Activity duration distribution - how to get it?

Problem:

We got distribution for 'time-of-day' choices for Home-Work trips and Work-Home trips. What we need is distribution of work-time duration. We needed it to populate activity chains. Here's the solution which I particulary like as a smart way to solve complex issue and exploit sources that are available.

 Background:

Our Uni (Cracow University of Technology) jointly with (Poznan University of Technology) gave another try for activity based models (it's a broader project incorporating MATsim + electric taxi dispatching, etc.), but our small part is to create damand input for agent-based model (MATsim).
What we had is an out-dated survey made for activities (year 2000) and quite recent classic travel survey. We need to make use of this as seamless as possible.
 

Input:

 

  

Algorithm:

Let’s define following linear program.
Inputs are HW and WH time-of-day vectors and variable is the duration distribution vector D_w.
Let’s define the mapping m(HW, D_w) à WH_t  as follows:

a)      let’s calculate cross product of vector HW  and transposed D_w shifted left by respective travel start time t(HW(i)):    WH_t = HW×((D_w + t(HW(i))^T

b)      Then compute output: theoretical  WH vector, as column sum of resulting matrix WH_t

c)      Objective function is likelihood between  WH_t  and WH (either r², either MRSE, or log- likelihood, whatever).

 

Solution of this problem gives results as follows:

 Summary:

We got what we needed: duration of time spent at work, from weel founded input that we had. We didn't need to make extra surveys - we just exploited what was available.