**How do manufacturing companies currently know when to order how much raw materials?**

*The goal of this article is to present and discuss the most widely used methods adopted by manufacturing companies for deciding when to order how much inbound materials. We also illustrate each method using a concrete example and evaluate the strengths and weaknesses of each method in the context of industrial purchasing. *

Manufacturing companies purchase large quantities of inbound materials and base components to produce finished goods, most of which are bought from external suppliers. As of today, manufacturing companies use a process named “**Material Requirements Planning**” to estimate the quantity of each material needed for production and when it is required.

The material planner (who is the person in charge of placing the order) must decide how large the order should be, as placing numerous orders each month could be very costly as well as inventory holding costs. Their decision should be made to find an **appropriate trade-off between the costs** to find a viable financial solution.

This optimization problem is called “**lot sizing**”. As more and more constraints are added to the equation such as quality control or perishability, it becomes quite thorny. In this article, we will present how lot sizing is done as of today and review the lot sizing techniques available to industrial companies to plan their purchases of raw materials and components.

**1. Material Requirements Planning: the first step before computing lot sizes**

Determining adequate lot sizes is needed to maintain acceptable inventory and service levels. Minimum order quantities demanded by suppliers often blow up inventory levels especially for items with long inventory turn cycles. To determine the lot size that will minimize the costs, industrial companies use Materials Requirements Planning (MRP).

Materials Requirements Planning (MRP) is a software-based solution used by every Entreprise Ressource Planned (ERP) system whose purpose is to:

**Ensure the availability**of materials, components, and products for planned production and for customer delivery,- Maintain the
**lowest possible levels of inventory**, **Plan production**,**delivery**schedules, and**purchasing**activities.

MRP is especially suited for manufacturing environments where the demand of many materials and components depend on products with external requirements.

- Demand for end products is independent.
- Demand for components depends on end products, which is why we call it dependent demand

The three major **inputs **of an MRP system are:

- the master production schedule, which expresses how much of each material is required and when it is required
- the product structure records, also known as the Bill of Material Records (
**BOM**), contains information on every item or assembly required to produce the end products - the inventory status records which contains the status of all materials in inventory

After running the Material Requirements Planning, industrial companies know the following:

- What items are required for the production
- How many of them are required
- When are they required (i.e. when they will be consumed in production)

However, they must now decide when to order what quantity of the desired material in order to minimize costs, which is when lot sizing comes into the equation.

**2. What are the different methods applied to determine optimal lot size?**

The different lot sizing techniques implemented across industrial companies can be categorized into **static, periodic, or dynamic**. **Static lot sizing** consists of placing a fixed order quantity or ordering exactly the amount that is needed to cover forecasted demand. **Periodic lot sizing** groups together the requirements that lie in a determined period. For **dynamic lot sizing**, the cumulative forecasted demand throughout the entire time horizon is taken into account to determine the optimal order quantities. As time progresses and the production requirements for the new time horizon adjusts, the previously developed planned orders might be adapted.

**Static lot sizing procedures**

Static lot sizing methods consist of ordering a fixed quantity or the exact amount of requirements for the date needed.

**1. Fixed Order Quantity**: This method involves ordering a fixed quantity when the reorder point is reached. The quantity often depends on the supplier-specific constraints.

→ To the example

**2. Economic Order Quantity**: This formula was developed in 1913 by Ford W. Harris which results in an order quantity that minimizes the total holding costs and ordering costs.

The formula is the following: Q= (2DS/H)0,5

where:

Q = Quantity to be ordered

D = Demand in units (typically on an annual basis)

S = Order cost (per purchase order)

H = Holding costs (per unit, per year)

→ To the example

**3. Lot for Lot (L4L)**: It consists of ordering the exact amount that matches the net requirement for each period.

→ To the example

**4. Single Lot**: It entails ordering the total requirement for the defined period in one go.

→ To the example

Static lot sizing procedures are easy to automate but do not provide much flexibility as a high demand variability could result in high inventory holding costs. The Lot for Lot procedure stands out as an exception as inventory is minimized which on the other hand results in extremely high ordering costs.

**Periodic Lot sizing procedures**

**Periodic lot sizing** groups several requirements within a time interval together to form a lot. Periodic lot sizing procedures are effective when used with cheap items when inventory cost is low.

**5. Period Of Supply (POS)**: A period of supply such as 3 weeks is specified, for which the net requirements across that period are ordered together each time.

