Capacitated solder sizing problem

Models with dynamic lot size determination

The Capacitated Lot-Sizing Problem ( CLSP ) is a model of dynamic lot size planning .

It assumes several products for which there is limited production capacity . It must be determined in which period which lots should be issued and how big they should be. The requirements of the individual periods are assumed to be known. A sequence in which the individual lots are to be produced within the periods is not determined.

In contrast to the Discrete Lot Sizing and Scheduling Problem (DLSP), it is possible to issue several lots in one period. It is therefore more suitable for medium-term planning, while the DLSP is suitable for short-term detailed planning, as it also determines a sequence of the lots.

Description of the model

Assumptions

The CLSP makes the following assumptions:

single-stage, multi-product production
finite production speed
demand that changes over time (dynamic demand)
the means of production (machines) are only available for a limited time within the individual periods. The available capacity is generally different in each period
for the laying of a batch falling fixed set-up costs of
the storage costs are proportional to the quantity stored
finite planning period

Mathematical description

A mathematical description looks like this:

Parameter:

${\ displaystyle b_ {jt}}$ Requirement of product j = 1,2, ..., n in period t = 1,2, ... T. The production quantity of a certain period can be used for satisfaction in that period.
${\ displaystyle p_ {j}}$ Manufacturing speed
${\ displaystyle f_ {j}}$ fixed setup costs per setup process
${\ displaystyle c_ {j}}$ Storage cost rate for the products still in stock at the end of a period
${\ displaystyle \ kappa _ {t}}$ Capacity in period t
${\ displaystyle \ kappa _ {j} ^ {p} = 1 / p_ {j}}$ Capacity requirement to produce a unit of measure of product j

Variables:

${\ displaystyle z_ {jt}}$ is equal to 1 if the product is manufactured in period t and 0 if not
${\ displaystyle l_ {jt}}$ Inventory of product j at the end of period t
${\ displaystyle q_ {jt}}$ Lot size of product j in period t

Objective function: ${\ displaystyle \ min K (z, l, q) = \ sum _ {j = 1} ^ {n} \ sum _ {t = 1} ^ {T} (f_ {j} z_ {jt} + c_ { j} l_ {jt})}$

under the constraints

${\ displaystyle l_ {j, t-1} + q_ {jt} -l_ {jt} = b_ {jt}}$ "Inventory balance equation"

${\ displaystyle q_ {jt} \ leqq z_ {jt} \ sum _ {\ tau = 1} ^ {T} b_ {jt}}$ ensures that the lot size is only included in the calculation when a lot is placed.

${\ displaystyle \ sum _ {j = 1} ^ {n} \ kappa _ {j} ^ {p} q_ {jt} \ leqq \ kappa _ {t}}$

${\ displaystyle l_ {jt} \ geqq 0}$

${\ displaystyle q_ {jt} \ geqq 0}$

${\ displaystyle z_ {jt} \ in \ {0,1 \}}$

${\ displaystyle l_ {j0} = l_ {jT} = 0}$

It differs from the well-known Wagner-Whitin model in that it includes several products, limited production capacity and finite production speed. The CLSP is one of the NP-hard problems and can be solved precisely using branch-and-bound algorithms. There are also heuristics .

Individual evidence

↑ Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, p. 133.
↑ Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, p. 134.
↑ Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, pp. 133-146.

[1] Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, p. 133.

[2] Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, p. 134.

[3] Domschke, Scholl, Voss: Production planning: process organizational aspects . 2nd edition, Springer, Berlin, 1997, pp. 133-146.