Modeling and Optimization of Chemical Fertilizers Supply Chain using Hybrid Whale Optimization and Simulated Annealing
محورهای موضوعی : Supply Chain Management
Motahareh Rabbani
1
,
Seyyed Mahammad Hadji Molana
2
,
Seyed Mojtaba Sajadi
3
,
Mohammad Hossein Davoodi
4
1 - Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
2 - Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran
3 - School of Strategy and Leadership, Faculty of Business and Law, Coventry University, Coventry, UK
4 - Soil and Water Research Institute, Karaj, Iran
کلید واژه: Simulated Annealing, phosphorus, chemical fertilizers, Whale Optimization Algorithm, Sustainable supply chain management,
چکیده مقاله :
Phosphorus is a basic constituent of chemical fertilizers and plays a pivotal role in crop yield enhancement in agriculture systems. Considering the growing demands for phosphorus and the limited resources of this vital substance, sustainable supply chain management (SCM) of chemical fertilizers is of great importance. In the present study, a mathematical model for sustainable chemical fertilizer SCM is presented. Taking into account the adverse environmental effects of the production and consumption of chemical fertilizers, the present study attempts to design a sustainable SCM concerning economic, environmental, and social factors. To solve the problem, a hybrid metaheuristic algorithm incorporating whale optimization and simulated annealing is used considering a multi-objective function. The simulation results obtained from a real case study of the chemical fertilizers supply chain network in Iran proved the effectiveness and applicability of the proposed model and solution method. Obtained results show the effectiveness of the proposed method compared with other algorithms with respect to economic, social, and environmental factors.
Phosphorus is a basic constituent of chemical fertilizers and plays a pivotal role in crop yield enhancement in agriculture systems. Considering the growing demands for phosphorus and the limited resources of this vital substance, sustainable supply chain management (SCM) of chemical fertilizers is of great importance. In the present study, a mathematical model for sustainable chemical fertilizer SCM is presented. Taking into account the adverse environmental effects of the production and consumption of chemical fertilizers, the present study attempts to design a sustainable SCM concerning economic, environmental, and social factors. To solve the problem, a hybrid metaheuristic algorithm incorporating whale optimization and simulated annealing is used considering a multi-objective function. The simulation results obtained from a real case study of the chemical fertilizers supply chain network in Iran proved the effectiveness and applicability of the proposed model and solution method. Obtained results show the effectiveness of the proposed method compared with other algorithms with respect to economic, social, and environmental factors.
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Modeling and Optimization of Chemical Fertilizers Supply Chain using Hybrid Whale Optimization and Simulated Annealing
A B S T R A C T
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Phosphorus is a basic constituent of chemical fertilizers and plays a pivotal role in crop yield enhancement in agriculture systems. Considering the growing demands for phosphorus and the limited resources of this vital substance, sustainable supply chain management (SCM) of chemical fertilizers is of great importance. In the present study, a mathematical model for sustainable chemical fertilizer SCM is presented. Taking into account the adverse environmental effects of the production and consumption of chemical fertilizers, the present study attempts to design a sustainable SCM concerning economic, environmental, and social factors. To solve the problem, a hybrid metaheuristic algorithm incorporating whale optimization and simulated annealing is used considering a multi-objective function. The simulation results obtained from a real case study of the chemical fertilizers supply chain network in Iran proved the effectiveness and applicability of the proposed model and solution method. Obtained results show the effectiveness of the proposed method compared with other algorithms with respect to economic, social, and environmental factors.
__________________________________________________________________________________________
Keywords:
Chemical fertilizers
Phosphorus
Sustainable supply chain management
Whale optimization algorithm
Simulated annealing
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1. Introduction
Phosphorus-based chemical fertilizers play a significant role in increasing productivity and product quality in the agriculture industry. Phosphorus (P) is regarded as the most widely used element in the chemical fertilizer industry. Phosphate rock (PR), as a limited and non-renewable material, is considered the main source of phosphorus (Scholz et al., 2013). Modern agriculture consumes a large quantity of phosphate in order to make crops grow, and as a result, with growing of the population in the world and the increased demand for agricultural products, phosphorus consumption is increasing (Roos et al., 2013). Given this increase in demand, some underground phosphorus mines are close to being run out. Also, the world's known PRs are located in limited countries, which has become a matter of concern (Gong et al., 2022a). In this regard, some countries have to import PR or ready-made chemical fertilizers from other countries.
