programming software to farm planning problems in agricultural management. ... objectives, i.e. seasonal labor, employment and business profitability, is finally obtained. ..... crease labor and gross margin (G-MARGIN) at the expense of the.
D. K. D E S P O T I S A N D J. S I S K O S
AGRICULTURAL
MANAGEMENT
USING
THE ADELAIS MULTIOBJECTIVE LINEAR PROGRAMMING
SOFTWARE:
A CASE APPLICATION
ABSTRACT. This paper presents an application of the ADELAIS multiobjective linear programming software to farm planning problems in agricultural management. This application is illustrated on a Spanish farm case study initially presented by Romero, Amador and Barco as an application of compromise programming. The method employed here generates a sequence of efficient solutions using interactive utility assessment and satisfaction levels. A compromise solution among three conflicting objectives, i.e. seasonal labor, employment and business profitability, is finally obtained. Keywords: Multiple criteria; linear programming; agricultural management.
1. I N T R O D U C T I O N
Most real-world decision problems involve multiple and conflicting objectives. Such problems are in fact semistructured in nature because simultaneous optimization of the objectives is usually unattainable due to their conflicting nature. In light of this situation, multiobjective mathematical programming (MOMP) provides an operational methodology for handling multiple objectives within the frame of traditional mathematical programming. Extensive surveys on MOMP methods can be found in Zeleny [26], Hwang and Masud [12], Chankong and Haimes [2], Evans [9], Cohon [3] and Cohon and Marks [4]. Agricultural planning is a wide management field including important problems of agricultural economics, such as land allocation and redistribution, irrigation, cropping pattern design, machine-hours allocation etc. In these problems MOMP is nearly self-imposed since the objectives under consideration, such as business profitability, employment level in the rural sector, seasonality of labor, environmental benefits and water resources saving, are often competitive and call for implicit or explicit trade-off decisions. An extensive review of agriculTheory and Decision 32: 113-131, 1992. 9 1992 Kluwer Academic Publishers. Printed in the Netherlands.
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D. K. D E S P O T I S A N D J. S I S K O S
tural planning problems and models, mainly at the farm level, is given in Glen [10]. Several applications of MOMP and related methodologies in agricultural planning problems have been reported in the literature. I-Iitchens, Thampapillai and Sinden [11] studied a land allocation problem in Australia considering money income and environmental benefits as objectives. They applied the weighting method according to which the original objectives are aggregated into an overall objective (the weighted sum), which is then used to generate the efficient alternatives by systematically changing the weights. In a similar problem Vedula and Rogers [23] considered economic benefits and irrigated cropped area as objectives. In order to define the opportunity cost curve between these two objectives they used the constraint method. According to this method only one objective is optimized at a time the others are being transformed into constraints. Then the efficient solutions and the trade-off curve between the objectives are determined by systematically changing the right-hand sides associated with the binding objectives. Romero and Rehman [18, 19] focus on the role of goal programming and other related approaches (multiobjective programming, compromise programming and generalized goal programming) in farm planning problems. Wheeler and Russell [24] suggest the use of goal programming in a 600 acre farm planning problem involving four objectives of the same priority level: gross margin; seasonal cash exposure; labor utilization in autumn and winter; labor utilization in spring and summer. Further applications of goal programming in agricultural management include, among others, the investigation of optimal fertilization alternatives (Minguez et al. [16]), the investigation and evaluation of agricultural development alternatives (Wit et al. [25]) and the optimal allocation of reclaimed lands to a variety of agricultural activities (E1-Shishiny [8]). In the frame of MOLP, Romero et al. [17] studied a real case concerning a cooperative farm planning in the frame of an agrarian reform program in Spain. The problem was to design the farm plans of a cooperative by taking into account three objectives: employment, labor seasonality and gross margin. Romero et al. first used the noninferior set estimation (NICE) method in order to determine the efficient solutions and to plot the trade-off curves for the objectives and then applied the compromise
