Inventory Under Production And Demand

From Production and Demand to Inventory

Inventory is a stock: it records how much material is on hand at a particular time. Production adds to that stock, while demand removes from it. A useful model therefore begins by identifying the rates of production and demand, the initial inventory, and the rule for what happens if demand exceeds the available stock.

Let \(I(t)\) denote the amount of inventory on hand at time \(t\). Let \(p(t)\) be the production rate and \(d(t)\) the demand rate. Both rates are nonnegative. If inventory is measured in items and time in hours, then \(p(t)\) and \(d(t)\) are measured in items per hour. The inventory itself cannot be negative: a negative value would describe a backlog or deficit, not physical stock on hand.

This model treats production as available to meet demand as it occurs. When inventory is positive, demand is met from production and stock. When inventory is zero and production is below demand, all production can be used immediately, but the remaining demand is unfilled. If instead unfilled orders are carried forward as a backlog, the state variable and its model are different.

The Balance Before Inventory Runs Out

Over a time interval on which inventory stays positive, conservation of stock gives a direct balance. The change in inventory equals the total amount produced minus the total amount demanded. In integral form, if \(I(s)>0\) throughout the interval from \(s\) to \(t\), then

Here the integrals are cumulative amounts: for example, \(\int_s^t p(u)\,du\) is the total production between times \(s\) and \(t\). If the rates are continuous on the interval, the Fundamental Theorem of Calculus gives the differential balance

This equation applies while there is inventory to remove. If its solution reaches zero, continuing the same equation could produce negative values. That continuation would be a backlog model, not an on-hand inventory model.

The boundary rule has a simple interpretation. At zero inventory, if production exceeds demand, the difference becomes new stock. If production is less than or equal to demand, no on-hand stock accumulates. The excess demand is unfilled. The phrase “almost everywhere” allows the rate equation to hold except possibly at isolated transition times where the left and right rates differ.

Constant Rates and the Depletion Time

Suppose production and demand are constant over an operating period, with rates \(p\) and \(d\), and inventory starts at \(I_0>0\). While stock remains positive, the net rate is \(r=p-d\), so the balance is \(I(t)=I_0+rt\), taking the start of the period as time zero. If \(r<0\), this expression reaches zero after \(I_0/(-r)\) time units. If \(r\geq 0\), inventory does not deplete during that period.

Inventory Across Changing Operating Periods

Production and demand often change at scheduled times. In each period, first use the current inventory and the period’s net rate to determine what happens. If the period’s net rate is negative, compare the time needed to deplete the current stock with the length of that period. If stock reaches zero partway through, set inventory to zero for the remainder of any interval in which demand still exceeds production. Do not carry a negative value into the next period.

Variable Rates and a Depletion Time

When rates vary continuously, integrate the net rate while inventory remains positive. A convenient way to organize the calculation is to define the cumulative production and cumulative demand from time zero:

Before any stockout, the balance is \(I(t)=I_0+P(t)-D(t)\). To find a candidate depletion time, set this expression equal to zero and solve. Then check that the inventory is positive before that time and that the resulting time lies within the period where the given rates apply.

Why the Boundary Rule Is Necessary

The boundary condition can be expressed as a conservation balance that includes unfilled demand. Let \(U(t)\) be the cumulative amount of demand that has not been met by time \(t\). Then

The correction \(U(t)\) is nondecreasing: unfilled demand can accumulate, but it cannot be undone by later production. It increases only when inventory is zero and demand exceeds production. This correction is exactly what prevents the on-hand stock from becoming negative. When inventory remains positive, \(U\) is constant, so the usual production-minus-demand balance applies without modification.

This also explains why an inventory model and a backlog model must not be confused. If orders that cannot be filled are recorded as a negative inventory, the balance \(I_0+P-D\) can continue below zero. That can be appropriate for a backlog variable, but it does not represent the quantity physically on hand. The model must match the meaning of the state variable.

A Common Mistake: Continuing Below Zero

For the constant-rate example, extending the unconstrained expression past the stockout gives \(I(7)=48-8(7)=-8\) units. This is a correct calculation for the unconstrained net balance, but it is not a valid on-hand inventory value. It treats all demand as met even after the available stock has been exhausted.

When calculating a depletion time, also check the direction of the net rate. Solving \(I_0+(p-d)t=0\) gives a positive time only when \(p-d<0\). If production is at least demand, that constant-rate period cannot deplete a positive starting inventory.

Summary

An inventory balance begins with a starting amount and the difference between production and demand. While inventory is positive, integrate that net rate to find the stock. If the balance reaches zero, enforce the nonnegative boundary: excess demand becomes unfilled demand rather than negative on-hand inventory. For changing rates, solve for the first zero of the integrated balance, verify the candidate against the time interval and the inventory path, and treat rate changes and stockouts as possible transition points where a derivative may fail to exist.