1 MAI Lab. Lot sizing and scheduling formulation in Single-level 발표자 : 정 현 종

2 Slide 2 of 33 MAI Lab. Planning & Scheduling Production Planning 시점, 품목, 수량 결정 + 자원 ( 생산시설, 공장 ) 년, 월, 주, 일 단위 Production Scheduling 시점, 품목, 수량 결정 + 자원 ( 라인, 사람, 기계 ) + 생산순서 일, 시 단위 언제, 무엇을, 얼마나 만들지 ? 언제, 무엇을, 얼마나, 어떤 자원 ( 사람, 기계 ) 를 사용하여, 어떤 순서로 만들지 ?

3 Slide 3 of 33 MAI Lab. Project Plan 논문 및 문헌 조사 Basic Modeling 문제의 Simplify  Single-level 에서의 다양한 formulation  Implementation and Test Realization  Single-level 을 multi-level 로 확장하여 formultaion  Implementation and Test

4 Slide 4 of 33 MAI Lab. Papers Lot sizing and scheduling – Survey and extensions A. Drexl, A.Kimms(1997) The Capacitated Lot-Sizing Problem with Linked Lot Sizes Christopher Suerie, Hartmut Stadtler(2003)

5 MAI Lab. Lot sizing and scheduling - Survey and extensions European Journal of Operation Research vol.99, 1997 Drexl, A.Kimms

6 Slide 6 of 33 MAI Lab. Contents 1.Background and motivation 2.Single-level lot sizing and scheduling 3.Continuous time lot sizing and scheduling 4.Multi-level lot sizing and scheduling 5.Further research opportunities

7 Slide 7 of 33 MAI Lab. Problem outline Opportunity costs Holding cost (Inventories) Setup cost Trade-off between low setup cost & low holding cost Decision about lot sizing and scheduling The key elements The precedence relations of operations The presence of scarce capacity

8 Slide 8 of 33 MAI Lab. The capacitated lot sizing problem Big bucket problem Decision variables for the CLSP I jt : Inventory for item j at the end of period t. q jt : Production quantity for item j in period t. x jt : Binary variable which indicates whether a setup for item j occurs in period t (x jt = 1) Parameters for the CLSP C t : Available capacity of the machine in period t. d jt : External demand for item j in period t. h j : Non-negative holding costs for item j. I j0 : Initial inventory for item j. J : Number of items. p j : Capacity needs for producing one unit of item j. s j : Non-negative setup costs for item j. T : Number of periods.

9 Slide 9 of 33 MAI Lab. Mathematical Programming Mixed-Integer Programming for CLSP Minimize: Subject to: Inventory constraint: Production constraint: Capacity constraint: Variable constraint: Lot-sizing 결정 Scheduling 결정 안함 *CLSP: The capacitated lot sizing problem

1010 Slide 10 of 33 MAI Lab. The discrete lot sizing & scheduling problem Small bucket problem Assumption All-or-nothing: Only one item may be produced per period A new decision variable for the DLSP y jt : Binary variable which indicates whether the machine is set up for item j in period t (y jt =1) y j0 : Binary value which indicates whether the machine is set up for item j at the beginning of period 1 (y j0 =1) *DLSP: The discrete lot sizing & scheduling problem

1 Slide 11 of 33 MAI Lab. Mathematical Programming Mixed-Integer Programming for DLSP Minimize: Subject to: Inventory constraint: All-or-Nothing: At most one item: Setup constraint: Variable constraint: Lot-sizing 결정 Scheduling 결정

1212 Slide 12 of 33 MAI Lab. The continuous setup lot sizing problem Small bucket problem The decision variables and the parameters equal those of the DLSP The exclusion of ‘all-or-nothing’ assumption No setup costs in idle periods between two lots of the same item

1313 Slide 13 of 33 MAI Lab. Mathematical Programming Mixed-Integer Programming for CSLP Minimize: Subject to: Inventory constraint: All-or-Nothing: At most one item: Setup constraint: Variable constraint: Lot-sizing 결정 Scheduling 결정 Unused capacity *CSLP: The continuous setup lot sizing problem

1414 Slide 14 of 33 MAI Lab. The proportional lot sizing & scheduling problem Small bucket problem To use remaining capacity for scheduling a second item in the particular period Assumption Setup state can be changed at most once per period y jt : the setup state of the machine at the end of a period

1515 Slide 15 of 33 MAI Lab. Mathematical Programming Mixed-Integer Programming for PLSP Minimize: Subject to: Inventory constraint: Production constraint: Capacity constraint: Setup state constraint: Variable constraint: *PLSP: The proportional lot sizing and scheduling problem

