A Class of Expected Value Bilevel Programming Problems with Random Coefficients Based on Rough Approximation and Its Application to a Production-Inventory System

Table 3

Optimal solutions under different confidence levels.

Variable	Item 1	Item 2	Item 3	Item 4	Item 5
𝜃=0.85	𝐹𝑈=35148.36,𝐹𝐿=5667.43
𝑥	182.5	0.54	1.62	0.84	0.073
𝑦1𝑗	210.7	0	0.58	0.84	6.29
𝑦2𝑗	0	0.56	0	0	480.51
𝜃=0.9	𝐹𝑈=41916.68,𝐹𝐿=7140.244
𝑥	190.51	0.15	0.87	0.38	0.13
𝑦1𝑗	307.81	0	0.481	0.38	4.48
𝑦2𝑗	0	0.61	0	0	489.71
𝜃=0.95	𝐹𝑈=49872.31,𝐹𝐿=8451.456
𝑥	198.05	0.058	0.42	0.33	0.081
𝑦1𝑗	395.29	0	0.409	0.33	2.58
𝑦2𝑗	0	0.42	0	0	495.37
𝜃=1	𝐹𝑈=50132.57,𝐹𝐿=7725.394
𝑥	199.95	0.032	0.35	0.12	0.057
𝑦1𝑗	394.39	0	0.345	0.12	2.231
𝑦2𝑗	0	0.348	0	0	498.45

Note: 𝐹_𝑈 and 𝐹_𝐿 denote objective values of the upper level and lower level, respectively. 𝑦_1𝑗 and 𝑦_2𝑗 denote the production quantity by producer 1 and produce 2, respectively.