The Stochastic Facility Layout Problem

Transcription

1 Lisa Turner Supervisors: Yefei Zhao and Stein W.Wallace 9th September, 2011

2 Background Introduction It is important to design production floors so they are: Flexible Robust Large amounts of money can be saved by producing a good layout.

3 Background Facility Layout Facility Layout looks at how to place machinery on the production floor Aim Minimise the moving costs between machines (a) Functional (b) Cellular (c) Maximal Distribution Different types of Layout

4 Basic problem Cost of a particular layout d ij is the distance machine i and j and is the equivalent to the cost between machines v ij = flow between machines in position i and j Total movement costs are n i=1 n j=1 d ijv ij m1 m2 m3 m4

5 The Optimisation Problem The Deterministic Model For a given order s: multiple copies of the same machine v ni m j = flow from nth machine type i to mth machines type j d lk = distance from position l to k 1 if nth machine of type i x ni k = is assigned to location k 0 otherwise The objective function Min z = N i N j K i=1 j=1 n i =1 m j =1 k=1 l=1 K v ni m j d kl x ni kx mj l

6 The Optimisation Problem The Stochastic model If more than one order is taken into consideration π s is the probability of order s The flow between machines under order s is v ni m j s The objective function Min z = S N i N j K s=1 i=1 j=1 n i =1 m j =1 k=1 l=1 K π s v ni m j sd kl x ni kx mj l (1)

7 The Optimisation Problem Constraints K x ni k = 1 (2) k=1 N i n=1 n i =1 x ni k = 1 (3) P N i p=1 i=1 n i =1 N i N j n i =1 m j =1 N i i=0 n i =1 v ni m j s c mj (4) v ni m j ps = v ijps (5) v ni m j s = N q q=0 r q=1 v mj r qs (6)

8 Simplifications Solving the Problem Using the Optimisation Program MPL Software, and existing code, I can model the problem. The Facility Layout Optimisation problem is an NP-hard problem. By looking at small number of machines, I can solve the problem optimally.

9 Simplifications My simplified model Constant machine capacity 4 machines (m11,m12,m21,m31) or 5 machines (m11,m12,m21,m31,m32) p1 p2 p1 p2 p3 p3 p4 p4 p5 (d) 4 Machines (e) 5 Machines No queuing Equal probabilities The positions machines can fill

10 Results and Applications Result Lowest costs are when two machines of the same type are visited second. m11 m31 m21 m12 Layout for order Application: Buy two machines with lower capacity

11 Results and Applications Result p2 is a key position machines type 2 is never in p3 p1 p2 p3 m21 m31 m11 p4 p5 m32 m12 (f) Positions machines can fill (g) Layout for order Application Place important machines in more central positions

12 Type of Layout Type of Layout? Machines of the same type are rarely placed next to each other m11 m32 m12 m31 m21 The layout for all 6 orders There appears to be a link between positions and machines. Applications: A Functional Layout is not optimal Maximal distribution may not be the best type of layout.

13 Conclusion Further work Look at a larger number of machines. m13 m32 m41 m21 m14 m42 m11 m12 m31 The layout for 9 machines with order Machines with different capacities.

14 Conclusion Conclusion Difficult to find a balance between the number of machines and insight. 5 machines was enough to relate to the bigger problem. A larger number of machines would take longer to solve and be harder to visualise. Any Questions?