Integration of Demand Planning and Manufacturing Planning

(1)

Task 879.2: Integration of Demand Planning

and Manufacturing Planning

Task leader: Yon-Chun Chou Co-PI: Argon Chen

National Taiwan University 2002.03.13

Progress report in liaison meeting

(2)

The Problem of Product Grouping

• Tool #6 in not a sensitive tool.

• Tool #7 is a sensitive tool across product families.

• Tool #3 is sensitive to both inter- and intra-family variation.

B F C J Family Tool ID 2 4 5 10 16 3 6 11 14 15 25 28 1 3.20 3.20 3.20 2.80 2.60 3.20 3.60 3.60 3.60 3.60 3.00 3.00 3 6.36 6.36 6.36 8.44 6.36 6.36 8.24 7.76 7.76 7.76 8.12 8.12 5 5.52 5.52 5.52 5.68 5.84 5.52 7.44 6.32 6.32 6.32 6.48 6.48 6 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 0.92 7 5.92 5.92 5.92 5.92 4.00 5.92 5.92 5.92 5.92 5.92 5.92 5.92 8 22.35 22.35 22.35 23.37 23.26 19.55 32.79 27.82 27.82 27.82 26.75 26.75 9 0.22 0.22 0.22 0.34 0.34 0.22 0.34 0.34 0.34 0.34 0.22 0.22

(3)

Example: 4 products and 3 tool sets

All products in 1 Group

Upper (Ug,k) Lower (Lg,k) Range (Rg,k)

Tool 1 6.33 5.15 1.18 Tool 3 20.99 16.17 4.82 Tool 4 15.61 7.93 7.68 Product Processing Time ₁ ₂ ₃ ₄ Tool 1 6.33 6.23 5.16 5.15 Tool 3 20.93 20.99 16.17 16.18 Tool 4 15.6 15.61 7.93 7.94

(4)

Notation

B C Family Tool ID 2 4 5 10 3 6 11 14 15 1 3.20 3.20 3.20 2.80 3.20 3.60 3.60 3.60 3.60 3 6.36 6.36 6.36 8.44 6.36 8.24 7.76 7.76 7.76 7 5.92 5.92 5.92 5.92 5.92 5.92 5.92 5.92 5.92 9 0.22 0.22 0.22 0.34 0.22 0.34 0.34 0.34 0.34 Tool: k Product group: g Product: i

Decision variable: X

_i,g

Range of processing times: R

_g,k

k g k g k g U L R _, = _, − _,

(5)

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(6)

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(7)

Disjunctive Constraints

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(8)

A Formulation for Product Grouping

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(9)

The Constraint for Empty Groups

B C J Family Tool ID 2 4 5 10 3 6 11 14 15 1 3.20 3.20 3.20 2.80 3.20 3.60 3.60 3.60 3.60 3 6.36 6.36 6.36 8.44 6.36 8.24 7.76 7.76 7.76 5 5.52 5.52 5.52 5.68 5.52 7.44 6.32 6.32 6.32

• Group J is empty.

• Moving any product from Groups B or C to Group J

will result in a grouping that is at least as good.

• Therefore, the no-empty-group constraint is not

necessary.

(10)

Work in Progress

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i

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k i i k i i

on tool

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demand

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is

where

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requiremen

Tool

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⋅

=

•

The above analysis has focused on W_i,k. Need to include the

effect of D_i.

•

The tool requirements of multiple product groups need to be combined.

Between sensitive tools

Between sensitive tools and non-sensitive tools Between non-sensitive tools

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Formal Proof for the Empty Group Constraint

•

Definition: Given a constrained minimization problem P, a problem Q is a quasi-relaxation of P if for every feasible

solution of P with objective function value v, there is a feasible solution of Q having an objective function values less than or equal to v.

•

The formulation with the no-empty group constraint will be called P. This constraint states that each group must contain at least 1 product. Removing this constraint has the effect of

legitimating solutions with one or more empty groups. It is

straightforward to say that Q is a quasi-relaxation of P, because Q is less constrained. We will prove that the reverse is also true, that is, P is also a quasi-relaxation of Q.

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P is a Quasi-relaxation of Q

•

To prove it, we must show that, for every feasible solution of Q, there is a corresponding feasible solution of P having a better or equal objective function value.

•

The feasible solutions of Q belong to either of two cases.

1.

In the first case, the solutions contain no empty groups. Therefore, those solutions are also solutions of P.

2.

In the second case, the solutions contain one or more empty

groups. Suppose that s is a feasible solution of Q. There exists a non-empty group g1 and an empty group g2. By moving a

product from g1 to g2, the resultant solution is a solution in P. Because g2 contains only one product, its contribution to the

objective function is 0. Because g1 now has one less product, its contribution to the objective function will be less or the same. Therefore, the resultant solution has a function value at least as good as that of solution s.