Current strategies for the management of P production and consumption suffer from the depletion of PRs, low PUE (P-use efficiency), and high P-induced environmental factors (Luo et al., 2017). One of the important concerns in this regard is to provide a strategy in order to conserve the environment (Simons et al., 2014). Considering the limited PR mines and environmental issues of the chemical fertilizers, especially on the ecosystem, as well as the adverse effects of excessive use of P, sustainable management of P-use should give particular attention to understanding the prerequisites for adequate P supply. In this regard, societies can prevent water pollution and ensure long-term food security as well as the potential negative effects of P crops on terrestrial biodiversity (Garske & Ekardt, 2021). Technically, in order to ensure the system’s survival, sustainability and flexibility should be taken into account simultaneously by exploiting synergies of these two issues. Furthermore, the supply chain must be flexible so that it can maintain its sustainable performance, especially considering the environmental dimension (Mehrjerdi & Shafiee, 2021). Despite the great importance of P supply chain network optimization, it has been a long time since this issue has not received enough attention from researchers, especially by mathematical modeling from the operations research point of view.
In recent years, researchers have paid special attention to phosphorus as a rare and critical resource. Generally, P-based chemical fertilizers including medium-low PR processing fertilizers such as single super P (SSP), triple super P (TSP) and calcium magnesium P (CMP), and high PR processing such as mono-ammonium P (MAP), di-ammonium P (DAP) and nitrogen P potassium (NPK) (Gong et al., 2022a). Most of the wastes caused by the application of fertilizers on the soil to the branch point are observed in the wastewater of agricultural lands (Scholz & Welmer, 2015). Research on phosphorus production and its use show that 80% of wastes originate from mine to the branch point, while 10% of the processed phosphorus fertilizer is consumed by humans (Cordell et al., 2009). According to UNWA (United Nations Wastewater Assessment) reports, insufficient wastewater imposes adverse effects on human health, the environment, and the economy (Zhongming et al., 2021). Phosphorus wastewater and untreated phosphorus sewage are considered the main causes of the development or expansion of "dead zones" (Nedelciu et al., 2020). This phenomenon imposes tangible adverse effects on the environment and the livelihood of people who live in these areas. P-rich wastewater is also considered as a threat to marine biodiversity (Martinez-Escobar & Mallela, 2019). The PR extraction and refinement for the production of phosphorus fertilizers can be regarded as the main cause of almost all impacts of the phosphorus supply chain on climate and air quality (Oberle et al., 2019).
1.1. Phosphorus fertilizer supply chain management
Modeling and optimization of the phosphorus supply chain network play an effective role in management’s decision-making on the reliable P stock flow among the different levels of the P supply chain, from the suppliers to the consumers. In particular, studies in the literature that consider important concerns of the P SCMs, e.g., flexibility and sustainability challenges, are worthy of attention. In addition, regional discordances of fertilizer requirements and environmental issues on P-SCM have not so far received enough attention. The literature in the field of phosphorus supply chain typically uses quantitative models to evaluate phosphorus deficiency. In these researches, it is assumed that the recycling of the phosphorus-containing waste in their model involved the reabsorption of phosphorus materials in food industry wastes. To address concerns about phosphorus depletion, Van Vuuren et al. (2010) developed a production and exchange model, concluding that phosphorus will not be depleted in the short or medium term. Mohr & Evans (2013) presented a model of demand-production interaction, which considers low, high, and best estimation modes of maximum phosphate production.
