1. Dynamic lot sizing and tool management in automated manufacturing systems M. Selim Aktürk, Siraceddin Önen presented by Zümbül Bulut

2. Content Problem Definition Literature Mathematical Formulation of the Problem Solution Approach Proposed Algorithms Results Conclusion

3. The Problem An automated machining environment, A single CNC turning machine, A finite planning horizon divided into T periods, In each period multiple items are produced, Deterministic, but time-varying, demand for every part in every period, 1.2. T.

4. The Problem Each operation can be performed by a set of alternative tool types from variety of available tool types, # of tools on hand is limited. Part Operation

5. The Problem Backlogging is not allowed, Initial and final inventory levels are assumed to be zero, For the machining operations, the cutting speed and the feed rate will be taken as a decision variables, The depth of the cut is given as an input, The CNC machine can work for a limited number of hours.

6. Statement of the Problem Satisfy the demand by producing all of the parts in each period, Perform all required operations to produce the parts, Use available tools only, Use the CNC machine without violating the availability hours constraint, Minimize the total cost

7. Literature Lot sizes are taken as predetermined input Decide on tool allocations and machining parameters Problems - empty feasible solution spaces - unnecessary limit on the number of alternatives possible for tool management problem

8. Objective Lot sizing Tool management Lot sizing + Tool management

9. Objective Solve lot sizing and tool management problems simultaneously to determine the following decision variables; 1. Lot sizing decisions, i.e. in what quantities each part will be produced, 2. Tool allocation, i.e. how tools will be allocated to parts in terms of quantities and allocation scheme, 3. Machining conditions selection, i.e. what the cutting speed and feed rate will be for each operation of each part.

10. Mathematical Formulation of the Problem Objective Function: Min Total Production Cost Total Production Cost = setup cost + inventory holding cost+ machining cost + non-machining cost + tooling cost

11. Mathematical Formulation of the Problem Constraints: 1. Production and inventory balance constraints: production quantity in period t + beginning inventory in period t - ending inventory in period t = demand for period t production quantity <= M*Y pt Y pt = 1 if part p is produced in period t, 0 otherwise.

12. Mathematical Formulation of the Problem 2. Machine hour availability constraints: machining time + non-machining time + setup time <= maximum available m/c hours 3. Tool assignment constraints: If part is produced in period t, then each operation of this part is assigned to a single tool-type of its candidate tool set If tool’s utilization rate is larger than 0, this tool must be assigned to an operation

13. Mathematical Formulation of the Problem 4. Tool availability constraints: total tool requirement <= the amount of tool on hand 5. Tool life constraint: machining time of operation <= available tool life 6. Machine power, surface roughness, non negativity and integrality constraints.

14. Mathematical Formulation of the Problem Non-linear MIP (Mixed Integer Program) formulation, An additional assumption: Wagner-Whitin property If production take place for part p in period t entering inventory of part p for period t must be 0. Lot sizes can assume values 0, D pt, D pt + D p,t+1 5 different joint solution algorithms to solve the problem.

15. Resource Directed Decomposition Procedure: 1. Relax the m/c hour availability constraint (coupling constraint among the parts) Several tool management subproblems, one for each part 2. For the reduced problems relax the set of tool availability constraints, Solution Approach

16. 3. Find the optimum machining conditions for every possible operation-tool pair and select the tool that gives the minimum cost by using SMOP (Single-Machining Operation Problem) 4. Impose the relaxed constraints to find the minimum combination of alternatives. Non-linear MIP is transferred to IP formulation

17. SMOP (Single Machining Operation Problem) For every possible operation-tool pair for a given lot size the following problem is solved, Min operating cost due to m/cing + nonmachining cost + + tooling cost Subject to tool life constraints, machine power constraints, surface roughness constraints. By formulating the dual problem, cutting speed and feed rate are determined.

18. Proposed Algorithms 1. Exact Algorithm: Step 1: Determination of alternative lot size values for all parts and periods, Ex: if planning horizon consists of 10 periods in period 3 alternative lot sizes for part 1 are D 13, D 13 + D 14, D 13 + D 14 + D 15 etc. Step 2:Determination of the optimum tool allocations and machining conditions for each alternative lot size. Step 2.1:Relax tool availability constraints and find the optimum machining conditions for every operation-tool pair of each part SMOP

19. Proposed Algorithms 1. Exact Algorithm: Step 2.2:Check tool availability constraints for each tool type. If tool availability constraints are satisfied move to Step 2.3. If tool availability constraints are violated - Generate a set of alternative tool allocations, - Solve SMOP for that alternatives, - Solve an IP to find the best allocation for every operation that satisfies the tool availability constraints

20. Proposed Algorithms 1. Exact Algorithm: Step 2.3:If a feasible solution is found for a lot size, the lot size is added to the set of feasible alternative lot sizes Step 3: Determination of total cost, m/c hour and tool requirements Step 4: Solve 0-1 formulation to find the optimum combination of alternatives.

21. Proposed Algorithms Min Total Cost Subject to 1. Exactly one alternative is selected for period 1 2. For each part the demand for each period is satisfied 3. Tool availability limits are not exceeded 4. Machine hour availability limits are not exceeded

22. Proposed Algorithms 2. Look Ahead Algorithms: Alternative lot sizes are generated and their corresponding optimum tool allocation and machining conditions are obtained until reaching a local minimum for: LPC (Least Period Cost): total relevant cost / unit time LUC (Least Unit Cost): total relevant cost / unit Reduced # of alternative lot sizes which simplifies the IP formulation

23. Proposed Algorithms 3. Single Pass Algorithms: Instead of an IP formulation, find the alternative with either least unit cost or the least period cost on the Look Ahead Algorithm Present the alternatives with minimum cost per unit and cost per period values as the final solution without solving an IP formulation

24. Results The proposed algorithms are compared with well-known uncapacitated lot sizing algorithms: - Wagner-Whitin (WW) - Least Unit Cost (LUC) - Least Period Cost (LPC) In this approaches: 1. Lot sizes that minimizes the sum of setup and inventory holding costs are determined, 2. An optimum tool allocation machining conditions are calculated for the given lot sizes. By utilizing a two-level approach

25. Results All of the proposed joint algorithms give better result (smaller cost values) than the two-level approaches

26. Results There are 7 experimental factors that can effect the efficiency of proposed algorithms - 2 to the power 7 full-factorial design corresponding to 128 com. - 5 replications for each combination is taken - 640 randomly generated runs

27. Results Number of infeasible cases and percent improvements Computation time (s) results for the joint algorithms

28. Results Two way analysis of variance (ANOVA) test on the performance measures of total cost, computation time and percent improvement: 1. All factors except demand variability are significant for the total production cost, 2. The most important factors on the computational times are number of parts, tooling cost, tool availability and assignment matrix, 3. All factors are statistically significant on the percent improvements.

29. Conclusion There is a close relationship between the lot sizing and the tool management decisions, These problems cannot be viewed in isolation, There are two advantages of proposed algorithms over the traditional two level approaches: 1. It is guaranteed that the lot sizing decisions will satisfy the tool management related constraints so that we ensure overall feasibility, 2. Total production cost is improved 6.9% on average compared to two-level approach which uses WW algorithm to calculate the lot sizes.