optimization under uncertainty in agricultural production...
TRANSCRIPT
1
Optimization under uncertainty in Agricultural production planning
Anjeli Garg , Shiva Raj Singh*
Department of Mathematics, Banaras Hindu University,
VARANASI- 221005, INDIA
Abstract
In this paper, a decision making model has been proposed for vegetable crop planning under
uncertainty. Vegetable crops are in general cost expensive with high risk in profitability due
to its fluctuating prices. The proposed method uses fuzzy multiobjective linear programming
(FMOLP) to model the problem under a situation of uncertainty i.e. crop prices be uncertain
(stochastic) for optimal land use planning’s with guaranteed return, best returns and high
weighted return for decision maker.
Keywords
Multiobjective linear programming, linear membership function, discrete random variables,
economic expectations, land use planning.
1. Introduction
In agriculture production system, a cropping pattern or allocation of land to different type of
crops varies with farmer’s perspective of his land holding. Further, it is observed that net
profit per acre is greater in vegetable crops (cash crops) rather than food crops. Thus for each
farmer, profit becomes an objective function which he wishes to maximize. These problems
of allocation of land for different crops, maximization of production of crops, maximization
of profit, minimization of cost of production are addressed in agricultural management
system. Initially, these problems of agriculture sector were modelled as single objective linear
programming problem by dealing with one objective at a time. But with changing scenario of
complex real life problem, several objectives need to be associated in the agricultural
planning and management. Thus some alternative methods are needed to handle this complex
problem of decision making, as the maximization of crop production can’t guarantee the
maximization of profit. In the agriculture sector, profit or loss also depend on fluctuating
* Corresponding author: [email protected]
2
demand, supply and pricing of a particular crop with minimization of cost of cultivation
needed for that crop. Thus the maximization of profit turns out to be a multiobjective decision
making problem.
2. Economic expectations, imprecision and planning
Every planning in financial management begins with economic expectations either qualitative
or quantitative. The success of a financial planning model depends on the fact that how nicely
it can sustain the expectations with market fluctuations. Thus the model development for
expectations must accommodate the conditions of uncertainty and complexity, while
handling imprecise information. As the basic equations in financial engineering is the study
of managing funds in bonds and stocks, with a view that bonds offer an absolutely guaranteed
rate of return where as the stock’s price can go up and down. In financial engineering the
stock price are dealt as random variable and the modelling is for providing optimal plans to
guaranteed return, best and worst rate of returns for decision makes. Similar to financial
management, the agribusiness management plans at farmer’s level need to provide crop
modelling. Here, the rate of return in food grain crops and vegetable crops much more behave
similar to bonds and stocks. In general food grains prices are not much volatile and give
almost guaranteed return, as in many countries (India) food grains have government support
prices, where as vegetable prices are mostly random variables and its cropping is also highly
cost effective. As a matter of fact vegetable cropping need capital investment in insecticides,
pesticides, fertilizers, frequent irrigation, more labours and high transportation cost. Another
factor responsible for its fluctuating prices is sudden availability in local market and
difficulty in its long tern storage. Here it is interesting to know that vegetable prices also vary
on day to day basis even in a season. Thus keeping in view of variable and uncertain prices
of vegetables, a proper land planning is needed for optimal return. For further details, one can
see the work of Tarrazzo and Gutierrez [20], Lee [9] related to the application of fuzzy sets
in economic expectations and financial planning. Thus, here we consider the quantitative
expectations as optimal profit considering profit coefficients as fuzzy random variables by
planning of resources: land and labour.
3. Literature Review
In management science several approaches have been developed to deal with multiobjective
decision making problem. Three of widely used methodologies for dealing with MOLP
problems are vector maximum method, goal programming and interactive techniques.
3
Bellmann and Zadeh [1] gave a basis for decision making in fuzzy environment based on
which a new approach to the problem definition for finding a compromise solution to MOLP
problem was initiated by Zimmermann [23]. The proposition of this regard was to explore a
compromise solution of MOLPP. The methodologies of obtaining compromise solution was
further developed in various directions by Buckley [3], Luhandjula [11], Sakawa and Yano
[13], Chanas [4] applying various type of membership functions.
In domain of agricultural production system, where uncertainty and vagueness play a major
role in decision making, several researchers such as Slowinski [18], Sinha et al. [17], Sher
and Amir [16], Sumpsi et.al. [19], Sarker et. al. [14], Pal and Moitra [12], Vasant [22],
Biswas and Pal [2] used fuzzy goal programming techniques for a farm planning problem.
