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| # https://www.wolframalpha.com/input?i=minimize+%28y%5E3%2B17y%5E2+-%2818%2B2c%29y%29+where+y+%3E%3D+1%2C+2c+%3E%3D+y%5E2%2B17y-18 | |
| import numpy as np | |
| import matplotlib.pyplot as plt | |
| plt.rcParams["figure.figsize"] = (10,8) | |
| # c -> y -> x -> f -> g | |
| def y(c): | |
| v = ((6*c+343)**0.5-17) / 3 | |
| return np.where(v>=1, v, 1) | |
| def x(c,y): | |
| return c - 0.5*(y*y+17*y-18) | |
| def f(x,y): | |
| return x*y | |
| def g(x,y): | |
| return x + (y*y+17*y)/2 - 9 | |
| def wait_for_upgrade(target_array): | |
| e = target_array[-1] | |
| tmp = [e:=min(e,target_array[i]) for i in range(len(target_array)-1,-1,-1)] | |
| tmp = np.array(tmp[::-1]) | |
| return tmp | |
| # 0~600 minerals | |
| c_limit = 601 | |
| cs = list(range(c_limit)) | |
| cs = np.array(cs) | |
| # real number | |
| ys = y(cs) | |
| xs = x(cs, ys) | |
| fs = f(xs, ys) | |
| gs = g(xs, ys) | |
| # integer (round up / round down) | |
| ys_floor = np.floor(ys) | |
| ys_ceil = np.ceil(ys) | |
| xs_floor = np.floor(x(cs, ys_floor)) | |
| xs_ceil = np.floor(x(cs, ys_ceil)) | |
| fs_floor = f(xs_floor,ys_floor) | |
| fs_ceil = f(xs_ceil,ys_ceil) | |
| gs_floor = g(xs_floor,ys_floor) | |
| gs_ceil = g(xs_ceil,ys_ceil) | |
| # choose bigger one | |
| fs_idx = np.where(fs_floor > fs_ceil, 1, 0) | |
| xs_int = np.where(fs_idx, xs_floor, xs_ceil) | |
| ys_int = np.where(fs_idx, ys_floor, ys_ceil) | |
| fs_int = np.where(fs_idx, fs_floor, fs_ceil) | |
| gs_int = np.where(fs_idx, gs_floor, gs_ceil) | |
| fig, axs = plt.subplots(3) | |
| plt.sca(axs[0]) | |
| #plt.xlabel('c') | |
| plt.ylabel('x') | |
| plt.plot(cs, xs, label='continuous') | |
| plt.plot(cs, xs_floor, label='floor') | |
| plt.plot(cs, xs_ceil, label='ceil') | |
| plt.plot(cs, xs_int, '--', label='optimal') | |
| #plt.legend() | |
| plt.sca(axs[1]) | |
| #plt.xlabel('c') | |
| plt.ylabel('y') | |
| plt.plot(cs, ys, label='continuous') | |
| plt.plot(cs, ys_floor, label='floor') | |
| plt.plot(cs, ys_ceil, label='ceil') | |
| plt.plot(cs, ys_int, '--', label='optimal') | |
| #plt.legend() | |
| plt.sca(axs[2]) | |
| plt.xlabel('c') | |
| plt.ylabel('f') | |
| plt.plot(cs, fs, label='continuous') | |
| plt.plot(cs, fs_floor, label='floor') | |
| plt.plot(cs, fs_ceil, label='ceil') | |
| plt.plot(cs, fs_int, '--', label='optimal') | |
| plt.legend() | |
| plt.savefig('fig1', bbox_inches='tight') | |
| plt.show() | |
| # cannot decrease x or y. | |
| # save mineral for upgrade | |
| # compare rounding too | |
| ys_round = np.round(ys) | |
| xs_round = np.floor(x(cs, ys_round)) | |
| xs_round = wait_for_upgrade(xs_round) | |
| xs_floor = wait_for_upgrade(xs_floor) | |
| xs_ceil = wait_for_upgrade(xs_ceil) | |
| fs_round = f(xs_round,ys_round) | |
| fs_floor = f(xs_floor,ys_floor) | |
| fs_ceil = f(xs_ceil,ys_ceil) | |
| gs_round = g(xs_round,ys_round) | |