AGRICULTURAL
MANAGEMENT
USING ADELAIS
115
programming approach [27] to obtain a compromise among the objectives. Another similar but large scale planning model for Spain has been developed by Cortes and Abello [5]. The authors applied lexicographic rules on the four objectives considered after having prioritized them as follows: 1st optimization level: Economic objectives (global gross margin; global added value); 2nd optimization level: Social objectives (total employment; employment stability); 3rd optimization level: Global risk. Other multiobjective programming applications in agricultural planning are those of Mendoza et al. [14, 15] and Romero et al. [20]. The purpose of this paper is to show how interactive multiobjective programming can be used as an effective tool for progressive articulation of preferences and rational decision making in agricultural planning. The case study concerns the application of the ADELAIS (Aide la DEcision pour syst~mes Lin6aires multicrit~res par Aide ~ la Structuration des pr6f&ences) multiobjective linear programming system to the farm planning problem discussed by Romero et al. The paper is organized as follows. In Section 2 the operational principles of ADELAIS are summarized. In Section 3 the case application is presented. Finally, a discussion and concluding remarks are provided in Section 4. 2. O P E R A T I O N A L
PRINCIPLES
OF ADELAIS
ADELAIS is a fully interactive and menu driven computer program which is designed to support decisions in MOLP problems of the general form: [max] gl (x) = c~x (1) [max] gn(x)
r : CnX
subject to x E A = {x~ R " : ~/x~0} where x = {x~ . . . . Xm} is the vector of the decision variables, ~/is the matrix of the technological coefficients, b is the right-hand side of ,
116
D. K. DESPOTIS AND J. SISKOS
the constraints and ej = (cjl . . . . . c#,) are the coefficients of the objective gj. ADELAIS consists of twelve independent software modules, which as a whole support extensive data management and realize a coherent MOLP methodology. Detailed information about the underlying methodology of ADELAIS, which is also presented briefly in the rest of this section, as well as its software structure and the user interface, are given in three papers: [21], [22] and [6], respectively. The MOLP method incorporated in ADELAIS operates in four stages (see also Figure 1).
Preliminary Stage In this stage upper and lower bounds for the objectives are obtained by maximizing and minimizing respectively each objective on the feasible set A. Particularly, if all or some of the minimization problems are unbounded, and this may happen even though the original MOLP problem has been well formulated in order to have a finite maximum, the lower bounds are computed with a heuristic (cf. Siskos and Despotis [22]). Afterwards, an initial efficient solution (i.e., a solution which is not dominated by any other feasible solution) is estimated in a way similar to that in Step Method (STEM) of Benayoun et al. [1]. This technique guarantees that the objective values which correspond to the estimated solution will be as close as possible to the upper bounds with respect to the weighted Tchebycheff norm. The iterative part of the method can be resolved in three successive stages.
Stage I At each iteration the system provides the DM with a new efficient solution and the corresponding objective values. These solutions, except the initial one which comes from the preliminary stage, are calculated in Stage III (see below). In Stage I the system screens the attained objective values, the achievement percentages with respect to the upper bounds and the satisfaction levels (i.e. the revised lower bounds) established in previous iterations. The DM compares the
AGRICULTURAL
MANAGEMENT
USING ADELAIS
117
Formulate thi MOLP problem
Calculate upper and lower bounds for the ob}cctives. Set the initial satisfaction levels.
Estimate an initial efficient solution.
No
].
I
Is there any intention\ Yes to modify the subjective weak order ? /
I
I N o / I s there any satisfactory \ objective value in the ) current solution ? /
Mayimi7e U to obtain a new compromise solution.
Ask the decision maker to indicate the ob|ectives he insists on increasing.
I
I
I Yes
Is there any i n t e n - \ ye~ tion to modify the assessed marginal I - utilities ? /
T No Is the weak order \ dlctatcd by U sufdentiy dose to the ) r~nking suggested b y / the decision maker 7 /
r
Can any of the remainder N of the obiectives be decresaed ? / No
[ The current solution is [ the best compromise. [
Assess the decision maker's utility function U in a piecewise linear form.
t I
Ask the decision
maker to rank the decision profiles according to his prcferances. Yes l
Modify the satisfaction ]
levels and the feasible region.
