Production Planning, Workforce Scheduling

Production Planning Basic Inventory Model Workforce Scheduling Enhance Modeling Skills Dynamic Models 1 Dynamic Inventory Model Modeling Time Modeling Inventory Unusual Network Example 2 Singapore Electric Generator Unit Costs Production $ Inventory $ Production Qty Production Limits Beginning Inventory Delivery Reqmts Ending Inventory Production Cost $ Inventory Cost $ Total Cost $ Singapore Electric Generator Production Jan Feb Mar Apr. 28.00 $ 27.00 $ 27.80 $ 29.00 0.30 $ 0.30 $ 0.30 $ 0.30 0 60 15 58 (43) 0 62 -43 36 (79) $ (4.20) $ (4.20) $ $ (18.30) $ (18.30) $ 0 64 -79 34 (113) May 0 66 -113 59 Minimum (172) 7 $ (28.80) $ (42.75) Total (28.80) $ (42.75) $ (94.05) 3 Inventory Balancing Your Checkbook Previous Balance + Income - Expenses = New Balance Modeling Dynamic Inventory Starting Inv. + Production - Shipments = Ending Inv. 4 Average Balances Assuming Smooth Cash Flows a Averages (Starting + Ending)/2 5 Challenge Formulate a Solver Model 6 Singapore Electric Generator Unit Costs Production $ Inventory $ Production Qty Production Limits Beginning Inventory Delivery Reqmts Ending Inventory Production Cost $ Inventory Cost $ Total Cost $ Singapore Electric Generator Production Jan Feb Mar Apr. 28.00 $ 27.00 $ 27.80 $ 29.00 0.30 $ 0.30 $ 0.30 $ 0.30 0 60 15 58 (43) 0 62 -43 36 (79) $ (4.20) $ (4.20) $ $ (18.30) $ (18.30) $ 0 64 -79 34 (113) May 0 66 -113 59 Minimum (172) 7 $ (28.80) $ (42.75) Total (28.80) $ (42.75) $ (94.05) 7 A Network Formulation Jan. mfg. Feb. mfg. Mar. mfg. Apr. mfg. Supply ≤ Prod. Limits Apr. Inv. May Inv. Production Variables Dec. Inv. Inventory Variables Jan. Inv. Feb. Inv. Mar. Inv. Shipment Quantities Jan. dem. Feb. dem. Mar. dem. Apr. Demand ≥ req dem. 8 A Network Formulation Singapore Electric Generator Production Unit Costs Dec Jan Feb Mar Apr. 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 Production Cost Inventory Cost Total Cost 0 60 58 -43 - 15 $ $ $ $ 2.25 $ 2.25 $ 0 62 36 (36) - 0 64 34 (34) $ $ $ - May 0 66 59 (59) Minimum 7 $ $ $ - Total $ 2.25 9 Another View s.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 10 Scheduling Postal Workers Each postal worker works for 5 consecutive days, followed by 2 days off, repeated weekly. Day Demand Mon Tues Wed Thurs Fri 17 13 15 19 14 Sat Sun 16 11 Minimize the number of postal workers (FTE’s) 11 Challenge Formulate a Solver Model 12 Formulating the LP Scheduling Postal Workers Shift Day Mon Tues Wed Thurs Fri Sat Sun Mon - Tues - Wed - Thurs - Fri - Sat - Sun Fri Sat Sun Mon Tues Wed Thurs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Demand 17 13 15 19 14 16 11 13 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 14 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 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. 16 Enhancement Some days we will have too many workers Excess Only concerned with the largest excess Minimize the largest Excess 17 Challenge Formulate a Solver Model 18 Formulating the LP Scheduling Postal Workers Shift Day Mon Tues Wed Thurs Fri Sat Sun Mon - Tues - Wed - Thurs - Fri - Sat - Sun Fri Sat Sun Mon Tues Wed Thurs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Demand 17 13 15 19 14 16 11 19 Minimize the Maximum Min Max{XS[Mon], XS[Tues], …} Min Z S.t. Z ≥ XS[Mon] S.t. Z ≥ XS[Tues] … S.t. MF + S.t. MF + TS + ThM + FT + SW + SuTh – XS[Mon] = 17 FT + SW + SuTh – XS[Tues] = 13 …. 20 Enhancement Ensure at least 30% of the workers have Sunday off Formulate a Solver Model 21 Formulating the LP Scheduling Postal Workers Shift Day Mon Tues Wed Thurs Fri Sat Sun Mon - Tues - Wed - Thurs - Fri - Sat - Sun Fri Sat Sun Mon Tues Wed Thurs 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Demand 17 13 15 19 14 16 11 22 The linear program Minimize z = MF + TS + WSu + ThM + FT + SW + SuTh 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 subject to Non-negativity 23 Summary More LP Modeling LPs are more general than Networks Modeling Time Clever choices of decision variables 24