For the three vectors shown in FIGURE EX3.20, A u + B u + C u = 1nj. What is vector B u ? a. Write B u in component form. b. Write B u as a magnitude and a direction.

Aggregate Production Planning Reading Pages with Linear Programming 622 – middle 623 The optimal solution for an aggregate production plan can be found with a linear programming model. Model Formulation Decision Variables R = units produced w/ regular production in period t t O t units produced w/overtime production in period t S t units produced w/ subcontracted production in period t It = inventory at end of period t W t workforce size for period t H t = # hired for period t Ft = # fired for period t Objective Function Minimize cost Constraints Demand, Capacity, Production, Workforce Decision Variables Consider a Trial and Error table for a mixed strategy of Level Production, Overtime, and Subcontracting: Regular Overtime Subcontract Quarter Demand Production Production Production Inventory BIT 3414 Copyright 2015 p. 1 Given the following information: Regular Prod. Capacity = 2000 units/qtr Regular Prod. Cost = $10/unit Quarter Demand 1 1200 Overtime Prod. Capacity = 500 units/qtr Overtime Prod. Cost = $20/unit 2 3100 Subcontracting Capacity = 1000 units/qtr Subcontracting Cost = $25/unit 3 2800 Inventory Capacity = 4000 units/qtr Inventory Cost = $5/unit/qtr 4 900 Beginning Inventory = 100 units Backorder Cost = $14/unit/qtr The company would like to determine the optimal mixture of the Level Production, Overtime, and Subcontracting strategies with a linear programming model. Decision Variables Objective Function Minimize Cost Z = + 25S + 25S + 25S + 25S 1 2 3 4 + 5I + 5I + 5I + 5I 1 2 3 4 Constraints Capacity Demand S: S 1 S 2 S ,3S ≤41000 3: R +3O + 3 + I 3 I 2 2803 I: I , I , I , I ≤ 4000 1 2 3 4 4: R +4O + 4 + I 4 I 3 9004 BIT 3414 Copyright 2015 p. 2 Given the following information: Hiring cost = $200 / worker Firing cost = $500 / worker Quarter Demand Regular Production Cost = $2 / unit Spring 30,000 Summer 10,000 Inventory cost = $0.50 / unit / quarter Production per employee = 1,000 units / quarter Fall 70,000 Winter 90,000 Beginning workforce = 50 workers Beginning inventory = 0 units The company would like to determine the optimal mixture of the Level Production and Chase Demand strategies with a linear programming model (assume no capacity limitations). Decision Variables Objective Function Minimize Cost Z = + 200H +1200H + 2002 + 200H 3 4 + 500F +1500F + 502F + 500F 3 4 Constraints Demand Production Workforce 3: R 3 I –2I =370,000 3: R –31000W = 0 3 3: W –3W – H2+ F =30 3 4: R 4 I –3I =490,000 4: R –41000W = 0 4 4: W –4W – H3+ F =40 4 BIT 3414 Copyright 2015 p. 3 Given the following information: Regular Prod. Capacity = 30,000 units/qtr Regular Prod. Cost = $10/unit Quarter Demand Subcontracting Capacity = 20,000 units/qtr Subcontracting Cost = $25/unit 1 30,000 2 60,000 Inventory Capacity = 10,000 units/qtr Inventory Cost = $5/unit/qtr Beginning Inventory = 4,000 units Backorder Cost = $10/unit/qtr 3 25,000 4 40,000 Production/employee = 2,000 units/qtr Hiring cost = $1000/ worker Beginning workforce = 10 workers Firing cost = $1500 / worker The company would like to determine the optimal mixture of the Level Production, Chase Demand, and Subcontracting strategies with a linear programming model. Decision Variables Objective Function Minimize Cost Z = 10R +110R + 102 + 10R 3 4 + 25S + 15S + 252 + 25S 3 4 + 5I 1 5I +25I + 3I 4 + 1000H + 1100H + 10002 + 1000H 3 4 + 1500F + 1500F + 15002 + 1500F 3 4 Constraints Capacity Production S: S 1 S 2 S 3 S 4 20,000 I: I1, 2 ,3I 4 I ≤ 10,000 2: R 2 2000W = 0 2 3: R – 2000W = 0 Demand 3 3 4: R 4 2000W = 0 4 Workforce 2: R + S + I – I = 60,000 2 2 1 2 3: R 3 S +3I – I2= 23,000 2: W –2W – H1+ F =20 2 4: R 4 S +4I – 3 = 44,000 3: W –3W – H2+ F =30 3 BIT 3414 Copyright 2015 4: W –4W – H3+ F =40 4 p. 4