# Time constraint in a simple linear programming model

 1 Based on my other question, here is a simpler hypothetical excerise that isolates that time cosntraint issue all together: A farmer has 1000 ha forest that is already at a mature age assumed to be 40 currently 200 ha are of wood type A and the rest are of wood type B Wood types A and B have the following m3 per ha Age yield_A yield_B 25 10 10 30 20 20 35 31 49 40 40 63 45 50 70  Wood types regenerate at the following rate (m3/ha): Age regen_a regen_b 5 3 4 10 11 19 15 24 35 20 35 48 25 50 69 30 75 86 35 101 121  wood type A sells for $100/ha while type B sells for$75/ha. Prices will increase 3% per period Optimization: Maximize the revenue of the farm over a 40 year period as well as period over period assuming 5 year periods and at that after 40 years , 70% of the forest is still planted. Attempted solution: Optimization functions: max_revenue_period_n = ($_a * vol_a +$_b * vol_b) for period n max_revenue_overall = ($_a * vol_a +$_b * vol_b)  Constraints: at age 40: vol_a_40 + vol_b_40 >= 0.6 (vol_a_0 + vol_a_40)  Question: How can I incorporate the time aspect of yields, regeneration, and optimizing over multiple periods into the problem? asked 18 Jul '13, 08:02 dassouki 89●3●9 accept rate: 0%

 2 First thoughts/clarifying question: You want to maximize revenue "overall" and every period. I'm not this is really what you want. I would think what you want to do is maximize the net present value over the 40 year time horizon starting now. Doing so requires a discount factor - a $now is not the same as a$ in 40 years. You can create a variable for the number of hectares indexed on: each tree type each 5 year interval each maturity level as well as the number of hectares harvested in a period. This is a bit redundant, but is probably easier to work with (I'm assuming the model is more complicated than you are saying, so I'd suggest very explicitly modelling everything, even though you could probably just solve this particular problem analytically). With that, you could easily write a single objective for the NPV, and the constraints would say something like "the number of 10 year old type A trees in period 5 is equal to the number of 5 year old type A trees in period 4 minus the number of hectares harvested". answered 18 Jul '13, 18:44 Iain Dunning 917●1●4●18 accept rate: 33% Thanks, I think you cleared up the problem for me enough to understand what I need to do. (19 Jul '13, 15:57) dassouki
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