, (which corresponds in the model to the total quantity to harvest (Totalpjl). A second component determines the distribution of the harvest in each week (Yieldphjl). For estimating the distribution of crops’ yields we assume only that the quantity harvested is stochastic and the percentage harvested per week remains constant regardless of the total production obtained. To estimate the values for different crops yields, parametric distributions were chosen, since they are an efficient option for estimating random variables when small samples are available (Just and Pope, 2002); furthermore, for our purposes, the most appropriate distributions are the normal and the beta distributions (Norwood et al., 2004). The beta distribution, which is flexible and capable of assuming different shapes depending on the fitted parameters, has been widely used for estimating crops’ yields (Law and Kelton, 2000). Likewise, the normal distribution has been widely used to model physical data, and for the case of yield distribution, has been used together with other functions to approximate the observed distribution (Just and Weninger, 1999). To determine the yield distribution, two sets of historical data were used for peppers and tomatoes. The first one is the historical statewide data that includes information from 1985 to 2006 (CIDH, 2006a) to estimate the parameters of the distribution for the farm yield. The second set is county level data from 1999 to 2006 (CIDH, 2006b) that we use as an approximation for estimating the yield for individual farms. Using information for statewide tomato yield, we could not reject the hypothesis that tomatoes yield came from the beta distribution. We also used the data to determine if there was any trend in the total yield, but there was no evident trend in the yield of tomato crops. In the case of peppers, there seemed to be a positive trend in the data collected, this conjecture was confirmed using regression analysis. For the stochastic components of the linear model, we could not reject the hypothesis that they come from a normal distribution. Thus, for the case of pepper’s yield, the best fit was a linear function with an error term following the normal distribution.

The development of the scenarios requires the combination of the joint distribution of prices and the crop’s yields. However, the main issue for estimating the joint probability distribution of prices and yields is the dependence between the estimation of prices and crops yield. If there is a bad season for all growers then the prices might tend to increase; on the other hand if the crops have high yields for all growers, then the prices might decrease. One simplifying assumption that we make for the present problem is that samples obtained from the distributions, affect only a single grower. This implies the results obtained are not representative of all the growers in the industry. However, we still need to investigate the issue of independence between the prices of crops and their respective yields. From the estimation of the correlation between the yields and prices for the portfolio of products (tomatoes and peppers) it is evident there are significant correlations among these variables. The existence of this correlation implies that when constructing scenarios, the samples cannot be assumed to be independent. To account for the existence of correlation when creating the scenarios, the multivariate normal distribution was used to sample from these variables, applying a transformation on the yield of tomatoes and the prices of both products to make them normally distributed. At this point, we have enough information on the behavior and parameters of the random variables to accurately create scenarios which are representative of crops yields and prices. The scenarios are developed using the joint distributions of prices (seasons 1999–2006) and yields (seasons 1985–2006), and the respective patterns detected for both distributions. 4.4. Regularized decomposition One of the difficulties with implementing algorithms based on Bender’s Decomposition is the convergence of the algorithm itself. This is due to the complexity of finding a feasible solution in the second phase of the algorithm (Birge and Louveaux, 1997). This has motivated several efforts to accelerate the convergence of these algorithms such as increasing the information available (cut strengthening), reducing the subproblems running times (bounding) and restricting the fluctuation of the solution space (regularization) (Santoso et al., 2005). This paper uses the latter, to reduce the running time and the number of iterations. The formulation follows the model proposed by Ruszczynski and Swietanowski (1997) for accelerating two-stage stochastic solution convergence. The model used is the following:

(

X max c x þ ps hs þ bkx  dk2 T

s

) ð18Þ

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

where d is a reference point from the previous solution and b > 0 is a factor that can be picked to determine the weight of penalization of deviations from the current solution. The solution algorithm uses the regularization formulation in conjunction with ‘‘good starting solutions’’. For example, when solving the stochastic program for the expected value, the deterministic solution is used as a starting point. The use of the regularized formulation, together with better starting points, reduced significantly the running times for finding optimal solutions.