→ To the example

**6. Period Order Quantity**: It consists of applying the EOQ model to a subset of the entire period under consideration at a time, where the demand is translated into the average requirement of each subset period.

→ To the example

**Dynamic lot sizing procedures**

**Dynamic lot sizing** considers the effect of cumulative needs across time to determine the best order quantities. As time advances and new production requirements for inbound materials are known, previously developed planned orders may end up changing. This could also be the result of forecast variability.

*In the examples for each technique provided at the end of this article, we oversimplify the situation for ease of comprehension, by assuming perfectly forecasted requirements which in reality is often not the case.*

**7. Least Unit Cost (LUC)**: An order size is determined such that the demand for the next “n” periods will be met, where “n” minimizes the average cost per unit.

→ To the example

**8. Least Total Cost (LTC)**: In this heuristic technique the optimal solution corresponds to the order plan where the order costs approximate the carrying costs.

→ To the example

**9. Part Period Balancing (PPB)**: This method represents a variation of the LTC approach. It converts the ordering cost to its equivalent in part periods, “the economic part period (EPP)”, by dividing the ordering cost by the cost of carrying one unit for one period. When “the cumulative parts period” which corresponds to the excess inventory x the number of weeks that it is carried, exceeds the EPP, we take it as the optimal lot size.

→ To the example

**10. Silver Meal (SM)**: Silver and Meal developed this heuristic in 1973, which determines the average cost per period. It first considers a lot that covers the demand for a period and calculates the costs. It then increases the lot size to cover the requirements for another period and calculates the average cost for that period. One period is added at a time until the average cost per period increases, after which the process stops.

→ To the example

**11. Uncapacitated Multi-Supplier Order Quantity Problem with Time-Varying All-units Discounts (UMSOQP****VAD****)**: Developed by Horst Tempelmeier in 2002, it is the model implemented by SAP APO (Advanced Planner and Optimizer) the software of SAP AG. The heuristic starts with a LUC solution and iterates on improvement steps until the solution cannot be further improved given the input parameters.

→ To the procedure

Lots of other methods could be applied to lot sizing like the McLaren’s order moment (MOM), Groff Reorder Procedure, Economic Order Interval (EOI), Maximum Part-period Gain algorithm (MPG), Wagner and Whitin’s (WW)… We will not cover all techniques as some are very difficult to illustrate easily and some are better applicable in the context of production planning.

On the table below we compared the different lot sizing procedures, evaluating their strengths and weaknesses according to 8 criteria.

**3. How lot sizing optimization should be done today?**

Many could think it would make more sense to only apply dynamic lot sizing procedures as static and periodic procedures allow a lower level of flexibility resulting in unnecessary high levels of inventory or too many orders.

Still, many **industrial companies continue to use static and periodic lot sizing techniques like EOQ **because of their simplicity to implement, although supplier constraints such as minimum order quantities or rounding values are not considered in the original formula. This implies that an optimum is difficult to reach, especially in terms of costs, **resulting in financial hidden losses** tied in the order planning processes (Nydick 1989).

For dynamic lot sizing techniques, every method could perform well depending on the environment and type of material (Collier 1980). **Some methods still managed to stand out as academic studies** have shown that the Silver Meal (SM) represents the best trade‐off between cost‐effectiveness and robustness (Jeunet 2000) and the Wagner Whitin (WW) algorithm performs best for the dynamic lot sizing problem and is often used as a benchmark for simpler heuristic techniques (Baciarello 2013, Beck 2015).

However, if these methods perform well in academic studies, there are some **clear real-world limitations**, as for example, the Silver-Meal is not easily expandable to include real-life constraints such as minimum order quantities or multiple quantity discounts (Benton 1996). Furthermore, the Wagner Whitin (WW) algorithm is not used in practice as it is not available on ERP systems due to its complexity and difficulty to implement (Bahl 2009).

Given the increased adoption of data analytics into business decision-making, **the problem has to be solved algorithmically** **in order to realize substantial savings **as it allows for more flexibility to integrate all the constraints relative to the complex lot sizing problem (Kulkarni 2019). The UMSOPQVAD provides good results integrating most supplier constraints but requires a high level of data requirement in order to perform well and does not consider material perishability.

