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Multi-period planning

Inventory carries forward.

Chapter: IX — Resource allocation · lesson id multi-period · practice lesson

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Add time. Demand changes period to period, and so does production capacity. You can pre-produce and carry inventory forward — at a cost per unit per period.

Demand {30, 50, 20} over three periods, capacity {60, 30, 25}. Period 2 demand (50) exceeds capacity (30) — you have to pre-produce 20 units in period 1 and carry them. Carrying cost = 2 per unit per period.

Minimum carry: 20 units from t1 to t2 → carry cost 2·20 = 40.

Predict before running

Period 2 demand 50 exceeds period 2 capacity 30. How does the solver cope?

• Pre-produce 20 in period 1 and carry them
• Demand drops to capacity (50 → 30)
• The model is infeasible

Starter code

model multi_period
set T = {1, 2, 3}
const demand = {30, 50, 20}, forall t in T
const capacity = {60, 30, 25}, forall t in T
const inv_cost = 2
var 0 <= prod, forall t in T
var 0 <= inv, forall t in T

constr prod_{t} <= capacity_{t}, forall t in T
constr inv_{1} = prod_{1} - demand_{1}
constr inv_{2} = inv_{1} + prod_{2} - demand_{2}
constr inv_{3} = inv_{2} + prod_{3} - demand_{3}

min sum_{t in T}{inv_cost * inv_{t}}
end

import elytica
def main():
  elytica.init_model("multi_period")
  elytica.run_model("multi_period")
  for i in [1,2,3]:
    print("prod"+str(i), "=", elytica.get_variable_value("multi_period", "prod"+str(i)))
  for i in [1,2,3]:
    print("inv"+str(i), "=", elytica.get_variable_value("multi_period", "inv"+str(i)))
  print("obj =", elytica.get_best_primal_bound("multi_period"))
  return 0

Hint

Inventory balance: inv_{t} = inv_{t-1} + prod_{t} − demand_{t}.

What counts as correct

• no-error — Model solves
• numeric-match — Minimum carrying cost is 40 (carry 20 units t1→t2)
• code-contains — Reads the objective from the solver (no hardcoding)

Mastery variant

Different demand, capacity, and a higher carrying cost. Where does the bottleneck sit?

model multi_period_v
set T = {1, 2, 3}
const demand = {20, 45, 30}, forall t in T
const capacity = {40, 25, 35}, forall t in T
const inv_cost = 3
var 0 <= prod, forall t in T
var 0 <= inv, forall t in T

constr prod_{t} <= capacity_{t}, forall t in T
constr inv_{1} = prod_{1} - demand_{1}
constr inv_{2} = inv_{1} + prod_{2} - demand_{2}
constr inv_{3} = inv_{2} + prod_{3} - demand_{3}

min sum_{t in T}{inv_cost * inv_{t}}
end

import elytica
def main():
  elytica.init_model("multi_period_v")
  elytica.run_model("multi_period_v")
  for i in [1,2,3]:
    print("prod"+str(i), "=", elytica.get_variable_value("multi_period_v", "prod"+str(i)))
  for i in [1,2,3]:
    print("inv"+str(i), "=", elytica.get_variable_value("multi_period_v", "inv"+str(i)))
  print("obj =", elytica.get_best_primal_bound("multi_period_v"))
  return 0

Read this through MCP: read with path: "lessons/multi-period", or resource hlpl://lessons/multi-period.