草算纸

$$f_i = min\{f_j+Tran_{i,j}+Cost_i\}\qquad j \in [1,i-1]$$ . $$Tran_{i,j} = \Sigma_{k=j+1}^{i-1}(dis_i-dis_k)*num_k$$ $$=dis_i*\Sigma_{k=j+1}^{i-1}num_k\ -\ \Sigma_{k=j+1}^{i-1}dis_k*num_k $$ . $$sum_x = \Sigma_{i=1}^{x}num_i$$ $$g_x = \Sigma_{i=1}^{x}dis_i*num_i$$ $$\therefore Tran_{i,j} = dis_i*(sum_{i-1}-sum_{j})\ -\ (g_{i-1}-g_j)$$ . $$\therefore f_i = min\{f_j+Tran_{i,j}+Cost_i\}\qquad j \in [1,i-1]$$ $$f_i = min\{f_j+dis_i*(sum_{i-1}-sum_{j})\ -\ (g_{i-1}-g_j)+Cost_i\}\qquad j \in [1,i-1]$$ $$= min\{f_j+dis_i*sum_{i-1}-dis_i*sum_{j}\ -\ g_{i-1}+g_j+Cost_i\}\qquad j \in [1,i-1]$$ $$= f_j-dis_i*sum_{j}+g_j+dis_i*sum_{i-1}-g_{i-1}+Cost_i\qquad j \in [1,i-1]$$ $$f_i+dis_i*sum_j = f_j+g_j$$ $$b\ \ \ \ +\ \ \ k\ \ *\ \ x \ \ \ =\ \ \ \ \ y$$