A simulation-based multi-echelon warehouse model

To simulate and analyze warehouse models with multiple echelons an easy to use, component based simulator was developed in MATLAB (SIMWARE), where the connection between warehouses can be dened by the help of a well-structured conguration le. As we will see in the following sections, using SIMWARE users are capable to model complex inventory models applying empirical data for consumption, and this simulation can be used for optimiza-tion. First, to present the capabilities of the new simulator based model, the optimization using Sequential Quadratic Programming (SQP) functionality of MATLAB's Optimization Toolbox is presented, then improved Particle Swarm Optimization (PSO) algorithms will be described.

During the optimization the minimum value of a restricted non-linear multi-variable function is calculated. In our case the reorder point as op-timization variable (parameter) is used. We are seeking for the minimum of the average days of inventory, while the restriction is the required value of the service level. In the basic case only one set of random consumption data is used for the optimization therefore the optimum is related to this dataset. To overcome this problem the Monte-Carlo process is applied. This is a robust methodology which generates empirical distribution functions of consumption. These data sets is used as input for the simulation model to simulate the stock movements, and many random paths like this is generated in every optimization step. Based on the proven convergence of the Monte-Carlo process the calculated stock movements, stock turnover and service level are good estimations of the actual process at a given reorder point.

In the current examples the optimization process executes 100 simulation runs for each parameter set and uses the average of the results as objective function or constraint. Results are demonstrated in the following sections.

Optimizing a 2-echelon warehouse model by SQP method

The main objective of the presented development is to propose a simulation method that can utilize the previously proposed building blocks to construct models of complex multi-echelon supply chain networks. In the following, we will describe a simple 2-echelon warehouse model to present the capabilities of our MATLAB simulator supporting the research of multi-level supply chains.

For demonstration purpose, we present a complete optimization process using our simulation method and well-known SEP algorithm.

Fig. 3.3 shows the supply chain, i.e. the structure of the analyzed 2-level system, where the objective function is given by equation 3.5. So the holding

Figure 3.3. Example of a 2-echelon supply chain with a distribution store and with 2 retailers. The supply from the manufacture is unlimited.

cost at the retailer is 30 percent higher than in the distribution store.

f(z) =mean(h₁) + 1.3·mean(h₂) (3.5) In Fig. 3.4, the values of the objective function (i.e. cost) is presented as a function of the reorder point of the two warehouses, while Fig. 3.5 shows

Figure 3.4. Values of the objective function for the 2-level system presented by equation 3.5.

the service level of Warehouse 1 in the 2-level system. The constraint for the service levels is 95% in this case.

Figure 3.5. Values of service levels in each warehouses in the investigated example (see Fig. 3.3.

As we will see int he next section, the simulator can be used to calculate the outputs of of the system's components simultaneously, as well as the compound objective value of the supply chain as a whole. The required values of service levels of the warehouses represent non-linear constraints, hence a nonlinear optimization algorithm has to be used to solve this complex optimization problem. In the following, the results of SQP method will be presented, followed by the discussion of our novel improvements of particle swarm optimization algorithms.

Figure 3.6. An example 3-level system depicted by SIMWARE.

In the SIMWARE program, users have to dene the structure of the problem, i.e. the connection between the warehouses. The parameters of each warehouse has to be dened also, e. g. the lead time or the average demand for a product as well as the cost values, like the holding cost. Fig. 3.6 represents an example for a 3-level system dened by the conguration le inside the simulator.

Fig. 3.7 represents the results of the simulation for a 3-level inventory model, where each inventory level is depicted by lines with dierent style.

Figure 3.7. Simulated inventory levels of the 3-level multi-echelon system presented in Fig. 3.6.

Note that we have used separate distribution functions for the central warehouse and the other regional warehouses to simulate the consumption.

We constructed these distribution functions based on actual data. The cen-tral warehouse consumption gures are calculated as a total of the down-stream warehouse consumptions. Using dierent distribution functions in a

multi-echelon supply chain, or using dierent consumption values for each warehouse is not a trivial, but necessary task if we want to construct a more realistic model or simulator. The SIMWARE program oers an easy-to-use interface to build even complex supply chains, and propose a novel component based structure. Using this program, users can easily optimize their supply chain by an optimization method (e.q. SQP), but the proposed methodology gives a chance to use more eective optimization algorithms also.

Fig. 3.8 shows the result of the optimization using the SQP method. Here, the constraint for service level was 95% in each warehouse. The optimal solution is highlighted with the green square, which satises the constraints and ensures the minimal holding cost in the warehouses. The result of the simulation runs is presented in Fig. 3.9. The uctuations in the average inventory levels after ten MC simulations are shown, and the investigated period is 50 weeks. The service levels of both warehouses are determined, they are 0.97 and 0.89, respectively. After the optimization, the adjusted parameters make sure that none of the warehouses running out of stock during the investigated period.

Figure 3.8. Result of the optimization of the 2-echelon system using SQP method.

Figure 3.9. Inventory levels in the optimized 2-level system.