INTRODUCTION

MODELING AND ANALYSIS OFMANUFACTURING SYSTEMS

Session 12 MACHINE SETUP AND

OPERATION SEQUENCING E. Gutierrez-Miravete

Spring 2001

INTRODUCTION• WHOLE SYSTEM DESIGN

STRATEGIES VS INDIVIDUAL CELL/WORKSTATION DESIGN STRATEGIES

• QUESTION: HOW TO TOOL THE MACHINE AND THEN SEQUENCE PRODUCTION ACTIVITIES IN IT?

• GOAL: MAXIMIZE PRODUCTIVITY

OPORTUNITIES

• SEQUENCE BATCHES SO AS TO MINIMIZE TOOLING CHANGEOVERS

• SEQUENCE ACTIVITIES SO AS TO MINIMIZE IDLE TIMES

• OPTIMIZE CELL LAYOUT SO AS TO MINIMIZE ASSEMBLY TIME

SEQUENCING AND OPTIMIZATION

MANY SEQUENCING PROBLEMS IN MANUFACTURING CELL PLANNING ARE CLASSIC OPTIMIZATION PROBLEMS

CELL VS SYSTEM• BOTH, CLEVER SEQUENCING OF

OPERATIONS AND OVERALL SYSTEM DESIGN ARE IMPORTANT FOR SUCCESS IN MANUFACTURING

• TIME FRAMES– FOR OVERALL SYSTEM DESIGN: WEEKS

OR MONTHS (LONG TERM)– FOR SEQUENCE DESIGN:

MINUTES/HOURS (SHORT TERM)

TASK ASSIGNMENT

LINEAR ASSIGNMENT PROBLEM

• GOAL:– TO DISTRIBUTE N TASKS

AMONG N WORKERS SO AS TO MINIMIZE COST

• CONSTRAINTS:– 1 TASK PER WORKER– 1 WORKER PER TASK

COST MATRIX C

• ROWS: WORKERS• COLUMNS: MACHINES• SUMMARIZES THE

ASSIGNMENT COSTS

LAP: MATHEMATICAL FORMULATION

• minimizeCi j cij xij

• subject to:i xij = 1 (for all tasks)

j xij = 1 (for all workers)

Facts

• IF A CONSTANT IS ADDED TO EVERY ELEMENT OF A ROW OR COLUMN OF C, THE OPTIMAL SOLUTION DOES NOT CHANGE BUT ITS VALUE CHANGES BY THE ADDED CONSTANT

• IF ALL cij > 0, ANY SOLUTION WITH COST = 0 MUST BE OPTIMAL

Hungarian Algorithm

• Proceeds by adding and substracting constants from rows and columns so as to maintan a non-negative cost matrix. When a feasible solution is found using only 0 cost cells, optimum has been found.

HA Steps

• STEP 1: COST REDUCTION BY CONSTRUCTION OF THE REDUCED COST MATRIX. THE RCM IS OBTAINED BY SUBSTRACTING FIRST THE MINIMUM ELEMENT IN EACH ROW FROM ALL ELEMENTS IN THE ROW THEN DOING LIKEWISE WITH COLUMNS

HA Steps cont’d

• STEP 2: SEARCH FOR A FEASIBLE SOLUTION USING ONLY THE 0’S IN THE RCM . IF THIS SUCCEEDS, OPTIMAL SOLUTION HAS BEEN FOUND. IF ALL 0’S CAN BE COVERED WITH LESS THAN n HORIZONTAL AND VERTICAL LINES, CONTINUE.

HA Steps cont’d

• STEP 3: FURTHER REDUCTION. FIND THE MINIMUM UNCOVERED ELEMENT. SUBSTRACT THIS REDUCED COST FROM EACH UNCOVERED ELEMENT AND ADD IT TO EACH TWICE-COVERED ELEMENT. GO TO 2.

