Economics of alternative simulated manual asparagus harvesting strategies

AGRICULTURAL SYSTEMS Agricultural Systems 92 (2007) 266–294 www.elsevier.com/locate/agsy Economics of alternative simulated manual asparagus harvesting strategies Tiziano Cembali *, Raymond J. Folwell, Ray G. Huffaker, Jill J. McCluskey, Phil R. Wandschneider School of Economic Sciences, Washington State University, P.O. Box 646210, Pullman, WA 99164-6210, United States Received 30 November 2004; received in revised form 23 March 2006; accepted 30 March 2006 Abstract Asparagus is harvested daily during the production season. The adoption of harvesting strategies less or more frequent than the traditional 24-h strategy has not occurred because of problems in hiring manual labor. A model that predicts daily harvest and the impact of different harvesting strategies was developed. This paper presents a bioeconomic model, capable of predicting daily asparagus harvests, composed by different mathematical functions: emergence, density dynamics, spear growth, diameter, weight, carbohydrates reserve dynamics, and profit. The bioeconomic model was used to simulate yield, number of harvests, profit, and the total cost of harvest for every year in the period 1989–2004. A simulation with the minimum wage harvesting constraint was developed and is labeled as the constrained model. The model was evaluated using data from different locations for four consecutive years in Washington State (USA) asparagus fields. The impact of the minimum wage requirements was estimated in terms of yield and profit for both processed and fresh asparagus. The traditional harvest interval of 24 h was compared to a more frequent (12 h) and a less frequent (48 h) interval. Manual harvest with the interval of 12 h showed the best results in terms of yields and profits for both processed and fresh asparagus. Gains in profits with the actual production conditions in Washington State were US$183.88/ha and US$210.60/ha for processed and fresh product, respectively. The 48-h strategy resulted in decreased yields and profits.  2006 Elsevier Ltd. All rights reserved. * Corresponding author. Tel.: +1 509 335 5556; fax: +1 509 335 1173. E-mail address: tizianocembali@yahoo.com (T. Cembali). 0308-521X/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.agsy.2006.03.009 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 267 Keywords: Asparagus; Harvest; Bioeconomic model; Mathematical model; Simulation 1. Introduction Asparagus is generally harvested daily during the production season. The daily harvesting decision depends upon whether or not sufficient spear growth has occurred in the asparagus bed to justify the harvesting expense since the last harvest. The actual harvesting usually occurs only once each day starting in the early morning and ending in the early afternoon. The yield maximizing harvesting strategy would be to cut a spear as it reaches the desirable length, so multiple daily harvests would be needed to maximize yields. This is because the energy used by the plant (crown) can be directed toward new spears rather than adding length to spears that are already at the required length for harvest and marketing. By law in the State of Washington (USA) as well as in the rest of the United States, manual labor is paid at least the minimum wage. Considering that asparagus growers pay a per unit amount to the manual labor to harvest asparagus, the harvesting becomes reality only when the revenue (quantity times the price per unit received) is at least equal to the minimum wage pay for the workers. The grower must make some monetary augmentations to guarantee the minimum wages if pay received for harvesting by manual labor does not meet this condition. This represents a constraint on the competitiveness of the Washington (USA) asparagus industry because the minimum wage is the highest in USA (US$7.16/h for 2003) (DOL, 2004). Daily harvests remain the common practice. Increasing the number of harvests per day would mean multiple cuttings per day. This has not been done because of the difficulty in recruiting manual labor willing to harvest in the afternoon when temperature are generally high. No research has been conducted to show the potential production using such a harvesting strategy. The adoption of a strategy with less frequent than daily harvests has not been considered profitable because an asparagus grower is paid on a given acceptable length and any additional length to the spear is trimmed. By not harvesting daily, the quantity of asparagus trimmed (not payable) is greater because spears tend to be longer than the required length. This creates a waste of carbohydrate (CHO) reserve that could be used to produce a marketable or payable product. The only research focused on harvest strategies was done by Lampert et al. (1980). They addressed the issue of harvesting strategies using a simulation model. Their approach considered the length of the harvesting season and the possibility to skip a harvesting season every nth year. Stout et al. (1967) addressed this issue of different frequency of harvest from an economic perspective, but they did not relate the study to the biological response of plants with the different strategy. Neither of these research efforts addressed the issue of predicting daily harvests of asparagus. Also, their modeling approach did not allow for evaluating different harvesting strategies within a season considering the biological impact of such strategies on the asparagus crown. 268 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 A bioeconomic growth simulation model for asparagus capable of predicting daily harvest is a necessary tool to analyze alternative harvesting strategies. Due to harvesting issues, asparagus field experiments can be extremely expensive. In addition, variability in weather conditions, pests, and weeds can affect the data of field trials. A bioeconomic growth simulation model would represent the solution for preliminary screening of different harvesting strategies. This paper represents the first attempt to address the issue of manual harvesting using a bioeconomic model. The specific objectives of this paper are: (1) to present an asparagus growth model capable to predict daily harvests; (2) to integrate the biological growth model with the economic decisions that the grower takes into account in the harvesting decisions; and (3) to determine the impact on profits of different harvesting strategies involving frequencies of manual harvests. This paper is organized into four sections: (1) model description; (2) methods; (3) results and discussion; and (4) conclusions. In general, the model description section is divided into: theoretical bioeconomic model, empirical bioeconomic model, biology and agronomy, and economics. The biology and agronomy subsection includes: (1) emergence and density dynamics; (2) spear growth, diameter, and weight; (3) CHO dynamics; and (4) production conditions. 2. Model description 2.1. The theoretical bioeconomic model In the model it was assumed that the manager would select a harvesting intensity that maximizes profit subject to the CHO constraint. In the model, harvesting is stopped at the minimum CHO level in order to not negatively impact the production in the following years. The number of asparagus spears harvested at time t (H(t)) represented the harvest intensity, that could be defined as the control variable in a dynamic optimization framework. The CHO level and the total number of asparagus in the field were the state variables of the model. In the theoretical model the payable weight of a spear was a function of the plant’s reserve of CHO, in accordance to Lampert et al. (1980). The harvesting costs were assumed decreasing as the number of spears available for harvesting increased. This last assumption was made because the higher the number of the spears available, the greater the efficiency of the manual labor. The theoretical model can be written mathematically as: max T 1 X ½pH ðtÞPW ðCRðtÞÞ  rðH ðtÞÞH ðtÞ1t t¼0 s:t: CRðt þ 1Þ  CRðtÞ ¼ H ðtÞW ðCRðtÞÞr ð1Þ N ðt þ 1Þ  N ðtÞ ¼ EðtÞ  H ðtÞ where p is the price per unit of asparagus, H(t) is the number of asparagus spears harvested, PW(CR(t)) is the payable weight of asparagus expressed as an increasing function of the CHO reserve at time t, r(H(t)) is the harvesting cost and is identified as a T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 269 decreasing function of the number of asparagus spears harvested, 1t is the discount 1 term defined as ð1þdÞ , where d is the daily discount rate, CR(t) is the CHO reserve at time t, W(CR(t)) is the total weight of the spear at time t expressed as an increasing function of the CHO reserve, r is the transformation coefficient of CHO into fresh asparagus weight, N(t) is the number of spears of asparagus at time t, and E(t) is the number of asparagus emerged at time t. Table 1 report the complete summary of variables symbols, definitions, and units for all the variables used in