Multi-period Inventory Balance in Gurobipy

Question

I would like to get the inventory balance in the oponed facility $\text{C}$ such that, the Inventory at time $t$ is equal to the previous one, plus the inflow minus the outflow. I have tried to create a list with zeroes, but I could not make it automated. The other issue is with indexing in python. For instance, if $t$ which is time in months, and it's range(0,3) then, $t-1$ is $(-1)$ which gives an error.

$INV_{c,t}$: is the inventory at time $t$.

$INV_{c,t-1}$: is the inventory at time $t-1$.

The equations: $INV_{c,t}=INV_{c,t-1}+\sum{QS}_{c,t}+\sum{QM}_{c,t}-\sum{QG}_{c,t}$

Inventory level decision variable

INV_c= m.addVars(storages,months, name="INV_ct")

Inventory Balance Constraints

m.addConstrs(INV_c[c,t] == INV_c[c,t-1] + sum(QD_wc[w,c,t,p] for w in range(warehouses) for p in range(products)) + sum(QH_hc[h,c,t]\
               for h in range(customers)) - sum(QG_co[c,o,t] for o in range(centers)) for c in range(storages) for t in range(months))

I have defined $INV0_{c,t}$ as parameter (an array of 3 by 2 dimension), in which $INV0_{c,t}$ could replace INV_c[c,t-1] in the above constraint.

I have tried to write a for loop to capture the cumulative inventory at each period, by defining the following code;

INV0_[c,t]=0
for t in range(months):
 for c in range(storages):
  INV0_[c,t]=INV0_[c,t]+INV_c[c,t]

Answer 1

A simple approach that does not require much thought would be to split this into two parts:

# the general case for t>0
m.addConstrs(INV_c[c,t] == INV_c[c,t-1] + sum(QD_wc[w,c,t,p] for w in range(warehouses) for p in range(products)) + sum(QH_hc[h,c,t]\
               for h in range(customers)) - sum(QG_co[c,o,t] for o in range(centers)) for c in range(storages) for t in range(1,months))

# special case for t=0
m.addConstrs(INV_c[c,0] == INV0[c] + sum(QD_wc[w,c,0,p] for w in range(warehouses) for p in range(products)) + sum(QH_hc[h,c,0]\
               for h in range(customers)) - sum(QG_co[c,o,0] for o in range(centers)) for c in range(storages))

where INV0[c] are exogenous values: inventory at t=-1.

IMHO, this is a little bit more straightforward than some of the other suggestions.

Answer 2

The standard inventory balance equation would be something like $I_{t-1} + x_{t} = d_{t} + I_{t}$ , where $x$ and $d$ are production quantity and demand respectively. Now, it seems you need to push the flow materials on both sides. Also, besides the inventory term, you have three additional terms in the RHS that needs more clarification. For the second error, I am not sure how it can be implemented in python, but for fixing the indexing error you have to define the inventory variable domain $\in \{0, \cdots, T \}$.

Answer 3

You can start your index from $0$ to $n+1$ if you have $n$ months for which you need to write the balance equations. Then you can define $I_{-1}=0$ and $I_{n+1}=0$ if you don't want to keep inventory at the end of planning period. While you still need to define $I_{-1}$ as a variable but you can fix this variable to have the value of $0$ by:

$I_{-1}.lb = 0$

$I_{-1}.ub = 0$

Answer 4

Ok I see you have updated your code because in the previous version I thought I saw cumulative inventory being used in the constraints. If not used in constraint then yes you can simply use it as expression as you have used in the code INV0_[c,t]=INV0_[c,t]+INV_c[c,t]. But need to change it a bit.
Declare INV0_[c,t] as a python matrix like

INV0_ = [[0,0,0,0],
          [0,0,0,]]

Possible if you define it as numpy matrix gurobi throws an error but native python matrix doesn't have a problem.
Then in code

for t in range(months):
   for c in range(storages):
     INV0_[c][t]=INV0_[c][t-1]+INV_c[c,t]

Then to get the values of INV0 use

for t in range(months):
       for c in range(storages):
         print(INV0_[c][t].getValue())