Wagner-Whitin algorithm

The Wagner-Whitin algorithm is an exact procedure presented in 1958 by Harvey M. Wagner and Thomson M. Whitin for determining the optimal lot size for a product with dynamic demand in single-stage production without taking into account capacity restrictions. The Wagner-Whitin problem is also referred to as the "Single-Level Uncapacitated Lot Sizing Problem" - SLULSP for short.

As with the classic batch size formula , it is assumed that the production speed is infinite and consumption is constant over the period. In the process, possible alternatives are determined in a forward calculation and then the optimal strategy is selected in a backward calculation. The Wagner-Whitin method also produces the optimum when fixed costs vary from period to period. The condition is that the respectively valid fixed costs are used in the periods.

An important implication of the Wagner-Whitin algorithm is the zero inventory property : production only takes place when the warehouse is empty. After that, the optimum requirement for a period is either completely covered by the stock or by the production of this period. A situation in which the demand is partly satisfied from production and partly from the warehouse, so that storage and set-up costs are incurred in one period, cannot be cost-minimal, because the set-up costs can be saved by moving production forward. An optimal lot always includes a total of complete period requirements. This general knowledge was taken into account as a basis for the heuristic procedure for determining the lot size.

The algorithm

The following example is presented in detail by Wallace J. Hopp and Mark Spearman.

${\ displaystyle t}$	Period (e.g. day, week, month) where is referred to as the planning horizon ${\ displaystyle T}$
${\ displaystyle D_ {t}}$	Demand in period t (in units)
${\ displaystyle c_ {t}}$	Production costs per unit in period t excluding setup and inventory costs
${\ displaystyle A_ {t}}$	Setup costs of a setup process in period t (i.e. per lot)
${\ displaystyle h_ {t}}$	Inventory management costs per unit of period t in period t + 1
${\ displaystyle I_ {t}}$	Remaining stock at the end of period t
${\ displaystyle Q_ {t}}$	Lot size in units in period t
${\ displaystyle Z_ {t}}$	Minimum cost of production in period t
${\ displaystyle j_ {t}}$	Last period in which, from the point of view of period t, production took place.

The following example is given:

t	1	2	3	4th	5	6th	7th	8th	9	10
${\ displaystyle D_ {t}}$	20th	50	10	50	50	10	20th	40	20th	30th
${\ displaystyle c_ {t}}$	10	10	10	10	10	10	10	10	10	10
${\ displaystyle A_ {t}}$	100	100	100	100	100	100	100	100	100	100
${\ displaystyle h_ {t}}$	1	1	1	1	1	1	1	1	1	1

Step 1

The algorithm first looks at a one-period problem. This is naturally quite simple:

${\ displaystyle Z_ {1} = A_ {1} = 100}$

and the last period in which production took place is

${\ displaystyle j_ {1} = 1}$

step 2

In the next step, the time horizon is set in the next period and a two-period problem is considered.

${\ displaystyle Z_ {2}}$ = the minimum of	${\ displaystyle = A_ {1} + h_ {1} D_ {2}}$	Produce in period 1
	${\ displaystyle Z_ {1} + A_ {2}}$	Produce in period 2

${\ displaystyle Z_ {2}}$ = the minimum of	${\ displaystyle 100 + 1 (50) = 150}$
	${\ displaystyle 100 + 100 = 200}$

Step 2 determines that production in period 1 results in lower total costs than production in periods 1 and 2. It remains to be noted that the last production takes place in week 1. ${\ displaystyle j_ {2} = 1}$

step 3

Step 3 expands the planning horizon by a further period, so that the minimum must now be determined from three calculation formulas

${\ displaystyle Z_ {3}}$ = the minimum of	${\ displaystyle = A_ {1} + h_ {1} D_ {2} + (h_ {1} + h_ {2}) D_ {3}}$	Produce in period 1
	${\ displaystyle Z_ {1} + A_ {2} + h_ {2} D_ {3}}$	Produce in period 2
	${\ displaystyle Z_ {2} + A_ {3}}$	Produce in period 3

${\ displaystyle Z_ {3}}$ = the minimum of	${\ displaystyle 100 + 1 (50) + (1 + 1) (10) = 170}$
	${\ displaystyle 100 + 100 + 1 (10) = 210}$
	${\ displaystyle 150 + 100 = 250}$

It is again cheaper to produce in period 1, so we

${\ displaystyle j_ {3} = 1}$

note.

