A computational approach for crop production of organic vegetables

Computers and Electronics in Agriculture 134 (2017) 33–42

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Computers and Electronics in Agriculture journal homepage: www.elsevier.com/locate/compag

Original papers

A computational approach for crop production of organic vegetables Peng-Sheng You a, Yi-Chih Hsieh b,⇑ a b

Department of Business Administration, National Chia-Yi University, ChiaYi 600, Taiwan Department of Industrial Management, National Formosa University, Huwei, Yunlin 632, Taiwan

a r t i c l e

i n f o

Article history: Received 15 February 2016 Received in revised form 23 September 2016 Accepted 1 November 2016

Keywords: Crop rotation Production Heuristic approach Mixed integer linear programming

a b s t r a c t This paper deals with a vegetable production-planning problem in which a farmer has to simultaneously determine the crop rotations and harvest schedules for a number of vegetables in his farmlands. The crop rotation decisions include the determination of crop sequence, cropping times, and cropping size in each farmland. The harvest decisions include the collection times and harvesting size for each planted crop. By taking into account the growing season and technical-ecological constraints, this paper formulates a mixed integer linear programming model to develop vegetable production decisions for organic farmers. Due to the complexity of the model, we proposed a two-phase heuristic approach to solve the model. We applied the proposed model to analyze an organic farm in Taiwan. Computational results show that the proposed heuristic is superior to CPLEX in terms of solution quality and computational times for large scale problems. In addition, some sensitivity analyses were also conducted to evaluate the impact of parameters in the model. Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction The demand for organic foods, such as organic vegetables, fruits, and coffees, increase in the recent years. The global organic food market grew from US$ 57.5 billion in 2010 to US$ 104.7 billion in 2015 with an estimated compound annual growth rate (CAGR) of 12.9%. It is expected to continue to steadily grow. According to the report of Global Organic Food Market Forecast & Opportunities by TechSci Research, the global organic food market is estimated to grow with a compound annual growth rate of 16% through 2020. Organic fruits and vegetables are the biggest-selling organic category of organic foods. In this paper, we focus on the organic vegetables. Usually farmers of organic vegetables must determine what kinds of vegetables to plant and how to design the growing schedule in line with guidelines such as natural law and the technical-ecological rules. The natural law requires that vegetables be planted in their designated growing season. With regard to the technical-ecological rules, two experiments are usually considered. The first is to avoid the cropping style of continuous growing two crops of the same botanic family on the same piece of land because this may have a dramatic impact on plant root zones and result in decreasing soil fertility or crops that are susceptible to diseases, plagues, or weeds (Haneveld and Stegeman, 2005). Thus, farmers usually adopt this ⇑ Corresponding author. E-mail address: [email protected] (Y.-C. Hsieh). http://dx.doi.org/10.1016/j.compag.2016.11.003 0168-1699/Ó 2016 Elsevier B.V. All rights reserved.

rule to provide essential chemical elements in quantities and proportions for the growth of specified plants. The second is to use fallow lands to build soil structure and reduce pest damage (Santos et al., 2010), or to use green manures to cover bare patches of soil between crops or during intervals between one crop and the next to maintain and improve soil fertility. To satisfy the above growing requirements and considerations, many farmers have used crop rotation techniques to make their cropping plans. Crop rotation is the process of growing different types of crops on a piece of farmland over each growing season. The purpose of the approach is to determine the cropping sequences and their cropping sizes in a piece of farmland (Santos et al., 2011). The decisions for crop rotations have a critical impact on crop yields over the long term. Over the past decade, many works have been proposed to deal with the crop rotation problems. Most of them focused on determining the right crop rotation for a specific field to maximize the crop yield. Plant (1997) pointed out that expert systems and numerical simulation models are two useful methods to develop crop decisions. (i) Expert systems use different methods such as artificial intelligences to convert data from complex eco-systems into available information to develop crop strategies and tactics (Chakraborty and Chakrabarti, 2008). Artificial intelligence focuses on how to combine computer equipment, software, and specialized information to imitate human experts in making decisions. In this area, Dogliotti et al. (2003, 2006) proposed a computational program,

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ROTAT, to produce crop rotations based on agronomic criteria. This approach firstly generates a crop-rotation pool composed of all possible permutations of crops, and then selects feasible crop rotations from the produced pool. (ii) Numerical simulation models adopt the simulated mechanical systems to mimic the natural behavior to reproduce the processes of a system in an analog or digital fashion to generate decisions. This approach numerically solves the equation without any turbulence model and applies a sort of numerical time-stepping procedures to generate natural behavior over time (e.g., Stockle et al., 2003). In addition to the above approaches, many mathematical models have also been proposed to study variant crop-production planning problems (e.g., Sarker et al., 1997; Sarker and Quaddus, 2002; Haneveld and Stegeman, 2005; Detlefsen and Jensen, 2007; Santos et al., 2010; Moghaddam and DePuy, 2011). These models use mathematical concepts and language to describe the behavior of a crop system. Sarker et al. (1997) formulated a linear programming model to deal with a crop planning problem for optimizing contributions through the determination of allocation of different land types, land availability and suitability, capital, and import bounds. Sarker and Quaddus (2002) proposed a goal programming approach to solve a multi-objective crop-planning problem. Haneveld and Stegeman (2005) developed a mathematical model to investigate an agricultural production-planning problem with the consideration of crop succession requirements. They assumed that the crop succession requirements are dependent on the types of crops grown and developed an algorithm to determine the crop sequences. Their paper aims to determine the crop sequences and their cropping sizes on a piece of land. However, their model does not consider the cropping location on the piece of land and the harvest schedule. Detlefsen and Jensen (2007) developed network flow models to study a crop rotation problem for a given selection of crops on a piece of land. Santos et al. (2010) considered a sustainable vegetable crop supply problem to maximize production volume or revenues by determining the best division of various heterogeneous pieces of land under known demands. They proposed a mathematical model to investigate this problem with known demand, ecologically based production constraints, and multi-planting areas. Their paper determined the division of the available heterogeneous arable areas in plots and crop rotation schedule for each plot to maximize revenues. However, the production lengths and the productivity of each harvesting for all crops are predetermined. Due to the computational complexity, most complex mathematical models are usually incapable of producing system parameters or exact solutions for completed systems within reasonable computational time. Therefore, many authors have devoted to finding compromised solutions within reasonable time instead of using incredible computational time to find optimum solutions. These approaches, including the already-mentioned numerical simulation (e.g., Stockle et al., 2003), approximation techniques (e.g., Haneveld and Stegeman, 2005), and the evolutionary computation approaches (e.g., Sarker and Ray, 2009), have been used to solve the variant problems. The evolutionary computation approach belongs to the artificial intelligence approach and wellknown algorithms include ant colony, particle swarm optimization, genetic algorithms, and so forth. Sarker and Ray (2009) formulated a multi-objective mathematical model to deal with a crop-planning problem. They proposed a genetic algorithm based on a multi-objective constrained algorithm to solve the problem. A vegetable’s harvestable periods may consist of several days or several weeks. Usually, the harvest schedule for ripe vegetables’ collection times and size is based on market demands. Since demands are usually dispersed evenly over time, farmers will harvest their ripe corps several times to match supply with demands