• Example 8.1, Table 8.1, Fig. 8.1, pp. 263-

TASK SEQUENCING

TWO CLASSES OF PROBLEMS

• EVALUATE CHANGEOVER COSTS WHEN COSTS ARE COMPLETELY DETERMINED BY THE CURRENT JOB SETUP AND THE NEXT JOB TO BE LOADED

• EVALUATE COSTS WHEN THE ENTIRE SEQUENCE OF JOBS MUST BE KNOWN

COMPLETE CHANGEOVERS• N JOBS ARE TO BE PERFORMED ON

ONE MACHINE• UNIT PROCESSING TIME AND BATCH

SIZE DETERMINE TOTAL PROCESSING TIME

• CHANGEOVER TIMES DEPEND ONLY ON CURRENT AND NEXT PRODUCT

• TOTAL SETUP TIME DEPENDS ON JOB SEQUENCE

TRAVELING SALESMAN PROBLEM

• A SALESMAN MUST VISIT EVERY CITY IN HIS/HER TERRITORY THEN RETURN HOME IN SUCH A WAY THAT THE SMALLEST POSSIBLE TOTAL DISTANCE IS TRAVELLED.

• TSP CAN BE VISUALIZED WITH A GRAPH OF NODES (CITIES) AND ARC LENGTHS (DISTANCES)

TRAVELING SALESMAN PROBLEM

• DISTANCE BETWEEN CITIES i & j = cij

• ALL ARCS ARE BIDIRECTIONAL• NO SUBTOURS ALLOWED• FIVE CITY COMPLETE TSP GRAPH

Fig 8.2a• FIVE CITY POSSIBLE TOUR GRAPH

Fig 8.2b

SOLUTION OF THE TSP• CLASSICAL OPTIMIZATION

TECHNIQUES ARE HARD TO APPLY WHEN N IS LARGE (See Eqn 8.5-)

• HEURISTIC METHODS ARE MOST FREQUENTLY USED– NO SUBTOUR CONSTRAINT IS

TEMPORARILY RELAXED– SOLVE RESULTING OPTIMIZATION

PROBLEM

TSP BY CLOSEST INSERTION ALGORITHM

1.- SELECT A STARTING CITY2.- PROCEED THROUGH N-1 STAGES

ADDING A NEW CITY AT EACH STAGE. THE NEW CITY IS SELECTED FROM THOSE CURRENTLY UNASSIGNED SUCH THAT IT IS CLOSEST TO ANY CITY IN THE ACCUMULATED PARTIAL SEQUENCE

• Example 8.2; Fig 8.3, pp. 268-

TSP BY MINIMUM SPANNING TREE

• A MST IS ANY SET OF N-1 ARCS THAT TOUCH EACH NODE AND HAVE THE SMALLEST SUM OF COSTS FOR ANY SUCH SET

• BY MODIFYING THE MST SO THAT EACH NODE IS CONNECTED EXACTLY BY TWO ARCS IN A CONNECTED TREE OBTAIN TSP SOLUTION

TSP BY SUBTOUR INTEGRATION

• START WITH A SOLUTION TO THE ASSIGNMENT PROBLEM

• TRY TO CONNECT TWO SUBTOURS AT A TIME BY SWITCHING ARCS

• ONCE ALL LAP SUBTOURS ARE COMBINED, GOT TSP SOLUTION

• Fig 8.4

PARTIAL CHANGEOVERS• N JOBS ARE TO BE PERFORMED ON ONE

MACHINE CAPABLE OF HOLDING M TOOLS• JOB j REQUIRES TOOLS Aj• TOTAL NUMBER OF TOOLS REQUIRED

EXCEEDS M (TOOL CHANGES REQUIRED)• NO JOB REQUIRES MORE THAN M TOOLS• AT EACH JOB COMPLETION SOME TOOLS

MAY HAVE TO BE REMOVED AND NEW TOOLS ADDED

• Fig 8.5

OBJECTIVE

TO ORDER JOBS AND TOOL CHANGEOVERS TO MINIMIZE THE TOTAL NUMBER OF TOOLS CHANGED ON THE MACHINE

NOTES

• ALWAYS KEEP M TOOLS ON MACHINE

• THE ORDERING FOR JOBS ON THE MACHINE IS GIVEN

• KEEP TOOL NEEDED SOONEST (KTNS) RULE IS OPTIMAL

• REMOVE ONLY “LONGEST UNTIL NEXT USE” TOOLS

JOB ORDERING

• IF JOB r USES ONLY A SUBSET OF TOOLS USED BY ITS PREDECESOR s CHANGEOVER IS NOT INCREASED (i.e. OPTIMAL SEQUENCE)

