One equation per period, Iₜ = Iₜ₋₁ + Pₜ − Dₜ, is what ties a plan together: the stock carried out of a period is the stock carried in, plus what was made, minus what was sold. The inventory variable is the thing that carries the link, so it is a decision of the model rather than a formula, and one aggregate row about total production is not the same statement.
Core ideas
The shape is one equality repeated. What makes it work is that the quantity linking two periods is a variable the model may choose.
Iₜ is the stock in the store at the end of period t, in units. The plan chooses it, subject to the balance equation and to Iₜ ≥ 0. Substituting a formula for it collapses the link into the objective and makes the constraint that forbids a negative store impossible to write.
Pₜ + Iₜ₋₁ − Iₜ = Dₜ for each t, with the first period taking the opening stock as data. The rows are equalities: the stock does not merely satisfy an inequality, it is determined by where it came from, and it appears in exactly two of the rows — the period that carries it out and the period that carries it in.
Demand in the second period must be met by stock that exists in the second period. The balance rows say so, period by period. A single row about total production over the horizon says something strictly weaker and cannot see the difference.
The material
The equation each period contributes, what the inventory variable carries, and the plan an aggregate constraint prefers instead.
A plant makes one product over four periods. Demand is 20, 35, 30 and 25 units; making one unit costs 9, 8, 7 and 6 as the periods go on; at most 60 units can be made in a period; carrying a unit into the next period costs 1; and there are 5 units in the store before the first period begins.
For each period t, with Pₜ made, Dₜ demanded and Iₜ left in the store at the end, the balance constraint is Iₜ = Iₜ₋₁ + Pₜ − Dₜ, where I₀ is the opening stock. Rearranged for a solver, with the data on the right, it is Pₜ + Iₜ₋₁ − Iₜ = Dₜ.
With Iₜ ≥ 0 for every t, the family of equations says exactly that demand is met in the period it occurs, out of stock that exists by then. Allowing Iₜ to go negative is a modelling decision with a name — a backorder — and it has to be made deliberately.
The cheapest plan makes 15, 35, 30 and 25 and carries nothing at any point, at a cost of 775. That is worth reading rather than accepting: making a unit gets cheaper every period and carrying one costs money, so there is no reason to build stock and every reason to make each unit as late as the balance rows allow — which is the period it is sold in.
Cut the capacity to 30 a period and raise the second period's demand to 40. Now the second period cannot make what it sells, so the plan makes 25 in the first period, carries 10 units forward, and makes 30, 30 and 25 after that, at a cost of 835. The first inventory variable is 10, and it is the only thing in the model that could have made the second period feasible at all.
The tempting simplification is one row instead of four: total production at least total demand. With 110 units demanded and 5 in the store that is P₁ + P₂ + P₃ + P₄ ≥ 105, and the inventory variables disappear along with the balance equations.
It solves, and it is cheaper: 675 against 775. The plan it prefers makes nothing at all in the first two periods and then 45 and 60 in the last two, because production is cheapest at the end and nothing in the model any longer says when a unit has to exist. Run that plan through the periods and 15 units of demand go unmet in the first period and 35 in the second: fifty units of demand unserved, by a plan the objective calls better.
This is the sharpest example on the course of a model being exactly wrong. Nothing is infeasible, no arithmetic is off, and the reported cost is a real number about a plan nobody would run. The aggregate row is a true statement about the plan; it is simply a much weaker one than the four rows it replaced, and the cost cannot show what was lost.
The same weakening is available inside the balance rows themselves. Drop I₂ ≥ 0 and the second period's row is satisfied by I₂ = −35 with nothing made, which reads as thirty-five units promised and not delivered. If backorders are what the situation allows, that is a variable to introduce and price on purpose; if they are not, the sign restriction is the row that says so.
Interactive
Inventory is the variable that carries one period into the next
Method
The variables come in two families and the rows come one per period. Doing the first period separately is what keeps the opening stock honest.
Pₜ is units made in period t and Iₜ is units in the store at the end of it. A four-period plan therefore has eight decisions, not four, and the second family is the one that carries the link.
Pₜ + Iₜ₋₁ − Iₜ = Dₜ. Get the signs right where the pattern is plain: what was carried in is added, what is carried out is subtracted, and the demand is on the right with the data.
There is no I₀ variable: the opening stock is data, so the first row is P₁ − I₁ = D₁ − (opening stock). Copying the middle pattern into the first period is the error this step exists to prevent.
Report Pₜ and Iₜ for every period, not just the total cost. A plan is a path through the horizon, and whether it ever holds stock is the thing the balance rows were written to decide.
The aggregate row's plan costs less than the balance plan. Running it forward period by period is what shows why the number is not the answer.
The aggregate plan satisfies its one row exactly — 105 units made — and misses fifty units of demand, then finishes with fifty units in the store. Both facts follow from the same thing: the row counts units over the whole horizon and says nothing about when they exist.
For a faded rehearsal, keep the balance rows and cut the capacity to 30 a period. The supplied first move is where the trouble must be: the second period demands 35 and can make at most 30, so its row can only be satisfied with stock carried in. Predict P₁ and I₁, then the rest of both paths and the cost, and check them against the lab — and say what happens to the model when the capacity goes to 20.
You should be able to declare two variables per period with units, write one balance equality per period with the opening stock handled separately, read the production and inventory paths off the solution, and say what an aggregate row would fail to say.
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