as fit as possible. Thus, some researchers have focused on harvesting-scheduling problems. Higgins et al. (1998) presented a mathematical model to discuss the harvesting-scheduling problem. The purpose of their model is to maximize the sugar yield and net revenue through the determination of harvest date and crop age. Muchow et al. (1998) investigated the problem of optimal harvesting time for sugar production. The decisions include (i) when to harvest different crop classes over the long harvesting period and (ii) the time at which crops are harvested (or planted) in the previous year. Moghaddam and DePuy (2011) investigated a farm-management problem for the horse breeding industry. The problem was formulated as a chance constrained mathematical model in which inventory is allowed. The paper aims to maximize the total profit by determining hay growing, buying, and selling decisions. They adopted the linear approximation technique and LINGO software to develop decisions. Foulds and Wilson (2005) developed an integer programming model to describe a harvesting-scheduling problem for renewable resources, and proposed heuristics to solve the model. Piewthongngam et al. (2009) developed a numerical simulated model to determine the best combination of cane cultivar, planting dates, and harvest dates. Although the above-mentioned literature deals with harvest scheduling problems, it does not consider crop rotation constraints. Most rotation works assumed that the length of crops’ harvest periods is fixed and does not consider vegetables’ inventories. That is, a crop is harvested when its growing periods reach the given maturity time length. Santos et al. (2010), the most recent and relevant study to our work, assume that each crop is harvested only one time and did not consider inventory. However, in practice, harvested vegetables can be stored fresh to satisfy future demands, and vegetables can be harvested during the interval of maturity date to their maximum permissible growing date beyond which crops are over-maturity and have lost their values. Thus, farmers do not necessarily harvest once, and they can stock harvested vegetables to satisfy future demands. Some farmers will attempt to satisfy demand by inventory. Thus, crop rotation production scheduling problems with variable harvesting times and size are worth researching. To our knowledge, no work simultaneously deals with the crop rotation and crop harvesting-scheduling problems. However, these two problems usually occur simultaneously. To fill the gap in the research, in this paper we assume that: (1) For multi-harvests, ripe vegetables can be collected during the entire harvestable interval. The harvest interval of a crop is from its maturity date to its maximum permissible growing period. (2) Vegetable inventories and their holding costs are considered. The operation process of this study includes planting, growing, harvesting, inventory, and supply. The purpose of this paper is to maximize the total expected revenues over a given planning interval through the determinations of (i) what kind of crops to plant, (ii) how many and when to crop, (iii) which farmlands to crop, and (iv) when to harvest under the condition that growing requirements are satisfied.

2. Model assumptions and formulation The manager of C organic farm has decided to plant vegetables in its L cultivation-houses over T planning periods. The area of cultivation-house-‘ is A‘ m2. The planting combination is selected from a candidate vegetable pool which consists of K types of vegetables. The K types of vegetables are divided into U botanical families. Vegetable type-k belongs to botanical family of qk . While planting vegetables, the farmer should obey some planting guidelines: (1) Season requirement: each type of vegetable should be planted during its suitable planting season. Let wkt = 1

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if vegetable type-k can be grown in period t and zero otherwise; (2) Land-consolidation/Land-clearing requirement: cultivation-house should be grounded before a new vegetable type is planted. If a new vegetable is to be planted after vegetable type-k, it should take ok periods to consolidate the land; (3) Rotation requirement: continuously growing two crops of the same botanic family on the same cultivation-house is not allowed. However, if two crops of the same botanic family on the same cultivation-house are planted, then a fallow or green manure is required. The periods required to make a land fallow and land consolidation or grow a

Decision variables the portion of the amount of type-k vegetable harvested xkjt at the end of period j distributed to satisfy demand of period t 1 if there is a harvest at the end of period j for vegetable y‘k ij type-k being planted at the start of period i in cultivation-house-‘, and 0 otherwise the harvested area at the end of period j for vegetable z‘k ij type-k, which is initially planted at the start of period i in cultivation-house-‘

k

green manure after vegetable type-k is f periods; and (4) Cultivation time limitations: each type of vegetable has a minimum and maximum growing time to ensure growth-to-maturity and to avoid over-maturity. The two time lengths for vegetable type-k are nk1 and nk2 , respectively. In addition, the holding cost per period k

for vegetable type-k is h per kg. The production in each cultivation-house is not always the same. The production in cultivation-house-‘ for vegetable type-k is r ‘k kg per square meter. Vegetable type-k will lose its value mk periods after its cultivation. k dt

and Demand and sale price for vegetable type-k in period t are pkt , respectively. The manager wants to maximize total expected profit by determining the planting decisions regarding vegetables’ cropping types, sizes, sequence and planting timings, and harvest decisions regarding harvest times and size. The notation is summarized as follows. Notation K the size of candidate vegetable pool L number of cultivation-houses T number of planning periods U number of crop families area of cultivation-house-‘ A‘

k

1 if vegetable type-k can be planted at the start of period t and 0 otherwise demand for vegetable type-k in period t

k

inventory holding cost per period for vegetable type-k

wkt dt

h mk nk1

the maximum stocking periods of vegetable type-k the minimum periods needed to grow vegetable type-k to maturity the maximum permissible growing period for vegetable nk2 type-k k the periods required to use a fallow/green manure and f land consolidation after vegetable type-k the periods needed to clearing a land after vegetable ok type-k sale price per kg of vegetable type-k in period t pkt the family type of vegetable type-k qk 1 if vegetable type-k belongs to family type u, and 0 qku otherwise the set of harvestable periods of vegetable type-k that is U‘k i began plowing at the start of period i in cultivationhouse-‘ ‘k the set of the planting periods from either one of which Xj vegetable type-k should be started to plant in cultivation-house-‘ such that vegetable type-k can be harvested at the end of period j r ‘k production per square meters in cultivation-house-‘ for vegetable type-k (in kg/square meters) B a very big positive number Function the inventory level of vegetable type-k at the end of Ikt period t

In addition, we define the following period set.

