In its simplest form, Minimum Order Quantities (MOQs) represent the smallest quantity that can be bought from a supplier. MOQs are prevalent among most businesses that don’t classify themselves as

**the product**, where each quantity associated with each product included in the purchase order needs to reach the MOQ.**the order**, where the sum of all the quantities associated with all the products included in the purchase order needs to reach the MOQ.

Product-level MOQs are frequent when there are economies of scale involved with the production of every distinct reference. For example, a company that specializes in book printing, unless it specializes in retail on-demand prints, is likely to have a MOQ associated with every print order.

Order-level MOQs are frequent when there are economies of scale involved with processing and delivering the order. For example, a FMCG company selling detergents might only accept an order if it is large enough to fill at least half a truck, with different products if need be. In this situation, an MOQ expressed in a monetary amount typically acts as a proxy to cover the shipment costs.

While MOQs may sometimes be negotiated with suppliers to some extent, it is frequently not the case. Indeed, the supplier may not even have the processes and workflows in place to deal with smaller orders. This function is typically delegated to retailers or distributors that are precisely focusing their added-value on their capability to service products in highly granular quantities. By removing small orders from their supply chain, those suppliers can focus on achieving a superior supply chain performance through economies of scale.

For example, clothing manufacturers frequently have complex MOQs that include several of the following constraints:

- the minimal quantity, in meters of fabric, is 3000 meters for every color present in any ordered product, for the order as a whole.
- the minimal quantity, in eaches, for every ordered product is 600 pieces.
- the minimal quantity, in dollars, for the whole order, is 20,000 USD.
- the minimal quantity, in units, for the whole order, is 2,000 pieces.

In this example, the first constraint reflects that the supplier is buying fabric in rolls of 3000 meters, and thus, through the complex MOQ, this supplier is pushing its own ordering constraints down the supply chain to its clients.

Then, the second constraint reflects economies of scale at the product level - as discussed in the previous section - but this constraint is supplemented by the third constraint which imposes a minimal order volume in dollars. This 3rd constraint is intended to prevent customers from passing low-value orders, for example ordering 10,000 pairs of socks priced at 0.30 USD per unit.

Finally, the last constraint is introduced as a proxy of the transportation cost, as the clothing producer is likely to use a truck shipment to make its delivery to the customer - who might, in turn, leverage a container (maritime) shipment.

**lot multipliers**, where the quantities ordered, by product, need to be a multiple of a given integer. This constraint frequently reflects a choice of packaging, where the product is packed in boxes or pallets of X units.**Economic Order Quantity**(EOQ), this reflects the customer-side friction of the order, whereas the MOQ reflects the supplier-side friction.**price breaks**, where the marginal unit price charged by the supplier varies, typically decreasing, with the quantity being ordered.

While MOQs can be seen as a pure

Conceptually, the most direct way of optimizing an order under MOQ constraints consists of first assessing the economic returns associated with every single quantity that could (reasonably) be ordered, second, filtering out all the

Even with suitable software tools, MOQs tend to be fairly technical to tackle, especially due to the retroactive impact that MOQs have on supply chain planning. Indeed, the greater the MOQ, the more infrequent the orders will be, which means greater ordering lead times. Thus, as the applicable lead time when considering a supplier purchase order is typically the sum of the supplier lead time and the ordering lead time, this value is dependent on the MOQ. This in turn impacts the lead demand.

For a manufacturer, the factors impacting the choice of MOQs are:

- production batch size, if any
- set-up times and fixed costs on every production cycle
- packaging formats (i.e. boxes, pallets)
- customer acquisition costs
- negotiated deals with key clients

Based on these economic factors, it is possible to optimize the MOQs and let them evolve over time to properly reflect the changing market conditions. However, in practice, MOQs should not be revisited too frequently, as constant variations, even minor, would negatively impact the clients’ ordering practices. Nevertheless, MOQs should be revisited regularly in order to remain closely aligned with the market and the manufacturer’s strategy.

Soft MOQs are conceptually a variant of the Economic Order Quantity. However, in practice, soft MOQs are typically not the result of any kind of econometric analysis, but rather an emergency practice that tends to arise “naturally”, when the purchasing team cannot cope with the volume of orders and/or deliveries that would occur if ordered quantities were fractionned to the maximum extent that ordering and transportation costs would allow.

Soft MOQs are frequently used in conjunction with a weekly or monthly ordering schedule, which is another approach to achieve the same goal, that is, reduce the pressure on the purchasing team to manage highly granular supplier orders.

