Silver-meal heuristic

The silver-meal heuristic is a heuristic method of dynamic lot size determination . In this way, lot sizes are determined for production that cover the requirements and at the same time minimize the total costs. The method was published in 1973 by Edward A. Silver and Harlan C. Meal.

Prerequisite and question

For future periods (weeks, months ...) the needs or demands of a product are known. To produce a lot, setup costs K [euros / lot] are incurred, while the storage costs are h [euros / piece / period].

What is needed is a production plan in which the total costs are as low as possible.

Conflict of goals

Cost history for batch production

You made with fewer, larger lots, then the reduced set-up costs , the increase for up storage costs . Conversely, more lots lead to lower storage costs, but higher set-up costs.

Silver meal heuristic

The silver meal heuristic first considers a lot, which covers the demand for a period, and determines the costs. It then increases the lot size in such a way that the demand for a further period is covered and calculates the average costs for these periods. A period is added until the average costs per period increase. Here the procedure breaks off.

If you produce for the requirements of periods t to t + n, then the average costs are calculated

${\ displaystyle Z (t, t + n) = {\ frac {1} {n + 1}} \ left (K + h \ sum _ {k = 0} ^ {n} {k \ cdot y_ {t + k}} \ right)}$

n = number of periods

K = setup costs for one lot [euros / lot]

h = storage costs in euros per unit and period [euros / unit / period]

{\ displaystyle y_ {t + k}}

= Production volume in period t + k

The method terminates if Z (t, t + n) <Z (t, t + (n + 1)) = Z (t, t + n + 1).

Through elementary transformations, the termination condition can also be written as

{\ displaystyle Z (t, t + n) <h \ cdot (n + 1) \ cdot y_ {t + n + 1}}

example

For periods 8 to 12, the following requirements arise in pieces: 27, 22, 13, 19, 12. The setup costs are K = 200 euros, the storage costs 10 euros / piece / period.

If you only produce for period 8 at the beginning of period 8, i.e. a lot with 27 pieces, then only set-up costs are incurred, but no storage costs. The costs are therefore Z (8.8) = 200 euros .

If you produce at the beginning of period 8 for periods 8 and 9, i.e. a lot with 49 pieces, then set-up costs of 200 euros. In addition, 22 pieces have to be stored for the following period, which leads to storage costs of 22 * 10 euros = 220 euros. The average costs for periods 8 and 9 are therefore Z (8.9) = (200 euros + 220 euros) / 2 = 210 euros .

Since Z (8,8) <Z (8,9), i.e. the average costs increase with two periods, a lot is formed that only covers the demand for period 8.

The remaining periods 9 to 12 must now be examined in the same way:

Z (9.9) = 200 euros

Z (9.10) = (200 euros + 13 * 10 euros) / 2 = 165 euros <Z (9.9)

Z (9.11) = (200 euros + 13 * 10 euros + 2 * 19 * 10 euros) / 3 = 710/3 euros = 236.66 euros > Z (9.10) 19 pieces are stored for two periods

Thus, at the beginning of period 9, production takes place for periods 9 and 10 with a lot size of 35 pieces.

For the remaining periods 11 to 12 this results.

Z (11.11) = 200 euros

Z (11.12) = (200 euros + 12 * 10 euros) / 2 = 160 euros <Z (11.11)

Production quantities and stocks
Period t	8th	9	10	11	12	Total costs
Opening balance l _t	0	0	13	0	12
Demand y _t	27	22nd	13	19th	12
Lot size x _t	27	35	0	31	0
Closing stock l _{t + 1}	0	13	0	12	0
Setup costs	200	200	0	200	0	600
storage costs	0	130	0	120	0	250

Three lots will be issued. 13 and 12 pieces are stored for one period each. The total cost is

Z = setup costs + storage costs = 3 * 200 euros + (13 units + 12 units) * 10 euros / unit = 850 euros.

Solution not always optimal

The solutions of the silver meal heuristic are not always optimal. This is because the process stops as soon as the average costs increase. However, these average costs could fall again if one adds more periods for lot formation.

The following example shows a non-optimal solution.

Production quantities and stocks according to Silver Meal
Period t	20th	21st	22nd	23	Total costs
Opening balance l _t	0	1	0	1
Demand y _t	3	1	7th	1
Lot size x _t	4th	0	8th	0
Closing stock l _{t + 1}	1	0	1	0
Setup costs	200	0	200	0	400
storage costs	10	0	10	0	20th

Here two lots are issued, the costs are

{\ displaystyle Z = {\ text {setup costs}} + {\ text {storage costs}} = 2 \ cdot 200 {\ text {Euro}} + (1 {\ text {pieces}} + 1 {\ text {pieces} }) \ cdot 10 {\ frac {\ text {euros}} {\ text {pieces}}} = 420 {\ text {euros}}}

.

If, on the other hand, one were to produce the entire requirement with a single lot in period 20, the following solution would result:

Better production program
Period t	20th	21st	22nd	23	Total costs
Opening balance l _t	0	9	8th	1
Demand y _t	3	1	7th	1
Lot size x _t	12	0	0	0
Closing stock l _{t + 1}	9	8th	1	0
Setup costs	200	0	0	0	200
storage costs	90	80	10	0	180

Here the costs are only

Z = set-up costs + storage costs = 1 * 200 euros + (9 pieces + 8 pieces + 1 piece) * 10 euros / piece = 380 euros .

Criticism of the silver meal process

The silver meal process does not take capacities into account. This can lead to unrealizable solutions, for example because the required storage space is not available or the calculated lot sizes cannot be produced due to a lack of capacity. The storage properties of the product, such as its shelf life, are also neglected.

literature

Edward A. Silver, Harlan C. Meal (1973): A heuristic for selecting lot size requirements for the case of a deterministic time-varying demand rate and discrete-opportunities for replenishment. Production and Inventory Management 14 (2), 64-74.
Christian Ortmann, Ingo Siebeking: Heuristics for lot size planning in PPS systems , Award-winning diploma thesis in the Faculty of Economics at the University of Osnabrück, June 2000 Online ( Memento from November 26, 2013 in the Internet Archive ) (accessed on February 7, 2016)

Individual evidence

↑ The example is taken from the lecture notes by Prof. Dr. Ulrich Thonemann on the subject of "Operations Management - Production Planning" in the 2011/2012 winter semester at the University of Cologne
↑ Prof. Dr. Ulrich Thonemann: Operations Management: Concepts, Methods and Applications , Publisher: Pearson Studium; Edition: 2nd, updated and expanded edition. (April 8, 2010), ISBN 978-3827373168 , pages 314–315 Online (accessed December 24, 2011)

[Thonemann-Vorlesung-1] The example is taken from the lecture notes by Prof. Dr. Ulrich Thonemann on the subject of "Operations Management - Production Planning" in the 2011/2012 winter semester at the University of Cologne

[Thonemann-G-books-2] Prof. Dr. Ulrich Thonemann: Operations Management: Concepts, Methods and Applications , Publisher: Pearson Studium; Edition: 2nd, updated and expanded edition. (April 8, 2010), ISBN 978-3827373168 , pages 314–315 Online (accessed December 24, 2011)