Duy Tan NGUYEN_Multi-objective optimization for inventory management systems under stochastic demand using grey wolf optimizer.pptx

optimization

1. Multi-objective optimization for inventory management systems under stochastic demand using grey wolf optimizer Nguyen Duy Tan Korea Maritime & Ocean University

2. Contents

3. I. Introduction Overview inventory model

4. II. Model formulation of an inventory management system Assumptions An item's demand rate is independent of its competitors and constant over time. It is possible to place only one order per period. There is a fraction of backorders and a fraction of lost sales due to shortages. All items have a zero-inventory level at the beginning. Quantity ordered on an item in one period exceeds or equals quantity short in the previous period. Parameters Stochastic parameters Decision variables Di,j : demand of the ith item in period j bi,j : shortage quantity of the ith item in period j Gi,j,k : a binary variable ( ), set 1 if item 𝑖 is purchased at price break point 𝑘 in period j, and 0 otherwise Ui,j : number of the packets for the ith item order in period j Wi,j : a binary variable ( ), set 1 if a purchase of item i is made in period j, and 0 otherwise Yi,j : the beginning positive inventory level of the ith item in period j Ei,j : a binary variable ( ), set 1 if a shortage for item 𝑖 occurs in period j, and 0 otherwise

5. II. Model formulation of an inventory management system Problem formulation The first target of the inventory problem is to optimize the required storage space. The total ordering cost is calculated as follows: The total holding cost is obtained as follows The total backorder cost will be given by The purchasing cost under AUD policy is obtained by The total purchasing cost under the IQD policy is obtained as follows The first objective function of the optimization problem can be expressed by  

6. II. Model formulation of an inventory management system Problem formulation The second target of the inventory problem is to optimize the required storage space. The inventory level at the beginning of a period is equal to that of the previous period. Since a quantity of order arrives to replenish the storage at each period, the objective function is described by:   Since PQi,j represents the purchase quantity of item 𝑖 in period 𝑗, by denoting the batch size as 𝐵𝑖 and the number of packets as Ui,j , then it becomes

7. III. Solution algorithm using multi object grey wolf optimizer The grey wolf algorithm mimicking wolves' social structures and hunting behavior in multi-objective search spaces is a novel swarm intelligence and metaheuristic algorithm ( Mirjalili et al., 2014). Single-objective function GWO 1. Action of surrounding the prey 2. Hunting mechanism The first step in hunting when the wolves detect the prey is encircling it. The mathematically model for surrounding actions is given as follows: where A and C are the vectors of coefficients, k is the iteration number, X(k) is the position vector of the grey wolves, Xi(k) is the goal position vector or the prey position vector of the grey wolves, D is the range from the prey to the grey wolf. Due to the robust prey detection ability, the social behavior of grey wolves is one of the highest success rates of hunting in the wildlife world. Alpha, beta , and delta are described as the leader and the best alternatives for alpha. In the algorithm, alpha is depicted as the optimal solution, while beta and delta could provide helpful prey knowledge to alpha. Their updated locations are described as follows: 1. Action of surrounding the prey 2. Hunting mechanism

8. III. Solution algorithm using multi object grey wolf optimizer Multi-objective function GWO In reality, the optimum solutions for the real-world problems pursue multiple goals. Several observations are made on how the MOGWO could be effective in providing the optimal solutions to multi-objective problems: The best non-outperform solutions obtained so far are saved by external archive. The random parameters A and C support candidate solutions to offer hyperspheres with various stochastic cases. The search agents are permitted to locate the expected position of the prey due to MOGWO becoming heir to the hunting mechanism. Exploration and exploitation are guaranteed by the adaptive values of a and A. Non-adaptive random values for C parameters during optimization enhance exploration and the local front immunity of the MOGWO algorithm concomitantly.

9. IV. Numerical simulation Data Case 1 Case 2 Case 3 Case 4 m (the number of item) 10 5 15 20 N (the number of periods) 13 20 19 7 Budget limit 370000 850000 1000000 934000 K (the number of item to apply policy) 4 3 7 5 Population 50 100 200 500 Basic information for inventory optimization As described in Table 1, the quantity of items varies 5 - 20 and the number of periods will be between 7 - 20. In addition, the total available budgets of the order quantity and the breakpoints to apply the discount policies are also provided. Depending on the business goals, there are several inventory model properties that might be useful in the optimization objective. A multi-objective GWO algorithm can provide the (near)-optimum solutions.

10. IV. Numerical simulation Optimum solutions for four different scenarios Optimal solutions Case 1 Case 2 Case 3 Case 4 Mean St. Dev Inventory cost (USD) 107743 86079 347822 140305 170487 104188 Storage space (m 2 ) 5235 4865 8336 11833 7567 2808 TFF ( ) 112978 90944 356158 152183 178066 105135 Optimal solutions Case 1 Case 2 Case 3 Case 4 Mean St. Dev Inventory cost (USD) 107743 86079 347822 140305 170487 104188 Storage space (m 2 ) 5235 4865 8336 11833 7567 2808 112978 90944 356158 152183 178066 105135 Table above shows the test results on the optimal solutions for each objective function aiming at the point of convergence. In this simulation, the different weights ( ) are employed for the calculation of the total fitness function (TFF) which describes the weighted sum of two objective  

11. IV. Numerical simulation Results explanation case 1 Boxplots of inventory cost after 100 iterations The box extends from the lower to upper quartile values of the data, with a line at the median. The whiskers extend from the box to show the range of the data. Flier points are those past the end of the whiskers.

12. IV. Numerical simulation Results explanation case 1 As can be seen in the first iteration, 100 solutions are randomly generated based on the parameters specified in Algorithm, and they are scatterly distributed. At this stage, a set of solutions does not indicate the optimum, like a pack of wolves starting to search for preys. After 20, 40, 60, and 80 iterations, the solutions gradually converge to a specific region as wolves finally catch their preys, as demonstrated in the 100th iteration. Then, the convergence point is considered as the (near) -optimal solution for this case. Convergence analysis for the optimal solutions in the search space

13. IV. Numerical simulation Optimum solutions for case 2 Boxplots of inventory cost after 100 iterations Boxplots of storage space after 100 iterations Convergence analysis for the optimal solutions in the search space

14. IV. Numerical simulation Optimum solutions for case 3 Boxplots of inventory cost after 100 iterations Boxplots of storage space after 100 iterations Convergence analysis for the optimal solutions in the search space

15. IV. Numerical simulation Optimum solutions for case 4 Boxplots of inventory cost after 100 iterations Boxplots of storage space after 100 iterations Convergence analysis for the optimal solutions in the search space

16. V. Conclusion In this article, a multi-item multi-period inventory management model under the limited budget with stochastic event is presented with multiple goals of optimizing the inventory cost and storage space. The multiple goals are to minimize both the total costs of inventory and the required storage space. The purpose of the study is to determine the optimal order quantity and shortage quantity of each product in each period so that the objective functions are optimized with the constraints. The developed model is an integer nonlinear programming model mixed with binary variables, and the multi-objective optimization algorithms have been applied to solve the inventory management problem. Finally, the proposed leader selection mechanism allows the MOGWO algorithm to guarantee superior coverage and convergence simultaneously. For specific real-word situation, real data can be employed to explore more practical marketing scenarios, and a comparison to other meta-heuristic algorithms should be considered in the future.

17. Thank you for your listening