5. Case study and results In this section, the stochastic version of the case study presented in Ahumada and Villalobos (2011) is presented. The case study includes two crops (tomatoes and peppers), a total area of 500 ha divided in two locations, two packing facilities, two potential warehouses, three customers and two distribution centers. For tomatoes, the maximum storage life is set to 1 week; and for peppers, 2 weeks. The maximum cycle time for delivery to the final customer is set to 1 week (both peppers and tomatoes). The stochastic version incorporates all the data and assumptions made in Ahumada and Villalobos (2011), with the addition of stochastic estimates of prices and yields, which are integrated using multiple scenarios. A large amount of scenarios was used with the aim of capturing the stochastic environment of producers. The problem was solved using a version of the multi-cut L-shaped method using AMPL and CPLEX v11.2 (ILOG, 2010). For this example, the scenarios were formed using the joint distribution for both prices and yields; since the relationship amongst these variables is known, we only sample the values for prices while accounting for the dependence. The results obtained from the stochastic model differ from the deterministic version significantly. We illustrate the difference between these models in Table 2, which compares the results for planting decisions obtained from the deterministic model (results presented in Ahumada and Villalobos, 2011) with the stochastic model presented in this paper. As it can be observed, the recommendations from both models change from 93 ha of peppers and 407 ha of tomatoes to 162 ha of peppers and 337 ha of tomatoes for the deterministic and stochastic (k = 1) models respectively. Table 3 presents the expected profit, costs and return on investment (ROI) of both models. These results were obtained by using training data for prices (seasons 1999–2006) and yield (seasons 1985–2006) and using testing data from the 2007–2008 season to estimate performance. As it can be observed there are significant benefits from using the stochastic model. Such a gap in the results obtained, could be caused by the interaction of the prices of tomatoes and peppers, since as it was found there is a significant correlation for the prices between these two crops. Furthermore the expected downside risk (Worst) is dramatically reduced when using the stochastic model. However, the higher performance

Table 2 Comparison of planting decisions (hectares). Week

Deterministic (Ahumada and Villalobos, 2009)

Stochastic k=0

Stochastic k=1

Peppers

Tomato

Peppers

Tomato

Peppers

Tomato

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

28 – – – – – 22 – – – – 23 – – – 20 – –

– – 113 20 – 29 24 – – – 85 – 47 – – – – 89

22 – – 20 – – – – 35 81 – – – – – – 20 –

– – 117 – – – 49 – – – 21 29 106 – – – – –

– 20 – – – – – 20 – 97 – – – – 26 – – –

– – 130 – – – 37 – 20 – – 27 123 – – – – –

Total

93

407

178

322

162

337

Table 3 Comparison of results from Stochastic Program. Model

k

Deterministic 0 Stochastic 0 Stochastic 1 Stochastic 10

Profit

Costs

ROI (%)

Worst

CPU/s

$3,255,643 $5,621,200 $5,619,360 $5,510,680

$14,439,900 $14,427,100 $14,434,100 $14,434,500

22.5 38.9 38.9 38.1

$37,598,000 $174,526 $138,500 $153,871

197.54 1325.41 1350.15 1485.52

comes with larger running times as shown in the last column of Table 3 (CPU/s); these running times, although higher, still allow for an efficient computation of the results. The results show that there are significant differences in the planning recommendations between the proposed stochastic approach and those rendered by deterministic models. For instance, for the same level of risk experimented by the producer, planning based on the proposed stochastic models rendered increases of expected profit of over 50%. Even with the increased running times and the larger requirements in terms of data and scenario generation, the results obtained from the stochastic program are worth the increased costs. Furthermore, the results for the runs with the penalty terms from Table 3 are consistent with the theory that predicates that reduced risk corresponds to lower expected returns. Nonetheless, one issue with the results obtained in Table 3 is that they are based on a single sample taken from a realization corresponding to the 2007–2008 season which was used to test the

Fig. 6. Procedure for simulation development.