the model. Eq. (1) can solved using the Lagrange multiplier method. The Lagrangian expression of the problem, following Clark (1990, p. 235), is: L¼  T 1  X ½pH ðtÞPW ðCRðtÞÞ  rðH ðtÞÞHðtÞ1t þ kðtÞ½CRðt þ 1Þ  CRðtÞ  H ðtÞW ðCRðtÞÞr þlðtÞ½N ðt þ 1Þ  NðtÞ  EðtÞ þ H ðtÞ t¼0 ð2Þ where k(t) and l(t) are shadow prices of a unit of CHO and a spear of asparagus, respectively. The shadow price represents the amount that the objective function would increase if the constraint were relaxed by one unit (e.g. an extra unit of CHO would be available for the production). The initial and terminal values of the state variable were given, CR(0) represented the initial CHO reserve (at time 0), and CR(T) represented the final CHO reserve (at time T). Therefore, harvest was interrupted when CR(T) was less or equal to the minimum value of CHO (CRmin). The necessary conditions for optimality are: oL ¼ ½pPW ðCRðtÞÞ  r0 H ðtÞ  rðH ðtÞÞbt  kðtÞW ðCRðtÞÞr þ lðtÞ ¼ 0 oH ðtÞ oL oPW ðCRðtÞÞ t ¼ pH ðtÞ b  kðtÞ  kðt  1Þ oCRðtÞ oCRðtÞ oW ðCRðtÞÞ H ðtÞr ¼ 0  kðtÞ oCRðtÞ oL ¼ CRðt þ 1Þ  CRðtÞ  H ðtÞW ðtÞr ¼ 0 okðtÞ oL ¼ lðtÞ þ lðt  1Þ ¼ 0 olðtÞ ð3Þ ð4Þ ð5Þ ð6Þ Solving for the shadow prices k(t), and l(t) and the adjoint equations, in equilibrium the following conditions exist: lðtÞ  lðt  1Þ ¼ 0 ð7Þ lðtÞ ¼ k ð8Þ t kðtÞ ¼ ½pPW ðCRðtÞÞ  r0 H ðtÞ  rðH ðtÞÞb k þ W ðCRðtÞÞr W ðCRðtÞÞr kðtÞ  kðt  1Þ ¼ pH ðtÞ oPW ðCRðtÞÞ t oW ðCRðtÞÞ b  kðtÞ rH ðtÞ oCRðtÞ oCRðtÞ ð9Þ ð10Þ 270 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 Table 1 Variable symbols, definitions, and units Variable symbol Definition H(t) Number of asparagus harvested at time t Price of asparagus CHO reserve at time t Payable weight expressed as a function of CHO reserve at time t Harvesting cost function of the number of asparagus harvested at time t Discount term Daily discount rate Constant Total weight of the spear at time t Transformation coefficient of CHO into fresh asparagus weight Number of asparagus at time t Number of asparagus emerged at time t Shadow price of the CHO reserve Shadow price of the number of asparagus Minimum level of CHO reserve Number of spears emerged at time t Average temperature in period t Number of plant per ha Parameter Parameter Parameter Number of spears of class ‘a’ a time t Number of spears of class ‘a’ harvested at time t Length of spears of class ‘a’ at time t Required length for harvest (RLf for fresh product and RLp for processed product’’) Underground part of the spear before its emergence from the ground Base temperature above which there is asparagus growth Response of elongation rates of the temperature above the base temperature Diameter of spears of class ‘a’ at time t p CR(t) PW(CR(t)) r(H(t)) 1 d k W(CR(t)) r N(t) E(t) k(t) l(t) CRmin Et Tt NP a b h Na,t Ha,t La,t RLy U Tb c Da,t Unit Equation number (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10) (1)–(10), (20) and (21) (1)–(10) (1)–(10) (2)–(10) (2)–(10) g/plant plant/ha (2)–(10) (11) (11) (11) (11) (11) (11) (12) (12), (22), (23) and (24) cm (12)–(14), (17)–(19) cm (12), (17) and (18) cm (13) and (19) C (13) (13) cm (15)–(17) (continued on next page) 271 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 Table 1 (continued) Variable symbol Definition Unit Equation number Dmax CLt Cmin Maximum spear diameter CHO level per plant at time t Minimum level of CHO for spear production Michaelis–Menten control parameter Payable weight of spears of class ‘a’ at time t Correction factor for approximation of spear volume to cylinder volume Density of the spear Maximum length at which a spear has commercial value Total weight of a spear of class ‘a’ at time t CHO reserve at time t CHO level at time t, that is equal to CRt subtracted by the emerged spears Seasonal profit per hectare with manual harvest Price of asparagus (Pf for fresh asparagus, and Pp for processed asparagus) Percent of spears harvested that are not marketable Manual harvesting cost per unit (Cf for fresh asparagus, and Cp for processed asparagus) Other cost involved in the manual harvest (housing for labor, and management costs) Fixed costs (except management fees, amortized establishment costs, and land rent) Variable costs except the harvesting costs Total harvesting time Walking time spent in harvesting 1 ha of asparagus Picking time for one asparagus Minimum wage per hour cm g/plant g/plant (15) (15) (15) Dk PWa,t f d Lmax Wa,t CRt CLt I Py w Cy OC CF CV Zt w pt r (15) g (17), (22) and (24) (17) kg/cm3 cm (17) (17) kg (19)–(21) g/plant g/plant (20) and (21) (21) $/ha (22) US$/kg (22) % (22) and (24) US$/kg (22) US$/ha (22) US$/ha (22) US$/ha (22) h h (23) and (24) (23) s US$/ha (23) and (24) (24) Eq. (7) shows that the value of the shadow price of the number of spears does not change over time. Therefore, the shadow price of an additional spear is considered as constant (k), as reported in Eq. (8). Eq. (9) represents the shadow price of a unit of CHO reserve at time t. Intuitively, the numerator of the first term of Eq. (9) identifies: 272 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 the value of a spear (term pPW(CR(t))), the marginal cost of harvest composed by r 0 H(t) which is negative because of decreasing costs, and r(H(t)). The shadow price of a unit of CHO, k(t), is represented by the value of a spear, the cost savings in harvesting it, its opportunity cost of leaving the spear for future harvests, and its cost of harvest, all deflated by the CHO used by the plant in producing it. In other words, k(t) is the net present value of a spear deflated by the units of CHO used for it. The change in shadow price of the CHO reserve is represented by Eq. (10). The first term represents the net present value of the change of the weight of payable spears due to a change in CHO reserve. The second term can be identified as the product between the shadow price of CHO and the change in quantity of CHO consumed to produce H(t) spears. More simply, it is the value of the change in CHO consumed (or saved) in producing the spears harvested at time t (H(t)) because of the change in CHO reserve. By solving the system of equations for H(t), it is possible to have the analytical solution of the number of asparagus spears harvested at each time t. Although a simulation model was used to describe the harvesting problem, the theoretical economic model represents the starting point for understanding the decision problem of an asparagus producer. By harvesting more frequently than the optimal rate the producer will maximize yield, but not the profit because the harvesting cost are decreasing as the per time amount harvested increases. On the other hand, by harvesting less frequent than the optimal frequency, the manager will benefit by the cost savings of the lower harvesting costs, but will loose part of the potential yield because of the increased waste in CHO due to the spear growth over the required length. 2.2. Empirical bioeconomic model description The economic model was used to develop a more pragmatic growth model. The asparagus growth model was build as a dynamic simulation model. The simulation framework was preferred to the optimization structure because of greater flexibility for model evaluation and to reproduce growers harvesting decisions. The model integrates biological and agronomic characteristics of asparagus. The time frame used in the model is the hour, in fact spear emergence and spear growth were considered hourly. This allows the schedule of the harvests at different times during the day. The model includes a number of parameters from recent publications and preliminary field trials conducted by Washington State University (USA). The asparagus bioeconomic model is a decision support tool to provide information and insights on hand harvesting, and to assist asparagus growers on the daily management practices during the production season. While other models attempted to incorporate the entire cycle of the asparagus field in the biological model (Lampert et al., 1980; Wilson et al., 2002a), in this bioeconomic model only the production part was considered. The underlying reason of this decision was to focus more on the daily harvesting decisions. Growers do not want to reduce their CHO content below a minimum level, because that would negatively affect future yields. Therefore, in the model the harvest would stop when the minimum level of CHO is reached. Growers in New Zealand and US used this approach following a recently introduced decision T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 273 management tool, AspireNZ (Wilson et al., 2002b) for New Zealand and AspireUS (Drost, 2003) for the United States. It was implicitly assumed that the plants were able to recover the CHO used in the production and have the recommended level restored by the beginning of the next harvesting season. This assumption was necessary to focus on the model evaluation and on the harvesting strategies. Cembali et al. (2006) focused on modeling the asparagus cycle and studied the inter-year impact of stopping the harvest at different CHO levels. The assumption in this paper is consistent with the finding from Cembali et al. (2006). By stopping the harvest at the minimum CHO level advised (200 g/plant), the plants are able to restore the CHO level in the remaining months before next production cycle. If the harvest is interrupted at a lower level, then the production of the following year would be impacted because the plants have less time to assimilate CHO. The asparagus growth model represents a single field of 1 ha. The harvest frequency and the harvest schedule can be chosen, as well as the density of plants per hectare, and the total energy reservoir per plant in percentage of CHO on root dry weight. This implies that the model is flexible in adapting to different production situations. For example, some fields may have a greater production potential because of the greater CHO reserve (Wilson et al., 1999a) and a higher number of plants than others (McCormick and Thomsen, 1990). The model considers a full production field that can produce 6160 kg/ha per year which is typical for Washington State (USA). 