Some studies focus on the analysis and improvement of the yield of phosphorus products and the management of environmental activities of the phosphorus supply chain (Gong et al., 2022a; Rabbani et al., 2022; Shokouhifar et al., 2023). Researchers have applied a series of integrated and single-factor methods in order to improve the efficiency of phosphorus fertilizer use in agricultural systems (Bai et al., 2013; Withers et al., 2014; Gong et al., 2022b). They have a yield response to phosphorus fertilizer to evaluate the optimized usage of phosphorus fertilizers for soil with phosphorus deficiencies (Bai et al., 2013). Also, Li et al. (2011) proposed a maintenance and compensation technique for management of phosphorus with P abundance. In another research, a practical approach has been examined to reduce environmental effects of phosphorus fertilizer (Gong et al., 2022b). In addition, the replacement strategy indicates that feeding the roots of crops instead of the soil can result in high efficiency of phosphorus fertilizer application (Withers et al., 2014).
In terms of sustainable approaches in agriculture, phosphorus supply chain management faces problems related to PR extraction, phosphorus fertilizer production, and crop production. The impact of different phosphorus fertilizers on PUE should be quantitatively calculated and analyzed from the supply chain perspective (Nedelciu et al., 2020). The nature and magnitude of the flow of phosphorus materials in the supply chain should be determined as well (Chowdhury et al., 2016). Recent studies have used these two approaches in order to identify general trends in P flow.
1.2. Our contributions to the literature
Despite extensive research in the field of chemical fertilizer production and consumption management, there are very limited studies with a focus on the issues related to the P-based chemical fertilizer sustainable SCM. To fill these gaps, we introduce a multi-objective model considering the economic, social, and environmental issues of the SCM. In addition to the economic costs (e.g., purchase of raw materials, production, maintenance, and transportation), the presented model evaluates the improvement of PUE (as a social objective) as well as the reduction of adverse environmental effects from the supply chain perspective. Finally, we propose a hybrid metaheuristic algorithm based on whale optimization and the simulation annealing to solve the model. This model allows the decision-maker to make a compromise between conflicting objectives. The most important specific innovations presented by this study versus existing research are summarized as follows:
· Modeling a sustainable supply chain for phosphorus-based chemical fertilizers considering the concerns of raw material supply, production, and chemical fertilizer distribution.
· Providing a multi-objective optimization model with mathematical modeling taking into account the economic, social, and environmental considerations.
· Presenting an ensemble metaheuristic utilizing whale optimization and simulation annealing (WOASA), to simultaneously take advantage of the local and global search capabilities of these two algorithms and as a result, increase the speed of convergence and achieve a better solution.
· Using a real case study on the chemical fertilizer supply chain conducted in Iran, which studies 3 PR mines, 9 ammonia supply factories, 7 phosphoric acid supply factories, 36 sulfuric acid supply factories, 40 chemical fertilizer production factories, and 32 distribution centers (provincial centers of Iran).
In the remainder of this study, the problem modeling and the proposed solution method are respectively introduced in Sections 2 and 3. The case study and simulation results are reported in Section 4. Finally, concluding remarks are discussed in Section 5.
2. Problem Modeling
In this section, we introduce a multi-objective mathematical model for sustainable chemical fertilizer supply chain management taking the economic, social, and environmental issues into account. In this regard, we consider not only the economic costs such as the purchase of raw materials, production, maintenance, and transportation costs, but also the PUE as a social issue and the reduction of adverse environmental effects of the supply chain. In the following, the problem statement is provided in Section 2.1, and then, the mathematical model is presented in Section 2.2.
2.1. Supply chain model
In this study, we consider a three-stage fertilizer SCM consisting of primary raw material suppliers, fertilizer producers, and distribution centers (provincial centers). The supply chain network model can be seen in Fig. 1. The proposed model is investigated in T time periods (months). Each distributor d may face the demand Ddpt for fertilizer type p in each month t. The list of notations in this paper is provided in Table 1. The following conditions are assumed in order to determine the obstacles of the model:
Ø Raw materials include phosphorus (P), ammonia (A), phosphoric acid (PA), and sulfuric acid (SA).
Ø Chemical fertilizers include TSP, SSP, and DAP.
Ø SSP consists of SA and P.
Ø TSP consists of PA and S.
Ø DAP consists of PA and A.
Ø Suppliers have limited capacity.
Ø The number of raw materials for a producer can be provided from different suppliers.