8 fi; i þ 1; . . . ; jg; > > > > > fi; i þ 1; . . . ; T; 1; . . . ; j  Tg; > > > < fi  T; . . . ; j  Tg; Aij ¼ > fi; i þ 1; . . . ; T; 1; . . . ; jg; > > > > > fj; j þ 1; . . . ; T; 1; . . . ; i  Tg; > > : fj  T; . . . ; i  Tg;

i 6 j 6 T; i 6 T < j; T < i < j;

ð1Þ

j < i 6 T; j 6 T < i; T < j < i:

The harvestable periods of vegetable type-k that begins plowing at the start of period i in cultivation-house-‘, U‘k i , is represented by 8 k k i 6 T  nk2 þ 1; > < fi þ n1  1; .. . ; i þ n2 g; k k U‘k ¼ fi þ n1  1; .. . ; T; 1; . .. ; n2  T þ i  1g; T  nk2 þ 1 < i 6 T  nk1 þ 1; i > : k i > T  nk1 þ 1: fn1  T þ i  1;. . . ; nk2  T þ i  1g;

ð2Þ Denote X the set of possible cropping periods where vegetable type-k can be harvested at the end of period j. The cropping periods for the vegetable must be the interval of nk1 periods before period j k j

to nk2 periods before period j. Thus, Xkj is represented by:

8 k k j < nk1 ; > < fT þ j  n2 þ 1; . . . ; T þ j  n1 þ 1g; X ¼ fT þ j  nk2 þ 1; . . . ; T; 1; . . . ; j  nk1 þ 1g; nk1 6 j < nk2 ; > : fj  nk2 þ 1; . . . ; j  nk1 þ 1g; j > nk2 : k j

ð3Þ Suppose that there is a harvest for vegetable type-k at the end of period j in cultivation-house-‘. For the land clearing and manure purpose, vegetables having the same botanical family with that of vegetable k cannot be planted over the following f Denote these periods by set

Gkj .

Then,

Gkj

k

periods.

is represented by:

8 k k > j6T f ; > < fj þ 1; j þ 2; . . . ; j þ f g; k Gj ¼ fj þ 1; . . . ; T; 1; . . . ; j  T þ f k g; T  f k < j < T; > > : k j ¼ T: f1; 2; . . . ; f g;

ð4Þ

Suppose that there is a harvest for vegetable type-k at the end of period j in cultivation-house-‘. For the land clearing purposes, vegetables differing from the botanical family of vegetable type-k cannot be planted over the following ok periods. Denote these periods by set Bkj . Then, Bkj is represented by:

8 k > j 6 T  ok ; < fj þ 1; j þ 2; . . . ; j þ o g; k k Bj ¼ fj þ 1; . . . ; T; 1; . . . ; j  T þ o g; T  ok < j < T; > : f1; 2; . . . ; ok g; j ¼ T:

ð5Þ

2.1. Mathematical model The objective function, F, is composed of sale revenues and inventory costs.

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max F ¼

K X T T K X T X X X k pkt xkjt  Ikt h k¼1 t¼1

ð6Þ

k¼1 t¼1

j¼1

in which

Ikt ¼

8 T t1 X X > > > xkjt þ xkjt ; 8k; t 6 mk ; > > < j¼Tþtmk j¼1 t1 > X > > > xkjt ; > :

ð7Þ

8k; t > m : k

cultivations in the same piece of a cultivation-house. Suppose that there is a harvest in period j for vegetable type-k that is planted in period i in a cultivation-house. Then, the following two events should be guaranteed: (i) No other vegetable can be planted in period i and no vegetable can be planted during periods i + 1 to j in the same piece of cultivation-house. (ii) No harvest is done during periods i to j for any vegetable planted in any period except for vegetable type-k planted in a period i. Eqs. (16) and (17) ensure events (i) and (ii), respectively.

j¼tmk

3. Solution procedure

where Ikt ¼ xkTT if t = 1 and mk ¼ 1.

subject to k y‘k ij 6 wi ;

8‘; k; i; j;

y‘k ij ¼ 0; z‘k ij

6

ð8Þ

8‘; k; i; j R U‘k i ;

ð9Þ

8‘; k; i; j;

ð10Þ

y‘k ij B;

T X K X ‘ z‘k ij 6 A ;

8‘; i;

ð11Þ

j¼1 k¼1 L X X

T X xkjt ;

‘k z‘k ij r P

‘¼1 i2X‘k

t¼1

T X k xkjt 6 dt ;

8k; t;

j

8k; j;

ð12Þ

ð13Þ

j¼1 T  0 2 XX 0 y‘k 6 qk  qk B þ ð1  y‘k ij ÞB; i0 j 0

8‘; k; k0 ; i; j;

0 i0 2Bkj j ¼1

ð14Þ T  0 2 XX 0 y‘k 6 qk  qk B þ ð1  y‘k ij ÞB; i0 j 0 0

i

8‘; k; k0 ; i; j;

0 2Gkj j ¼1

ð15Þ K T K X X T   X X X 0 0 y‘k þ y‘k 6 1  y‘k ij B; i0 j 0 i0 j 0

k0 ¼1;k0 –k j0 ¼i

k0 ¼1i0 2A j

iþ1

8‘; k; i; j;

j0 ¼i

ð16Þ K T X X X 0

0

0

0

k ¼1i ¼1;i –ij

0

y‘k 6 ð1  y‘k ij ÞB; i0 j0

8‘; k; i; j:

ð17Þ

2Aij

Eq. (6) represents the objective function of the problem. Eq. (7) represents the inventory level for each vegetable type in each period. Eq. (8) ensures that vegetables are planted in growing season and Eq. (9) ensures that vegetables are not harvested in nonharvestable season and should be planted within cultivation time limits. Eq. (10) ensures that the harvest area is zero if no harvest action is made. Eq. (11) ensures that the total harvest area for any kind of vegetable planted at a certain period in a cultivationhouse does not exceed the cultivation-house’s area. Eq. (12) is a production-supply conservation constraint. It ensures that the vegetable yield in a period must be no less than what is used to satisfy demands. Eq. (13) ensures that supply does not exceed demand. Eq. (14) ensures that there is enough time for land arrangement after a vegetable cultivation and ensures rotation requirement for vegetables with different botanical family. In addition, if vegetable type-k is planted in cultivation-house-‘ in period i and has a harvest in period j (y‘k ij = 1), this constraint will ensure that no vegetable is over