2.3. Biology and agronomy 2.3.1. Emergence and density dynamics The first spear emergence was predetermined in the model by a set day (5 April). This approach is similar to the model of Lampert et al. (1980). In the literature, researchers have tried to predict the first spear emergence of an asparagus field using degree days. Although Dufault (1996) suggests that soil temperatures should be used to predict the first emergence, researchers prefer to use the ambient air temperature. Base temperatures adopted ranged from 4.4 C (Blumenfield et al., 1961; Bouwkamp and McCully, 1972) to 7.1 C (Wilson et al., 1999b). Results of simulations using the approach on first spear emergence from Wilson et al. (2002a) were not consistent with the commercial practices in the state of Washington (USA). Using the base temperature would allow for first emergences 15–20 days earlier than when usually occur. In relation to the number of spears that emerged, both models from Wilson et al. (2002a) and Lampert et al. (1980) assumed that each plant of asparagus carries a certain amount of spears that are growing simultaneously. The spears emerge throughout the growing season. Although the results from Lampert et al. (1980) (25.6 spears per plant) agreed with a previous work by Ellison and Scheer (1959), they do not reflect the dynamics of asparagus field in high density plantings. For example, McCormick and Thomsen (1990) reported that the number of spears per plant ranges from 9.5 to 5.7 for density of 19,000–44,000 plants/ha, respectively. Wilson et al. (2002a) did not report the plant density assumed in their study, so a comparison with this model is not possible. Lampert et al. (1980) simulated five plants, and 274 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 by comparing the yield per plant with a commercial production level, it would be equivalent to a density of 7500 plant/ha, lower than the densities currently adopted. In this model, to determine the number of spears emerged in each period (hour) the following transcendental emergence function was adopted: Et ¼ aT ht expðbT t ÞNP ð11Þ where Et is the number of spears emerged in the period t, Tt is the average air temperature in the period t, NP is the number of plants, a, h, and b are parameters of the function. The values of NP and the other parameters are reported in Table 2. The values of the parameters were determined using the results of field trials conducted in Prosser, Washington (USA) (Dean, 1999). The two components of the density dynamics are spears emerged and spears harvested. Although, the number of spears in a field might be affected by environmental Table 2 Parameter’s values for the equations Parameter Equation number Value Source 10 a h b NP RLf RLp U Tb c L0,t Dmax Cmin Dk CL0 f d Lmax r = bset/dw bset dw CR0 = CR(0) CRmin = CR(T) Pf Pp (11) (11) (11) (11) (12), (17) and (18) (12), (17) and (18) (13) and (19) (13) and (14) (13) (13) and (14) (15) (15) (15) (15) (17) and (19) (17) and (19) (17) and (18) (20) and (21) (20) and (21) (20) and (21) (20) (20) (22) (22) 5.95 · 10 5 0.21 42,000 22.86 cm 19.05 cm 12 cm 7.1 C 0.02232 1.27 cm 2.8 cm 168.5 g 55 270 g 0.75 0.95 34.29 cm 7.78 0.7 0.09 270 g 200 g US$0.99/kg US$1.19/kg w Cf = Cp CF CV OC r w pt (22) and (24) (22) and (24) (22) (22) (22) (24) (23) (23) 50% US$0.51/kg US$388.36/ha US$837.64/ ha US$407.73/ ha US$7.16/h 1.8 h 1.31 s Curve fitting from Dean (1999) Curve fitting from Dean (1999) Curve fitting from Dean (1999) Ball et al. (2002) USDA (1996) Seneca Foods Corporation (2002) Wilson et al. (1999a) Wilson et al. (1999a) Wilson et al. (1999a) Cembali (unpublished data, 2003) Lampert et al. (1980) Scott et al. (1939) Calibrated value Drost (personal communication, 2003) Value fitting data Hopper and Folwell (1999) Holmes (personal communication, 2004) Calculated value Penning de Vries et al. (1974) Wilson et al. (2002a,b) Drost (personal communication, 2003) Drost (personal communication, 2003) Schreiber (personal communication, 2004) Seneca Food Corporation (personal communication, 2004) Value fitting field data Ball (personal communication, 2004) Ball et al. (2002) Ball et al. (2002) Holmes (personal communications, 2004) DOL (2004) Calculated value Calculated value T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 275 factors as wind, insects, and temporary lack of moisture, those adverse factors were not included in the model. The model accounts for harvested and marketable spears. The marketable spears are expressed as a percentage of the total spears in the field. The total spears in the field are represented by the spears emerged, spears that are below the marketable length (not ready to be harvested), and the spears that are above the marketable length and therefore ready to be harvested. After emergence, the dynamics of the number of spears is only affected by the harvest. Spears are harvested once their length is above the minimum length required in the fresh or processed market. Spear number dynamics is then ruled by the following equation: N a;t ¼ N a1;t1  H a;t for a P 1 if La;t P RLy ; ð12Þ where Na,t is the number of spears of class ‘a’ at time t, (note that N0,t1 = Et1), Na1,t1 is the number of spears of class ‘a  1’ at time t  1, Ha,t is the number of spears of class ‘a’ harvested in period t, La,t is the length of the spears of class ‘a’ at time t, RLy is the required length (RLf is the required length for the fresh market, and RLp is the required length for the processed market). Recall that Ha,t is positive if the spears’ length of class ‘a’ at time t are greater than the required length (RLh) for harvest. More consideration on Ha,t were made in the economics section. The class indicates age and is expressed in hours of life since emergence. For example, N61,t indicates the number of spears of 61 h of age at time t. The values of the parameters RLf, and RLp are reported in Table 2. Variable symbols, definitions, and units are reported in Table 1. 2.3.2. Spear growth, diameter, and weight The asparagus growth model utilizes the spear growth model developed by Wilson et al. (1999b). Eqs. (13) and (14) report the growth function for a spear of class ‘a’ in the period t: La;t ¼ ðLa1;t1 þ U Þ expðcðT t  TbÞÞ  U ð13Þ if T t 6 Tb ð14Þ La;t ¼ La1;t1 where La,t is the length of ‘a’ spear of class a at time t, U is the underground part of the spear before its emergence from the ground, Tt is the average air temperature for period t, Tb is the base temperature above which there is asparagus spear growth, and c is the response of elongation rates of the temperature (Tt) above the base temperature (Tb). The length for spears just emerged, class 0, (L0,t) was predetermined, its value is reported in Table 2. Eq. (14) represents the spear growth constraint, and shows that if temperatures are equal or lower the base temperature, there is no spear growth. The values of the parameters U, c, Tb, and L0,t are reported in Table 2. The base temperature (Tb) reported by Wilson et al. (1999b) was considered a reliable measure in determining spear length because it was estimated with field data, but it was not used in determining first emergence. Using hourly temperature, the model accounts for frosting period by interrupting the growth when the temperature is below Tb. For asparagus, frosting damages are not common. Rajeev and Wisniewski (1992) reported frost hardiness (defined as the temperature at which 50% injury 276 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 occurred) to temperature lower than 2.8 C. Some frost damages can occur when the temperature is lower than 1 C for 4–5 h. However, in Washington temperatures below 2 C for 2 h only occurred in one year over the 16 years of weather data available. In the model, frost damages were accounted together with other damages that may occur to spears (e.g. wind, and insects) in Eq. (22) with the term w (percent of asparagus that