basic inventory model workforce scheduling enhance ... · inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30...
TRANSCRIPT
![Page 1: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/1.jpg)
Production Planning
Basic Inventory Model Workforce Scheduling
Enhance Modeling Skills
Dynamic Models
15.057 Spring 03 Vande Vate 1
![Page 2: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/2.jpg)
Dynamic Inventory Model
Modeling Time
Modeling Inventory
Unusual Network Example
15.057 Spring 03 Vande Vate 2
![Page 3: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/3.jpg)
Singapore Electric Generator Singapore Electric Generator Production
Unit Costs Jan Feb Mar Apr. May Production $ 28.00 $ 27.00 $ 27.80 $ 29.00
Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30
Production Qty 0 0 0 0 Production Limits 60 62 64 66
Beginning Inventory 15 -43 -79 -113 Delivery Reqmts 58 36 34 59 Minimum Ending Inventory (43) (79) (113) (172) 7
Production Cost $ - $ - $ - $ -Inventory Cost $ (4.20) $ (18.30) $ (28.80) $ (42.75) Total
Total Cost $ (4.20) $ (18.30) $ (28.80) $ (42.75) $ (94.05)
15.057 Spring 03 Vande Vate 3
![Page 4: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/4.jpg)
Inventory
Balancing Your Checkbook Previous Balance + Income - Expenses = New Balance
Modeling Dynamic Inventory Starting Inv. + Production - Shipments = Ending Inv.
15.057 Spring 03 Vande Vate 4
![Page 5: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/5.jpg)
Average Balances
Assuming Smooth Cash Flows
a Averages (Starting + Ending)/2
15.057 Spring 03 Vande Vate 5
![Page 6: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/6.jpg)
Challenge
Formulate a Solver Model
15.057 Spring 03 Vande Vate 6
![Page 7: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/7.jpg)
Singapore Electric Generator Singapore Electric Generator Production
Unit Costs Jan Feb Mar Apr. May Production $ 28.00 $ 27.00 $ 27.80 $ 29.00
Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30
Production Qty 0 0 0 0 Production Limits 60 62 64 66
Beginning Inventory 15 -43 -79 -113 Delivery Reqmts 58 36 34 59 Minimum Ending Inventory (43) (79) (113) (172) 7
Production Cost $ - $ - $ - $ -Inventory Cost $ (4.20) $ (18.30) $ (28.80) $ (42.75) Total
Total Cost $ (4.20) $ (18.30) $ (28.80) $ (42.75) $ (94.05)
15.057 Spring 03 Vande Vate 7
![Page 8: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/8.jpg)
A Network Formulation
Dec. Inv.
Jan. Inv.
Feb. Inv.
Mar. Inv.
Apr. Inv.
May Inv.
Jan. mfg.
Feb. mfg.
Mar. mfg.
Apr. mfg.
Jan. dem.
Feb. dem.
Mar. dem.
Apr. dem.
Supply ≤Prod. Limits
Demand ≥ req
Production Variables
Inventory Variables
Shipment Quantities
15.057 Spring 03 Vande Vate 8
![Page 9: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/9.jpg)
A Network Formulation Singapore Electric Generator Production
Unit Costs Dec Jan Feb Mar Apr. May Production $ 28.00 $ 27.00 $ 27.80 $ 29.00
Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30
Production Qty Production Limits Delivery Reqmts
Calc. Ending Inv. Ending Inventory 15 7
Production Cost $ - $ - $ - $ -Inventory Cost $ 2.25 $ - $ - $ - Total
Total Cost $ 2.25 $ - $ - $ - $ 2.25
0 0 0 0 60 62 64 66 58 36 34 59
-43 (36) (34) (59) Minimum - - - -
15.057 Spring 03 Vande Vate 9
![Page 10: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/10.jpg)
Another Views.t. InitialBalance:
Production['Jan'] - EndingInv['Jan'] = 43
s.t. MonthlyBalances['Feb']: Production['Feb'] + EndingInv['Jan'] - EndingInv['Feb'] = 36
s.t. MonthlyBalances['Mar']: Production['Mar'] + EndingInv['Feb'] - EndingInv['Mar'] = 34
s.t. MonthlyBalances['Apr']: Production['Apr'] + EndingInv['Mar'] - EndingInv['Apr'] = 59
s.t. FinalBalance: EndingInv['Apr'] >= 7
15.057 Spring 03 Vande Vate 10
![Page 11: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/11.jpg)
Scheduling Postal Workers Each postal worker works for 5 consecutive days, followed by 2 days off, repeated weekly.