The developed model is a combinational optimization problem in which the farmer must determine (i) the planting schedules, (ii) the harvest schedules for the corresponding planting schedules, and (iii) the supply-distribution plans. For each planting schedule, the decisions include what vegetables to be planted, in which cultivation-houses to plant the vegetables, and when to plant the vegetables. For each harvest schedule, the farmer must determine the harvest times for each vegetable planted in each cultivationhouse. For the supply-distribution plan, the farmer must determine how to distribute the harvested vegetable to satisfy demand. ~ 1 ¼ maxfnk1 ; 8kg, Suppose that the largest one among all nk1 is n and the largest one among all vegetables’ cultivation time gaps is k ~ 1 þ f Þ times to f ¼ maxff ; 8kg. Then, there are at least n1 ¼ T=ðn plant vegetables in planning cycle, T, in cultivation-house-‘. Suppose that the minimum harvest frequency in a planning cycle, T, in cultivation-house-‘ is n‘1 . Then, there are K n‘1 possible outcomes to plant vegetables in the cultivation-house. The overall possible outcomes in all cultivation-houses are PL‘¼1 K n‘1 . Consider a small scale case with K = 5, L = 5 and n‘1 ¼ 2 for all ‘. In this case, there are 255 = 9,765,625 possible outcomes. Note that it is up to 2510 possible outcomes when K = 5, L = 10 and n‘1 ¼ 28‘. That is, the number of possible outcomes for this problem increases dramatically as the number of plan periods, the number of cultivationhouses, and the number of vegetables increases. Moreover, the proposed mathematical model contains KT2 + 2LKT2 decision variables. Since there are KT, LKT2, P LT Kk¼1 ðnk2  nk1 þ 1Þ, LKT2, LT, KT, KT, LK2T2, LK2T2, LKT2 and LKT2, constraints in Constraints (7)–(17), respectively. Thus, the total number of decision variables and constraints increases drastically in the mathematical model. For example, the number of decision variables is up to 698,112 and the number of constraints is up to 52,899,106 when (L,K,T) = (50, 27, 16) and (nk1 , nk1 ) = (3, 5). It implies that the considered problem is a highly complicated mixed integer linear programming problem. Thus, the computational time using enumeration method to find optimal solution is impractical. Although some commercial optimization software, such as Lingo solver and CPLEX, can be used to solve the combinational optimization problems. However, they cannot guarantee to produce compromised solutions within reasonable computational time for a medium or large problem. For larger problems, their performance becomes worse since total number of variables and constraints is very large. To solve the considered problem within reasonable computational time, we develop a two-phase heuristic approach to find a compromised solution in this paper. We refer to the first and the second phases as initial solution approach and solution improvement approach, respectively. The structure of the heuristic approach is as follows.

clearing periods Bkj . Moreover, if y‘k ij ¼ 0, this constraint becomes in-active since the right-hand-side is a very larger number. Eq. (15) ensures that there is enough time for green manures and rotation requirement between two vegetables of the same botanic family on the same piece of land. Eqs. (16) and (17) avoid overlapping

3.1. Structure of the heuristic approach (Two-phase approach) Parameter setting Do the followings while stopping criteria is not reached.

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(1) Initial solution approach (Phase-I) Do the following procedure while unsatisfied demands and available cultivation-houses exist. (a) Selection of the vegetable and cultivation time: Select the vegetable and the cropping starting time for the selected vegetable. (b) Selection of cultivation-house: Select the cultivation-house with greatest yield from the available cultivation-house to plant the selected vegetable in the selected time. (c) Harvest schedule: Determine harvest time and size to satisfy demands over harvestable periods. For each cultivation-house, execute solution improvement model to improve harvest time, harvest amount, and productionsupply decisions. (3) Solution update if a better solution is found. (4) Report the best result. 3.2. Initial solution approach (Phase-I) Before addressing the details of the heuristic approach, we define the following notation. bkm = the mth greatest productivity cultivation-house for growing vegetable type-k. k

bt = the earliest harvestable period for vegetable type-k planted from the start of period t. ekt = the latest harvestable period for vegetable type-k planted from the start of period t. g kt = portion of the demand for vegetable type-k in period t that has not been assigned supply. g‘k t = 1 if vegetable k can be planted at the start of period i in cultivation-house-‘, and zero otherwise.

Heuristic solution-1 (by Initial solution approach) will produce initial solutions for the decisions associated with (i) the planting schedules and (ii) the harvest schedules. Initial solution approach, one at a time, determines the vegetable to be planted, the vegetable’s planting time, the planting cultivation-house, and harvest  times. For each planning, let k , t  , and ‘ be the vegetable selected, the vegetable’s planning time and the vegetable’s planting cultivation-house, respectively. 3.1.1. The planting schedule For the planting schedule, Phase-I first determines the values of  ðk ; t Þ, then the value of ‘ . We refer to the decision process associated with the former and later parts as the schema for the vegetable and planting time selection, and the schema for  cultivation-house selection, respectively. The values of ðk ; t Þ are determined by the following rule. (1) Initially, generate N distinct numbers of v n , 1 6 n 6 N from the pool of f2n=NðN þ 1Þ; 1 6 n 6 Ng. These numbers are determined by the appearing sequence of N distinct integer numbers of {1,2, . . . , N}. For example, suppose N = 4 and the appearing sequence of the four numbers is (2, 4, 3, 1). Then, the values of fv 1 ; v 2 ; v 3 ; v 4 g are {2/10, 4/10, 3/10, 1/10}. The appearing sequence of the distinct integer numbers is updated according to the evolution rule. (2) Sort the values of Skt s and let the first N largest ones be Skt 1 ; Skt 2 ; . . . ; Skt N . (3) Determine

the

value



ðk ; t  Þ

of

by

6 n 6 Ng. After the value of ðk ; t Þ ¼ maxfðk; tÞjv  ðk ; t Þ is determined, we determine the value of ‘ by   ‘ ¼ maxf‘jr‘k ; ‘ 2 rkt g. 



n kn St ; 1

3.1.2. The harvest schedule  After the value of ðk ; t ; ‘ Þ is determined, we determine har

 

vesting times according to the value of Skt ; t 2 U‘t k . We determine ‘ k t



for vegetable-k planted in the harvest amounts over t 2 U cultivation-house-‘ . These values are determined one at a time

dkt = the set of cultivation-houses wherein vegetable type-k can be planted at the start of period t. = the mth greatest productivity cultivation-house among akm t all available cultivation-houses for growing vegetable type-k in period t. rkt = the set of cultivation-houses wherein vegetable type-k can be planted at the start of period t. IðEÞ = an indicator function. IðEÞ ¼ 1 if event E is true and zero otherwise. R‘t = the remaining area of cultivation-house-‘ in period t.

where z‘t tk ¼ minfg kt =r ‘ k ; R‘t g. The process stops if either one of the following two cases occurs: (1) all demands have been confirmed their supply source or (2) no available cultivation-house can be used to plant vegetables. For a given pair of (k, t), the decision processes of the vegetable cultivation-house assignment and the harvest schedule

skte = the eth element in set U‘ki .

