Location-Inventory-Routing Model with Considering Urban Road Networks

Purpose: To develop LIRP (location-inventory-routing problem) model with considering multiple links and solve it using method of heuristic based on algorithm of simulated annealing. Method of heuristic for the LIRP model is applied in city of Jakarta to improve effectiveness and efficiency of the food supply chain. Design/methodology/approach: The LIRP model is developed using main references. To solve the model, this paper develops two methods, namely method of optimal and method of heuristic. Computational experiments are performed to obtain the efficiency of the method of heuristic. New design of food supply chain is resulted from the application of the method of heuristic in city of Jakarta. Findings: The new design of food supply chain resulted from the application of LIRP model in city of Jakarta reduces total cost by 18%, increases availability from 76% to 95%, and reduces the number of vehicles by 73%. This paper also shows that distance is not the only consideration to decide the traversed links in cities. Research limitations/implications: Average gap between method of heuristic and method of optimal in terms of total cost is 3.1%. Practical implications: Government of city of Jakarta can improve effectiveness and efficiency of the food supply chain by implementing the LIRP model. Social implications: Citizens of Jakarta are well provided with their needs of vegetables and fruits. Originality/value: The first LIRP model that considers multiple links to represent road networks in cities. The LIRP model developed in this paper consists of probabilistic demands, multi products, and multi echelons. Traditional markets, UCCs (urban consolidation centers) and province of suppliers are the places where decisions of inventory made.


Introduction
There are multiple road networks that connect places in urban areas. To apply the model of LIRP (location-inventory-routing problem) in urban areas, one needs to consider multiple links to represent the road networks. There is no model of LIRP that considers multiple links. It can be known from literature review of LIRP that has been done in Table 1; therefore, there is a research gap that needs to address related to this issue. To solve the model, this paper develops two methods, namely method of optimal and method of heuristic. The method of heuristic is developed based on algorithm of simulated annealing and is applied in city of Jakarta to design a new supply chain for food and improve the effectiveness and efficiency.
In the last 10 years, here are papers related to LIRP model. Sajjadi and Cheraghi (2011) developed LIRP model and solved it using simulated annealing. The model consisted of probabilistic demands, multi products, and multi echelons. Depots and customers were the places where decisions of inventory made. Guerrero, Prodhon, Velasco and Amaya (2013) developed LIRP model and solved it using local search and algorithm of randomized extended Clarke and Wright. The model consisted of deterministic demand, single product, and single echelon. In Guerrero et al. (2013), depots and customers were the places where decisions of inventory made. Nekooghadirli, Tavakkoli-Moghaddam, Ghezavati and Javanmard (2014) developed LIRP model and solved it using algorithm of multi-objective meta-heuristic. The model consisted of probabilistic demands, multi products, and multi echelons. Distribution centers were the places where decisions of inventory made. Zhang, Qi, Miao and Liu (2014) developed LIRP model where customers were the places where decisions of inventory made. Zhang et al. (2014) solved the model using simulated annealing and local search. The model consisted of deterministic demand, single product, and multi echelons. Tang, Ji and Jiang (2016) developed LIRP model and solved it using multi-objective particle swarm optimization. Warehouses were the places where decisions of inventory made. The model in Tang et al. (2016) consisted of probabilistic demand, single product, and multi echelons. Ghorbani & Akbari-Jokar (2016) developed LIRP model and solved it using imperialist competitive and simulated annealing. The model consisted of probabilistic demands, multi products, and multi echelons. Depots and customers were the places where decisions of inventory made. Zhalechian, Tavakkoli-Moghaddam, Zahiri and Mohammadi (2016) developed LIRP model and solved it using variable neighborhood search and self-adaptive genetic algorithm. The model consisted of probabilistic demands, multi products, and multi echelons. Distribution centers were the places where decisions of inventory made. Hiassat, Diabat and Rahwan (2017) developed LIRP model that consisted of deterministic demands, single product, and single echelon. The model solved using genetic algorithm. Customers were the places where decisions of inventory made. Rayat, Musavi and Bozorgi-Amiri (2017) developed LIRP model and solved it using archived multi-objective simulated annealing. Distribution centers were the places where decisions of inventory made. The model consisted of deterministic demands, multi products, and multi echelons.
Rafie-Majd, Pasandideh and Naderi (2018) developed LIRP model that consisted of deterministic demands, multi products, and multi echelons. The model was solved using algorithm of Lagrangian relaxation. Customers were the places where decisions of inventory made. Vahdani, Veysmoradi, Noori and Mansour (2018) developed LIRP model and solved it using multi-objective particle swarm optimization and non-dominated sorting genetic algorithm II. The model consisted of probabilistic demands, multi products, and multi echelons. Distribution centers/warehouses were the places where decisions of inventory made. Saragih, Bahagia, Suprayogi and Syabri (2019) developed LIRP model and solved it using simulated annealing. The model consisted of probabilistic demand, single product, and multi echelons. Retailers, depots and supplier were the places where decisions of inventory made. Bagherinejad & Najafi-Ghobadi (2019) developed LIRP model and solved it using genetic algorithm and algorithm of evolutionary simulated annealing. The model consisted of deterministic demand, single product, and multi echelons. Retailers and warehouses were the places where decisions of inventory made.
Farias, Hadj-Hamou and Yugma (2020) developed LIRP model and solved it using algorithm of Branch-and-Cut and two-phase heuristic. The model consisted of deterministic demand, single product, and multi echelons. Customers, distribution centers, and supplier were the places where decisions of inventory made. Rahbari, Razavi-Hajiagha, Raeei-Dehaghi, Moallem and Riahi-Dorcheh (2020) developed LIRP model that consisted of deterministic demand, single product, and multi echelons. The model was solved using general algebraic modeling system. Decisions of inventory made at all entities. Karakostas, Sifaleras and Georgiadis (2020)  The LIRP model developed in this paper consists of probabilistic demands, multi products, and multi echelons. Traditional markets, UCCs (urban consolidation centers) and province of suppliers are the places where decisions of inventory made. UCCs are logistics facilities where consolidation and coordination activities are performed (Saragih et al., 2015). As for the main contribution, this paper considers multiple links to represent the road networks in urban areas, such as the city of Jakarta. The model developed in this paper represents more realistic real systems where there are multiple road networks that connect places in urban areas. From Table 1, this model of LIRP has never been done before. This paper fills the gap by developing the LIRP model.