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

performance of the model. To improve the statistical validity of the results and to test the robustness of the stochastic model, a new sample of 50 scenarios was generated from the price distributions. This set was used to simulate the impact of the first stage decisions obtained from the stochastic and deterministic model solutions on the second stage decisions and final profit. To test the behavior of the distribution of profits for each model, the first stage decisions (planting and labor planning) were fixed, leaving as possible changes only those related to distribution. Therefore, under optimal allocation we obtain the expected profit for each model and each scenario simulated. This procedure, which is summarized in Fig. 6 below, yields a simulation which approximates the expected profit and costs for the deterministic and stochastic (0, 1, 10) scenarios. Finally, we run the optimization problem for each individual scenario assuming that perfect information is available for the yields and prices of the next season; we call this the wait and see (WS) solution in accordance to (Birge and Louveaux, 1997). This deterministic idealization of the problem provides a measure for how much the value of perfect information is under a given setting. The results obtained from the simulation, as well as the WS solutions are summarized in Table 4, which shows the mean results obtained from the 50 samples (Mean). Other fields include: the risk-aversion factor (k), the number of scenarios used to build the model (Scen), the standard deviation of the profits obtained (Std. dev.), the worst scenario (Worst) and the best scenario (Best). From the results in Table 4, it can be observed that the differences between the average results for the deterministic model, the risk neutral model (k = 0) and the more risk averse models are close to those obtained in Table 3, with the variation caused because the results from Table 3 come from a single scenario whereas those from Table 4 come from a sample of 50. Furthermore, we observed that the stochastic model solutions perform

Table 4 Results of the experiments for 50 different samples. k

Scen Mean

Std. dev.

Worst

Best

Deterministic 0 1 10 Perfect information

1 50 50 50 –

$2,500,621 $2,615,054 $2,781,673 $2,648,494 $3,320,269

$1,741,163 $268,011 $223,124 $221,385 $3,005,492

$9,139,454 $11,523,316 $12,203,016 $12,038,200 $21,148,775

$3,978,744 $6,070,825 $6,194,092 $6,119,366 $8,321,542

significantly better than the deterministic model in over 90% of the individual scenarios. Particularly, we also notice that the deterministic model may perform well in scenarios which are close to the expected value; nonetheless, it fails to take advantage of favorable scenarios while also performing significantly worse on less favorable instances as seen in Fig. 7 below. The distribution of profits for the deterministic solution and the risk neutral solution can be seen in Figs. 7 and 8 respectively. For more traditional measures, the value of the stochastic solution (VSS) and the expected value of perfect information (EVPI) can be calculated from the difference between the stochastic and deterministic solution or the WS and stochastic solutions respectively (Table 4). From the simulation results, we also notice that the distinction in the behavior of the risk neutral and risk-averse models for the new scenarios still behaves in accordance to theory, although the distinction becomes less clear. For instance, the worst scenario becomes less extreme as the risk penalization term is increased. However, the expected mean for the risk neutral model is smaller than that of the risk averse models for the simulation results, suggesting that in practice a risk averse strategy may yield better results in terms of expected profit, although, this could be caused in part by variability on the performance of the other scenarios.

Fig. 7. Distribution of profits (deterministic model).

Fig. 8. Distribution of profits (stochastic model k = 0).

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

Table 5 Downside risk for different target and risk penalty combinations. Target profit (MM)

Target ROI (%)

Deterministic

Stochastic (k = 0)

Exp. down

Prob (%)

Exp. down

Prob (%)

Exp. down

Prob (%)

Exp. down

Prob (%)