are not marketable). Spear diameter is highly influenced by CHO reserve in the roots (Tiedjens, 1924; Norton, 1913; Ellison and Scheer, 1959). Therefore, it was decided to adopt the Michaelis–Menten functional form used by Lampert et al. (1980) to account for the change in diameter over the season. Eq. (15) represents the relationship between spear diameter and CHO reserve in the root. Eq. (16) represents the dynamics of spear diameter as the spear becomes older. Dmax ðCLt1  C min Þ Dk þ CLt1 þ C min ¼ Da1;t1 for a P 2 D1;t ¼ ð15Þ Da;t ð16Þ where D1,t is the diameter of spears of class ‘1’ at time t, Dmax is the maximum spear diameter, CLt1 is the CHO level per plant at time t  1 (when the spear emerged), Cmin is the minimum level of CHO level for spear production, and Dk is a Michaelis– Menten control parameter. The values of the parameters Dmax, Cmin, Dk, and the initial value of CHO level per plant (CL0) are presented in Table 2. The Michaelis–Menten control parameter used by Lampert et al. (1980) has been adjusted to obtain diameter lower values that are more representatives of the commercial production conditions in Washington State (USA). Eq. (15) does not take into account directly other factors that may affect the spear size (e.g. heat stress, over harvest, drought). Those factors influence indirectly the CHO level of the plant, and consequently the spear diameter. With the assumption of restoring the original CHO level, these factors do not play a relevant role in the model. The impact of low level of CHO at the beginning of the production season can be found in Cembali et al. (2006) were the impact of over and under harvesting is estimated. The weight of each spear was calculated using a weight function as in Lampert et al. (1980). In the model each spear is harvested only if its length is greater than the required length (RLy). Therefore, in calculating the product harvested the model only considered the portion of spear of the payable length. On the other hand, the remaining portion of the spear (called trimmed part) consumed CHO, and this consumption was considered in the use of CHO. In addition, the underground portion of the spear (the portion from the root to the ground) was accounted for in the CHO usage. The model also considered that as the spear length reached a certain height (Lmax), it did not have any commercial value because of low quality. In fact, when a spear continues to grow over Lmax it starts to develop open branches (crooked) that make it unmarketable. The value of the limiting length (Lmax) is reported in Table 2. Eqs. (17) and (18) describe the payable product, while Eq. (19) explain the effective weight of the asparagus for the CHO balance. T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 PW a;t  2 Da;t ¼ ðRL Þ pðf ÞðdÞ 2 y if RLy < La;t < Lmax if La;t < RLy or La;t > Lmax  2 Da;t pðf ÞðdÞ ¼ ðU þ La;t Þ 2 277 ð17Þ PW a;t ¼ 0 ð18Þ W a;t ð19Þ where PWa,t is the payable weight of a spear of class ‘a’ at time t, RLy is the required length, Da,t is the diameter of the spear of class ‘a’ at time t, f is the correction factor for the approximation of spear volume to cylinder volume, d is the density of the spear, and Wa,t is the total weight of a spear of class ‘a’ at time t. The values of the parameters used in Eqs. (17)–(19) are reported in Table 2. Variable symbols, definitions, and units are reported in Table 1. 2.3.3. CHO reserve dynamics Asparagus yields depend on the CHO reserve. As mentioned before, recent research has focused on using the CHO root content as an indicator for crop management purposes (Wilson et al., 2002b). The idea underlying this asparagus decision support system was to ensure a high level of CHO during the harvest, and to preserve CHO reserve for the following year. In the model, when plants reach the minimum CHO level the harvest is stopped for the year under consideration. The initial and the minimum optimal level of CHO content during the production period were defined using values from Drost (personal communication, 2003) and assuming an average dry weight of 600 g per plant (Wilson et al., 2002a). In the model, the consumption of CHO was adopted from Wilson et al. (2002a). The variable that accounts for the CHO reserve at time t was defined as CRt. For computational purposes another variable that accounts for the level of CHO was defined in the model as CLt. In this way, the consumption of CHO for spears not yet harvested was considered in calculating the diameter of the new spears emerging. Eqs. (20) and (21) defined those two variables X CRt ¼ CRt1  r H a;t W a;t ð20Þ a CLt ¼ CRt  r X N a;t W a;t ð21Þ a where r is the transformation coefficient of CHO in asparagus fresh weight, r = dw/ bset, and bset is the biosynthetic efficiency of transforming CHO in asparagus dry matter, and dw is the dry weight content of asparagus. Values of these last two parameters are presented in Table 2. 2.3.4. Production conditions The model was developed for the 1-ha asparagus field with a plants density of 42,000 plants/ha in full production and the row spacing assumed was 1.37 m. The field was assumed to be cultivated according the accepted practices in the State of Washington (USA). The production level of an asparagus field for this area varies 278 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 from year to year and location to location, but, on average is around 6160 kg/ha/yr (Ball et al., 2002). The model was intended to predict the daily production of an asparagus field as described above. The model might also be used to predict daily productions for fields with a variable plant stand by changing the value of the variable NP in Eq. (11). For example, the model could be used to determine the production during the lifecycle by changing the plant density to account for mortality. Asparagus production can be for the fresh or processed market. These two different markets have different grading requirements in terms of length. The fresh market prefers all green spears of 22.86 cm length and the processing market requires spears of 19.05 cm in length. Growers in both markets are allowed to bring in asparagus with some basal white portion (underground portion) for a maximum of 2.54 cm length. In the model, it was assumed that the product for both markets was a green spear. The reason of this assumption was because those are the harvesting practices commonly adopted (Holmes, personal communications, 2003). It was assumed that the asparagus field responded in the same manner for those two different cutting heights and that the production was driven by temperature and by CHO reserve. The CHO reserve value of 450 mg/g was considered as the starting value (CR(0)), while the terminal value (CR(T)) was 330 mg/g of dry roots (Drost, personal communication, 2003). Those levels are equivalent to 270 g/plant and 200 g/plant of CHO assuming an average dry weight per plant of 600 g. No mortality of the plants was assumed. The first emergence was assumed to be on 5 April at 1:00 am. The weather data utilized was from Mathew Corner, a weather station located in the main asparagus production area of Washington State (USA). The hourly temperature was used to model the biodynamics of the asparagus field. 2.4. Economics Profits generated by the manual harvest were calculated using the following profit function: X X P ¼ Py ðH a;t PW a;t ÞC y  OC  CF  CV ð22Þ ðH a;t PW a;t Þð1  wÞ  a;t a;t where P is the seasonal profit per hectare with manual harvest, Py is the price of asparagus (Pf indicates fresh asparagus, and Pp processed asparagus), Ha,t is the number of spears of class ‘a’ harvested at time t, PWa,t is the payable weight of the spear of class ‘a’ harvested at time t, w represents the percent of harvested spears that are not marketable, Cy represents the harvesting cost per unit of product with the manual harvest (Cf indicates fresh asparagus, and Cp processed asparagus), OC represents other costs involved in the manual harvest (housing for labor, and management costs), CF represents the fixed costs (except management fees, amortized establishment costs, and land rent), and CV represents the variable costs except the harvest. The values of the parameters Pf, Pp, w, Cf, Cp, OC, CF, and CV used in the simulation model are reported in Table 2. Two models with different constraints on the harvesting were considered. An unconstrained model, where harvest was allowed, as described in the emergence T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 279 and density dynamics subsection, when there were spears above the required length (RLy). A constrained model was also considered to account for the minimum wage requirements in the harvesting decisions. Not all the farmers in Washington State are willing to pay the monetary augmentation on the harvesting cost. Those farmers prefer to harvest only when the minimum wage is assured to manual labor, because they have belief that the expected profit will be greater. This economic constraint was integrated with the length requirements for harvesting. To account for the wage constraint, the time spent in manual harvesting was estimated. The time of manual harvest was assumed to be a function of the time spent for walking, cutting and handling the spears, and the number of spears ready for harvest present in the field. In the constrained bioeconomic model, if the costs of manual harvest (considered as the product of the quantity harvested and the per unit cost of harvesting) were lower than the potential minimum wage pay rate for harvesting, then there was no harvest. Harvest only occurred if it was possible to guarantee the minimum wage requirement. Eq. (23) reports the function used to estimate the time spent for manual harvesting and Eq. (24) the harvesting constraint. X Z t ¼ w þ pt H a;t ð23Þ a X H a;t a ( >0 if C y P ðH a;t PW a;t Þð1  wÞ P rZ t a ¼0 ð24Þ otherwise where Zt is the total harvesting time, Pw is the walking time spent in harvesting 1 ha of asparagus, pt is the cutting time, a H a;t is the sum in number of the spears harvested, and r is the minimum wage per hour. The values of the parameters w, pt, and r adopted in the simulation are reported in Table 2. Variable symbols, definitions, and units are reported in Table 1. 