The membership function for each fuzzy constraint can be defined as follows
In this study FGP models are developed using different fuzzy solution methods to provide the DM for a more confident solution set for policy decision making. In this regard, three different solution approaches are employed, namely “min operator” [12], “fuzzy and operator” [13] and “weighted additive approach” [14]. These approaches are introduced briefly in the following.
2.3.1. Bellman and Zadeh's min operator Zimmermann [15] proposed Bellman and Zadeh's [12] “min
operator” approach to the multiobjective LP model. This operator is usually applied due to its easy computation and several nice properties. However, the solution generated by min operator is not compensatory and does not guarantee pareto-optimal [16].
According to min operator approach the decision problem is choosing x
By introducing the auxiliary variable l, which is the overall satisfactory level of compromise, the following conventional LP problem can be obtained:
Maximize l subject to l m
2.3.2. Werners' “fuzzy and” operator As Bellman and Zadeh's “min operator” focuses only on the
maximization of the minimum membership grade, it is not a compensatory operator, which means that goals with a high degree of membership are not traded off against goals with a low degree of membership. As the most of the decision problems in real world, energy planning problems are neither noncompensatory nor fully compensatory. Because of the noncompensatory nature of its aggregation operator, “min operator” is not very suitable for modeling the real world problems. For some computationally efficient compensatory operators, Lai and Hwang [17] can be referred. Werners' “fuzzy and” operator is one of the most powerful compensatory operators that can be used in setting the objective function in fuzzy programming to investigate better results.
Werners [13] formulates the “fuzzy and” operator as follows:
ðxÞ is the membership function of goal k, and g is the coefficient of compensation defined within the interval [0,1]. By adopting “min operator” into Eq. (33), the following LP problem can be formed:
In this study, we use a modified version of “fuzzy and” operator which was developed by Selim [18]. In this version the following equation is used as the objective function to reflect the relative importance of l
to the objective function. In this manner, relative weights of the objectives are considered and integrated in the objective function.
2.3.3. Tiwari et al.'s weighed additive approach We use Tiwari et al.'s [14] weighted additive approach to reflect
the relative importance of fuzzy goals. Formulation of this approach can be stated as in the following:
3.1. Case study region
To explore the viability of the proposed model, computational experiments are performed on a real world problemin Turkey. In this regard, we aimto design a biomass to energy supply chain network in
_ Izmir, which is the third largest city in Turkey with a population of
around 4 million. As agriculture and stockbreeding are among the most common economic activities in
_ Izmir, there exists a rich waste
biomass resource which can be used as feedstock in bioenergy facilities. More specifically,
_ Izmir is the second largest producer in Turkish
poultry sector and has the fourth largest biogas production potential from animal wastes in Turkey. In spite of this high biomass potential, biomass utilization in energy production in
_ Izmir is scarcely any.
The proposed model is applied to all 20 counties of _ Izmir. All of
the counties of _ Izmir are considered as potential sites for supply of
biomass, bioenergy plants and biomass storages in the model. The map of the case study region is depicted in Fig. 2. Each county is represented by a number in the model as depicted in Fig. 2. The starred counties on the map constitute the city centre, therefore considered as one county in the model.
The length of the time period used in our computational experiments is one month and the planning horizon is three years.
3.2. Biomass sources
The proposed model includes two types of feedstock to be transformed into energy: waste biomass in the form of animal manure and energy crops. In this regard, three types of waste biomass, namely cattle manure, laying chicken manure and broiler chicken manure, and an energy crop, corn silage, are considered as feedstock to be used by bioenergy plants. Available biomass amount is assumed to increase gradually with a prespecified rate. Data about the biomass potential from husbandry and agriculture is gath