Ø Each producer may save some of the fertilizers at end of every month.
Ø Each distributor is able to buy from different producers until their need for different fertilizers is met.
Ø Each distributor is able to store some of the fertilizers that are surplus to consumption at end of each month.
Ø The delivery lead time between each supplier and producer as well as between each producer and distributor is ignored (zero). In other words, raw materials/fertilizers ordered at the beginning of month t are delivered to the producer/distributor in the same month t.
Ø Each producer may have one or more producing lines for the production of TSP, SSP, or DAP.
Ø Producers have limited capacities for the production of each fertilizer.
Ø Quantity of raw materials supplied by each supplier as well as the quantity demanded by each distributor for each period of time is assumed to be definite and scheduled.
Figure 1. Three-level model of chemical fertilizer supply chain.
Table 1. Notations.
Indices and sets: i∈I suppliers j∈J producers d∈D distributors r∈R raw materials (A, P, SA, PA) p∈P chemical fertilizers (TSP, SSP, DAP) t∈T months (time periods)
Parameters: Air equal to 1 if material r is provided by supplier i and equal to 0 otherwise. Urp equal to 1 if material r is used for production of fertilizer p and equal to 0 otherwise. CapSirt quantity of material r supplied by supplier i in month t (tons) CapPjp capacity of fertilizer p in production center j in each month for each month (tons) WSjr storage capacity of producer j in each month for material r (tons) WPjp storage capacity of producer j in each month for fertilizer p (tons) WDdp capacity of distribution center d in each month for fertilizer p (tons) αrp quantity of material r needed for production of fertilizer p (%) Ddpt demand of distribution center d for fertilizer p in month t (tons) TSr truck size for transporting material r (tons) TPp truck size for transporting fertilizer p (tons) dSij distance between supplier i and production center j (kilometers) dPjd distance between production center j and distribution center d (kilometers) FTCSijr fixed transporting cost of material r between supplier i and producer j ($ per truck) FTCPjdp fixed transporting cost of fertilizer p between country k and distribution center d ($ per truck) VTCSijr variable transporting cost of material r between supplier i and production center j ($ per truck per kilometer) VTCPjdp variable transporting cost of fertilizer p between producer j to distribution center d ($ per truck per kilometer) BCSir purchase cost of material r from supplier i ($ per ton) PCjp producing cost of fertilizer p by producer j ($ per ton) ICSjr cost of holding inventory of material r in producer j for each month ($ per ton) ICPjp cost of holding inventory of fertilizer p in producer j for each month ($ per ton) ICDdp cost of holding inventory of fertilizer p in distribution center d for each month ($ per ton) PYp increase in the average yield of crops with fertilizer p (%) G fuel consumption of vehicles (liters per kilometer) et quantity of greenhouse gas emissions (tons) resulting from fuel per liter of gasoline ep greenhouse gas emissions (tons) resulting from production of type p products er greenhouse gas emissions (tons) resulting from production of type r material
Direct decision variables: Ojp equal to 1 if producer j generates fertilizer p, otherwise equal to 0 Pjpt quantity of fertilizer production in producer j in month t (tons) XSijrt equal to 1 if supplier i supplies material r to producer j in month t and equal to 0 otherwise. XPjdpt equal to 1 if producer j distributes fertilizer p to distribution center d in month t and equals 0 otherwise. YSijrt quantity of material r transported from supplier i to production center j in month t (tons) YPjdpt quantity of fertilizer p delivered from production center j to distributor d in month t (tons)
Indirect decision variables: SDdpt delivered demands of distribution center d for fertilizer p in month t (tons) SMjrt delivered demands of producer j for material r in month t (tons) USjrt equal to 0 if the demand of producer j for material r is completely delivered in month t, and 1 otherwise. UDdpt equal to 0 if the demand of distribution center d for fertilizer p is completely delivered at month t and 1 otherwise. ISjrt quantity of inventory of material r kept in producer j in month t (tons) IPjpt quantity of inventory of fertilizer p kept in producer j in month t (tons) IDdpt quantity of inventory of fertilizer p kept in distribution center d in month t (tons) |
2.2. Objective function
2.2.1. Economic function
Overall economic costs of the model contain the cost of raw material purchased from suppliers (CB), producers' production cost (CP), raw material and fertilizer inventory holding costs at producers and distributors (CI), and transportation costs (CT). These costs are respectively presented by Eqs. (1) - (4).