To summarize, many techniques consider the requirements as a starting point to compute an optimal lot size, however perfectly forecasted requirements are not possible. Moreover, supplier constraints such as minimum and maximum order quantities or rounding values often have a significant impact on the final order quantity (Enns 2005). Adding material constraints such as perishability, we realize that **constraints should be integrated in the optimization problem from the beginning to ensure optimal purchasing**.

For all the above reasons, at GenLots*, we undertook the challenge to develop a proprietary algorithm built with machine learning especially for industrial use, considering every real-world constraint as a starting point to find the solution that will minimize the Total Cost of Ownership in less than 5 seconds. The algorithm is easy to integrate with most known ERPs and has already been tested in production with Merck (pharmaceutical industry) effectively saving up to 10% of the total cost per material.

** GenLots is the first company fully dedicated to order planning, optimizing lot sizing with machine learning concepts. If you are interested in learning more about GenLots and evaluating how our software can meet your needs, don’t hesitate to get in touch with us at contact@genlots.com.*

**4. Illustrations and examples of classical lot sizing techniques**

For all examples we take the following variables:

Order cost (per purchase order): 100€

Carrying rate: 17% yearly

Cost of the material: 25€

#### Static Lot Sizing Techniques

**1. Fixed Order Quantity **

It specifies the number of units arbitrarily to be ordered each time an order is placed.

**Reorder Point:**

For the first two methods, the question of when to place the order was not relevant, but for other methods like the Fixed Order Quantity, the supply planner must decide when to place the order. In this case, the planner looks at when the projected available inventory is anticipated to go below the safety stock level, and must hence place an order “X” weeks in advance depending on the total lead time for materials to arrive on time to avoid a potential stock-out.

In the example below, the supply planner may arbitrarily decide that an acceptable order quantity is 500 units (which might represent a pallet’s volume) according to their business knowledge. In the first two weeks no orders can be received as it takes two weeks to be received if we place an order as early as in the first period. Our starting inventory is 500 units. In the fourth week, we have requirements of 260 units that will cause the available inventory to go below the safety stock level if we do nothing (280-260<200), that is why we plan an order of 500 units in the first week that will arrive in the third week. We plan another one to be received in week 5 to cover the demand until the end of the horizon.

Lead time | 2 weeks | Week | |||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | |

Projected Available inventory | Initial stock: 500 | 400 | 200 | 580 | 320 | 690 | 570 | 385 | 270 |

Planned receipt | 500 | 500 | |||||||

Planned release | 500 | 500 | |||||||

Order costs | 100 | 100 | |||||||

Inventory costs | 32.7 | 16.3 | 47.4 | 26.1 | 56.3 | 46.6 | 31.4 | 22 |

Order costs: 200€

Inventory costs: 252.7€

Purchasing costs: (500 + 500) x 25€ = 25000€

Total cost: **25’452.7€**

→ Back to Fixed Order Quantity description

**2. Economic Order Quantity (EOQ) **

Q= ((2*7995*100)/(0,17 x 25)0,5 = ( 1’599’000 / 4.25 )0.5 = **613**

In the example below, we will need to order this quantity in the first week for it to be delivered in the third week and avoid going below the safety stock level. Similarly, we have to reorder in the fifth week so as to not go under the safety stock level during week 7.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 693 | 433 | 303 | 796 | 611 | 496 | |

Planned receipt | 613 | 613 | ||||||||

Planned release | 613 | 613 | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 56.6 | 35.4 | 24.8 | 65 | 49.7 | 40.5 |

Order costs: 200€

Inventory costs: 321€

Purchasing costs: (613+613) x 25€ = 30’650€

Total cost:** 31’171€**

→ Back to Economic Order Quantity description

**3. Lot for Lot (L4L)**

For the Lot for Lot procedure, we must order the exact amount that matches the net requirements in each period. In the simple example below, we have a lead time of two weeks which means that we cannot order during the first two weeks but we assume that we had already placed some orders before to cover for the demand. As in the third week, we plan to consume 120 units, we are going to plan an order in week 1 for the third week, of 120 units which correspond to the expected requirements of this period.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 200 | 200 | 200 | 200 | 200 | 200 | |

