# Prioritized Ordering Definition

By Joannès Vermorel, last updated December 2015

Supply chain literature classically focuses on ordering policies where each separate item is treated in complete isolation to all the other items: the decision to order more units of item A is strictly independent from the decision to order more units of item B. However, this approach entails significant intrinsic limitations. In contrast, the prioritized ordering policy emphasizes multi-item decisions, where each item competes for capital allocation with all the other items. In practice, prioritized ordering provides much more fine-grained control over inventory performance, and, when the proper predictive technologies are available, prioritized ordering allows to reach superior inventory performance. Lokad systematically recommends adopting a prioritized ordering policy whenever possible.

## The four classic inventory policies

A typical inventory policy seeks answers to the following questions: When should a replenishment order be placed? and How large should the replenishment order be? However, before quantitatively addressing these issues, we need to decide on the form of the inventory control policy. There are numerous possible control systems, and the most common ones are listed below (1):
• Order-Point, Order-Quantity (s,Q) System: A fixed quantity Q is ordered whenever the inventory position drops to the reorder point s or lower. The inventory position takes into account the net stock and the on-order stock; that is the materials requested but not yet received from the supplier.
• Order-Point, Order-Up-To-Level (s,S) System: Like the (s,Q) system, a quantity of goods is reordered whenever the inventory position drops to the reorder point s or lower. However, instead of reordering a constant quantity, the order is sized to raise the inventory position to the order-up-to-level S.
• Periodic-Review, Order-Up-To-Level (R,S) System: Every R units of time, this system proceeds like the (s,S) system and raises the inventory position to the level S. This policy is typically used when ordering is not fully automated and goes through manual and scheduled validation by the Purchasing Manager.
• Periodic-Review (R,s,S) System: This is a combination of the (s,S) and (R,S) systems. In this case, the inventory position is checked every R units of time. If it is below the reorder point s, a sufficient quantity is ordered to raise it to S. In particular, the (s,S) is a special case of (R,s,S) with R=0.

Supply chain literature provides ample theoretical proofs - some dating back from the 1960s - where, given certain assumptions, certain policies are superior to others. However, from a more modern perspective, those proofs bear little practical significance because they drastically frame the ordering process under simple (simplistic) assumptions, which fail to properly account for the actual economic drivers.

## Economic drivers in constant flux

The classic supply chain viewpoint separates all the items in order to process them in isolation. However, one of the key supply chain insights gathered over hundreds of companies serviced by Lokad is that considering items in isolation makes little sense in practice:
• New items enter the market all the time, while older items exit the market all the time as well.
• Items may have substitutes of varying quality, ranging from perfect substitutes to loose ersatz.
• Fulfilling the demand may require a combinations of items in stock, thus intensifying the impact of stock-outs of an individual item when this item is frequently bundled or kitted.
• The access to a new lower-priced supplier for one item can vastly reshuffle the inventory strategy towards stocking more in order to intensify sales on a certain segment.
• The access to a new, faster operating supplier for one item can largely deprioritize the ordering of this item compared to other items where only slow and unreliable suppliers are available.

From a very practical perspective it does not make sense to spend 1€ more on an item while there is an expected return of 2€ within one year, as long as there are other alternative items with 3€ of expected returns within one year.

Inventory is optimized only when the capital allocation for inventory maximizes the market potential of the company while taking all inventory risks into account. Within this capital allocation, all items are in constant competition with each other for every marginal investment. Each item should be assessed against its expected returns and its expected costs for the next additional unit to be ordered.

However, the variables that drive the expected returns and the expected costs are themselves also in a constant state of flux. Examples are plentiful: the cost of capital varies as the company gains or losses access to liquidity, items’ gross margins vary as competitors change their prices and force other companies to adjust their prices as well, storage space pressure varies at different times during the year as the constraint of the fixed-space warehouse becomes a much more significant constraint during the high-season compared to the low-season.

## The prioritized ordering policy

The prioritized ordering policy, as the name suggests, provides a prioritized list of items to be purchased. Each line is associated with the minimum quantity that can be ordered for a given item, which we assume here to be 1 unit for the sake of simplicity (see constraints on priority list for more complex situations). Every item appears many times on the list, and in practice, items are frequently interleaved. This implies that once 1 extra unit is purchased for a given item, the next most profitable extra unit to be purchased is unlikely to be for the very same item - although this might happen from time to time.

In practice, we recommend using the stock reward function to access the economic value of each unit to be purchased. This function indicates the expected returns (in dollars or euros) if the unit were to be purchased. The stock reward function is used to build the score for each unit which actually define the prioritization.

Conceptually, the priority list is unbounded: it goes to infinity, with each subsequent line being less profitable than the previous one. In practice however, the list stops when it reaches zero profitability, and probably well before that point as a minimal amount of marginal profitability is required to cover all fixed costs. And it goes without saying that there is no point in wasting processing power to even consider vastly unprofitable scenarios.

From a formal perspective, the purchasing priority list is built with a score function $s(u,k)$ where $u$ is a SKU and $k$ the kth unit to be purchased. The score is usually decreasing with $k$, that is $s(u,k) > s(u,k+1)$, because extra inventory typically comes with diminishing returns. However in certain situations, the score may locally increase with $k$ (ex: teacher purchasing books for a classroom with a requirement of 20 units). Then, the list of all possible pairs $(u,k)$ are ordered in decreasing order against their score. This ordered list is the actual purchase priority list.