0.00 0.75 1.50 2.00 2.40

0 3 6 8 10

946,790.29 1,391,046.70 1,784,764.63 1,712,578.99 1,344,893.96

6 8 10 14 24

268,011.47 1,018,011.47 1,205,791.67 1,755,791.67 1,167,451.25

2 2 4 4 8

223,123.58 623,263.75 1,323,263.75 1,341,493.64 1,397,654.61

2 4 4 6 8

221,385.00 971,385.00 1,125,407.00 1,190,634.67 1,246,241.00

2 2 4 6 8

In the case of minimizing the expected downside risk and the probability of having a scenario below target, the stochastic model performs significantly better than the deterministic model. Moreover, comparing the behavior of the stochastic model for different risk parameter combinations we see that the most appropriate value of k is dependent on the target that has been set (Table 5). For instance, take the case in which the grower sets the target on making a ROI anywhere between 0% and 6%. In this case, the most riskaverse solution would be most appropriate since it has the lowest deviation from target, with an expected downside deviation that is at least 7% better than the risk neutral solution. However, also note that if the farmers target is set on getting a higher ROI, then the risk neutral approach becomes more desirable. Table 5 below details the contrast between different targets (for return on investment and minimum profit in millions) and risk penalty combinations, showing the expected number of scenarios below target (Prob), and the expected lower deviation from target (Exp. Down) for each solution. One of the advantages of the current risk-averse model is that the level of risk can be changed according to the requirements of growers. For example, the users of this model could get the results for different levels of risk and present these results to growers so they could select the plan that best fit their risk and revenue expectations. Moreover, some additional benefits can be obtained by assessing strategies and policies with the stochastic program as an aid to check benefits and costs incurred. For example, it is possible to compare between two contracting policies and determine the best one given the expected markets and the environment of the producer. Furthermore, the model can be expanded to estimate the risks incurred when determining contracting policies, which could also be left as decision variables. 6. Conclusions We presented an initial stochastic model that can be used for basic tactical planning for a fresh produce grower that handles production and distribution decisions. The resulting model yields a

Stochastic (k = 1)

Stochastic (k = 10)

solution which is significantly different than the deterministic model and provides considerably better results when tested with real data from the following season. Furthermore, the model also provides better results when tested using a simulation of 50 scenarios which gives more confidence on the robustness of the stochastic formulation. Moreover the model can be tuned to different risk preferences, resulting in a more robust plan that can be customized to the risk preferences of different growers. As shown in the results section, this tuning parameter (k) affects the models expected profit while also reducing the probability of losing the investment. Therefore, the choice of the parameter can be left to the farmer to determine its exposure to risk depending on how willing to lose the investment they are if this can be compensated through higher profits on average. Although the formulation, data collection and solution approach of the stochastic model increase in complexity and time investment, this increased cost yields a solution that is superior to that of the deterministic model. Clearly a solution approach that considers the stochastic nature of agricultural tactical planning has significant benefits and would prove to be a useful support tool for growers. Furthermore, taking the current tactical planning tool as a base, the model can be expanded to include additional features that might increase the benefits of the proposed model for grower/shippers. For instance, this model can be used directly for a portfolio selection of products; by adding more products as options in the model, farmers can select the products that are best to include in the portfolio based on the tradeoffs of revenue increase and risk reduction. Moreover, another application can be related to assessing the impact of variability reducing methods such as new production techniques, better seed varieties and protected agriculture. Finally, some topics left for future research and expansion of the model are the addition of further decisions such as including contracts in the planting decision process which can be accepted or rejected. A related issue would be the determination of the optimal ratio of production committed to contracts to that left for open markets.

Appendix A Indexes and sets: l2L t2T j2J p 2 TPðj; lÞ # T h 2 THðj; lÞ # T k 2 KðjÞ q2Q w2W i2I d2D f 2 PF

Locations available for planting Planning periods (weeks) Potential crops and/or varieties to plant Set of feasible planting weeks for crop j in location l Set of feasible harvesting weeks for crop j in location l Products obtained from crop j Quality of crop at harvest (Color) Warehouses available for storage Customers Distribution centers Packaging facilities

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r 2 TM s2S Parameters Waterj LAl Yieldphjl s

Totalpjl Podphjk Weightk Maxj Minj Totlabor Colhkq W Inv pr s Q st T k Cost parameters Pricestki Cplant jl Ctemp Clabor Chire Transportation parameters CT fir CTW wir CTDdir CTPW fwr CTPDfdr CTWDwdr