‘k will stop if (i) g kj ¼ 0 for all j 2 U‘k t or (ii) gt ¼ 0 for all ‘. (i) means

k

from the vegetable’s earliest harvestable period bt to the latest      harvestable period ekt . The harvest amount in period t is z‘t tk r ‘ k  



 



u‘k i = 1 if planting vegetable type-k in cultivation-house-‘ in period i, and zero otherwise. u‘k ij = 1 if vegetable type-k planted in cultivation-house-‘ in per-

have been assigned their supply that demands in periods U‘k t source and (ii) means that no available cultivation-house can be used to plant vegetable at the start of period t. Since either one will

iod i can be harvested in period j, and zero otherwise.

result in the situation of Skt ¼ 0, the initial solution process is

Skt

= possible sales revenue for demands over the interval from earliest to latest harvestable periods when there is cultivationhouse in which to plant vegetable type-k in period t. qkn = the period corresponding to the nth greatest in fSk1 ; Sk1 ; . . . ; SkT g. x‘k ij = the portion of harvest from type-k vegetable planted in cultivation-house-‘ in period i used to satisfy demand of period j.

Function Skt is defined by the following equation.

! ekt L X X Skt ¼ g kj pkj I g‘kt > 0 ; j¼bkt

‘¼1

8k; t:

stopped when all of Skt become zero. 3.1.3. Summarized solution procedure for the initial solution approach (Phase-I) (1) If TS ¼

PK PT k¼1

k t¼1 St

> 0, go to step 2, else go to step 11.

n (2) ðk ; t Þ ¼ maxfðt; kÞjSkn t v ; 1 6 n 6 Ng. (3) m = 1 and e = 1. 



 k m . (4) if m > jdk t j, go to Step 2, let ‘ ¼ at

ð18Þ

‘ k t

¼ 0, let m = m + 1 and go back to step 4, else go to (5) if g step 6.

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P.-S. You, Y.-C. Hsieh / Computers and Electronics in Agriculture 134 (2017) 33–42 

‘ k (6) If e > jUt j, go to step 1, else go to step 7.

(7) If

¼ 0 where j is the eth element in U

g k j

back to Step 6, else let 

 

(8) If R‘t r‘





 g kj

 g kj

R‘t ¼ R‘t  z ¼

‘ k t ,



‘ k t j

,



else

    z‘t jk r‘ k

and

let  R‘t

let e = e + 1 and

 



x

‘ k t j

 



k

 

¼ R‘t r ‘

k



, g kj ¼ 0 and ,

‘ k t j

z



¼ R‘t ,

(9) Update g and  (10) If g kj ¼ 0, let e = e + 1 and go back to Step 6 else let m = m + 1 and back to Step 4. (11) Initial solution output. Before executing the initial solution approach, we set the initial k ‘ k ‘k k values of g kt , g‘k t , St and Rt as: g t ¼ dt ; 8k; t, gt ¼ wt ; 8‘; k; t, Pekt PL ‘ k ‘ k k ‘k St ¼ k g j pj Ið ‘¼1 gt > 0Þ, 8k; t and Rt ¼ A ; 8‘; t. k

j¼bt

3.2. Solution improvement approach (Phase-II) The initial solution approach can quickly determine the values PL ‘k ‘¼1 xjt . The distribution resource

‘k k of xkjt , y‘k ij and zij where xjt ¼

planning of the approach is based on demand for vegetable typePL PT ‘k k in period j is satisfied by ‘¼1 i¼1 xij . Although this approach can quickly find feasible solutions, it still has some drawbacks. The procedure determines planting areas after decisions regarding vegetable, planting time, and planting cultivation-house are made. Thus, each produced vegetable is independently planned on its distribution list. In general, it is better to use the aggregate planning technique to establish the distribution list. That is, simultaneously designing the distribution list of all produced vegetables for all demands. Once vegetables’ cropping cultivation-houses are known, the solution improvement procedure will use the aggregate planning technique to establish a linear programming model and to improve the solutions obtained by initial solution procedure. We express the programming model in Model-2. 3.2.1. Model-2

max F k ¼

T T mk T T X X X X X k pkt xkjt  ð xkjt þ xkjt Þh t¼1



T X

t¼1 j¼Tþtmk

j¼1 t1 X

xkjt h

j¼1

k

ð19Þ

t¼mk þ1j¼tmk

subject to T X ‘k ‘ z‘k ij 6 ui A ; 8‘; i;

ð20Þ

j¼1 ‘k z‘k ij 6 ui B; 8‘; i; j; L X X ‘¼1 i2X‘k j

u‘k ij

‘k j¼1 yij

¼

u‘k i ;

‘k > 0, let u‘k i = 1, else let uij = 0.

8‘; k; i; j 2 U‘k i .

Since Model-2 is a linear model, it can be optimally solved by linear programming approach.

¼ 0.

 Skt .

‘ k t

(2)

PT

¼ 1 and go to Step 8.

 

> g kj , let x‘t jk ¼ g kj , z‘t jk ¼ g kj =r ‘

k



  y‘t jk

(1) If



T X ‘k z‘k xkjt ; 8j; ij r ¼

ð21Þ ð22Þ

t¼1

T X k xkjt 6 dt ; 8k; t;

ð23Þ

j¼1

xkjt ¼ 0; 8j; t R Hkj :

ð24Þ

In Eq. (24), Hkj represents the set of retention periods of vegetable ‘k type-k harvested in period j. In addition, parameters u‘k i and uij

are obtained according the value of y‘k ij obtained from initial solutions, and they are determined by the followings rule.

4. Numerical examples In this section, we provide numerical examples to illustrate the performance of the proposed approach and some characteristics of the investigated vegetable rotation problem. The data used in these examples are obtained from C organic vegetable company in Taiwan.