| (1) | |||||
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| (12) | |||||
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| (14) | |||||
| (15) | |||||
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| (17) | |||||
| (18) | |||||
| (19) | |||||
| (20) | |||||
| (21) | |||||
| (22) | |||||
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| (26) | |||||
| (27) | |||||
| (28) | |||||
| (29) | |||||
| (30) | |||||
| (31) | |||||
| (32) | |||||
Algorithm 1. Proposed WOASA algorithm. |
WOA: |
1. Generation of a random population of whales: Sw (w=1,2,…,PopSize) |
2. Calculation of OBJ for each solution using Eq. (8) |
3. i = 0 |
4. while (i ≤ MaxIterWOA) |
5. for each whale w |
6. Update a, p, and l |
7. if (p > 0.5) |
8. Update solution w by bubble-net attacking |
9. else |
10. if |A| > 1, update solution w by the search for prey, otherwise, using encircling prey |
11. end 12. Amend whale w if goes beyond the search space 13. end |
14. Calculation of OBJ for each solution using Eq. (8) |
15. Updating GBS (global best solution) |
16. i = i + 1 |
17. end |
SA: |
1. Considering GBS as initial solution of SA: S* |
2. j = 0 |
3. while (j ≤ MaxIterSA) |
4. Generation of a neighbor S*new |
5. Replace S* by S*new, if OBJ*new<OBJ* or rand<Pacp |
6. Updating GBS |
7. Update T |
8. j = j + 1 |
9. end |
Return GBS as the optimized supply chain model |
3.3.1. Representation of a solution
As seen in Fig. 2, a possible solution to the problem is encoded as a hybrid structure consisting of 3 binary and 3 integer matrices. The decision variables comprise S.O to specify the lines of fertilizer productions in producers, S.P to determine the quantity of producing fertilizers by different production centers for each month, two binary matrices to determine the connections in the supply chain, and two integer matrices to specify the quantity of material/fertilizer transportation between different levels of the supply chain.
Figure 2. Probabilistic solution coding.
3.3.2. Global search using WOA
At the beginning of the WOA algorithm, a population of whales is randomly generated. In each WOA iteration, the current population is updated using the search for prey, encircling prey, and bubble-net attacking operators. In order to update a whale, a uniform random parameter p in the interval of [0,1] is generated. If p≥1, the solution would be updated through the bubble-net attack operator, and when p<1, a vector A is randomly constructed. Then, if |A|<1, encircling prey is run, and otherwise, the solution would be updated by using search for prey.
3.3.3. Local search using SA
As stated above, in the proposed hybrid WOASA algorithm, the final global solution of the WOA is provided to the SA as the initial solution. In each iteration, a solution is constructed in the vicinity of the old solution. Then, SA may consider the new solution or not by checking the acceptance rule. At the beginning of SA, a large value is assumed for the temperature T so that worse solutions can be accepted with a higher probability. As the temperature gradually decreases during the algorithm running, the probability of accepting worse solutions decreases. In this study, the temperature value decreases linearly from Tinitial to Tfinal during the algorithm running. In order to develop a neighbor solution in the vicinity of the old one, the first one of the 6 structures in the solution is randomly selected. Then, depending on the selected structure, a local binary or integer operator is performed. Examples of binary and integer local search operators are shown in Figs. 3 and 4, respectively.
Figure 3. Binary local search operator.
Figure 4. Integer local search operator.
4. Simulation
The data required for the case study of this research was collected from the statistical reports of the Soil and Water Research Institute (SWRI) of Iran and the experts’ opinions. SSP, TSP, and DAP are the three types of phosphorus-based chemical fertilizers having the most application in Iran's agriculture. The raw materials for the production of these fertilizer types include A, P, SA, and PA. The details of the case study dataset are provided in Appendix 1.