Step 4

Now a four-period problem is calculated:

${\ displaystyle Z_ {4}}$ = the minimum of	${\ displaystyle = A_ {1} + h_ {1} D_ {2} + (h_ {1} + h_ {2}) D_ {3} + (h_ {1} + h_ {2} + h_ {3}) D_ {4}}$	Produce in period 1
	${\ displaystyle Z_ {1} + A_ {2} + h_ {2} D_ {3} + (h_ {2} + h_ {3}) D_ {4}}$	Produce in period 2
	${\ displaystyle Z_ {2} + A_ {3} + h_ {3} D_ {4}}$	Produce in period 3
	${\ displaystyle Z_ {3} + A_ {4}}$	Produce in period 4

${\ displaystyle Z_ {4}}$ = the minimum of	${\ displaystyle 100 + 1 (50) + (1 + 1) (10) + (1 + 1 + 1) (50) = 320}$
	${\ displaystyle 100 + 100 + 1 (10) + (1 + 1) (50) = 310}$
	${\ displaystyle 150 + 100 + (1) (50) = 300}$
	${\ displaystyle 170 + 100 = 270}$

This time the optimum is no longer in period 1. It is therefore better to only meet the demand for period 4 in period 4. It follows: . ${\ displaystyle j_ {4} = 4}$

Step 5 and beyond

Since the last production takes place in period 4, periods 1–3 are not taken into account in the following. The problem of period 4 is therefore treated and calculated as a one-period problem. Then the period is increased by 1 and a two-period problem is solved. This reduces the number of invoices to a fairly manageable number. The solution to the problem shown above is as follows

Last period of production	Planning horizon t
Last period of production	1	2	3	4th	5	6th	7th	8th	9	10
1	100	150	170	320
2		200	210	310
3			250	300
4th				270	320	340	400	560
5					370	380	420	540
6th						420	440	520
7th							440	480	520	610
8th								500	520	580
9									580	610
10										620
${\ displaystyle Z_ {t}}$	100	150	170	270	320	340	400	480	520	580
${\ displaystyle j_ {t}}$	1	1	1	4th	4th	4th	4th	7th	7 8 ${\ displaystyle \ lor}$	8th

rating

Although the process can be used with comparatively little effort, it has hardly found widespread use. This is mainly due to the uncertainty about the plan figures in practice. Like most optimizing methods, this also reacts to changes in the input values with mostly strongly changed output values. For this reason, the rolling planning that is widespread in practice , in which only one time window is taken into account in each period, is incompatible with the Wagner-Within algorithm and, in this respect, is subject to heuristic procedures. On the other hand, in terms of value, an optimum is often only slightly better than a good solution found using heuristics .

Another problem lies in the black box nature of the process: Due to the mistrust of the decision-makers in solutions that are difficult to understand, the more stable solutions in their development, for example the unit cost method or the period cost method or the cost compensation method, are often preferred.

A serious disadvantage is the failure to observe capacity restrictions, which can lead to non-executable production plans. Even with the widespread successive planning in subsequent stages of time and capacity planning, this deficiency cannot be completely compensated for.

swell

^ Harvey M. Wagner, Thomson M. Whitin: Dynamic Version of the Economic Lot Size Model. In: Management Science . Vol. 5, No. 1, 1958, pp. 89-96, JSTOR 2626974 .

^ Horst Tempelmeier : Material logistics. Models and algorithms for production planning and control in advanced planning systems. 6th, revised edition. Springer, Berlin et al. 2006, ISBN 3-540-28425-7 , p. 138.

^ Wallace J. Hopp, Mark L. Spearman: Factory Physics. Foundations of manufacturing management. 2nd Edition. McGraw-Hill, Boston MA et al. 2001, ISBN 0-256-24795-1 .

↑ Wolfgang Tysiak: Introduction to the manufacturing economy . Hanser, Munich et al. 2000, ISBN 3-446-21522-0 , pp. 94-104.

[Wagner1958-1] Harvey M. Wagner, Thomson M. Whitin: Dynamic Version of the Economic Lot Size Model. In: Management Science . Vol. 5, No. 1, 1958, pp. 89-96, JSTOR 2626974 .

[Tempelmeier2006-2] Horst Tempelmeier : Material logistics. Models and algorithms for production planning and control in advanced planning systems. 6th, revised edition. Springer, Berlin et al. 2006, ISBN 3-540-28425-7 , p. 138.

[Hopp-3] Wallace J. Hopp, Mark L. Spearman: Factory Physics. Foundations of manufacturing management. 2nd Edition. McGraw-Hill, Boston MA et al. 2001, ISBN 0-256-24795-1 .