Variables Plant pjl Har v estsphjl Packhkqf Hiretl Firetl SP sphjlf SC stkqfir

Transportation mode Scenarios

Water required per acre of crop j in cubic meters (in cubic meters) Land available at location l (in ha) Yield of crop j planted in location l at time p and harvested in week h (percentage of total) Total production of crop j planted in location l at time p (pounds per hectare) Percentage of product k from crop planted in location l at time p harvested in week h (percentage) Quantity of crop j required to pack a box of product k (in pounds) Maximum amount to plant of crop j (in ha) Minimum amount to plant of crop j (in ha) Maximum number of contracted seasonal workers available (Men-week) Percentage of harvested fruits with color q from product k at period h Water restriction (in cubic meters) Investment quantity available Estimated probability of scenario s Second stage expected revenues from scenario s and period t Target income Weight of the risk metric Price for the grower of a box of product k sold to customer i at time t Cost per hectare of production for crop j planted in location l Cost of one man-day for day-laborers Cost of seasonal laborers per man-week Fixed cost of hiring a seasonal laborer at the field site Cost of transportation from packing facility f to customer i using transportation mode r Cost of transportation from warehouse w to customer i using transportation mode r Cost of transportation from DC d to customer i using transportation mode r Cost of transportation from packing facility f to warehouse w using transportation mode r Cost of transportation from packing facility f to DC d using transportation mode r Cost of transportation from warehouse w to DC d using transportation mode r

Area to plant of crop j, in period p at location l (in ha) Harvest (pounds) of crop j in period h and planted in period p at location l Quantity of product k with color q packed at facility f in period h (in boxes) Number of workers hired at period t in location l Number of workers terminated at period t in location l Pounds of crop j to ship from location l to facility f in period h Boxes of product k with color q shipped to customer i from facility f in period t by transportation mode r (continued on next page)

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O. Ahumada et al. / Agricultural Systems 112 (2012) 17–26

SPDshtkqfdr

Boxes of product k harvested in period h with color q shipped from facility f to DC d in period t by transportation mode r Boxes of product k harvested in period h with color q shipped from facility f to warehouse w in period t by transportation mode r Boxes of product k harvested in period h with quality q shipped from facility f to DC Boxes of crop k harvested in period h with color q shipped from warehouse w to DC d in period t by transportation mode r Boxes of product k harvested in period h with color q shipped from warehouse w to customer i in period t by transportation mode r 1 if crop j is planted at period p at location l, 0 otherwise

SPW shtkqfwr SDshtkqdir SWDshtkqwdr SW shtkqwir

Y jpl Where Y 2 Bn Plant; Harv est; Pack; Opl; Opf ; Hire; Fire; Opt; SP; SPD; SPW; SD; SWD; SW 2 Rnþ

Appendix B The proposed algorithm to solve SP is as follows: (1) Set i = 1 (2) Solve the following master problem:

(

X X LB ¼ max c x þ ps hs þ k ps ds  kg

)

T

s

s

Ax ¼ b hs 6

X

pTs ðhs  xs Þ

ð19Þ

s

X ps mTs ðt s  xs Þ P 0

ð20Þ

s

g P ds ; hs P ds

ð21Þ

ds ¼ minfqy; gg

ð22Þ

i

his

(3) Let x , be an optimal solution. If no constraint (19) is present then hi = 1 and is not considered in computation of xi. (4) For s = 1 to S solve the subproblem:

max w ¼ qy Wy ¼ hs  xi

ð23Þ

(5) If the subproblem is infeasible then obtain extreme ray m. Add constraint 20, set i = i + 1 and return to step 2. P (6) If hs > s Ps pTs ðhs  xs Þ then let ps be dual variable from solution. Then add constraint 19, set i = i + 1 and return to step 2. (7) If no cut is added in step 6, then stop. Optimal solution is xi.

References Ahumada, O., 2008. Models for Planning the Supply Chain of Agricultural Perishable Products. PhD Dissertation, Arizona State University.

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