I
I
Construct a set of decision profiles within the new satisfaction levels.
Fig. 1. The flow chart of the method.
118
D. K. D E S P O T I S
A N D J. S I S K O S
attained objective values with the upper bounds and then he/she is asked to indicate which objectives he/she insists on increasing and if he intends to decrease some of the others in compensation. The DM's answers are combined with relative answers of previous iterations and then are used by the system for the establishment of new satisfaction levels. These new satisfaction levels limit the decision space but the DM can relax them, whenever he/she wants, by analysing the local trade-offs among the objectives. This possibility allows the DM to remove the consequences of previous answers which eventually contradict his/her current desires. That is to say the DM can dilate the decision space in order to reexamine solutions that had been rejected in previous iterations. The iterative process terminates within Stage I when a best compromise is achieved, i.e. when the DM is not willing to decrease any objective.
Stage H Stage II constitutes a learning process of the DM's preferences. At first, a simple technique is set 'up to construct a reference set of decisions profiles (i.e. a set of n vectors that might be assumed by the n objective functions). These reference alternatives are presented in pairs to the DM, who is asked to rank order them according to his/her preferences. Then a concave additive utility function, which is as consistent as possible with the DM's ranking, is assessed by a modified version of the U T A ordinal regression algorithm (cf. Jacquet-Lagr~ze and Siskos [13] and Despotis and Yannacopoulos [7]). The system plots the curves of the assessed marginal utilities and then analyses the inconsistencies that may appear between the DM's preference ranking and the ranking rendered by the utility model on a utility-ranking regression curve. The DM then is invited to interact with the model in order to remove all or part of these inconsistencies. The utility assessment process is terminated by the system when full consistency is achieved or by the DM himself when acceptable consistency is achieved.
Stage III The DM's utility function is maximized over the set A of the accept-
AGRICULTURAL MANAGEMENT USING ADELAIS
119
able solutions; a new efficient solution is obtained and the process is repeated from Stage I. For the maximization of the DM's utility function a piecewise linear programming technique is employed. 3. THE CASE STUDY
The Farm Planning Model The general farm planning model considered in this section is a multiobjective linear programming model concerning the farm land allocation among different crops within a cooperative. Data
i~I={l,...,m}: k EK={1,...,l): T
:
t E {1 . . . . , T} b~
: :
B
:
air
:
Rt Di
: :
qi,
:
Ei
:
hit
:
the m different crop combinations considered in the cooperative; the l agricultural products considered; the number of time periods considered in a year; time periods; maximum farm land available for product k (hectares (ha)); total farm land available in the cooperative (ha); mean tractor-hours requirement in crop i during period t (hours/ha); tractor-hours available in period t; average labor utilization for crop i (hours/ha/period); labor requirement for crop i in period t (hours/ha); average working capital requirement for crop i (thousand pesetas/ha/ period); working capital requirement for crop i in period t (th.pts./ha);
120
D. K. DESPOTIS AND J. SISKOS average labor requirement for crop i in a year (hours/ha), (r~ = D~T); average annual gross margin with respect to crop i (th.pts./ha).
ri
ci
Variables farm land to be allocated to crop i (ha); under- and over-achievements with respect to a null deviation from the average labor utilization in period t (hours); possible surplus of working capital at the end of period t (thousand pts.).
Xi
v+, , v ? .