3. Methods The model was evaluated using different statistics to determine its ability in predicting daily productions. Then, two scenario situations were modeled. In each case, a hectare of asparagus in ‘‘normal production conditions’’ was assumed. The first scenario was a production simulation for the constrained and unconstrained bioeconomic model. The second scenario was a simulation of different harvesting frequencies. Historical hourly weather data from 1989 to 2004 for the location of Mathew Corner, Washington (USA) were used to simulate daily production for both scenarios (PAWS, 2004). 3.1. Evaluation of the production model Model evaluation was based upon aggregate daily production data from receiving stations from a major asparagus processor in Washington State (USA). Receiving stations at different locations were used to test the model robustness in simulating 280 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 the daily production in different areas. Data from 2000 throughout 2003 were used in the evaluation procedure. Table 3 lists the receiving stations used, the representative weather station for that area, the contracted area (ha), the contracted and actual production (kg/ha) for each year. These data were used to compare the simulation results with actual data. The unconstrained model was used in the simulation. The aggregated data included a variety of production situations, from young bearing fields to old asparagus fields. The simulation results are compared to an aggregated large sample to avoid having specific production site characteristics to influence the model evaluation. These results in assessing the validity of the model were applied to a variety of commercial production situations as a result of the method followed. Data of daily aggregated production for four consecutive years for several locations (four locations for 2002 and 2003, and three locations for 2000 and 2001) were used in the evaluation process. An actual series of daily production was considered the production recorded for one year by a location. Then the simulated series from the model were compared to the actual series and the two series were compared to determine how similar or how different they were. The closer the two series (simulated and actual) were, the better was the model able to predict daily productions for a certain location in a certain year. The model was evaluated using 14 independent series. Each location was evaluated using a simulation with weather data of the closest weather station available. Table 3 Receiving stations used for model validation: locations, names, contracted areas, contracted yields, and actual yields received during the years 2000 thorough 2003 Weather station Contracted area (ha) Contracted yield (kg/ha) Actual yield received (kg/ha) 2000 Pasco (Unit 15) Pasco (Gibbons) Sunnyside (Sunnyside) Pasco (Ice Harbor) CBC Pasco Mathews Corner Sunnyside Fishhook 1246 1272 330 454 4008 4519 3815 4425 4056 5107 4164 4121 2001 Pasco (Unit 15) Pasco (Gibbons) Sunnyside (Sunnyside) Pasco (Ice Harbor) CBC Pasco Mathews Corner Sunnyside Fishhook 935 1384 327 396 4875 4659 4591 4335 6047 4849 5189 5089 2002 Pasco (Unit 15) Pasco (Gibbons) Sunnyside (Sunnyside) CBC Pasco Mathews Corner Sunnyside 1106 1178 585 5119 4779 4148 5592 5042 4778 2003 Pasco (Unit 15) Pasco (Gibbons) Sunnyside (Sunnyside) CBC Pasco Mathews Corner Sunnyside 1168 1236 567 4932 4692 3889 5621 5082 5205 Receiving station location (name) T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 281 The 14 series had different input, and the model was evaluated on its ability to predict daily productions. The model was evaluated using 10 different statistics: mean square error (MSE), mean absolute deviation (MAD), mean absolute percent error (MAPE), Pearson correlation coefficient (PCC), v2 test (CHI-SQUARE) (Goldsmith and Hebert, 2004), autocorrelation function test for period comparison (ACF-T), cross-correlation function test for phase lag detection (CCF-T), mean comparison (MC), percent of error in variation (PEV), and discrepancy coefficient (DC), (Barlas, 1989). The ACF-T, and CHI-SQUARE are statistical tests, while the others are statistics that represented different aspects of the variation between the actual and the simulated daily asparagus production. The smaller the calculated values for MSE, MAD, MAPE, MC, and PEV, the smaller were the differences between the actual and the simulated daily production. The calculated value for PCC can range between 1 and 1, it represents the level and the direction (positive or negative) of correlation. A calculated value close to 1 indicated a strong positive correlation between the two series. The DC represents the relative discrepancy between the simulated and the actual production and it ranges between 0 and 1. A value close to zero represents low relative discrepancy (Goldsmith and Hebert, 2004). The CHI-SQUARE tested the joint hypothesis that each individual simulated outcome (Si) was equal to the actual value (Ai) for that time period. The null hypothesis (Ho) would be rejected if at least one predicted value were statistically different than the actual value (Ho: Ai = Si for all i). The ACF-T and the CCF-T tests focus on the behavior pattern evaluation (Barlas, 1989). In particular the ACF-T can be used to detect errors in the periods of behavior patterns. The autocorrelation function was then calculated for lag k = 0, 1, . . ., n  1, where n is the number of simulated daily productions. Individual tests for each lag value were performed to determine whether the autocorrelation function of the actual data is equal to the autocorrelation function for the simulated sequence. The percentages of the cases where the simulated values were not statistically different from the actual were considered. The CCF-T is similar to the ACF-T. However, its focus is to check how the actual and the simulated data are correlated at different time lags (Barlas, 1989), or more simply if they are out of phase or not. If the cross-correlation function has its maximum value at the lag equal to zero, then the two series are completely in phase. Appendix A reports the formulas and additional information on the statistics used. 3.2. Scenario 1: production simulation Scenario 1 simulated the production of a hectare of asparagus to show the outcome of the simulation model for 16 years. This scenario was chosen to highlight the profit performances of manual harvesting for the fresh and processed product using both the constrained and unconstrained model. In the unconstrained model it was assumed that harvesting occurs each day at 5 a.m. if there were spears longer than the required minimum length (RLh). The constrained model, as described before, presented an additional constraint (Eq. (24)). The cost of harvesting had 282 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 to provide minimum wage to the manual labor. For both models, if their respective constraints did not hold, then harvesting would take place the following day. Detailed results were obtained for each year: (1) yield (kg/ha); (2) number of harvests; (3) profit for the manual harvest (US$/ha); and cost of harvesting (US$/ha). 3.3. Scenario 2: comparison of harvesting schedules Because of the lack of information in the literature for different asparagus harvesting strategies, scenario 2 was used to determine yield, number of harvests, profits, and costs of harvesting at different time intervals of 12, 24 (control), and 48 h. The listed harvesting intervals were chosen because the harvesting was always during the day. In fact, with the 12 h interval the model allowed harvest to occur at 5.00, and 17.00; with the 24 h interval at 5.00; and with the 48 h interval the model allowed harvest to occur at 5.00 of alternating days. Both the constrained and the unconstrained models were simulated at different harvesting schedules. Statistical differences were tested for each combination of results. 