4.1. Settings
The presented model was coded in the MATLAB R2020b environment and was solved with the help of the WOASA algorithm. All the simulations were performed on a PC with an i7, 2.56 GHz processor, 16 GB of memory, and Windows 10. Different values were evaluated in order to set each controllable parameter in the proposed algorithm, and finally, the best value in terms of convergence speed and accuracy was considered for the final simulations. Table 2 lists the controllable parameters of the proposed algorithm.
Table 2. Parameters of WOASA.
Parameter | value |
Iterations of WOA | 100 |
Population of WOA | 50 |
b in WOA | 1 |
Iterations of SA | 5000 |
Tinitial in SA | 2 |
Tfinal in SA | 0 |
Economic weight (wEC) | 0.5 |
Social weight (wSC) | 0.25 |
Environmental weight (wEN) | 0.25 |
4.2. Results
4.2.1. Comparison with exact search
In this section, we compare the WOASA algorithm with the results derived using the exact search for different test examples presented in Table 3 with different problem sizes. Table 4 presents the comparison of the results derived using the exact method and the WOASA algorithm (in 10 consecutive runs). According to this table, the average deviation of the solution of WOASA from the optimal solution ranges between 0 and 0.343% from small to medium problem sizes. Although the running time of the exact search for test problems 1 and 2 is lower than that of the WOASA algorithm, the running time of the exact search exponentially increases as the dimension of the problem increases. As can be seen, the exact method cannot find the optimal solution for large problems in an acceptable running time. However, the computational time to reach the near-optimal solution increases almost linearly as the problem size in the proposed method increases.
Table 3. Dimensions of the problems.
Problem | No. materials | No. products | No. suppliers | No. producers | No. distributors | No. months |
1 | 2 | 1 | 1 | 1 | 1 | 3 |
2 | 2 | 1 | 2 | 2 | 2 | 3 |
3 | 2 | 2 | 2 | 3 | 2 | 6 |
4 | 2 | 2 | 3 | 5 | 3 | 6 |
5 | 3 | 2 | 5 | 10 | 3 | 12 |
Case study | 4 | 3 | 45 | 40 | 32 | 12 |
Table 4. Comparison of WOASA with exact search method for different problems.
Data set | Exact Method (optimal solution) | WOASA (near-optimal solution) | |||
OF | Time (s) CPU | OF | Time (s) CPU | Error (%) | |
1 | 0.6125 | 0.9 | 0.6125 | 7.3 | 0.00 |
2 | 0.582 | 4.7 | 0.582 | 12.1 | 0.00 |
3 | 0.5451 | 198 | 0.5456 | 38.4 | 0.092 |
4 | 0.6123 | 27350 | 0.6144 | 70.2 | 0.343 |
5 | N/A | N/A | 0.5246 | 182 | N/A |
Case study | N/A | N/A | 0.5935 | 712 | N/A |
4.2.2. Model analysis
Table 5 presents the results of WOASA to justify its performance for the case study dataset. Given the stochastic nature of metaheuristic algorithms, the results of running WOASA for 10 consecutive runs are reported in this table along with the mean and standard deviation of the results for all the runs. The small value of the standard deviation of the results in 10 consecutive runs indicates the suitable reliability of the proposed algorithm.
Figures 5 and 6 depict the convergence diagrams of WOASA in the running phases of the WOA and SA in 10 consecutive runs, respectively. As shown in Figure 5, the WOA has a very high convergence speed at the beginning, so the value of OBJ decreases from 0.7029 in the initial iteration to about 0.647 in the 40th iteration. However, after about 40 iterations of the WOA running, the convergence speed gradually decreases and no significant improvement is observed in the objective function value. Ultimately, the WOA phase ends with an objective function value of 0.6452 in the last iteration. In addition, as shown in Figure 6, the local search operators in the SA algorithm once again make a significant improvement in the value of the objective function, resulting in the decrease of the value of the objective function to 0.5935.
Table 5. Optimal value of objectives for case study dataset, obtained by WOASA.