Methodology
To develop the LIRP model, this paper uses Saragih et al. (2018) and Taniguchi, Noritake, Yamada and Izumitani (1999) as the refences. Saragih et al. (2018) only considered one link or road to connect two locations in the LIRP model, whereas in urban areas, two locations can be connected by more than one link. This weakness is improved using Taniguchi et al. (1999). The approach used to develop the LIRP model in this paper can be seen in Figure 1.

Development of Model
The LIRP model developed in this paper is described as follows.

Index sets
K set of traditional markets J set of potential UCCs    (1) Subject to: (2) (4) (8) Equation (1) is the total cost. Constraints (2) guarantee that each traditional market is served exactly once by a vehicle. Constraints (3) ensure that products sent in one vehicle route may not exceed the vehicle's capacity. Constraints (4) are the subtour elimination. Constraints (5) are the flow conservation. Constraints (6) guarantee that only one UCC per route. Constraints (7) connect allocation and routing decisions. Constraints (8) guarantee that each vehicle can only traverse one link. Constraints (9) guarantee that each UCC can only have one level capacity. Constraints (10) guarantee that a UCC must not supply traditional markets beyond its capacity. Constraints (11) guarantee that demands in a UCC for each product is the sum of the demands for the traditional markets served for each product. Constraints (12) guarantee that each traditional market is supplied exactly once by a UCC. Constraints (13) guarantee each province of supplier can service more than one UCC and one product. Constraints (14) guarantee that demands at province of supplier are the demands for each product that can be supplied to UCC without exceeding its capacity. Constraints (15) guarantee that demands of each UCC for each product must be fulfilled by province of supplier. Constraints (16) guarantee that demands at province of suppliers are the demands for each product supplied to UCC served. Constraints (22)  The LIRP model developed in this paper is MINLP (mixed integer nonlinear programming) model. Since LIRP belongs to the class of NP-hard problems, the application of the MINLP model is limited to small data (Ahmadi-Javid & Azad, 2010). To apply the model in a real system case which is Jakarta, it needs method of heuristic. The method of heuristic based on algorithm of SA (simulated annealing) is developed in this paper. SA is algorithm of local search (meta-heuristic) that is capable of escaping from local optima. It is used to solve discrete and to a lesser extent, continuous optimization problems (Henderson, Jacobson & Johnson, 2003). The LIRP consists of location and routing problems which are discrete optimization problems and inventory problem which is continuous optimization problems. SA is appropriate to solve the model in this paper.

