You asked moreless the same question almost a month ago and it seems the answers you received weren't satisfying to you. Could you provide us with some example parameters of the problem?

My understanding is you are saying you want to solve a dynamic problem, right? So as Przemyslaw said, provide more detail about what your main problem is. However, the first idea that came to me is:

I assume you have assigned each silo with index i: S(i). If S(i)=1 then it means the silo is selected. Now let`s add a time index to specify the period that silo is being used. So you need a if constraint saying:

for t=T if S(i)(t)=1 then run the allocation. 

I don`t have enough information about the parameters so I just provided a simple example.

Yes, Hossein, it looks as a dynamic problem in which time must be considered. I suppose it may be a problem of multi-stage optimization whose stages are connected  with selecting particular silos.

This has much in common with reservoir operating problems, in which we enforce the continuity equation for all time periods subject to a capacity constraint.  Could you clarify (a) what are your decision variables; and (b) what is your time step relative to the filling rate & capacity of the silo?  i.e. once you select a silo how many time periods does it take to fill?

Hi Stuart,Przemysław,Hossein,

My decision variables are

1. Silobinvar(s,t) - which tells what silo is selected

2. Recipebinvar (r,t) - which tells what recipe is currently being produced

The idea you have said is what I have been trying to implement. My time period of filling is different for different silo based on distance of that silo.

But the average time period is about 6-8. 

Can you provide any link to papers which I can refer to get an idea of how to implement your idea?

Thanks in advance

Regards

Pradyumna Krishnan

Hi Pradyumna:

OK, so I gather Silobinvar(S,t) =(0,1) if silo s is selected to receive product in time t

and Recipebinvar(r,t) =(0,1) if recipe r is being produced in time t.    You want to only select one silo in any time period so, SUM_s(Silobinvar(s,t) =1 for all t.

You also want to keep silo s selected for as may consecutive periods as it takes to fill silo. another way to say that is you want to minimize the cumulative transitions from non-filling to filling states for all silos.  

Suppose you consider Silo(s,t+1)-Silo(s,t) =u(s,t+1) - v(s,t+1). and add min epsilon*Sum_s,t (u(s,t)). to the objective function.  epsilon is small enough so it otherwise does not affect the optimal solution, but all else equal this chooses fill patterns that only start to fill (u(s,t+1)>0) once.  (Check this logic.  on first blush it does not appear to me you need to min eps*SUM_s,t (u(s,t)+v(s,t)), but adding this does not appear to matter, since their difference will equal Silobinvar(s,t,) for all s and t, and minimizing one should minimize the other.  since their difference will always be 0 or 1, they should not even need to be declared integer.. Check the logic on this to confirm this works.  Probably need some special treatment of first and/or last period as well.

I presume you have other constraints that drive the solution to completely filled silos - i.e. no partially filled silos are feasible/optimal. so this would drive solutions that fill the silos but only start to fill once - and therefore once they start filling they don't stop until the silo is full.   

If on the other hand there are no other constraints that force the silos to be full, this will not be sufficient.  If the latter you may need to think about more binary variables X(r,s,t) = (0,1) if recipe r is being put into silo s at time t.

Alternatively you could look at the so-called unit commitment problem for electric utilities.  this problem selects the generating units that are available and unavailable, and some units (like large thermal/nuclear plants have startup times and shut down times that need to be enforced, and once they are online they stay online for consecutive time periods - unlike peaking units or hydropower that can be dispatched quickly with little to no startup and then shut down without a cooling period.

There is a huge literature on this with many different algorithns - primal-dual methods have received much attention.  Note the unit commitment problem is typically closely coupled with the economic dispatch problem:  given the units currently committed (available) distribute the current period demand across the available units to minimize generating costs (and other secondary objectives).  These problems obviously link across time periods based on projected demand.

One of Dantzig's last papers was on the stochastic version of this problem with Gerd Infanger in the European journal of operations research:  

Intelligent control and optimization under uncertainty with application to hydro power  By:Dantzig, GB (Dantzig, GB); Infanger, G (Infanger, G)  EUROPEAN JOURNAL OF OPERATIONAL RESEARCH.  Volume: 97. Issue: 2. Pages: 396-407

DOI: 10.1016/S0377-2217(96)00206-8

Hope this is of some help

Thanks Stuart for your  explanation. I will try to use the tricks provided by you and also will search the literature to translate my formulation.

Hi Pradyumna,

Dynamic and multi-stage programming are concepts that are used widely in different fields. I do not have any concrete reference to introduce it to you because the basics are the same but our fields are different.  I think Stuart has provided a very good answer but feel free to post your codes also here then we can help you more in details.