4. Results and discussion 4.1. Model evaluation Model evaluation of predicted asparagus daily yield productions were based on simulations performed across years and locations. For purposes of demonstration the results for 2001 at the location of Unit 15 in Pasco, Washington (USA) are presented in Fig. 1. The results of the model evaluation procedures used are presented in Table 4. Four years of actual data for a total of fourteen different years and locations combinations were compared to the simulated series. Each location differed from the others because of the different hourly temperatures. The series were independent by each other. Daily values of the actual harvested asparagus ranged between 0 (no harvest) to 270 kg/ha/day, and the average value in a season was approximate to 110 kg/ha/ day. Values for MSE ranged between a minimum of 1232.47 and a maximum of 2776.05. The MAD represents more intuitively the real error of the simulation model because it is expressed in kg/ha/day. The lowest value of MAD recorded was 24.49 kg/ha/day, while the highest was 36.55 kg/ha/day. The result of the MAD indicated that the model, on average predicted values, was quite close to the real observations. Only in five situations the MAPE was below 50%. In the two worse situations the MAPE was above 100%. These cases were both in the same year (2002) and their location was quite close, which suggests there might have been the influence of other weather variables (e.g. frost, wind). The PCC calculated were mostly over 0.70, except for three cases and two of them were the same as the high MAPE. There was a failure to reject the Ho of the CHI-SQUARE test. This result was expected. Goldsmith and Hebert (2004) obtained similar results in their model vali- 283 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 300 Actual Simulated 250 200 kg/ha per day 150 100 50 6/ 7 /2 00 1 01 31 / 5/ 24 / 5/ 17 / 5/ 20 20 01 01 20 01 20 10 / 5/ /2 00 1 01 5/ 3 20 26 / 4/ 19 / 4/ 4/ 12 / 20 20 01 01 0 Date Fig. 1. Actual versus simulated daily asparagus production for the Unit 15, 2001. Table 4 Evaluation results 2000 thorough 2003 Year and receiving station MSE (kg/ha/day)2 MAD (kg/ha/day) 2000 Unit 15 Gibbons Sunnyside Ice Harbor 2776.05 1253.66 2043.37 1902.88 2001 Unit 15 Gibbons Sunnyside Ice Harbor MAPE (%) PCC ACF-T (%) MC (%) PEV (%) DC 38.45 27.87 36.55 34.22 94.70 53.95 69.26 78.28 0.67 0.81 0.78 0.84 93.44 97.06 88.41 94.12 50.85 20.22 48.09 50.12 65.88 35.87 90.28 65.56 0.46 0.34 0.44 0.37 1617.11 2269.36 2133.00 2270.51 31.47 35.20 33.53 34.72 40.54 57.24 47.80 56.15 0.77 0.74 0.74 0.72 85.96 96.49 75.86 77.59 2.30 15.88 7.96 12.15 16.97 44.98 53.49 46.85 0.34 0.40 0.41 0.41 2002 Unit 15 Gibbons Sunnyside 2236.38 1508.87 1406.28 29.13 24.49 25.26 142.19 224.87 60.26 0.49 0.69 0.70 75.81 85.25 85.48 8.90 13.60 18.44 4.11 0.73 19.88 0.50 0.40 0.39 2003 Unit 15 Gibbons Sunnyside 1297.02 1410.45 1232.47 26.79 28.77 25.44 46.97 49.14 46.10 0.73 0.73 0.74 87.30 83.33 91.80 2.64 2.86 3.53 5.44 8.39 11.76 0.37 0.37 0.36 284 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 dation. Despite this, it can be argued that the model is still consistent and robust. This test only stated that at least one predicted value was statistically different from the actual value. Recalling that the number of values predicted ranged from 58 to 70 per case, it is expected that some of the predicted values will not be statistically equal to the actual production. ACF-T indicated that in twelve of the fourteen cases 80% or more of the autocorrelation functions (n  1 for every case) of the actual data were not statistically different from the corresponding autocorrelation functions of the simulated data. In the remaining two cases this percentage was above 70%, supporting that the simulated series did not differ from the actual data. The results for the CCF-T were also positive in validating the model in trend patterns for all cases. In six cases the results of the MC were below the 10% and in three cases were around 50%. This difference was mainly due to the lower production capability of the contracted asparagus at a certain receiving station with respect to the potential in production assumed in the model (6164 kg/ha per season). Table 4 contains the expected yields for each receiving station and year. In all cases the production was lower than the production assumed in the model. Weather or agronomic reasons influenced the difference in the asparagus harvested per day between the actual and the simulated series (e.g. frosts, windy weather, pests, etc.). The PEV had contrasting results, although in one case the actual and the simulated data had almost the same PEV value with a difference of only 0.73%. In seven cases the values of PEV were lower than 20%. In two of the four years examined the simulation seemed not to perform well in terms of PEV. The DC values calculated ranged from 0.342 to 0.504 and indicated some relative discrepancy between the actual and the simulated series. The simulations showed a similarity in trends and correlation with the actual production of asparagus. The dissimilarities were due mainly a difference in potential yield between the model and the aggregated data. This finding associated with the other tests supported the prediction capability of the simulation model of daily harvesting of asparagus. Lampert et al. (1980) did not discuss the validity of its model in predicting daily production levels. Only the yearly average production per plant was reported. In addition, they did not use hourly temperatures as input for the model, but daily averages. This difference approach is critical in determining the ability to predict daily production because it has lower precision in predicting growth and emergence. The work from Wilson et al. (2002a) does not report the results and does not specify if the daily or hourly temperature is used. Another difference between our model and the ones developed by Lampert et al. (1980) and Wilson et al. (2002a) is the integration of the economic constraint to the harvest. This aspect adds more accuracy to the predicted daily production because it simulates the decision process a producer has to face in harvesting asparagus. The validity of the model can be visually determined from Fig. 1. At the beginning, the model was able to predict the daily production following the same pattern as the actual yield. Then around the end of April (30 April) the model predicted lower daily productions for 3 days. On 5 May the predicted and actual values were almost the same. After that period, the predicted daily production had the same 285 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 pattern as the actual values did except in two periods (12–14 May, and 22–25 April) where the actual yield are lower than the predicted. When the predicted yields were higher than the actual, it could be due to extensive wind damages that were not accounted directly by the model, but by a proxy constant (w) during the season. Also, emergence might have been affected by adverse conditions that were not accounted. On the other hand, when the model predicted lower yield than the actual recorded data, it might be due to a lower number of spear emerged in previous periods or due to the fact that the model is not able to react rapidly to the changing weather conditions. The model could benefit from ad hoc field trials that focus on modeling spear emergence. 4.2. Scenario 1: production simulation The production simulation was performed for both the processed and the fresh asparagus. Detail results for the constrained model for the processed and the fresh product are reported in Tables 5 and 6, while average results for both models and both asparagus products are reported in Table 7. The yield generated by the unconstrained model was always higher than the constrained model. Intuitively, because the constraint on the harvest lowered the number of harvests, then the losses of CHO to produce non-payable product (spear growth exceeding the RLy) were greater, with a negative impact on the overall potential yield. The yield obtained for the fresh product was always higher than for the processed product. Table 5 Yearly results of the constrained simulated daily harvest of processed product Year Yield (kg/ha) Number of harvests (#) Profit for manual harvest (US$/ha) Total cost of harvesting (US$/ha) 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 5774.80 6024.77 6116.29 6000.64 6000.72 5930.99 6100.19 5705.92 6010.21 5970.98 6088.47 6018.68 5887.75 6100.89 6109.11 5962.61 46 55 58 49 52 47 51 58 52 50 58 53 49 54 56 52 2313.00 2483.83 2546.38 2467.34 2467.39 2419.74 2535.38 2265.93 2473.88 2447.07 2527.37 2479.67 2390.19 2535.86 2541.47 2441.35 3335.88 3462.62 3509.03 3450.39 3450.43 3415.07 3500.87 3300.95 3455.24 3435.35 3494.92 3459.54 3393.15 