Efficient Algorithm for Maximizing the Expected Profit from a Serial Production Line with Inspection Stations and Rework

Transcription

1 Discrete Optimization Day 5 August 2008, Technion Efficient Algorithm for Maximizing the Expected Profit from a Serial Production Line with Inspection Stations and Rework Tal Raviv Department of Industrial Engineering Tel Aviv University 1

2 Serial Production Line M 1 M 2 QC M 3 M 2 QC 3 4 Implications Labor, energy, capital and material costs Throughput Flowtime and WIP 2

3 Talk outline Problem definition Literature survey O(N 2 ) time algorithm for the cost minimization problem O(N 4 ) algorithm for the profit maximization problem Conclusion and directions for further research 3

4 Problem definition A Serial Production Line with N machines Identical jobs arrive according to some stationary process Each operation may fail with known probability Unlimited buffers A QCS may be installed after each machine A QCS detects any non conforming item and send it for rework 4

5 Notation Optional M 1 QC 1 M 2 QC 2 r G Revenue r B Penalty x i Mean processing time c i Variable processing cost p i Successes probability x ij Mean inspection time c ij Variable inspection cost f ij Fixed cost (per time unit) Y Set of installed QCSs a - Throughput 5

6 Objectives Cost minimization: For a given production rate (of conforming products), select a QCS configuration so as to minimize cost per unit of time at steady state. Profit maximization: Determine production rate and select QCS configuration so as to maximize the expected profit per unit of time at steady state. 6

7 What have been done? First QCS model presented by Lindsey and Bishop (1964). Non conforming items are discarded. A DP algorithm to minimize cost per job Yum and McDowell (1987) allow rework. MILP approach. Emmons and Rabinowitz (2002) - multistage deteriorating production systems Penn and Raviv (2007, 2008) Profit maximization 7

8 Segment a a M i+1 M j QC j QC i M i+2 Stability? 8

9 Talk outline Problem definition Literature survey Some observations O(N 2 ) time algorithm for the cost minimization problem O(N 4 ) algorithm for the profit maximization problem Conclusion and directions for further research 9

10 Solving the cost minimization problem The problem can be cast as the shortest path problem Node set {0,1,,N, N+1} Edge set all 0 i < j N+1 such that {i+1,i+2,,j} is a stable segment for a given throughput a Edge lengths - total cost (including rework) of production and inspection along the corresponding segment For last segments also include expected penalty cost A shortest path from node 0 to node N+1 represents the configuration that minimize the production cost for arrival (production) rate a 10

11 Example 3 Machines

12 Example Construction of edge set

13 Example Edge lengths

14 Example Edge lengths

15 Example Reduction to the Shortest Path Problem Optimal Solution: Y = {2,3}, Value =

16 Complexity The time and space complexity of our algorithm is O(N 2 ). Proof: Clearly the shortest path problem can be solved, using Dijkstra s algorithm in O(N 2 ). We also show how the graph can be constructed in O(N 2 ) 16

17 Talk outline Problem definition Literature survey Some observations O(N 2 ) time algorithm for the cost minimization problem O(N 4 ) algorithm for the profit maximization problem Conclusion and directions for further research 17

18 Observation If (Y,a) is an optimal solution for the profit maximization problem then a is either the maximal stable rate for the QCS configuration Y or a = 0. 18

19 19

20 Theorem: The cardinality of the set of all maximal stable production rates over all the possible QCS configurations is bounded from above by 20

21 Theorem: The cardinality of the set of all maximal stable production rates over all the possible QCS configurations is bounded from above by Proof Sketch: 1. The maximal stable production rate of each configuration is determined by the most loaded segment. 2. The maximal stable rate of a segment is independent of QCSs outside the segment 3. The total number of distinct segments is 21

22 Profit Maximization Algorithm Calculate the maximum rate of each segment (u,v) T (u; v) = Q v k=u+1 p k max (x u+1 ;:::;x v ;x 0 u(v)) Solve the cost minimization problem for each such rate and calculate the total profit rate Return the maximal profit rate obtained 22

23 Conclusions Efficient algorithm for the inspection allocation problem with rework First efficient algorithm for the profit maximization version of the problem Using similar methodology to Penn and Raviv (2007) Easily extendable to more complex notion of rework Rework may be different from first time production both in terms of time, cost Rework may be carried out by different sequence of operations 23

24 Further Research Reentrant line We have an hardness proof Mixed rework / discarding QA policy Complexity status? Other production modes E.g., in-tree assembly line 24