1616 Slide 16 of 33 MAI Lab. The general lot sizing and scheduling problem A critique against small bucket models The number of periods Big bucket problem Consideration for lot sizing and scheduling simultenously The underlying idea for the GLSP Each lot is uniquely assigned to a position number in order to define a sequence *GLSP: The general lot sizing and scheduling problem

1717 Slide 17 of 33 MAI Lab. A new parameter for the GLSP N t : Maximum number of lots in period t. Decision variables for the GLSP I jt : Inventory for item j at the end of period t. q jn : Production quantity for item j at position n. x jn : Binary variable which indicates whether a setup for item j occurs at position n (x jn =1) y jn : Binary variable which indicates whether the machine is ready to produce item j at position n (y jn =1) ex) the first position the last position

1818 Slide 18 of 33 MAI Lab. Mathematical Programming Mixed-Integer Programming for GLSP Minimize: Subject to: Inventory constraint: Production constraint: Capacity constraint: Setup constraint: Variable constraint: N t =1 => CSLP

1919 MAI Lab. The Capacitated Lot-Sizing Problem with Linked Lot Sizes Management Science Vol.49, No.8, August 2003 Christopher Suerie, Hartmut Stadtler

2020 Slide 20 of 33 MAI Lab. Contents 1.Introduction 2.Literature Review 3.Model Formulation 4.Extended Model Formulation and Valid Inequalities 5.Solution Approaches 6.Computational Results 7.Conclusion

2121 Slide 21 of 33 MAI Lab. Big Bucket vs. Small Bucket Big bucket problem Capacitated lot-sizing problem Capacitated lot-sizing problem with linked lot sizes Small bucket problem Discrete lot sizing and scheduling problem Continuous setup lot-sizing problem Proportional lot sizing and scheduling problem

2 Slide 22 of 33 MAI Lab. Characterization of Models

2323 Slide 23 of 33 MAI Lab. Model Formulation Indices and index sets j : Products or items, j=1,…,J m : Resources (e.g. personnel, machines, production lines), m=1,…,M t : Periods, t=1,…T; R m : Set of Products j produced on resource m Data a mj : Capacity needed on resource m to produce one unit of item j B jt : Large number, not limiting feasible lot sizes of product j in period t; C mt : Available capacity of resource m in period t h j : Holding cost for one unit of product j per period P jt : Primary, gross demand for item j in period t sc j : Setup cost for product j st j : Setup time for product j Variables I jt : Inventory of item j at the end of period t X jt : Production amount of item j in period t (lot size) Y jt : Binary setup variable (=1, if a setup for item j is performed in period t) Starting Base: CLSP

2424 Slide 24 of 33 MAI Lab. CLSP model formulation Minimize: Subject to: Inventory constraint: Capacity constraint: Production constraint: Variable constraint:

2525 Slide 25 of 33 MAI Lab. Simple Plant Location Representation To obtain a tight model formulation D n jt : net demand for product j in period t Z jst : the portion of demand of product j in period t fulfilled by production in period s New constraint

2626 Slide 26 of 33 MAI Lab. SPL representation for CLSP Minimize: Subject to:

2727 Slide 27 of 33 MAI Lab. Linked Lot sizes Two new sets of variables for the linkage property W jt : indicate whether a setup state for product j is carried over from period t-1 to period t (=1) Q mt : indicate that production on resource m in period t is limited to at most one product for which no setup has to be performed (the setup state for this specific product is linked to the preceding and following period)

2828 Slide 28 of 33 MAI Lab. Additional constraints

2929 Slide 29 of 33 MAI Lab. Extended Model Formulation To strengthen the model formulation Q mt (resource dependent) = > QQ jt ( product dependent)

3030 Slide 30 of 33 MAI Lab. New constraint

3131 Slide 31 of 33 MAI Lab. Valid Inequalities Preprocessing - Inequalities It further restricts the range of QQ jt ex)

3232 Slide 32 of 33 MAI Lab. Valid Inequalities Inventory/Setup - Inequalities Capacity/Single-Item Production – Inequalities RS : Subset of set of products j produced on resource m, RS ⊂ R m

3 Slide 33 of 33 MAI Lab. Discussion Formulation 의 현실성 검증 풀고자 하는 현실 문제 Specify Simplified formulation 의 현실적인 제약 추가 Solution Approach 차이에 따른 성능 비교 CLSPL Solution Approach  Branch and Cut  Time-oriented decomposition approach