3501.22 3505.39 3431.10 Average Minimum Maximum 5987.69 5705.92 6116.29 52.50 46.00 58.00 2458.49 2265.93 2546.38 3443.82 3300.95 3509.03 286 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 Table 6 Yearly results of the constrained simulated daily harvest of fresh product Year Yield (kg/ha) Number of harvests (#) Profit for manual harvest (US$/ha) Total cost of harvesting (US$/ha) 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 6213.76 6617.59 6637.99 6172.55 6054.79 6153.26 6327.60 6264.09 6295.74 6403.96 6421.85 6521.53 6011.81 6469.75 6451.08 6430.86 47 55 54 45 47 45 47 56 49 48 56 52 45 53 52 51 1352.70 1546.78 1556.58 1332.89 1276.30 1323.62 1407.41 1376.89 1392.10 1444.11 1452.71 1500.61 1255.64 1475.73 1466.76 1457.04 3558.45 3763.22 3773.56 3537.56 3477.84 3527.78 3616.18 3583.97 3600.02 3654.90 3663.97 3714.51 3456.05 3688.26 3678.79 3668.53 Average Minimum Maximum 6340.51 6011.81 6637.99 50.13 45.00 56.00 1413.62 1255.64 1556.58 3622.72 3456.05 3773.56 Table 7 Average simulated yearly results for the unconstrained and constrained simulation model of manual harvest for both the processed and fresh asparagus (1989–2003) Asparagus utilization Model used Yield (kg/ha) Number of harvests (#) Profit for manual harvest (US$/ha) Total cost of harvesting (US$/ha) Processed Processed Fresh Fresh Unconstrained Constrained Unconstrained Constrained 6140.86bA 5987.69c 6394.19a 6340.51a 58.94aA 52.50b 52.63b 50.13b 2563.17aA 2458.49b 1439.41c 1413.62c 3521.49bA 3443.82c 3649.94a 3622.72a A Average values followed by same lower case letter are not significantly different at P 6 0.05 according to LSD test. Processed asparagus required a shorter spear length to be harvested; consequently a higher number of spears were harvested. Each time a spear was harvested, the underground portion did not account for as payable product (it was accounted as a loss), but it did consume CHO affecting negatively the potential yield. Either the work of Lampert et al. (1980) or Wilson et al. (2002a) did not address the difference in potential yield between fresh and processed asparagus. Profits for processed product were higher despite the lower production because of its higher price. The average profit simulated per ha with the constrained model for manual harvesting was US$2458.49 for processed asparagus, and US$1413.62 for fresh products. Yields simulated with the constrained model were similar to the T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 287 common production levels in Washington (6160 kg/ha), and were 5987.69 for the process asparagus and 6340.51 kg/ha for fresh. The numbers of harvests simulated with the constrained model were not statistically different. The number of harvests was 52.63 for processed asparagus and 50.13, for fresh product. Costs were higher for the fresh product because of the cost structure adopted (US$/kg of asparagus harvested) and the higher production for the fresh product. The total cost of harvesting included the term defined in Eq. (22) as OC that account for housing for labor and management costs. The constraint accounting for minimum wage generated differences for both processed and fresh asparagus with respect to the unconstrained model. This result confirmed the fact that minimum wage represents an additional cost for Washington asparagus growers. The impact of the minimum wage constraint is US$104.68/ha for the processed asparagus, and US$25.79/ha for the fresh asparagus. Intuitively, the time spent in walking and picking up asparagus is almost the same for processed and fresh asparagus, but processed asparagus spears are smaller and weight less. The constraint for minimum wage is statistically significant only for processed asparagus. Tables 5 and 6 show the variability in predicting yield, that shows how sensible is the model to the hourly temperature in forecasting daily productions. The economic constraint has a direct impact on the number of harvest. As described before, if the expected pay for the manual worker does not guarantee the minimum wage, harvest is postponed to the next day. That, associated with the different temperatures, causes the differences in number of harvests. Previous literature approach the issue of harvesting asparagus either from a biological view (Lampert et al., 1980; Wilson et al., 2002a) or from an economic perspective (Stout et al., 1967; Michalson and Thomas, 1972), but there is not a study that integrated the biological and economic implications of harvesting asparagus. There is no literature examining the impact of the wage constraint on asparagus production. The harvesting constraint used in this model represents the harvesting conditions for the State of Washington (USA). However, different production areas may have different economic conditions or contracts for harvesting asparagus. The model presented can be modified and different harvesting constraint can be set to determine the impact on the daily production from both an economic and agronomic perspective. 4.3. Scenario 2: comparison of harvesting schedules The constrained model yields for both the processed and fresh product were statistically higher for the 12 h interval of harvests (Table 8). Gains in yield by increasing the frequency of harvest to the 12 h interval were 269.05 and 438.21 kg/ha, for the processed and fresh product. The main reason of this result was that the fresh product has a taller spear that, because of the spear growth function (Eq. (13)), grows faster. Therefore, by increasing the harvesting interval, there would be less trimmed product that consumed CHO. Increasing the frequency of harvest would 288 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 Table 8 Average simulated yearly results for the unconstrained and constrained simulated daily manual harvests for both the processed and fresh asparagus at different frequencies (1989–2003) Asparagus utilization Model used Frequency of harvest (h) Yield (kg/ha) Number of harvests (#) Profit for manual harvest (US$/ha) Total cost of harvest (US$/ha) Processed Processed Processed Unconstrained Unconstrained Unconstrained 12 24 48 6490.32aA 6140.86b 4987.56c 123.44aA 58.94b 26.94c 2802.00aA 2563.17b 1774.98c 3698.68aA 3521.49b 2936.70c Processed Processed Processed Constrained Constrained Constrained 12 24 48 6256.74aA 5987.69b 4901.83c 81.19aA 52.50b 26.13c 2642.37aA 2458.49b 1716.38c 3580.25aA 3443.82b 2893.22c Fresh Fresh Fresh Unconstrained Unconstrained Unconstrained 12 24 48 6915.33aA 6394.19b 4598.67c 109.81aA 52.63b 24.13c 1689.88aA 1439.41b 576.48c 3914.19aA 3649.94b 2739.51c Fresh Fresh Fresh Constrained Constrained Constrained 12 24 48 6778.72aA 6340.51b 4521.77c 84.94aA 50.13b 23.50c 1624.22aA 1413.62b 539.53c 3844.92aA 3622.72b 2700.52c A Average values followed by same lower case letter are not significantly different at P 6 0.05 according to LSD test. increase the potential yield production of the asparagus field. Assuming the same cost structure of the classic 24 h interval, there would be an equivalent to an increase in profits. Increases in profit calculated with the constrained model adopting the 12 h interval harvesting strategy instead of the 24 h were US$183.88/ha for the processed product and US$210.60/ha for fresh asparagus. These results showed that multiple daily harvests might represent a way to increase yields and profits without negatively affecting the production of the following year. The 48 h harvesting interval had yields and profit levels significantly lower than the control interval (24 h). The 48 h interval harvesting strategy, using the constrained model, generated yields of 4901.83 for processed asparagus and 4521.77 kg/ha for fresh product (Table 8). Those values represented a reduction in yields of 1085.86 and 1818.74 kg/ha for the processed and fresh product, respectively. Results in terms of profits were similar. Reductions in profit were US$742.11/ha for the processed asparagus and US$874.09/ha for fresh product. Results with the unconstrained model in increasing the frequency of harvesting resulted in an increase in yield of 349.46 for the processed asparagus and 521.14 kg/ha for fresh. The gain in yield by increasing the harvesting frequency was greater in the unconstrained model. Similar results were found in terms of the profits. The 12 h harvesting interval had a gain in profit of US$238.83/ha and US$250.47/ha for the processed and fresh product, respectively. The profit levels by moving from the 24 h interval to the 12 h for the processed product increased by US$54.95/ha for the constrained model. This indicated that if there were no T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 289 economic constraint on the harvest, growers could benefit an extra US$50.06/ha by moving to the 12 h strategy. The constrained simulation model is more relevant in supplying information on harvesting interval decisions to the grower in Washington. The unconstrained model, on the other hand, had no economic constraints on the harvest, therefore its results were