Day Tues Wed Thurs Fri Sat Sun
Demand 17 13 15 14 16
Mon
19 11
Minimize the number of postal workers (FTE’s)
15.057 Spring 03 Vande Vate 11
![Page 12: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/12.jpg)
Challenge
Formulate a Solver Model
15.057 Spring 03 Vande Vate 12
![Page 13: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/13.jpg)
Formulating the LPScheduling Postal Workers
Shift Mon - Tues - Wed - Thurs - Fri - Sat - Sun -Fri Sat Sun Mon Tues Wed Thurs
Day Demand Mon 1 1 1 1 1 17 Tues 1 1 1 1 1 13 Wed 1 1 1 1 1 15 Thurs 1 1 1 1 1 19
Fri 1 1 1 1 1 14 Sat 1 1 1 1 1 16 Sun 1 1 1 1 1 11
15.057 Spring 03 Vande Vate 13
![Page 14: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/14.jpg)
Formulating as an LP
The Objective Total Workers Required Minimize $I$5
The decision variables The number of workers assigned to each shift $B$5:$H$5
The Constraints Enough workers each day $I$6:$I$12 >= $J$6:$J$12
15.057 Spring 03 Vande Vate 14
![Page 15: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/15.jpg)
The linear program
Minimize z = MF + TS + WSu + ThM + FT + SW + SuTh
subject to MF + ThM + FT + SW + SuTh ≥ 17 MF + TS + FT + SW + SuTh ≥ 13 MF + TS + WSu + SW + SuTh ≥ 15 MF + TS + WSu + ThM + SuTh ≥ 19 MF + TS + WSu + ThM + FT ≥ 14
TS + WSu + ThM + FT + SW ≥ 16 WSu + ThM + FT + SW + SuTh ≥ 11
Non-negativity 15.057 Spring 03 Vande Vate 15
![Page 16: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/16.jpg)
The Decision Variable Decision
Would it be possible to have the variables be the number of workers on each day? Conclusion: sometimes the decision variables incorporate constraints of the problem.
Hard to do this well, but worth keeping in mind We will see more of this in integer programming.
15.057 Spring 03 Vande Vate 16
![Page 17: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/17.jpg)
Enhancement
Some days we will have too many workers
Excess Only concerned with the largest excess
Minimize the largest Excess
15.057 Spring 03 Vande Vate 17
![Page 18: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/18.jpg)
Challenge
Formulate a Solver Model
15.057 Spring 03 Vande Vate 18
![Page 19: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/19.jpg)
Formulating the LPScheduling Postal Workers
Shift Mon - Tues - Wed - Thurs - Fri - Sat - Sun -Fri Sat Sun Mon Tues Wed Thurs
Day Demand Mon 1 1 1 1 1 17 Tues 1 1 1 1 1 13 Wed 1 1 1 1 1 15 Thurs 1 1 1 1 1 19
Fri 1 1 1 1 1 14 Sat 1 1 1 1 1 16 Sun 1 1 1 1 1 11
15.057 Spring 03 Vande Vate 19
![Page 20: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/20.jpg)
Minimize the Maximum
Min Max{XS[Mon], XS[Tues], …}
Min Z
S.t. Z ≥ XS[Mon]S.t. Z ≥ XS[Tues]…
S.t. MF + ThM + FT + SW + SuTh – XS[Mon] = 17 S.t. MF + TS + FT + SW + SuTh – XS[Tues] = 13
….
15.057 Spring 03 Vande Vate 20
![Page 21: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/21.jpg)
Enhancement Ensure at least 30% of the workers have Sunday offFormulate a Solver Model
15.057 Spring 03 Vande Vate 21
![Page 22: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/22.jpg)
Formulating the LPScheduling Postal Workers
Shift Mon - Tues - Wed - Thurs - Fri - Sat - Sun -Fri Sat Sun Mon Tues Wed Thurs
Day Demand Mon 1 1 1 1 1 17 Tues 1 1 1 1 1 13 Wed 1 1 1 1 1 15 Thurs 1 1 1 1 1 19
Fri 1 1 1 1 1 14 Sat 1 1 1 1 1 16 Sun 1 1 1 1 1 11
15.057 Spring 03 Vande Vate 22
![Page 23: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/23.jpg)
The linear program
Minimize z = MF + TS + WSu + ThM + FT + SW + SuTh
subject to MF + ThM + FT + SW + SuTh ≥ 17 MF + TS + FT + SW + SuTh ≥ 13 MF + TS + WSu + SW + SuTh ≥ 15 MF + TS + WSu + ThM + SuTh ≥ 19 MF + TS + WSu + ThM + FT ≥ 14
TS + WSu + ThM + FT + SW ≥ 16 WSu + ThM + FT + SW + SuTh ≥ 11
.7(MF + TS) - 0.3*(WSu + ThM + FT + SW + SuTh) ≥ 0
Non-negativity15.057 Spring 03 Vande Vate 23
![Page 24: Basic Inventory Model Workforce Scheduling Enhance ... · Inventory $ 0.30 $ 0.30 $ 0.30 $ 0.30 Production Qty 0 0 0 0 Production Limits 60 62 64 66 Beginning Inventory 15 -43 -79](https://reader030.vdocuments.net/reader030/viewer/2022040309/5f254abb21b45320d4541f12/html5/thumbnails/24.jpg)
Summary
More LP Modeling
LPs are more general than Networks
Modeling Time
Clever choices of decision variables
15.057 Spring 03 Vande Vate 24