Kruse and Meyer [8] attracted researchers to study agricultural crop planning with stochastic
values as stochastic linear programming problem to address such problems. Hulsurkar et. al.
[5] studied the fuzzy programming approach to multiobjective stochastic linear programming
problem. Lodwick et. al.[10] made a comparison of fuzzy, stochastic and deterministic
methods in a case of crop planning problem followed by a study of Itoh and Ishii [6] based on
possibility measure. Itoh et. al. [7] considered a problem of crop planning under uncertainty
assuming profit coefficients are discrete random variables and proposed a model to obtain
maximum and minimum value of gains for decision maker. Toyonaga et. al.[21] studied a
crop planning problem with fuzzy random profit coefficients. Sharma et. al.[15] studied FGP
for agricultural land allocation problem and proposed an annual agricultural plan for different
crops.
In subsequent section of this paper, we considered multiobjective linear programming
problem of crop planning with profit as discrete random variable. The problem has been
transformed into a deterministic model and fuzzy programming has been used to find a model
crop planning with an optimal profit to support decision maker. The work has been presented
in the following sections as: Section-4 describes a problem of agricultural production
planning, Section-5 deals with the basics of fuzzy programming together with of
development of computational algorithm and the numerical illustration of developed
algorithm is placed in Section-6. Section-7 consists of conclusion and the references are
placed in the last.
4. Problem description
4
We consider the problem [7] in which number of producible kinds of crops are n and
respective profit coefficients for these crops are per unit area with respective
probability . The decision variable and element denote cultivation area for crop j and
the work time in labour hours for growing crop j at the unit area respectively. As the land of a
farm is limited, has to be less than or equal to L (the gross of farm’s land)
and we call it “land constraint”. The total labour hours of working time is limited and thus
has to be less than or equal to a certain W and we call it “labour
constraint”. Under these constraints and discrete crisp and fuzzy random profit coefficients,
we want to find the decision variables so as to maximize the profit(R).
Maximize R
such that (Land constraint)
(Labour constraint)
(4.1)
..................
5. Fuzzy Programming
In fuzzy environment a decision is considered as a fuzzy objective function, characterized by
its membership function. The similar approach is also applied to the constraints. In case of
several objectives, a procedure of selection of activities comes into existence which
simultaneously satisfies all the objective functions and the constraints. This process can be
viewed as intersection of fuzzy constraints and fuzzy objective functions. Further, the
membership function of the solution set is used to maximize decision to a level of
satisfaction. The present work on optimization under uncertainty in agriculture production
planning has been studied in a situation when profit coefficients are crisp discrete random
variables using max-min approach of fuzzy programming developed by Zimmermann.
5.1 Max-Min approach
A multiobjective decision making (MODM) problem is defined as
Max/Min ( ) (5.1)
subject to ( )
5
where ( ) is the vector of profit/cost coefficients of the objective
function and , - is the vector of total resources
available. , - is parameter of decision variable and [ ]
is matrix
of coefficients. Zimmermann first used the max-min operator to solve MOLP problem and
considered the equation as
find x
such that ( ) (5.2)
where are corresponding goals, and all objective functions are to be maximized. Here
objective functions of equation (5.1) are considered as fuzzy constraints. If the tolerance of
fuzzy constraints are given, one can establish their membership function ( ) , and then
a feasible solution set is characterized by its membership function
( ) * ( ) ( )+. (5.3)
Furthermore, a decision maker makes a decision with a maximum value in the feasible
region. The solution can then be obtained by solving the problem of maximizing ( )
subject to .i.e.
, ( )-
such that (5.4)
Now, let ( ) be the overall satisfactory level of compromise.We obtain the
following equivalent model
such that ( ) (5.5)
Here, membership functions of objective functions is estimated by obtaining payoff table of
Positive Ideal Solution (PIS) and assume that membership functions are of type non
decreasing linear/hyperbolic etc.
5.2 Computational algorithm
Here we propose a computational algorithm for a scenario when profit coefficients are crisp
discrete random variables by using fuzzy multiobjective linear programming approach, as
follows
6
Step1 Solve the problem in different probabilistic cases one by one using linear programming
techniques taking one objective function with constraints at a time while ignoring the other
probabilistic cases.
Step 2 Using the solution obtained in step1, find the corresponding value of all the objective
functions for each of solutions.
Step 3 From step 2, obtain the lower and upper bounds and
for each objective
functions and construct a table of PIS.
Step 4 Consider a linear and nondecreasing membership function between and
as
( )
{
( )
[ ( ) ]
,
-
( )
( )
(5.6)
The above membership function is essentially based on the concept of preference or
satisfaction.