relevant in growing conditions where manual labor does not represent a limitation to the asparagus crop. A change in the minimum wage requirement, as well as any other economic variable might change these results. Among the results presented, the only ones that would be unchanged if economic conditions would change are the yields from the unconstrained simulation model. These results indicated a potential gain with manual harvest for asparagus growers in reducing the interval of harvest from 24 to 12 h. The gain was in both yields and profits. The 48 h interval strategy presented the lowest profit and yield performances. There are several studies in the literature that examine the issue of different harvesting strategies for asparagus. Lampert et al. (1980) examined the impact of harvesting every other year, two years out of three, and three years out of four, and they compared those findings with the every year results and concluding that yield is higher if harvest occurs every year. Stout et al. (1967) considered different harvesting strategies for non-selective mechanical harvester for asparagus (daily, one harvest every two days, one every three days, and one every four days) concluding that the right interval depends on the capacity of the mechanical harvester. With manual harvesting, the availability of labor might be hard to find in case of multiple daily harvests because of high temperatures during the day. Despite it is more profitable harvesting asparagus at the 12 h interval (when the harvesting constraint is satisfied), it might not be feasible to embrace by asparagus producers. This paper adds to the existing literature the idea to explore multiple daily harvests for asparagus. If mechanical harvesting is adopted this could represent an opportunity. The model shows flexibility for changing assumptions that could be used to further investigate those aspects. This allow to determine faster if a harvesting strategy might or might not increase yields. Using the traditional field research it could have been taken years, but the model allow identifying harvesting strategies that could increase profitability in a shorter time. 5. Conclusions This paper represents a contribution to the existing literature of harvesting asparagus. It is the first to incorporate economics to the decision of harvesting asparagus using a bioeconomic model. In addition, it is the only attempt to predict the daily production of asparagus. The bioeconomic model developed was able to calculate the impact of harvesting decisions and economic constraints. The outcomes of different harvesting strategies and the impact of the minimum wage constraint were identified. 290 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 The bioeconomic model was developed and validated using 10 different statistical methods to test its prediction capabilities in forecasting the daily harvest for several locations in Washington State (from five to three locations) in four different years (2000–2003). The testing procedure adopted proved that the model was able to predict the daily production of asparagus in different locations with a good degree of precision. The model was used to simulate yield, number of harvests, profits, and the total costs of harvesting for every year in the period 1989–2004 using the weather data from a location in Washington. By comparing the results of the unconstrained and constrained model, it was possible to evaluate the impact of the minimum wage requirements for Washington on the yields and profits for both processed and fresh asparagus. The impact of different harvesting intervals was identified with the bioeconomic model. The traditional harvest interval of 24 h was compared to a more frequent (12 h) and a less frequent interval (48 h). Manual harvest with the interval of 12 h showed the best results in terms of yields and profits for both the processed and fresh asparagus. Gains in profits with the actual production conditions in Washington were US$183.88/ha and US$210.60/ha for processed and fresh product, respectively. Although it might not be possible to hire manual labor for multiple daily harvests, these results showed that there is a potential gain also for the manual labor involved. Appendix A In this appendix the formulas used to calculate the statistics and the tests described in the Model evaluation section are discussed. 1. MSE (mean square error) n P ðAi  S i Þ2 MSE ¼ i¼1 ðGoldsmith and Hebert; 2004Þ n ðA1Þ where Ai is the actual production for day i, Si is the simulated production for day i, and n is the number of days in which production occurred. 2. MAD (mean absolute deviation) n P jAi  S i j i¼1 ðGoldsmith and Hebert; 2004Þ ðA2Þ MAD ¼ n 3. MAPE (mean absolute percent error)  n  P jAi S i j  100 Ai i¼1 MSE ¼ ðGoldsmith and Hebert; 2004Þ n 4. PCC (Pearson correlation coefficient) ðA3Þ T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 n P ðAi  AÞðS i  SÞ PCC ¼ i¼1 ðn  1ÞS A S E ðGoldsmith and Hebert; 2004Þ 291 ðA4Þ where A is the average actual production per day, S is the average simulated production per day, SA is the sample standard deviation for the actual data, and SS is the sample standard deviation for the simulated production. The value of the Pearson coefficient is between 1 and 1. The value 0 indicates no correlation, the value 1and 1 indicate perfect positive and negative correlation, respectively. Closer to 1 is the PCC, more correlated are the two series, and for validation purposes the better is the simulation model. 5. CHI-SQUARE (joint Chi-square test) v2 ¼ 2 n X ðS i  Ai Þ Ai i¼1 ðGoldsmith and Hebert; 2004Þ ðA5Þ The chi-square statistics (v2) is found with the above equation. It is tested whether each predicted (or simulated) production value is equal to the actual value recorded for that day. This is a joint test, therefore the null hypothesis (Ho) will be reject if at least one of the values is statistically different than its prediction (Ha). In the case of validation a failure to reject the Ho will be a sign of no statistical difference between the actual and the simulated series. Details of the joint hypothesis testing are reported below. Ho: Si = Ai for i = 1, 2, 3, . . ., n. Ha: Si 6¼ Ai for at least one i. 6. ACF-T (autocorrelation function test) The autocorrelation function (r(k)) of the actual and simulated series should be the same for lag values k = 0, 1, 2, . . ., n. Each value of both the actual and the simulated series were indicated generically by xi. The autocorrelation function is equal to: rðkÞ ¼ CovðkÞ Varðxi Þ ðBarlas; 1989Þ ðA6Þ where the Cov(k) is expressed by CovðkÞ ¼ nk P ðxi  xÞðxiþk  xÞ i¼1 n ðA7Þ The hypothesis testing will be similar to the one seen for the chi-square testing. However, in this case an individual test is carried out. The null hypothesis (Ho) will be that the autocorrelation functions for the simulated and actual series are equal (given the same lag considered) or similarly that there is no difference in the periods of the two behavior patterns test (Barlas, 1989). The confidence interval was calculated for each value of lag k using the variance of the autocorrelation function (Var(r(k))) and the standard error (Se(dk)) of the distance between rS(k) (autocorrelation function of the simulated series) and rA(k) (autocorrelation function of the actual series) using the following formulas: 292 T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 nP 1 ðn  iÞðrðk  iÞ þ rðk þ iÞ  2rðkÞrðiÞÞ2 i¼1 VarðrðkÞÞ ¼ nðn þ 2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Seðd k Þ ¼ VarðrS ðkÞÞ þ VarðrA ðkÞÞ ðA8Þ ðA9Þ The individual hypothesis are: Ho: rA(k) = rS(k) for k = 1, 2, 3, . . ., n  1. Ha: rA(k) 6¼ rS(k) for k = 1, 2, 3, . . ., n  1. 7. CCF-T (cross-correlation function test)The cross-correlation function test detects if the two series are perfectly in phase. If that is the case the crosscorrelation function has the highest value with a phase lag equal to zero. The two cross-correlation functions considered were: CovSA ðkÞ ¼ n P ðS i  SÞðAik  AÞ i¼k nS S S A for k ¼ 0; 1; 2; . . . ; n  1 ðBarlas; 1989Þ CovSA ðkÞ ¼ n P ðA10Þ ðAi  AÞðS iþk  SÞ i¼k nS S S A for k ¼ 0; 1; 2; . . . ; n þ 1 ðBarlas; 1989Þ ðA11Þ 8. MC (mean comparison) MC ¼ jS  Aj jAj ðBarlas; 1989Þ ðA12Þ The MC indicates the percent of error in the difference between means. 9. PEV (percent error of variation) PEV ¼ jS S  S A j jS A j ðBarlas; 1989Þ ðA13Þ where SS is the standard deviation of the simulated series, and SA is the standard deviation of the actual series. The PEV indicates the difference in the variation among the two series. It expresses the percent of error in variations of the sample estimate. 10. DC (discrepancy coefficient) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n P 2 ðS i  S  Ai þ AÞ i¼1 DC ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n n P P ðS i  SÞ2 ðAi  AÞ2 i¼1 i¼1 ðBarlas; 1989Þ ðA14Þ T. Cembali et al. / Agricultural Systems 92 (2007) 266–294 293 The DC ranges from 0 to 1. 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