Run # | ZEC (106 $) | ZSC | ZEN (106) | PF | OF |
1 | 573.6 | 0.39 | 8.91 | 0 | 0.5932 |
2 | 571.3 | 0.412 | 8.66 | 0 | 0.5999 |
3 | 565.7 | 0.395 | 8.41 | 0 | 0.5855 |
4 | 571 | 0.423 | 9.1 | 0 | 0.6108 |
5 | 570.5 | 0.388 | 9.18 | 0 | 0.594 |
6 | 573.8 | 0.396 | 9.03 | 0 | 0.5978 |
7 | 567.1 | 0.385 | 8.47 | 0 | 0.5819 |
8 | 566.4 | 0.387 | 8.49 | 0 | 0.5828 |
9 | 569.7 | 0.391 | 8.7 | 0 | 0.5891 |
10 | 572 | 0.413 | 8.58 | 0 | 0.5997 |
Average | 570.1 | 0.398 | 8.75 | 0 | 0.5935 |
STD % | 0.5 | 3.3 | 3.2 | 0 | 1.52 |
Figure 5. Convergence diagram of the WOA phase.
Figure 6. Convergence diagram of the SA phase.
4.2.3. Sensitivity Analysis
The resources that are available in different situations can lead the decision-maker to make different decisions. As a result, sustainability goals (economic, environmental, and social issues) may have different effects (weights). Table 6 reports the results of WOASA according to the variations of these weights, which are used to investigate the effect of different weights on the objective function. The first row of this table shows the default value of these weights. The first three lines are related to the algorithm running with three objective functions. The second three lines correspond to the algorithm running with two objective functions. Finally, the last three lines correspond to the single-objective running. According to this table, there are correlations between different goals. The social objective function tries to use more DAP fertilizers in order to increase the PUE, which, on the other hand, increases the economic costs. Moreover, the environmental objective seeks to reduce transportation and production quantities to decrease environmental pollution, which in turn demolishes the social objective.
Table 6. Sensitivity analysis of the proposed method by changing the objective function weights.
𝑤EC, 𝑤SC, 𝑤EN | ZEC (106 $) | ZSC | ZEN (106) |
0.5,0.25,0.25 | 570.1 | 0.398 | 8.75 |
0.25,0.5,0.25 | 623.8 | 0.475 | 9.23 |
0.25,0.25,0.5 | 601.7 | 0.422 | 8.11 |
0,0.5,0.5 | 745.5 | 0.483 | 8.2 |
0.5,0,0.5 | 561.3 | 0.314 | 7.97 |
0.5,0.5,0 | 586 | 0.455 | 9.53 |
1,0,0 | 512.6 | 0.321 | 9.44 |
0,1,0 | 769.2 | 0.51 | 9.83 |
0,0,1 | 611.9 | 0.355 | 7.81 |
5. Conclusion
In the present study, a mathematical model for the management of a chemical fertilizer sustainable supply chain network is presented and a hybrid metaheuristic algorithm encompassing whale optimization and simulated annealing is introduced to maximize model-solving accuracy. Finally, the designed model and the ensemble metaheuristic algorithm have been evaluated using a real case study dataset. The simulation results are indicative of higher accuracy of the proposed solution method compared to population-based and solution-based algorithms. The presented model helps managers to consider social and environmental issues in addition to economic costs, and try to enhance performance as one of their permanent responsibilities. Since sustainable phosphorus supply chain management has not received adequate attention in the literature, further studies can focus on the optimization of the supply chain of phosphorus fertilizers using modeling approaches and solution-based methods. The need to establish new factories to solve the issues related to the shortage of fertilizers could also make a perfect and interesting area of study. Similarly, phosphorus recycling could make a research idea to reduce phosphorus dissipation rates and enhance network efficiency. Other metaheuristic algorithms such as Ant Colony Optimization, Aquila Optimizer, Grey Wolf Optimizer, Pareto-based techniques, and hyper-heuristic algorithms can also be recommended to solve the proposed model.
Appendix 1: Case Study
Table 7 presents the required composition of these materials to produce different chemical fertilizers. Table 8 lists the average monthly demands of distribution centers. Tables 9 and 10 present the supply capacities of the raw materials and production capacities of the chemical fertilizers, respectively.