Planned receipt | 120 | 260 | 130 | 120 | 185 | 115 | ||||

Planned release | 120 | 260 | 130 | 120 | 185 | 115 | ||||

Order costs | 100 | 100 | 100 | 100 | 100 | 100 | ||||

Inventory costs | 32.7 | 16.3 | 16.3 | 16.3 | 16.3 | 16.3 | 16.3 | 16.3 |

Order costs: 600€

Inventory costs: 146.8€

Purchasing costs: 930 x 25€ = 23’250€

Total cost:** 23’997€**

→ Back to Lot for Lot description

**4. Single Lot**

For the Single Lot procedure, the order quantity is equal to the total requirement and only one order is to be placed. In the example below we place in the first week an order for the third week of 930 units which corresponds to the total requirements for weeks 3-4-5-6-7-8.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 1010 | 750 | 620 | 500 | 315 | 200 | |

Planned receipt | 930 | |||||||||

Planned release | 930 | |||||||||

Order costs | 100 | |||||||||

Inventory costs | 32.7 | 16.3 | 82.5 | 61.3 | 50.7 | 40.9 | 25.7 | 16.3 |

Order costs: 100€

Inventory costs: 326.4€

Purchasing costs: 930 x 25€ = 23’250€

Total cost:** 23’676.4€
**→ Back to Single Lot description

#### Periodic Lot Sizing Techniques

**5. Period Of Supply (POS)**

The Period Of Supply procedure computes the lot size will be equal to the net requirements for a given period of time in the future.

We select 3 weeks as the default supply period for this example. The requirements for weeks 3 to 5 amount to 510 units. We hence order 510 units in the first week so that they are delivered by the third week to cover the weeks 3-4-5. We then order 420 units during the fourth week to cover demand for the last 3 weeks.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 590 | 330 | 200 | 500 | 310 | 200 | |

Planned receipt | 510 | 420 | ||||||||

Planned release | 510 | 420 | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 48.2 | 27 | 16.3 | 40.9 | 25.3 | 16.3 |

Order costs: 200€

Inventory costs: 223€

Purchasing costs: (510+420) x 25€ = 23’250€

Total cost:** 23’673€**

→ Back to Period Of Supply description

**6. Period Order Quantity (POQ)**

The Period Order Quantity uses the EOQ model for a limited period where the demand is represented as the average requirements per week. The formula used is the following: EOQ / Avg. Period Usage.

EOQ: 613 (as computed before)

Avg Period Usage: (1230)/8 = 153.75

POQ = EOQ / Avg Period Usage = 613 / 153.75 = **4 periods**

In the example below, the POQ computed above is 4 periods which means we have to consider the demand for four periods for each order. For weeks 3,4,5 and 6 the total projected gross requirements is 815 units, this is why we schedule an order of 630 units in the first week. Available inventory will then go below safety stock in week 7 if we do not reorder, but we need to know what the requirements are beyond the end of the horizon.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 710 | 450 | 320 | 200 | ? | ? | |

Planned receipt | 630 | ? | ||||||||

Planned release | 630 | ? | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 58 | 36.7 | 26.2 | 16.3 | ? | ? |

Order costs: 200€

Inventory costs: 186.2€ + (inventory costs of the last two periods (unknown))

Purchasing costs: (630 + ?) x 25€

Total cost: we need the requirements of the future periods to estimate the total cost.

→ Back to Period Order Quantity description

#### Dynamic Lot Sizing Techniques

**7. Least Unit Cost (LUC)**

The goal of the Least Unit Cost method is to minimize the average cost per unit. Unit is defined as “one piece of equipment/raw material/component”.

We find the order size that will cover the next “n” periods, where “n” is set to minimize the average cost per unit.

The Least Unit Cost method is a dynamic lot-sizing technique that adds ordering and inventory carrying cost for each trial lot size and divides by the number of units in each lot size, picking the lot size with the lowest unit cost.

For this example we’ll use a different table to compute the optimal lot size according to this technique.

Gross requirements | Week | Cum order quantity | Excess inventory | Week carried | Order cost | Carrying cost | Cum carrying cost | Total cost | Unit cost |

120 | 3 | 120 | 0 | 0 | 100 | 0 | 0 | 100 | 0.83 |

260 | 4 | 380 | 260 | 1 | 100 | 21.2 | 21.2 | 121.2 | 0.32 |

130 | 5 | 510 | 130 | 2 | 100 | 21.2 | 42.4 | 142.4 | 0.28 |

120 | 6 | 630 | 120 | 3 | 100 | 29.4* | 71.8 | 171.8 | 0.27 |

185 | 7 | 815 | 185 | 4 | 100 | 60.4 | 132.2 | 232.2 | 0.28 |

115 | 8 | 930 | 115 | 5 | 100 | 47 | 179.2 | 279.2 | 0.30 |

Cum order quantity: cumulative quantity to order each week

Excess inventory: quantity carried that is not needed in period n but will be needed further on