Zt
Objectives T
minimize seasonal labor (hours) : m i n G l : • ( V
++vt)
t=l
maximize employment (hours) : max G2 = ~ rix i i=1
maximize gross margin (th.pts.) : max G3 = ~ cix i i=l
Constraints Farm land availability per product
• PikXi
~
bk ,
k E K
i=1
where Pik = 1 if crop i includes product k, 0 otherwise. Total farm land availability
~xi=B
i=1
Tractor-hours availability per period
AGRICULTURAL MANAGEMENT USING ADELAIS ~aitx i ~ Rt,
121
t ~ { 1 , . . . , T}
i=1
Seasonal labor per period m
+ v;
- v;
T}
= o,
Positive cash flows from period to period (hit - Ei)x i - z t -k 0.25z r >/0
for the first period (t = 1), ( h i t - Ei)x i + zt_ 1 - zt>~O
for t = 2 , . . . , T
The cash flow constraint corresponding to the first period is separated from the others as for every period the whole surplus is transferable to the next period except the surplus of the last period, for which only 25% can be transferred to the next year. This model was developed in the context of an agrarian reform program, concerning Andalusia, Spain, among the objectives of which was the reduction of the high rate of unemployment ascertained in the rural sector. In this context, the objective of maximizing the employment level, when considered in combination with the objective of minimizing the seasonal labor, enables the decision maker to achieve a satisfactory level of stable employment. The objective of maximizing the gross margin mainly concerned the members of the cooperative. The model is applied to the case of a cooperative with B = 100 ha, m = 13, l = 9 (nine products are considered, namely: cotton, wheat, corn, soybean, potatoes, sugarbeet, sorghum, lettuces and peach trees), T = 4. (See Table I for an exact presentation of the LP matrix.) The Decision Framework
Let us assume a decision maker (DM) operating within the Institute of Agrarian Reform, which is responsible for the recommendation of the
1
x2
I
3 8 2.5 1 -8.06 -7.03 20.80 -5.71
1
x4
1
9.5 10
x5
5.25 15 1 31.24 -21.53 90.13 -9928 -68.085 -37.961 231,992 -3,780
5
1
x6
1
i
2 5 8 11.5 1 -12.64 -0.30 -21.88 -9,25 -3,207 77,872 -70.440 111,982
x7
26,25 114.40 32.25 375.50 434 58,50 5 3 . 7 8 0 7 9 . 3 8 5 50.815 315.973 117,551 104.595
17 1 1 -25.32 -23.33 55.91 13924 -8.21 -87.57 -22.42 -28.35 -2,822 -40,612 -72.653 -2.078 405.823 I58.5(16 6 1 . 8 6 1 -3.631 -5.320 -4,748 -37.715
12 6 3
1
x3
multiobjective
1 1 1
1
489.90 394,424
21.5 16 3 17 1 -48.65 195.15 -95.78 -50.77 -43.434 509.210 -21.165 133.005
x9
110 55.298
790 1%,2~)
46
1
x12
1
l
VI+
904.40 1.365 275.511 33%877
1
1
-1
V1-
1
1
V2+
(source: Rornero
1 4.5 4.5 4 6 1 -35.94 1 571.68 -331.71 -235.05 -87.457 721.797 -10.761 -30.827
x13
model
58 6 45 48 52 52 1 1 150.10 124.78 -197.50 -141.59 -71.1 -79.31 118.5 96.79 8 8 . 8 6 3 86,040 -77,.653 -49.737 1~.319 145.656 140.336
xll
I
planning
TABLE farm
1 -24.35 53.76 -7.89 -16.65 -2.764 -71.153 142.828 -5.210
6 7 17
xl0
programming
407.75 365.918
9,5 13 8 19.5 1 -31.88 132.21 -66.77 -34.06 -40,612 403254 -61.770 77.537
x8
linear
1
-1
V2-
1
1
V3+
1
-1
V3-
et al, [ 1 7 ] ) .
1
1
V4+
1
-1
V4-
xl: cotton; x2:wheat; x3: corn; x4: soybean; x5: potatoes; x6: sugarbeet; xT: soybean & wheat; x8: potatoes & soybean; xg: potatoes & corn; xl0: sorghum; xll: lettuces; x12: lettuces & corn; x13: peach trees.
254.50 107.128
10 2 5 2 3.5 2 9 1 1 -59.49 -4,57 -5.40 633 43.59 1.0~ 21.29 -3.24 -2,953 -3.207 -41.450 79.950 -76,342 -12,301 252,043 -3,270
xl
The
Z2 Z3 Z4 R.H.S.
G1 G2 G3
`