Conceptually, prioritized ordering leaves two questions open:
• It does not specify when to stop ordering within the list.
• It does not specify the frequency of the reviews for ordering.

The issue of deciding when to stop ordering is treated in more detail in the following section.

Concerning the frequency of reviewing the ordering, for more modern supply chain situations, the answer is as follows: a daily periodic review is necessary for any day that is open for review. In fact, refreshing the priority list can be completely automated, and therefore, if it is correctly implemented, a daily ordering review should only incur very marginal costs, minutes or work, or even less if the system is on auto-pilot. Apart from this, while sub-day reviews are possible in theory, the reality of transportation of physical goods typically involves irreducible delays that would not be shortened by reviewing inventory multiple times per day (2).

## Truncating the priority list

As seen above, the priority list has no end, at least in theory. However, in order to make an order, the practitioner needs to truncate the list, so that it reflects a definite list of quantities to be ordered at the very moment the order is to be carried out. The truncation itself requires a halting criterion of some kind. Let’s review the most obvious ones:
• Up-To-Capital-Level: items are purchased up to a certain threshold for the entire amount of capital allocated to inventory, counting both current inventory and the extra inventory being considered for the purchase. On the plus side, this approach always fits with the cash flow constraints of the company. On the negative side, this approach says nothing about the performance of the threshold itself.
• Up-To-ROI-Level: items are purchased as long as the full amortized ROI is higher than a given threshold; as ROI decreases steadily as we go further down the list (higher stock levels have strong diminishing returns), the threshold always ensures that the list gets truncated.

No matter which threshold is considered, another function $g$ needs to be introduced. The function $g$ is assumed to be decreasing when moving down the priority list. Let $g_\text{min}$ be the stopping threshold. Then, we should be moving down to the $N$th line in the list as follows: $$N = \text{argmax}_n \{ g(\mathbf{s}_n) \geq g_\text{min} \}$$ Where $\mathbf{s}_n$ is the inventory position after purchasing all the $n$ first lines of the priority list.

## Superiority of prioritized ordering

Only empirical evaluations can prove the superiority of one inventory policy over another. Indeed, purely theoretical considerations are typically misleading because while it’s fine to have the assumptions violated to some degree (e.g. demand can be approximated as stationary in the very short term), it’s very difficult to quantitatively assess the full impact of such violations. Some approaches prove to be very resilient to real-world conditions, other much less so.

At Lokad, we have observed that when a probabilistic forecasting technology is available, that is, a forecasting technology capable of forecasting the respective probabilities for the entire future demand level – and not merely forecasting the future mean or median demand level – then, approaches that rely on the purchase priority list systematically demonstrate superior inventory performance.

The reasons that explain this superior performance are numerous. Below, we list the positive aspects of using a purchase priority list that we observe the most often:
• Prioritization is quite robust against all sorts of biases. When many items are over-forecast they may remain correctly prioritized. Thus, a (relatively) systematic bias has little impact on the actual ordering. A little amount of bias is not sufficient to sink a top item very far down the list, and conversely, it is not sufficient to bubble-up a tail item up to the top.
• Prioritization accommodates well all sorts of non-linear multi-item constraints. With classic ordering policies, it’s a challenge to integrate something as basic as a warehouse storage constraint into the ordering policy. With purchasing prioritization, it’s as natural as truncating the list when the warehouse is full.
• Prioritization is much more local to the current inventory level. When a company adjusts its inventory strategy, for example switching to much higher service levels, classic ordering policies generate big “jumps” in stock levels which disorganize the supplier chain. In contrast, the purchase priority list offers the possibility to make the transition as smooth as desired as it is just an adjustment in the truncation threshold.
• Prioritization better accommodates loose scheduling. If a company orders one container every two weeks on average, using classic ordering policy tends to come with a lot of friction as one needs to monitor the exact date when the quantity to be ordered reaches a threshold that is just below the container capacity. Once the container capacity is exceeded, it’s up to the Purchasing Manager to manually remove overflowing quantities to fit the container. In contrast, the prioritization approach always provides the next most profitable container at any time.

In fact, whenever Lokad had the chance to compare a classic ordering policy (as listed above) with a prioritized ordering policy, the prioritized ordering policy was such an obvious and immediate winner that further benchmarking the two methods was considered a moot point by both Lokad and our client.

Also, these comparisons were fair in the sense that the forecasting technology powering the classic approach and the prioritized approach were both developed by Lokad with similar degree of technology refinement - i.e. it would have been unfair to compare an ordering policy powered by an advanced forecasting engine, against another policy powered by only a basic forecasting engine.

## Notes

(1) Inventory Management and Production Planning and Scheduling, Third Edition, Edward A. Silver, David F. Pyke, Rein Peterson

(2) In some industries, such as aerospace, urgent requests for spare parts, colloquially referred to as AOG requests (Aircraft On Ground), benefit from specific supply chain circuits geared towards swift deliveries, where every minute counts. The pharmaceutical industry also benefits from similar “emergency” circuits. For these kinds of urgent circuits, it is typically not recommended to examine the situation from an inventory optimization viewpoint, as maintaining the circuit itself (with night shifts and similar processes) drives the bulk of the costs.