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### IE-573 Bicriteria Robotic Cell Scheduling

Serdar YILDIZ

Outline

- Introduction
- Problem definition
- Solution Procedure
- Different Cost Structures
- Conclusion
- References

Introduction

- Minimizing Production Time
- Has the highest pirority on production planning
- Minimizing Production Costs
- Has the highest priority in process planning
- Demand is limited in most industries and cost’s importance is increased
- So these two objectives are crucial in today’s industry

Introduction

- Why bicriteria optimization?
- Most of the real life problems are related more than one objective
- There are trade-offs between these objectives
- A solution which minimizes one of the objectives may perform poorly in other objectives
- So bicriteria optimization is useful in solving such problems

Problem Definition

- In-line robotic cell
- m-machine robotic cell
- No buffers
- identical parts
- Each part is assumed to have a number of operations
- represents the operation to be performed on each machine i
- denotes the corresponding processing time to

Problem Definition

- Objective
- Minimize production time
- Maximizing throughput
- Minimize production cost
- There is a trade-off between machining and tooling cost
- Reducing the processing time reduces the machining cost
- However it also reduces the tool life which in turn increases the tooling cost

Problem Definiton

- Kayan and Akturk (2005) determined lower and upper bounds for processing times
- These costs are convex cost functions
- The lower bound corresponds to the minimum processing time-maximum cost case
- Whereas the upper bound corresponds to the maximum processing time-minimum cost case

Problem Definition

- Now we can formulate upper bound of processing as and respectively lower bound as for operation
- So
- denotes the convex and differentiable manufacturing cost for operation
- The total manufacturing cost by all operations is

Problem Definition

- We can see the figure representation of our cell as below

Problem Definition

- We can see manufacturing cost with respect to processing time as in the figure

Problem Definition

- Obviously, the total manufacturing cost does not depend on the robot move cycle
- It depends on processing times of the operations
- We can see from figure 2 that the manufacturing cost between and is decreasing
- We denote the processing time vector as
- A feasible processing time vector have to satisfy the upper and lower bounds, it can be represented as:

Problem Definition

- We will use the feasible robot move cycles defined by Crama and Van de Klundert(1997)
- In these feasible cycles, the robot does not unload an empty machine and does not load a loaded machine
- Now we will present some definitions and notations as follows:

: the load unload times of machines by robot, same for all

: time taken by robot between two consecutive machines

: Cycle time

: operating costs of the machines, same for all machines

: set of all feasible robot move cycles in an m-machine cell

Problem Definition

: Cost of tool i used

: Total manufacturing cost that depends only on processing times

: Cycle time corresponding to robot move cycle S and processing time vector P

Now, we can present our optimization problem as

We can see that the first objective is minimizing

The second objective is minimizing

Problem Definition

- The problem definition we used minimizes two conflicting objectives simultaneously
- However there are some practical ways of dealig with bicriteria problem
- Now, we will show some ways of dealing bicriteria problem in literature:
- Let and represent the two peformance measures
- The first method minimizes a linear composite objective function in

and with unknown relative weights

We denote it as,

Problem Definition

- The second way is called the hierarchical optimization or lexicographical optimization

It is denoted as

In this approach, performance measure is assumed to be more important then

So, this approach minimizes subject to the minimum value of

- The third way is the epsilon-constraint method

It is denoted as

This method has been used in literature widely

Decision maker can interactively specify and modify the bounds and analyse the influence on solution

Nondominated points are found by solving a series of problems that minimizes given an upper bound on

Problem Definition

- The last approach minimizes a composite function of

and

It is denoted as,

All nondominated points are generated where is nondecreasing

We will also use this method in this study

Problem Definition

- In this study, we wil solve bicriteria problem for each 1-unit cycle
- By this, we obtain the processing time vector and compare with the others
- First, we will solve to determine the conditions for processing time vectors minimizing manufacturing cost for a given level of cycle time.
- Thus we will be able to construct the composite objective of
- So we can determine the manufacturing cost corresponding to a cycle time

Problem Definition

- The Epsilon Constraint Problem (ECP) that we will solve can be represented as follows:
- A solution for the bicriteria problem corresponds to, both a feasible robot move cycle and a feasible processing time vector

Problem Definition

- The definition below represents a solution of bicriteria problem

Problem Definition

- The second definition shows us how we can say that a solution

dominates another solution

Problem Definition

- We try to find the robot move sequence and the procesing times of the parts on the machines
- To achieve this objective, we will fix the robot move cycles and for each cycle we will find the nondominated solutions
- The definition below shows the nondominated processing time vectors for a cycle

Problem Definition

- In the slide before we presented how we find nondominated solutions
- Now we will show how a cycle dominates another cycle
- A dominance relation between two cycles is found by as follows:
- comparing the minimum cost values of the two cycles corresponding to the same cycle time value