Step5 Transform multiobjective linear programming into LPP as
( ) (
)
(
) , (5.7)
where ( )
which can be equivalently written as
( ) (
)
i.e. ( ) (
)
i.e. (
) (5.8)
Thus the problem (4.1) reduces to LPP given as
Such that
7
(
)
(
) (5.9)
...
(
)
It can be easily solved by using two phase simplex method.
6. Numerical illustration
A farmer is to grow carrot, radish, cabbage and Chinese cabbage in a season in areas
be , and (unit 10 acres=1000 ) respectively. The farmer has a total land of 10
acres and a max labour work time available to him is 260 hours. The profit coefficients (unit
10,000 Japanese Yen) and work time for the crops are given in the Table-1 as considered in
[7].
Table 1
Random variable Carrot Radish Cabbage C. Cabbage Probability %
29.8 10.4 13.8 19.8 10
23.9 21.4 49.2 32.8 50
37.0 16.0 3.6 9.7 10
19.3 26.6 48.4 75.6 30
Work time 6.9 71 2 33
Here, we illustrate solution of the problem by our developed algorithm 5.2. The undertaken
problem is to solve
{
(6.1-6.4)
Subject to constraints
(6.5)
(6.6)
Now, we proceed as
such that
8
The optimal solution to this crisp LP problem is
( )
And the positive ideal solution of other objective functions at this solution are calculated as
( ) =239
and similarly ( ) ( )
Similar process is to be repeated for taking other objective functions , , . Thus the
Positive Ideal Solution (PIS) obtained are placed in Table 2
Table 2
Max 370
Max 484
Max 298 239 193
Max 184.44 365.06 83.21
( ) ( ) denote the maximum and minimum values for the respective columns.
Then reformulating the problem, using (5.7), it reduces to a LPP as
Maximize
such that
(6.7)
Solving the above LPP problem using MATLAB®, we get following solution
Finally, the values of profit in different probabilistic scenarios are calculated and are placed
in Table 3.
: profit coefficients having 10% probability of occurrence.
9
: profit coefficients having 50% probability of occurrence.
: profit coefficients having 10% probability of occurrence.
: profit coefficients having 30% probability of occurrence.
Table 3
Probabilistic cases Profit by proposed
method
Profit by Itoh method
[7]with d=30
207.37 268.4
348.58 280.9
181.07 303.5
410.54 274.8
Weighted profit 336.29 280.08
7. Conclusion
In real world problem of decision making like agricultural production system certain goals
are to be considered for compromise solution, not necessarily a maximal solution. The
present study aimed to tackle some of the above mentioned factors for modelling the
agricultural production system. Here, we developed an algorithm for solving such stochastic
programming problem using the MOLP approach in a fuzzy environment and have
implemented it on the problem undertaken by Itoh [7]. The results obtained by our algorithms
and that of Itoh are placed in Table -3, which clearly show superiority of our method.
Table 4
d R Weighted
profit
10 8.02
0.00
1.14 0.84
280.35
271.35 275.35 308.99 273.35 277.71
30 7.88
0.00
1.38 0.73
295.53
268.53 280.53 303.77 274.53 280.08
60 7.68 0.00
1.75
0.58
318.29 264.30 288.30 295.94 276.30 283.06
100 7.40 0.00
2.23
0.37 348.65
258.65 298.65 285.49 278.65 287.33
Further, the proposed method to solve the stochastic programming problem is also free from
any arbitrary parameter d as taken by Itoh. Clearly considering the different values to the
parameter d as 10, 30, 60, 100 different solutions by Itoh method are obtained and are
illustrated in Table 4. The presented method has an advantage of providing unique optimal
solution with better weighted profit than Itoh method and thus may be considered a superior
10
crop model for optimal profit. The proposed model of land planning for vegetable cropping
show best expected returns in high probabilistic cases of 50% and 30% of occurrence, more
over our weighted profit is much higher than that of Itoh for every value of decision maker
parameter d.
The scope of present study of application of fuzzy sets in land use planning for high
economic expectations is to incorporate uncertainty and imprecision. The reason behind it is
that the markets are made up of people and thus influenced by their feelings and events.
Further, as the financial problems involve multiple objectives, depending on complex
financial relations, that too of conflicting in nature, the financial world in moving towards
more and more mathematical models. More over many corporate organizations are entering
in agribusiness management, which is developing as a potential sector. In view of developing
a supply chain of network, these organizations are financing the growers for smooth
functioning of their supply chain, thus a proper land utilization and proper cropping pattern is
needed at their farmers level.