| gs_floor = g(xs_floor,ys_floor) | |
| gs_ceil = g(xs_ceil,ys_ceil) | |
| fig, axs = plt.subplots(3) | |
| plt.sca(axs[0]) | |
| #plt.xlabel('c') | |
| plt.ylabel('x') | |
| plt.plot(cs, xs, label='continuous') | |
| plt.plot(cs, xs_floor, label='floor') | |
| plt.plot(cs, xs_ceil, label='ceil') | |
| plt.plot(cs, xs_round, label='round') | |
| #plt.legend() | |
| plt.sca(axs[1]) | |
| #plt.xlabel('c') | |
| plt.ylabel('y') | |
| plt.plot(cs, ys, label='continuous') | |
| plt.plot(cs, ys_floor, label='floor') | |
| plt.plot(cs, ys_ceil, label='ceil') | |
| plt.plot(cs, ys_round, label='round') | |
| #plt.legend() | |
| plt.sca(axs[2]) | |
| plt.xlabel('c') | |
| plt.ylabel('f') | |
| plt.plot(cs, fs, label='continuous') | |
| plt.plot(cs, fs_floor, label='floor') | |
| plt.plot(cs, fs_ceil, label='ceil') | |
| plt.plot(cs, fs_round, label='round') | |
| plt.legend() | |
| plt.savefig('fig2', bbox_inches='tight') | |
| plt.show() | |
| for ys_arr in [ys_floor, ys_round, ys_ceil]: | |
| for i in range(600): | |
| if ys_arr[i]!=ys_arr[i+1]: | |
| print(int(xs_floor[i]), end=" ") | |
| print() | |
| # then just pick any strategy. | |
| ys_int2 = ys_round | |
| xs_int2 = xs_round | |
| fs_int2 = fs_round | |
| gs_int2 = gs_round | |
| # apply x limit | |
| x_limit=238 | |
| # yy + 17y - 18 - 2c + 2x =0 | |
| def y_238(c,x): | |
| # assert c >= x | |
| y = ((17*17-4*(-18-2*c+2*x))**0.5 - 17) / 2 | |
| return np.where(y>=1, y, 1) | |
| xs_x238 = np.array(list(range(x_limit))+[x_limit]*(len(cs)-x_limit)) | |
| ys_x238 = np.floor(y_238(cs, xs_x238)) | |
| fs_x238 = f(xs_x238, ys_x238) | |
| gs_x238 = g(xs_x238, ys_x238) | |
| fs_idx = np.where(xs_int2 <= x_limit, 1, 0) | |
| xs_actual = np.where(fs_idx, xs_int2, xs_x238) | |
| ys_actual = np.where(fs_idx, ys_int2, ys_x238) | |
| fs_actual = np.where(fs_idx, fs_int2, fs_x238) | |
| gs_actual = np.where(fs_idx, gs_int2, gs_x238) | |
| fig, axs = plt.subplots(3) | |
| plt.sca(axs[0]) | |
| #plt.xlabel('c') | |
| plt.ylabel('x') | |
| plt.plot(cs, xs, label='continuous') | |
| plt.plot(cs, xs_int2, label='discrete') | |
| plt.plot(cs, xs_x238, label='tower first') | |
| plt.plot(cs, xs_actual, '--', label='actual') | |
| #plt.legend() | |
| plt.sca(axs[1]) | |
| #plt.xlabel('c') | |
| plt.ylabel('y') | |
| plt.plot(cs, ys, label='continuous') | |
| plt.plot(cs, ys_int2, label='discrete') | |
| plt.plot(cs, ys_x238, label='tower first') | |
| plt.plot(cs, ys_actual, '--', label='actual') | |
| #plt.legend() | |
| plt.sca(axs[2]) | |
| plt.xlabel('c') | |
| plt.ylabel('f') | |
| plt.plot(cs, fs, label='continuous') | |
| plt.plot(cs, fs_int2, label='discrete') | |
| plt.plot(cs, fs_x238, label='tower first') | |
| plt.plot(cs, fs_actual, '--', label='actual') | |
| plt.legend() | |
| plt.savefig('fig3', bbox_inches='tight') | |
| plt.show() |
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