Pulp: slack variable to identify & measure extent infeasible in supply problem

Question

I am currently modelling a supply problem that attempts to optimise a rail schedule which moves products from a production plant, to a warehouse to satisfy sales.

The model is working fine (thanks in part to OR!) however I would like to create a variable that shows the extent of a supply shortfall. Currently if the current inventory is 0, and demand is > 0, then the model is unable to satisfy the constraints and returns 'Infeasible'.

I would like to define a slack/soft constraint that captures the shortfall so that the end result is always 'optimal' even though a shortfall is realised, and shows this extent.

Unfortunately my current soft constraint is not recognised by the model and the output is still 'infeasible'

Desired outcome:

desired_output_df.head(10)

>>>

date            product      current_inventory     sales_demand     inventory_shortfall
'2020-01-01'    'AFM'        10000                 5000             0
'2020-01-02'    'AFM'        5000                  5000             0
'2020-01-03'    'AFM'        0                     6000             6000


# Sales Demand
# Storage levels must meet sales demand
for date, grade in storage_inventory_vars:
  model += storage_inventory_vars[date, product] \
      + insufficient_storage_supply[(date, product)] \
      >= sales_demand[date][product]

The variable insufficient_supply[(date, product)] is the key slack constraint here that I would like to measure as it should prevent an infeasible solution (owing to insufficient supply to meet demand).

You can observe in the sales demand data on the 2020-05-18 and 19 that there is a very large spike in demand for AFE so that it greatly exceeds supply.

Here, if storage_inventory_vars['2020-05-18, 'AFE'] == 50,000 then insufficient_supply[('2020-05-18, 'AFE')] should == -50,000. The sum should then produce an optimal solution is the sum is greater than the demand.

All help very gratefully received, thank you.

Answer 1

The following constraints are infeasible :

 _C129: Rail_Loadings_From_Washplant_('2020_05_22',_'ABC',_'PRE')
        + Rail_Loadings_From_Washplant_('2020_05_22',_'ABC',_'ZBF')  = 25200

 _C134: Rail_Loadings_From_Washplant_('2020_05_23',_'ABC',_'PRE')
        + Rail_Loadings_From_Washplant_('2020_05_23',_'ABC',_'ZBF')  = 25200

 _C161: Rail_Loadings_From_Washplant_('2020_05_22',_'ABC',_'ZBF')
        + Port_Inventory_Levels_('2020_05_21',_'ZBF')
        - Port_Inventory_Levels_('2020_05_22',_'ZBF')  = 200000

 _C165: Rail_Loadings_From_Washplant_('2020_05_23',_'ABC',_'ZBF')
        + Port_Inventory_Levels_('2020_05_22',_'ZBF')
        - Port_Inventory_Levels_('2020_05_23',_'ZBF')  = 200000

 _C241: Port_Inventory_Levels_('2020_05_21',_'ZBF') <= 200000

I think there is a problem with your inventory equations. Not exactly sure where yet.

Finding the exact error is not that easy. Either there is a typo, either the model is not written correctly. My suggestion : back to basics, write the equations of the linear problem, and before anything, lets see if the model is properly written.

Answer 2

I found the solution.

1- The storage inventory definition is as follows:

model += storage_stockpile_current[product] \
              + pulp.lpSum(
                    train_consignment_variables[(date, plant, product)] 
                    for plant in _plants_combo) \
              - sales_demand[date][product] \
              == storage_inventory_vars[(date, product)]

2 - Given the addition of the slack constraint:

for date, grade in storage_inventory_vars:
  model += storage_inventory_vars[date, product] \
      + insufficient_storage_supply[(date, product)] \
      >= sales_demand[date][product]

Whenever sales_demand greatly exceeds the storage variable, the equation becomes inbalanced, because it has a lower bound of 0, i.e. cannot be negative. Therefore the definition needs to reflect this slack constraint:

model += storage_stockpile_current[product] \
              + pulp.lpSum(
                    train_consignment_variables[(date, plant, product)] 
                    for plant in _plants_combo) \
              - sales_demand[date][product] \
              + insufficient_storage_supply[(date, product)] \
              == storage_inventory_vars[(date, product)]

A big thank you to Kuifje for their help.