4.1. Data setting Organic vegetable company C is in Taiwan and has L = 50 cultivation-houses with cultivation areas between 13 and 30 (m2). The vegetables planted are selected from a candidate vegetable pool with vegetable codes 1–27. These vegetables are divided into U = 7 botanic families, namely, Cruciferous, Composite, Goosefoot, Umbelliferae, Basellaceae, Convolvulaceae, and Amaranthaceae. The botanic family for each vegetable is shown in Columns 2, 8 and 14 in Table 1. Additionally, it takes ok ¼ 1 period (two weeks) to consolidate the land after the cultivation of vegetable type-k in a cultivation-house is finished. If two crops of the same botanic family are grown on the same piece of cultivationhouse, the cultivation-house will use green manure or fallow to supply organic matter to the soil between the two crops. The periods required to consolidate and apply green manure or fallow after k

vegetable type-k is f ¼ 1 periods. For vegetable type-k, nk1 , nk2 , pk and mk are shown in Table 1, and k wt is shown in Table 2. The inventory holding cost per kg per perk

iod for vegetable-k is roughly h = NT$1. The minimum, maximum, and average production rates r‘k (kg/m2) are shown in Table 3. In this section, we designed five problem categories in terms of number of cultivation-houses L, size of candidate vegetable pool K, and planning periods T. For each problem category, ten instances were generated. The first two categories are used to exam the performance of the proposed heuristic approach, and the other three categories are used to analyze the impact of parameters on the problem. The combination of (L, K, T) for the first two problem categories are (4, 5, 12) and (12, 6, 12), respectively. For problem category 3, the values of (K, T) are fixed at (10, 26) and the value of L is changed from 2 to 11 for cases 1 to 10. For problem category 4, the values of (L, T) are fixed at (10, 26) and the size of candidate vegetables pool is changed from 2 to 20. For problem category 5, the values of (L, K, T) are fixed at (50, 27, 26). The candidate vegetable pool considered in problem categories 1, 2, 3 and 5 are vegetable codes {1–2, 8, 16, 24}, {1–3, 8, 16, 24}, {1–3, 8, 14, 16, 24–27}, and {1–27}, respectively. For problem category 4, the size of the candidate vegetable pool varies from 2 to 20, for instance, in cases 1–10, respectively. The candidate vegetable pool for the ten sets are respectively {1, 16}, {1, 6, 16, 24}, {1–3, 8, 16, 24}, {1–3, 8, 14, 16, 24, 26}, {1–3, 8, 14, 16, 24–27}, {1–3, 8, 14, 16, 24–27}, {1–5, 8, 14, 16, 24–27}, {1–6, 8, 9, 14, 16, 24–27}, {1–9, 14–17, 21, 24–27} and {1–9, 14–17, 21–27}. The detailed demand forecast can provide a reasonable basis for planning decisions. However, such decisions are not the subject of this paper. Therefore, demands for all vegetables in all periods were randomly generated at 15–105 kg per period. We set the product of the generated demands and problems’ instance number-n as the demands for instance number-n of problem categories 1, 2

39

P.-S. You, Y.-C. Hsieh / Computers and Electronics in Agriculture 134 (2017) 33–42 Table 1 Parameters of qk , nk1 , nk2 , mk , pk . k

qk

nk1

nk2

mk

pk

k

qk

nk1

nk2

mk

pk

k

qk

nk1

nk2

mk

pk

1 2 3 4 5 6 7 8 9

1 1 1 1 1 1 1 1 1

3 3 3 3 3 3 3 3 3

5 5 5 5 5 5 5 5 5

3 3 2 2 2 2 2 2 2

96 96 96 80 96 96 80 80 96

10 11 12 13 14 15 16 17 18

2 2 2 3 3 4 5 6 7

3 3 4 2 4 4 4 4 4

5 5 5 3 6 6 6 6 6

2 2 2 1 2 2 2 2 2

96 96 96 96 96 96 96 96 96

19 20 21 22 23 24 25 26 27

2 2 2 3 3 4 5 6 7

5 4 3 5 4 6 2 2 2

6 6 5 6 6 7 4 4 4

1 2 2 2 2 2 1 2 2

120 96 80 96 80 96 96 96 96

Table 2 Parameters of wkt . k

Wk

k

Wk

k

Wk

k

Wk

k

Wk

1 2 3 4 5 6

1–26 1–26 1–26 1–26 1–26 4–26

7 8 9 10 11 12

1–26 1–26 1–26 1–26 4–24 1–10, 19–25

13 14 15 16 17 18

1–8 1–26 4–11, 15–21 1–26 1–8 1–8

19 20 21 22 23 24

1–7 1–7 1–10 1–6 1–8, 16–26 2–21

25 26 27

4–20 4–19 4–19

wkt is defined as: wkt ¼ 18t 2 W k and wkt ¼ 08t R W k .

Table 3 Ranges of parameter r ‘k . k

min

max

avg.

k

min

max

avg.

k

min

max

avg.

k

min

max

avg.

1 2 3 4 5 6 7

33.0 41.3 24.8 14.9 24.8 19.8 16.5

39.6 49.5 29.7 17.8 29.7 23.8 19.8

37.1 46.4 27.8 16.7 27.8 22.3 18.6

8 9 10 11 12 13 14

19.8 24.8 24.8 24.8 16.5 13.2 19.8

23.8 29.7 29.7 29.7 19.8 15.8 23.8

22.3 27.8 27.8 27.8 18.6 14.9 22.3

15 16 17 18 19 20 21

19.8 19.8 19.8 19.8 9.9 13.2 23.1

23.8 23.8 23.8 23.8 11.9 15.8 27.7

22.3 22.3 22.3 22.3 11.1 14.9 26.0

22 23 24 25 26 27

19.8 19.8 16.5 16.5 18.2 18.2

23.8 23.8 19.8 19.8 21.8 21.8

22.3 22.3 18.6 18.6 20.4 20.4

and 2 only. However, both the proposed Phase-I and Phase-II solution approaches can produce feasible solutions for all five problem categories. The detailed computational results are discussed below.

and 5; and we set the values of four times the generated demands as demand parameters for all instances of problem categories 3–4. The solution procedure was coded in Visual C++ programming language to solve test problems and to perform the sensitivity analyses. The computational results were also used to test the performance of the proposed heuristic approach. All tests were implemented on an Intel(R) Core2 Quad CPU 2.4 GHz notebook computer with 3.95 GB RAM and were terminated if the execution time exceeded six hours. The computational results are shown in Tables 3–7 for problem categories 1–5, respectively. Computational results show that within time limit, GAMS solver can produce feasible solutions for problem categories 1

4.2. Performance of the two-phase approach In this subsection, we compared the computational results of problem categories 1 and 2 obtained through our heuristic approach to those generated by the GAMS solver to evaluate the performance of the proposed approach.

Table 4 Computational results for problem category-1. No

1 2 3 4 5 6 7 8 9 10

GAMS

Phase-I

Phase-II

Obj

Time

HC

Obj

Time

Obj

HC

Time

75,990 165,570 207,270 281,420 395,260 362,828 378,956 399,688 416,022 419,842

11,406 11,438 11,406 11,414 11,408 11,441 11,407 11,448 11,448 11,448

5975 4350 5850 5300 9976 5410 10,302 8457 8617 5157

72,000 150,080 203,040 295,680 398,400 425,280 461,440 514,560 532,800 547,200

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4

117,105 243,120 313,260 449,440 573,995 642226.3 663535.8 732993.7 816508.8 854172.1

2575 5200 6900 8800 10,837 12,448 13,562 14,353 15,984 16,867

1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.6

Average HC = Total inventory holding cost. P1G = (Obj of Phase-I–Obj of GAMS)/Obj of GAMS. P2G = (Obj of Phase-II–Obj of GAMS)/Obj of GAMS.