The concepts relevant for the general MOQ problem. We have:

- the
**items**which represent what can actually be purchased. Item quantities are frequently integral numbers; although there is no restriction here for them to be. - the
**ordered quantities**for every item (possibly zero) which represent a potential solution to the MOQ problem. - the economic
**rewards**associated to each extra unit for each item. - the
**costs**associated to the units to be acquired. The goal is indeed to maximize the reward for a given spending budget expressed in*costs*. Costs are typically expected to be flat per unit, but here we don't make assumptions; thus, price breaks may be taken into account. - the
**targets**which represent a way of specifying a stopping criteria which may not be the actual costs.

Ex: Frank the supply chain manager puts a target at 90% fill rate. The solving the MOQ problem consists of computing the smallest order - in costs - while *maximizing the rewards* that delivers a 90% fill rate. This order is NOT the smallest order possible to achieve the 90% fill rate - as this would be a pure fill-rate prioritization. Instead, it's the smallest order that, while prioritizing the rewards, is large enough to deliver a 90% fill rate. A pure fill-rate prioritization would have been a mistake because, unlike the stock reward, it does not take into account the cost associated to the generation of dead stock.

Let $I$ be the set of items being considered for ordering.

Let $q_i$ with $i \in I$ the quantity to be ordered for the item $i$.

Then, we define a series of functions.

- Let $r_i(q)$ be the
*reward*when holding $q$ units of the item $i$. - Let $c_i(q)$ be the
*cost*when buying $q$ units of the item $i$. - Let $t_i(q)$ be the
*target*when holding $q$ units of the item $i$.

The reward function can return positive or negative values, however both the cost and target functions are strictly positive: $$\forall i, \forall q, c_i(q) > 0 \text{ and } t_i(q) >0$$ Let $M$ be the set of MOQ constraints. For each $m \in M$, we have $I_m$ the list of items that belongs to the constraint $m$ and $Q_m$ the minimal quantity that should be reached to satisfy the constraint. Let $m_i(q)$ the function that defines the contribution of the item $i$ to the MOQ constraint $m$ when $q$ units are purchased. The constraint $m$ is said to be satisfied if: $$\forall i \in I_m, q_i = 0 \text{ or } \sum_{i \in I_m}m_i(q_i) \geq Q_m$$ Thus, all MOQ constraints can be satisfied in two ways: either by reaching the MOQ threshold, or by having all item's quantities at zero.

Then, let $C$ be the maximal cost that can be afforded for the purchase order. We define $\textbf{q}_C=(q_i)_i$ the best purchase order as: $$\textbf{q}_C = \underset{q}{\operatorname{argmax}} \left\{ \sum_i r_i(q_i) \text{ with $m$ satisfied } \forall m\in M \right\}$$ The purchase order is the "best" in the sense that is maximizes the reward for a given budget. The solution $\textbf{q}_C$ is not unique, however, this consideration is rather theoretical because the MOQ problem is too hard for an exact resolution anyway. For the sake of simplicity, we proceed as if the solution was unique in the following.

Finally, let $T$ be a target minimum, we define $\textbf{q}^T$ with $$C^T = \underset{C}{\operatorname{min}} \left\{ \left(\sum_{q_i \in \textbf{q}_C} t_i(q_i) \right) \geq T \right\}$$ and $$\mathbf{q}^T = \textbf{q}_{C^T}$$ The solution $\mathbf{q}^T$ is built on top of $\textbf{q}_C$, that is, it's the smallest optimal (budget-wise) ROI-maximizing solution that is good enough to fulfill the target.

On the purchasing side, too frequently, orders are simply rounded-up to the next MOQ, without quantifying the inventory risks involved with a large order, and without quantifying the impact on the ordering schedule either. Seasoned and experienced supply chain practitioners are aware of this problem and typically resort to extensive Excel spreadsheets, lacking any better alternative.

On the manufacturer side, MOQs tend to be too frequently driven by the batch sizes and/or the choice of packaging, based on a “guesstimation” rather than a quantitative analysis weighing the pros and cons. Once MOQs are adopted, many manufacturers rigidify their processes around those values that become an unchallenged

Finally, as MOQs are a tough - and often misunderstood - numerical challenge, many software vendors rely on simplistic rules to accommodate them, treating them as an edge case, even for those MOQs that are involved with the majority of orders either passed or received. Lokad has developed dedicated numerical solvers that are precisely tailored for the resolution of MOQ problems.

- (1) The terminology of hard vs soft MOQs was actually coined at Lokad a few years back to shed light on supply chain problems, as hard and soft MOQs reflect two fairly distinct intents. However, some of our clients are still using the term MOQ to refer indifferently to both situations.