Week carried: number of weeks the excess inventory is carried

Carrying cost per week per unit = 0,17 x 25€ / 52 weeks = 0,08173

Carrying cost week 6 = 120 x 0.08173 x 3 = 29.4***
**Unit cost = Total cost / Cum order quantity

In the example below the Least Unit Cost technique would recommend ordering 630 units which would cover the projected demand until week 6 as the cost per unit would be 0,27€ per unit per week. After week 6, we start the process once more when we have more visibility over the projected demand of the next few weeks.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 710 | 450 | 320 | 200 | ? | ? | |

Planned receipt | 630 | ? | ||||||||

Planned release | 630 | ? | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 58 | 36.7 | 26.2 | 16.3 | ? | ? |

Order costs: 200€

Inventory costs: 186.2€ + (inventory costs of the last two periods (unknown))

Purchasing costs: (630 + ?) x 25€ = ?

Total cost: we need the requirements of the future periods to estimate the total cost.

→ Back to Least Unit Cost description

**8. Least Total Cost (LTC)**

The Least Total Cost method is a dynamic lot sizing technique that calculates the order quantity by comparing the carrying cost and the ordering cost for various lot sizes and selects the lot in which these are most nearly equal (Chase 2002). To obtain a result close to the optimum, lots of different scenarios must be analyzed to see which one will minimize the total cost, this is why we will only look at three different examples.

**a. Two weeks coverage scenario: order less often to balance order costs and carrying costs **

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 460 | 200 | 320 | 200 | 315 | 200 | |

Planned receipt | 380 | 250 | 300 | |||||||

Planned release | 380 | 250 | 300 | |||||||

Order costs | 100 | 100 | 100 | |||||||

Inventory costs | 32.7 | 16.3 | 37.6 | 16.3 | 26.2 | 16.3 | 25.7 | 16.3 |

Order costs = 300€

Inventory costs = 167.4 €

Purchasing costs: 930 x 25€ = 23’250€

**Total Cost = 23’717.4€ **

**b. Two orders **

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 590 | 330 | 200 | 500 | 315 | 200 | |

Planned receipt | 510 | 420 | ||||||||

Planned release | 510 | 420 | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 48.2 | 27 | 16.3 | 40.9 | 25.3 | 16.3 |

Order costs = 200€

Inventory costs = 223 €

Purchasing costs: 930 x 25€ = 23’250€

**Total Cost = 23’673€ **

**c. Two orders placed differently**

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 710 | 450 | 320 | 200 | 315 | 200 | |

Planned receipt | 630 | 300 | ||||||||

Planned release | 630 | 300 | ||||||||

Order costs | 100 | |||||||||

Inventory costs | 32.7 | 16.3 | 58 | 36.8 | 26.2 | 16.3 | 24.5 | 16.3 |

The ideal lot size would be 510 units for the third week and 420 units for the sixth week as it is the solution where the order costs get as close as possible to the carrying costs.

Order costs = 200€

Inventory costs = 227.1 €

Purchasing costs: 930 x 25€ = 23’250€

**Total Cost = 23’677.1€ **

→ Back to Least Total Cost description

**9. Part Period Balancing (PPB)**

This method represents a variation of the LTC approach. It converts the ordering cost to its equivalence in part periods, “the economic part period (EPP)”, by dividing the ordering cost by the cost of carrying one unit for one period. When “the cumulative parts period” which corresponds to the excess inventory x the number of weeks that it is carried, exceeds the EPP, we take it as the optimal lot size.

EPP = Order cost / carrying cost per period per unit

EPP = 100 / 0,08173 = **1223**

Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | |

Cumulated order qty | 120 | 380 | 510 | 630 | 815 | 930 | |||

Excess inventory | 260 | 130 | 120 | 185 | 930 | ||||

Week carried | 0 | 1 | 2 | 3 | 4 | 5 | |||

Parts period | 260 | 260 | 360 | 740 | 575 | ||||

Cum parts period | 0 | 260 | 520 | 880 | 1620 | 2195 |

**1620 **> **1223 **-> order **815 **to cover demand until week 7

On the example below, we place an order during the first week of 815 units which will arrive in the third week. We’ll have to reorder during the sixth week but as we don’t know future requirements, there is no point computing the lot size.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 895 | 635 | 505 | 385 | 200 | ? | |

Planned receipt | 815 | ? | ||||||||

Planned release | 815 | ? | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 73.1 | 51.9 | 41.3 | 31.4 | 16.3 | ? |

Order costs = 200€

Inventory costs = 253 € (+ inventory costs of the last period (unknown))

Purchasing costs: (815+?) x 25€ = ?