• JOB SEQUENCING PROBLEM IS SIMILAR TO GROUP FINDING IN GT

• THE TOOL-JOB MATRIX

JOB ORDERING

• SOLUTION METHODS– BINARY CLUSTERING

• Ex 8.3, Tables 8.3, 8.4

– TSP• Table 8.5

SOLUTION OF THE PARTIAL CHANGEOVER

PROBLEM

• STEP 1: JOB COMBINATION (REDUCTION)

• STEP 2: JOB ORDERING• STEP 3: TOOL SETUP PLANNING

BY KTNS

INTEGRATED ASSIGNMENT AND SEQUENCING

QUESTION

• WHAT TO DO WHEN CELL SETUP (TOOLING) AND JOB SEQUENCING ARE RELATED BY NEITHER DICTATES THE OTHER?

• NEED TO:– MAKE A SEQUENCING DECISION and– MAKE A SETUP DECISION

NOTES

• PROBLEM 1: PLAN SETUP AND OPERATION OF ASSEMBLY CELLS

• PROBLEM 2: SETUP AND SEQUENCE OF A MACHINE (INTERDEPENDENT TOOLS)

• GOAL: TO MINIMIZE CYCLE TIME• SOLUTION TECHNIQUES:

HEURISTICS

ASSEMBLY CELL LAYOUT AND SEQUENCING

• ASSEMBLY ENVIRONMENTS– MASS PRODUCED SINGLE PRODUCT– MULTIPLE PRODUCTS PRODUCED IN

ALTERNATING LOTS WITH CHANGEOVER REQUIRED

– MIX OF PART TYPES ASSEMBLED SIMULTANEOUSLY IN CELL WITHOUT CHANGEOVER

REQUIRED

• FIND HOW TO PERFORM THE SET OF TASKS THAT HAVE BEEN ASSIGNED TO A WORKSTATION

SINGLE PART TYPE

• A SINGLE PRODUCT FRAME• PRODUCT FRAME HAS N

LOCATIONS WHERE PARTS ARE TO BE ADDED

• WORKSTATION HAS N BINS WHERE PART FEEDERS ARE PLACED

• Fig 8.6

REQUIRED

• ASSIGN FEEDERS TO BINS• DETERMINE THE ORDER IN

WHICH PARTS ARE TO BE ADDED TO FRAME

• PROBLEM CAN BE MODELED AS A 2N-CITY TSP

BIN ASSIGNMENT AND INSERTION SEQUENCING

• BIN ASSIGNMENT GOAL: MINIMIZE THE LOADED TRAVEL TIME (DISTANCE) FOR THE ASSEMBLER

• INSERTION SEQUENCING GOAL: MINIMIZE THE UNLOADED TRAVEL TIME (DISTANCE) FOR THE ASSEMBLER

• Ex 8.4, Tables 8.6 , 8.7; Ex 8.5, Table 8.8

MIXED PRODUCTS

• TYPICALLY, SEVERAL PRODUCT TYPES ARE BEING PRODUCED SIMULTANEOUSLY IN THE SAME CELL

• DEMAND PROPORTIONS DETERMINED BY BILL OF MATERIALS FOR END PRODUCTS

MIXED PRODUCT: UNPACED LINE

• ASSUME – RELATIVE DEMANDS ARE KNOWN– M FRAMES TO BE MADE– pm IS THE PROPORTION OF TYPE m

FRAMES– BINS KEPT ON SAME PLACE

• GOAL: MINIMIZE AVERAGE ASSEMBLY TIME OF PRODUCT

MIXED PRODUCT: UNPACED LINE

• FOR EACH PART TYPE COMPILE TABLE OF TRAVEL TIMES (Table 8.7)

• COMBINE TABLES BY WEIGHTED AVERAGE

cij = m pm cijm

– pm proportion of type m frames– cijm total travel time per type m frame if part i

is assigned to bin j– Ex. 8.6, Tables 8.9, 8.10

CELL LAYOUT AND SEQUENCING:

INTERDEPENDENT TOOLS• THE NC PUNCH PRESS

– 36 TOOL TOOL TURRET (Fig 8.7)– PARTS MAY REQUIRE UP TO 200 HITS– HIT SEQUENCE MAY BE SUBJECT TO

PRECEDENCE CONSTRAINTS• GOAL: LOAD TOOLS AND SEQUENCE

HITS TO MINIMIZE PRODUCTION CYCLE• Ex 8.7, Fig. 8.8, Tables 8.11, 8.12