P1G

P2G

5.25% 9.36% 2.04% 5.07% 0.79% 17.21% 21.77% 28.74% 28.07% 30.33%

54.11% 46.84% 51.14% 59.70% 45.22% 77.01% 75.10% 83.39% 96.27% 103.45%

11.53%

69.22%

40

P.-S. You, Y.-C. Hsieh / Computers and Electronics in Agriculture 134 (2017) 33–42

Table 5 Computational results for problem category-2. No

GAMS

1 2 3 4 5 6 7 8 9 10

Phase-I

Phase-II

Obj

Time

HC

Obj

Time

Obj

HC

Time

170,195 363,213 497,271 628,312 797,524 824,675 793,576 903,860 909,607 890,873

3624 3621 3620 11,452 11,456 11,449 11,432 11,467 11,448 11,448

3725 9847 13,997 17,707 17,918 13,791 14,846 8284 9430 2317

157,120 335,680 449,760 661,120 783,642 913,860 889,060 1,045,040 1,129,956 1,183,529

3 3 3 3 3 3 3 3 3 3

193,715 437,640 577,962 853,823 1,074,982 1,202,751 1,251,432 1,486,771 1,520,131 1,638,979

2125 6200 8917 11,908 17,850 15,712 21,634 24,374 21,976 27,712

8.6 8.6 8.5 8.4 8.5 7.1 8.1 7.9 7.2 8.5

Average

P1G

P2G

7.68% 7.58% 9.55% 5.22% 1.74% 10.81% 12.03% 15.62% 24.22% 32.85%

13.82% 20.49% 16.23% 35.89% 34.79% 45.85% 57.70% 64.49% 67.12% 83.97%

7.42%

44.03%

Table 6 Computational results for problem category 3 (Sensitivity analysis on L). No

L

Obj(P-I)

Obj(P-II)

Gap

HC

CPU(P-I)

CPU(P-II)

CPU

NC

PC

ACH

AVT

1 2 3 4 5 6 7 8 9 10

2 3 4 5 6 7 8 9 10 11

456,960 609,920 773,760 951,680 1,062,400 1,166,531 1,294,669 1,345,286 1,422,288 1,459,200

609,428 817,132 1,021,308 1,204,744 1,386,243 1,494,384 1,586,929 1,669,642 1,719,034 1,761,048

33.37% 33.97% 31.99% 26.59% 30.48% 28.10% 22.57% 24.11% 20.86% 20.69%

9918 11,455 15,958 16,120 19,516 21,616 24,936 21,690 22,709 19,373

6 13 24 34 47 49 115 97 116 131

7 11 19 28 36 0 84 76 92 113

13 24 43 62 84 49 199 172 208 244

9 12 17 18 22 26 30 31 34 43

3–6, 8, 9, 10 1, 2, 4–6, 9, 10 1–6, 9–10 1–6 ,9, 10 1–6, 8–10 1–6, 8–10 1–10 1–10 1–10 1–10

1102 955 939 896 887 831 831 700 668 451

1.50 1.50 2.13 2.25 2.75 2.89 3.00 3.10 3.40 3.91

Average

27.27%

NC = number of planting frequency, PC = planting combination, P-I = Phase-I, P-II = Phase-II.

Table 7 Computational results for problem category 4 (Sensitivity analysis on K). No

K

Obj(P-I)

Obj(P-II)

Gap

HC

CPU(P-I)

CPU(P-II)

CPU

NC

NP

PC

AVT

1 2 3 4 5 6 7 8 9 10

2 4 6 8 10 12 14 16 18 20

568,320 919,680 1,110,666 1,311,239 1,422,288 1,429,052 1,458,205 1,471,224 1,477,414 1,478,486

568,320 1,070,421 1,355,843 1,620,045 1,719,034 1,743,869 1,818,385 1,865,606 1,884,969 1,913,855

0.00% 16.39% 22.07% 23.55% 20.86% 22.03% 24.70% 26.81% 27.59% 29.45%

0 8686 16,118 19,987 22,709 23,615 24,036 27,072 26,554 30,596

5 16 35 65 155 173 214 294 351 425

26 38 59 75 113 117 143 149 169 232

31 54 94 139 268 290 357 443 520 657

20 28 29 34 34 35 35 36 37 39

2 4 6 8 10 12 13 14 15 16

1,2 1–4 1–6 1–8 1–10 1–12 1–10, 12–14 1–12, 14, 15 1–10, 12, 16–18 1–6, 8–10, 12–14, 16, 18–20

10.00 7.00 4.83 4.25 3.40 2.92 2.69 2.57 2.47 2.44

Average

21.34%

(1) Solution quality: Column 10 of Tables 4 and 5 shows that the solutions obtained by initial solution approach (Phase-I) are superior to those by GAMS in 7 out of 10 cases and 6 out of 10 cases for Problem categories 1 and 2, respectively. Column 11 of the two tables also shows that the GAMS solutions are all inferior to those by the improved solution approach (Phase-II). The average gaps of solutions by GAMS and initial solution approach (Phase-I), and by GAMS and improved solution approach (Phase-II) are respectively 11.53% and 69.22% for Problem category 1, and respectively 7.42% and 44.03% for Problem category 2. (2) Computational times: Both the initial and improved solution approaches take little time to obtain compromised solutions. However, the GAMS program converged to poor solutions or did not converge within time limit. These results show that the initial solution approach can quickly produce good initial solutions that are close to the solutions by GAMS with the use of a longer time.

(3) The improving effect of the improved solution approach: Tables 4 and 5 show that the improved solution approach (Phase-II) can improve the solutions found by the initial solution approach (Phase-I) up to over 20%. This implies that the improved solution approach is very effective. In addition, from Columns 10 and 11 of Tables 4 and 5, we found that the improvement is more significant as demands increase.

4.3. Sensitivity analyses To further evaluate the impact of parameters for the problem, including the number of cultivation-houses, number of vegetables, and demands, we define ACT (the average cropping times per cultivation-house), AVT (the average cropping times per vegetable), ACH (the average inventory holding cost per cropping), and AHH (the average cropping inventory cost) as follows.