One of the interesting features of the present study is that like financial planning’s, where
expectations for returns are considered for present quarter, next quarter etc. , in vegetable
cropping in a season be considered in several quarters/phases. The farmer must grow the
vegetable crops in a way that it should be harvested and be marketed in whole season to find
at least best weighted return in view of fluctuating prices as a guaranteed profit. Thus the
developed fuzzy set based quantitative methodology is capable to incorporate uncertainty for
planning model.
Acknowledgements
The authors are highly thankful to the University Grants Commission,Govt. of India, New
Delhi, INDIA, for the provided financial assistance.
References
1. R.E. Bellman, L. A. Zadeh, Decision making in a fuzzy environment, Management
Science 17(1970), B141-164.
2. A. Biswas, B. B. Pal, Application of fuzzy goal programming technique to land use
planning in agriculture system, Omega33(2005) 391-398.
3. J.J. Buckley, Fuzzy Programming and the Pareto Optimal Set, Fuzzy Sets and Systems
10(1983) 57-63.
4. S. Chanas, Fuzzy Programming in multiobjective linear programming-A parametric
approach, Fuzzy Sets and Systems 29(1989) 303-313.
5. S. Hulsurkar, M.P. Biswal, S.B. Sinha, Fuzzy programming approach to
multiobjective stochastic linear programming problems, Fuzzy Sets and Systems
88(1997) 173-181.
11
6. T. Itoh, H. Ishii, Fuzzy crop planning problem based on possibility measure, IEEE
international fuzzy system conference, 2001.
7. T. Itoh, H. Ishii, T. Nanseki, A model of crop planning under uncertainty in
agriculture management, Int. J. Production Economics 81-82(2003) 555-558.
8. R. Kruse, K.D. Meyer, Statistics with vague data, D. Riedal Publishing Company,
1987.
9. C.F. Lee, Financial Analysis and Pananning: Theory and Applications, Addison-
Wesley Reading MA. (1985)
10. W. Lodwick, D. Jamison and S. Russell, A comparison of fuzzy stochastic and
deterministic methods in Linear Programming, Proceeding of IEEE, 321-325, 2000.
11. M.K. Luhandjula, Compensatory operators in fuzzy linear programming with multiple
objectives, Fuzzy Sets and Systems 8(1982) 245-252.
12. B. B. Pal, B.N. Moitra, fuzzy goal programming approach to long term land allocation
planning problem in agricultural systems: A case study, Proceedings of the fifth
International Conference on Advances in Pattern recognition, Allied Publishers Pvt.
Ltd., 441-447, 2003.
13. M. Sakawa and H. Yano, Interactive fuzzy decision making for multiobjective
nonlinear programming using augmented minimax problems, Fuzzy Sets and Systems
20(1986) 31-43.
14. R. A. Sarker, S. Talukdar and A.F. M. Anwarul Haque, Determination of optimum
crop mix for crop cultivation in Bangladesh, Applied Mathematical Modelling
21(1997) 621-632.
15. Dinesh K. Sharma, R.K. Jana, A. Gaur, Fuzzy Goal programming for agricultural land
allocation problems, Yugoslav Journal of Operation Research, 17(2007) N0.1 31-42.
16. A. Sher and I. Amir, Optimization with fuzzy constraints in Agriculture production
planning, Agricultural Systems 45(1994) 421-441.
17. S.B. Sinha, K.A. Rao, and B.K. Mangaraj, Fuzzy goal programming in multicriteria
decision systems –A Case study in agriculture planning, Socio-Economic Planning
Sciences, 22(2)(1988) 93-101.
18. R. Slowinski, A multicriteria fuzzy linear programming method for water supply
system development planning, Fuzzy Sets and Systems 19(1986) 217-237.
19. J.M. Sumpsi, Francisco Amador, Carlos Romero, On Farmer’s Objectives: A multi
criteria approach, European Journal of Operation Research 96(1996) 64-71.
12
20. M. Tarrazo, L. Gutierrez, Economic expectations, fuzzy sets and financial planning,
European Journal of Operational Research. 126(2000)89-105.
21. T. Toyonaga, T. Itoh, H. Ishii, A crop planning problem with fuzzy random profit
coefficients, Fuzzy Optimization and decision making, 4(2005) 51-59.
22. P. M. Vasant, Application of fuzzy linear programming in production planning, Fuzzy
Optimization and Decision Making, 3(2003) 229-241.
23. H.J. Zimmermann, Fuzzy programming and linear programming with several
objective functions, Fuzzy Sets and Systems 1(1978) 45-55.