Table 7. Characteristics of chemical fertilizers based on the composition of the materials (%).
Fertilizer | P | A | PA | SA |
TSP | 40% | 0 | 34% | 0 |
SSP | 64% | 0 | 0 | 37% |
DAP | 0 | 23% | 47% | 0 |
Table 8. Average monthly demand of distribution centers (provinces) by fertilizer type (tons).
Distribution center | SSP | TSP | DAP |
1 | 1679 | 1633 | 838 |
2 | 1518 | 1513 | 910 |
3 | 1518 | 1627 | 915 |
4 | 1257 | 1338 | 643 |
5 | 563 | 639 | 313 |
6 | 623 | 666 | 385 |
7 | 200 | 215 | 113 |
8 | 496 | 589 | 297 |
9 | 326 | 397 | 197 |
10 | 328 | 387 | 189 |
11 | 2745 | 2930 | 1584 |
12 | 671 | 767 | 389 |
13 | 3624 | 3761 | 2038 |
14 | 769 | 1018 | 485 |
15 | 263 | 343 | 153 |
16 | 667 | 753 | 397 |
17 | 2434 | 2977 | 1543 |
18 | 956 | 1153 | 487 |
19 | 270 | 337 | 132 |
20 | 1181 | 1363 | 732 |
21 | 1137 | 1152 | 623 |
22 | 1663 | 1706 | 1018 |
23 | 289 | 318 | 165 |
24 | 1775 | 1913 | 940 |
25 | 1026 | 1130 | 661 |
26 | 1156 | 1458 | 689 |
27 | 782 | 799 | 438 |
28 | 674 | 618 | 353 |
29 | 493 | 494 | 277 |
30 | 1440 | 1320 | 763 |
31 | 232 | 257 | 133 |
32 | 815 | 1003 | 487 |
Total | 33,570 | 36,574 | 19,287 |
Table 9. Monthly capacity of suppliers (tons).
Material | Number of suppliers | Capacity | Total capacity |
A | 9 | (1,350-125,00) | 500,000 |
P | 3 | (5,000-35,000) | 65,000 |
PA | 7 | (1,000-21,650) | 75,000 |
SA | 26 | (1,250-54,150) | 300,000 |
Table 10. Monthly capacity of producers (tons).
Producer No. | SSP | TSP | DAP |
1 | 4650 | 1530 | 3090 |
2 | 630 | 510 | 990 |
3 | 2460 | 3750 | 3150 |
4 | 780 | 180 | 0 |
5 | 750 | 390 | 120 |
6 | 990 | 690 | 330 |
7 | 1740 | 780 | 630 |
8 | 600 | 345 | 300 |
9 | 960 | 330 | 0 |
10 | 1140 | 1230 | 0 |
11 | 870 | 450 | 240 |
12 | 1650 | 0 | 0 |
13 | 720 | 540 | 480 |
14 | 1350 | 420 | 360 |
15 | 1440 | 0 | 1020 |
16 | 990 | 240 | 195 |
17 | 720 | 435 | 0 |
18 | 3090 | 1710 | 0 |
19 | 750 | 600 | 330 |
20 | 0 | 4650 | 0 |
21 | 1560 | 0 | 720 |
22 | 6300 | 0 | 4650 |
23 | 0 | 4500 | 2850 |
24 | 2700 | 3450 | 1350 |
25 | 5550 | 3720 | 360 |
26 | 1050 | 900 | 690 |
27 | 630 | 180 | 0 |
28 | 3240 | 3690 | 1650 |
29 | 720 | 300 | 0 |
30 | 1350 | 195 | 540 |
31 | 1260 | 0 | 990 |
32 | 2850 | 600 | 0 |
33 | 510 | 360 | 300 |
34 | 2850 | 1440 | 0 |
35 | 540 | 330 | 150 |
36 | 1050 | 1200 | 1080 |
37 | 420 | 195 | 0 |
38 | 660 | 0 | 270 |
39 | 570 | 360 | 0 |
40 | 1050 | 360 | 300 |
Total | 61,140 | 40,560 | 27,135 |
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