41

P.-S. You, Y.-C. Hsieh / Computers and Electronics in Agriculture 134 (2017) 33–42 Table 8 Computational results for problem category 5 (sensitivity analysis on d). No

d

Obj(P-I)

Obj(P-II)

Gap

HC

CPU(P-I)

CPU(P-II)

CPU

NC

ACH

1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10

1,247,840 2,427,520 3,676,800 4,509,060 5,093,176 5,612,966 6,235,905 7,136,120 7,220,717 7,545,963

1,421,160 2,766,660 4,197,884 5,239,989 6,164,540 6,685,032 7,613,719 8,522,663 8,853,498 9,297,054

13.89% 13.97% 14.17% 16.21% 21.04% 19.10% 22.09% 19.43% 22.61% 23.21%

10,400 19,500 33,321 48,955 66,006 68,385 85,442 96,409 97,469 109,671

17,389 16,834 16,976 16,370 15,345 14,423 14,531 14,103 13,498 13,468

5981 6243 6311 6258 6369 6045 6122 6231 6373 6310

23,371 23,077 23,287 22,628 21,714 20,468 20,653 20,333 19,871 19,778

156 157 158 161 165 168 169 171 172 173

67 124 211 304 400 407 506 564 567 634

Average

18.57%

ACT = the total number of cropping times/the total number of cultivation-houses. AVT = the total number of cropping times/the total number of vegetables in a copping combination. ACH = the total inventory holding cost/the total number of cropping. AHH = the total inventory holding cost/the total number of cultivation-houses. (1) Sensitivity analysis of the number of cultivation-houses: In problem category 3, we set the candidate vegetable pool at the set of {1–3, 8, 14, 16, 24–27} and changed the number of cultivation-houses incrementally from 2 to 11. (a) Column 4 of Table 6 shows that profits improve as the number of cultivation-houses increases. However, with the increase in the number of cultivation-houses, the marginal profit improvements have a decreasing tendency. It is because that the farmer wants to earn as high of a profit as possible by selecting a better vegetable planting combination for each cultivation-house. Basically, a combination with a higher profit has a higher priority to be selected. Thus, when cultivation-houses are tight, the farmer will eventually plant a planting combination with a higher profit. With the increase of the number of cultivation-houses, the tight situation will gradually ease. Less profitable vegetables are then chosen in the planting combination. For example, Column 11 of Table 6 shows that the 7th vegetable is not included in planting combination when the number of cultivation-houses is seven. However, it is included when the number of cultivation-houses is increased to eight. (b) Columns 12 and 13 of Table 6 show that ACH decreases and AVT roughly increases as the number of cultivationhouse increases. It is because that: (i) When cultivationhouses are tight, in order to provide more types of vegetables to satisfy market demands, the farmer will try to plant more types of vegetables on the same piece of land. Thus, the farmer may have urgent needs to collect reaped crops to increase each cultivation-house’s cropping frequency. Consequently, more vegetables are harvested before the demand time and thus needed to be stocked to satisfy future demands. (ii) When cultivation-houses become loose, the farmer will try to collect reaped vegetables near their demand times. (2) Sensitivity analysis of the size of candidate vegetable pool: In problem category four, we fixed L = 10 and changed K from 2 to 20. (a) Columns 3–4 of Table 7 show that the solution gaps between the initial solution approach (Phase-I) and the improved solution approach (Phase-II) increase as K (the size of candidate pool) increase.

(b) Column 10 of Table 7 shows that the cropping frequency (NC) ascents from 20 to 39. The values have an increasing tendency as the size of candidate pool increases. This is because that the farmer wants to plant more vegetables to satisfy demand. (c) Column 11 of Table 7 show the size of planting combinations (NP) has an increasing tendency as K (the size of candidate pool) increases. The reason may be addressed as follows. When K is not so high, the farmer has sufficient cultivation-houses to include all candidate-pool vegetables in the planting combination, and thus the revenues and cropping times can quickly increase. When K is expanded to a certain level, the cultivation-houses will become tight. Under this situation, the farmer cannot put all candidate-pool vegetables in the planting combination. Thus, farmers should crop suitable vegetables on their cultivation-house. When including some new vegetables in the planting combination, the farmer has to remove some existing vegetables simultaneously. Thus, the improvement of revenue will become slow. For example, when K = 12, the farmer crops all vegetables. However, Column 12 shows that the farmer replaces vegetable type-11 with vegetables 13–14 when K increases from 12 to 14. This reflects that the heuristic approach will replace the less-profitable planting combination with a more-profitable planting combination. (d) Column 13 of Table 7 shows that AVT descends from 10 to 2.44. This shows that AVT has decreasing tendency as K increases. This is because that more vegetables share the limited planting chances. (3) Sensitivity analysis of demands: In problem category 5, the demands are expanded from 1 to 10 times of the initial generated demands (d = 1–10). We fixed L = 50, K = 27, and all other parameters have the values specified in Section 4.1. (a) Columns 3–6 of Table 8 show that the solution gaps between the initial solution approach (Phase-I) and the improved solution approach (Phase-II), and the total inventory holding costs (HC) increase as the demands increase. (b) Columns 10–11 of Table 8 show the number of planting frequency (NC), and the average inventory holding cost per cropping (ACH) increase as the demands increase. The reason may be addressed as follows. When demands are not so high, since cultivation-houses is looser, the farmer can adopt the cropping plan with lower inventory to reduce unnecessary inventory costs. When demands become high and cultivation-houses becomes tight, the farmer may increase the cropping frequency and take a supply plan with higher inventory to satisfy as much demands as possible to generate more revenues.

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P.-S. You, Y.-C. Hsieh / Computers and Electronics in Agriculture 134 (2017) 33–42

5. Conclusions The crop rotation problem is a combinational optimization problem. Most researches assumed that crop rotation is allowed for only a single harvest for a real crop, and do not consider inventory holding costs. However, it is a common practice for a farmer to take several times to harvest a reaped vegetable and to stock the harvested vegetables for future demands. Thus, taking the two phenomena into account, this paper developed a mathematical model in which a reaped crop can be harvested several times and harvested vegetables can be stocked for future demands. In addition, due to computational complexity, it is impractical to use the enumeration approach to deal with this problem. In this paper, to shorten the demand-supply gap and improve the total profits of planting organic vegetables, we have proposed a twophase approach to derive the vegetable rotation strategies within reasonable computational times. Our approach might provide an alternative strategy for the farm (company) to maximize their profit. The performance of the proposed approach has the following characteristics. (1) Initial solution approach (Phase-I) can produce a good initial solutions closing to the solutions found by the commercial optimization software GAMS, and improved solution approach (Phase-II) can effectively improve the solutions of Phase-I approach. (2) The two-phase approach performs better than the GAMS solver in terms of solution quality and computational time. The proposed approach can produce vegetable rotation decisions for rotation problems with practical sizes that the GAMS solver cannot find feasible solutions within a reasonable time. (3) Numerical examples have shown that the two-phase approach can select a better planting combination to improve revenues when the number of cultivation-houses is tight. In the future research, one may investigate a more generalized problem with the consideration of climate change or weather into

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