Solution Procedure

- We will solve the 2-machine case first:
- The results obtained from 2-machine case is as follows:
- The cycle time of is
- So the first constraint of ECP becomes:
- Now denotes the optimum solution for ECP problem of
- In the next slide we will see the results of

Solution Procedure

- It can be easily seen from first condition that K takes its lowest value when and and this is the only possible solution
- In addition, in the second case K takes maximum value so

and is the only possible solution

Solution Procedure

- However, most of the points are not one of the extreme points that we have described in the previous slide
- To find these feasible points we use KTT (Karush Kuhn Tucker)

because there is a single function and constaints are convex

- These points are described in lemma-1.3

Solution Procedure

- The solution procedure we have described in lemma 1 finds the processing time vector which gives the minimum cost for a given cycle time.
- In this procedure, we increase the processing time which has the highest contribution to the cost.
- We can present results from lemma 1 with cost values also.
- Since and is the cost of operating machines, then machining cost is
- In addition, the tooling cost can be found as
- So we can modify the results of lemma 1

Solution Procedure

- Now we will observe the cycle
- The cycle time of is
- So the first constraint of ECP can be replaced with the linear constraints:
- We can find the procesing time vector that minimizes cost by lemma 2

Solution Procedure

- We see from lemma 2 that increasing processing times without exceeding upper bound of processing times decreases the total cost
- In addition, gives less cost than with same cycle time
- The reason for this result is both machines are balanced under
- Now we can represent theorem 1 as follows:
- It is obvious that both and gives the same cycle time
- However manufacturing cost of is smaller so it is preferred to

Solution Procedure

- We have been analysing only two sequence and
- However, 1-unit cycles may not be optimal in some regions.
- The only 2-unit cycle in 2-machine cell is and this cycle performs better than the 1-unit cycles and
- Now we wil describe in which regions 1-unit cycles are optimal in theorem-2 as follows:
- 1-unit cycles need not to be optimal in the region of

Solution Procedure

- We have solved the two machine case
- Now we will solve the three machine case
- Increasing the number of machines increases the feasible robot move cycles drastically
- The is very similar to in two machine case
- In addition, is very similar to in two machine case
- We will see that the results obtained from three machine case are similar to two machine case
- Now we are going the present results of three machine case
- We can find the optimal processing time vectors by using the procedure in Lemma-3 as can be seen in the paper

Solution Procedure

- In lemma-4 we can see the optimal region of cycle as follows:

Solution Procedure

- Now we will prove some dominance relaions that we have described in definition-4
- Crama and Van de Klunder proved that the set of pyramidal permutations contains optimal solutions
- The pyramidal permutations can be described as follows:
- So we can present theorem-3:

Solution Procedure

- The cycles and are nonpyramidal cycles so we can eiminate these cycles because they are dominated
- While comparing the other remaining cycles with , we see that dominates all other cycles. So we can present the theorem-4 as follows:
- It can be seen that this theorem is similar to the result of theorem-1 for two machine case

Solution Procedure

- In lemma-5 the solution procedure that finds optmal points in cycle and in the lemma-6 solution procedure for cycle can be found in a similar manner
- So we do not go into deep details for these lemmas
- However we cannot find any dominance relation among cycles

and

- This is an important results that makes this study different from the other studies since Sethi et al.(1) compared all cycles according to cycle time objective
- However in bicriteria optimization we could not compare all cycles

Different Cost Structures

- In this section, we will show different assumptions on the machining cost and on the robot operating cost
- A different cost structure assumes that
- the machining cost can be in terms of cycle time
- In this approach, we consider the cost of the robot operations included
- Lemma-1 , lemma-2 and theorem-1 is still feasible in this approach

Different Cost Structures

- Another point of view is to include the exact working time of robot as an additional cost
- Then the theorem-1 is not still valid in this condition
- So we can see that cycle time as the only peformance measure hinders the other characteristics of the problem
- Even the basic rules of Sethi et al.(1) becomes invalid

Conclusion

- We considered robotic cell scheduling with identical parts in two and three machine robotic cells
- In metal cutting industry highly flexible CNC machines are used
- So we can adjust the parameters such as processing times and decrease the total manufacturing costs
- However, these adjustments effects the tool life and cost
- So we have used cycle time and total manufacturing cost as bicriteria decisions
- In the future the m-machine case of this problem can be studied
- Another future extension is to study on other cycles than 1-unit cycles

References

- Bicriteria robotic cell scheduling
- Hakan Gultekin*, M. Selim Akturk, Oya Ekin Karasan

Department of Industrial Engineering, Bilkent University, 06800 Bilkent, Ankara, Turkey

Thank you for listening

- Questions?

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