Solution Method
As it was mentioned previously, to solve the LIRP model, this paper develops method of heuristic based on algorithm of SA. Solution resulted from the method of heuristic is compared to the solution resulted from method of optimal (the MINLP). The method of heuristic uses parameters as follows. Pseudo code of the method of heuristic is given in Figure 3.

Parameters of the method of heuristic
Take β 0 Set the SA parameters θ, C max , I a , I 0 Initial β a = β 0 , I c = I 0 , i = 1 While I c > I a For i = i: C max r ← U(0,1)

Computational Experiments
Example of solution for the LIRP model is given in Figure 4. The solution is resulted from method of optimal by running the model in LINGO 12.0. Data used for the problem in Figure 4 are described as follows. Opened UCC is UCC 1 with capacity level 2. UCC 1 is supplied by province of supplier 1 and 2. Province of supplier 1 only supplies product 1 because the product that is available at province of supplier 1 is only product 1. Province of supplier 2 can supply product 1 and product 2 because on province of supplier 2 has both of the products. Opened UCC 1 serves all traditional markets by forming vehicle routes. There are 2 tours of vehicle routes established to serve the traditional markets. Tour 1 consists of UCC 1 -TM 3 -TM 4 -UCC 1. From UCC 1 to TM 3, the vehicle traverses link 1, from TM 3 to TM 4, the vehicle traverses link 1, and from TM 4 to UCC 1 traversed link is link 2. Tour 2 consists of UCC 1 -TM 5 -TM 6 -UCC 1. From UCC 1 to TM 5, the traversed link is link 1, from TM 5 to TM 6, the traversed link is link 2, and from TM 6 to UCC 1, the traversed link is link 1. Comparison for method of heuristic and method of optimal can be seen in Table 7. The efficiency for method of heuristic is 3.1% on average.

No. # province of supplier # UCC # traditional markets
Method of optimal Method of heuristic Gap (%)

Application in City of Jakarta
Method of heuristic for the LIRP model is applied in city of Jakarta. There are two links considered. Link 1 is road with the longest distance and link 2 is road with the shortest distance. Illustration of link 1 and link 2 can be seen in Figure 5. Figure 5 is location map of Cempaka Putih Market and Gembrong Inpres Market which are connected by 2 links. Link 1 is taken through Jl. Letjen Suprapto and Jl. Pangkalan Asem Raya. Link 2 is taken through Jl. Cempaka Putih Barat III and Jl. Cempaka Raya.

Performance Analysis
In Table 13, it can be seen that the new design of the food supply chain provides a more efficient total cost by 18%. This is due to benefit provided by the UCCs in terms of availability which can be maintained at 95% level. The availability of the existing system only 76%. As it was mentioned previously, UCCs are logistics facilities where consolidation and coordination activities are performed. In the existing supply chain system, there is no both activities. Without consolidation activity, inventory policies are carried out independently in each traditional market and in each province of supplier. In addition, without coordination activity, there is no vehicle routes formed since vehicles used to deliver goods are the vehicles originating directly from each province of supplier.
The new design also reduces the number of vehicles by 73% due to the use of vehicles together at UCCs. Moreover, with the new design of supply chain in Jakarta, there is opportunity to use environmentally friendly and fuel-efficient goods vehicles (green vehicles) that can be used at UCCs to deliver goods to traditional markets in urban areas.

Conclusion
This paper has successfully developed a model of LIRP that considers multiple links to represent the road networks in cities. There is no model of LIRP that considers multiple links; therefore, this paper gives contribution in the LIRP models that were developed before. The model developed in this paper represents more realistic real systems where there are multiple road networks that connect places in urban areas.
To apply the model in a real system, which is Jakarta, this paper develops method of heuristic based on algorithm of simulated annealing. The new design resulted from the application of the model improves both the effectiveness and efficiency of Jakarta's food supply chain. Total cost reduces by 18%, availability/service level increases from 76% to 95%, and number of vehicles reduces by 73%.
Number of links considered in the real system is 2 links which represent the longest (link 1) and the shortest distance (link 2). It can be seen from the results (Table 11) that the traversed links are not always the shortest ones. This is due to cost of congestion which also considered when deciding the vehicle routes. In cities, distance is not the only consideration to decide the traversed links. This paper shows that the distance can be short, but if the traffic is congested then the link is not traversed.
Since the average gap in terms of total cost between method of heuristic and method of optimal is 3.1%, this paper also provides opportunities for future work to develop more efficient method of heuristic to solve the model in large data.