**Total Cost = **we need the requirements of the future periods to estimate the total cost.

→ Back to Part Period Balancing description

**10. Silver Meal (SM)**

The Silver Meal heuristic developed in 1973 by Silver and Meal requires determining the average cost per period as a function of the number of periods the current order is to span and stopping the computation when this function first increases.

K = order cost = 100€

h = inventory holding cost per unit per week = 0,08173€

rn = requirements

C (1) = K

C (2) = (K + hr2 )/2

C (3) = (K + hr2 + 2hr3 ) / 3

….

Until C(t) > C(t-1)

Period | 3 | 4 | 5 | 6 | 7 | 8 | C(t) | |

Gross requirements | 120 | 260 | 130 | 120 | 185 | 115 | ||

Excess inventory at the end of the week | Q | Per period costs | ||||||

1 week, week 3 | 120 | 0 | 100 | |||||

2 weeks, weeks 3 to 4 | 380 | 260 | 0 | 60.6 | ||||

3 weeks, weeks 3 to 5 | 510 | 390 | 130 | 0 | 44.3 | |||

4 weeks, weeks 3 to 6 | 630 | 510 | 250 | 120 | 0 | 53 | ||

1 week, week 6 | 120 | 0 | 100 | |||||

2 weeks, week 6 to 7 | 305 | 185 | 0 | 62.4 | ||||

3 weeks, week 6 to 8 | 420 | 300 | 115 | 0 | 47.8 |

C(1) = 100

C(2) = (100 + 260 x 0.08173 ) / 2 = 60.6

C(3) = (100 + 390 x 0.08173 + 2 x 130 x 0.08173) / 3 = 44.3

C(4) = (100 + 510 x 0.08173 + 2 x 250 x 0.08173 + 3 x 120 x 0.08173) / 4 = 53

**53 > 44.3**

In week 6:

C(1) = 100

C(2) = (100 + 305 x 0.08173) / 2 = 62.4

C(3) = (100 + 300 x 0,08173 + 2 x 115 x 0,08173) = 47.8

In the example below, we order 510 units in the first week and 420 in the sixth week.

Lead time | 2 weeks | Week | ||||||||

Safety stock | 200 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |

Gross requirements | 100 | 200 | 120 | 260 | 130 | 120 | 185 | 115 | ||

Projected Available inventory | Initial stock: 500 | 400 | 200 | 590 | 330 | 200 | 500 | 315 | 200 | |

Planned receipt | 510 | 420 | ||||||||

Planned release | 510 | 420 | ||||||||

Order costs | 100 | 100 | ||||||||

Inventory costs | 32.7 | 16.3 | 48.2 | 27 | 16.3 | 40.9 | 25.7 | 16.3 |

Order costs = 200€

Inventory costs = 223.4 €

Purchasing costs = 930 x 25€ = 23’250€

**Total Cost = 23’673.4€**

→ Back to Silver Meal description

**11. SAP APO heuristics – UMSOQPVAD**

UMSOQPVAD stands for Uncapacitated Multi-Supplier Order Quantity Problem with Time-Varying All-units Discounts. This particular implementation was developed by Horst Tempelmeier in 2002. It is the model implemented in SAP’s APO (Advanced Planner and Optimizer) software.

The heuristic starts with a LUC solution and iterates on improvement steps. The iteration stops when the solution cannot be improved given the input parameters.

The improvement steps are about:

- Price changes: if a price is planned to be increased, it might be worth ordering it earlier
- Splitting of orders: evaluate splitting and deletion of orders
- Combine orders: Combining small orders could substantially reduce ordering costs.
- Postponement of partial order quantities: for products with high demand variability it may happen that in several consecutive periods there are low demand periods followed by one large demand period. If the large demand quantity is included in a low demand interval, it may be favorable to make one extra order later at the cost of ordering costs as it may save a lot in inventory costs.