Math discussion question

To formulate the linear model, we first need to find out the crash costs per week. This is calculated using the formula (Crash cost - Normal Cost) / (Normal time - Crash time). This is calculated for each of the activities as shown below The variables that considered for are below Let xi (i = 1 to 8) be the earliest start time of each activity andyi (i = 1 to 8) be the crash times for each activity. xs and xf are the start and finish time. The objective function is to minimize the total crash cost. This can be represented as Minimize : 6y1 + 12y2 + 4y3 + 6.67y4 + 10y5 + 7.3y6 + 5.75y7 + 8y8 subject to constraints y1 < 2 (2 is the maximum time activity 1 can be crashed = 5-3 =2. similarly for all activities) y2 < 1 y3 < 2 y4 < 3 y5 < 1 y6 < 3 y7 < 4 y8 < 2 xf < 15 (The constraint for finish time after crashing) xs = 0 (Constraint for start time = 0) The constraint of crashing can be represented as x1 - xs + y1 >= 5 x2 - xs + y2 >= 3 x3 - x1 + y3 >= 4 x4 - x1 + y4 >= 6 x5 - x2 + y5 >= 5 x6 - x2 + y6 >= 7 x7 - x3 + y7 >= 9 x7 - x5 + y7 >= 9 x8 - x4 + y8 >= 9 x8 - x6 + y8 >= 8