Using Binary Integer Programming to Deal with Yesor-No. Chapter 7: Hillier and Hillier

Transcription

1 Using Binary Integer Programming to Deal with Yesor-No Decisions Chapter 7: Hillier and Hillier 1

2 Agenda Case Study: California Manufacturing Company Wyndor Case Revisited Variation of Wyndor s Problem 2

3 Applications of Binary Variables Binary variables only allow two choices This makes them suited for problems that are characterized by variables that can take on only two possibilities. Examples: Do a project or not do a project? To hire or not to hire? To build or not to build? To Sell or not to sell? 3

4 Case Study: California Manufacturing Company (CMC) The California Manufacturing Company is a company with factories and warehouses throughout California. It is currently considering whether to build a new factory in Los Angeles and/or San Francisco. Management is also considering building one new warehouse where a new factory has been recently built. Should the CMC build factories and/or warehouses in Los Angeles and/or San Francisco? 4

5 Case Study: CMC Cont. Binary Decision Decision Variable NPV (Millions) Capital Needed (Millions) Build a factory in Los Angeles FLA $8 $6 Build a factory in San Francisco Build a warehouse in Los Angeles Build a warehouse in San Francisco Building Money Available: FSF 5 3 WLA 6 5 WSF 4 2 $10 million 5

6 Case Study: CMC Cont. FLA, FSF, WLA,WSF are all binary variables which take on the value of 1 if the specific item is done and zero if it is not done. We also need to make sure that at most one warehouse is built and it is built where a factory is built. 6

7 Mathematical Model for CMC FLA, FSF, WLA, WSF Subject to : 6* FLA + 3* FSF + 5* WLA + 2* WSF 10 WLA WSF Max FLA WSF WLA + WSF 1 FLA, FSF, WLA, WSF 8* FLA + 5* FSF + 6* WLA + {0,1} 4* WSF 7

8 Wyndor Case Revisited Two new products have been developed: An 8-foot glass door A 4x6 foot glass window Wyndor has three production plants Production of the door utilizes Plants 1 and 3 Production of the window utilizes Plants 2 and 3 Objective is to find the optimal mix of these two new products. 8

9 Wyndor Case Revisited Cont. Production Time Used for Each Unit Produced Plant Doors Windows Available Per Week 1 1 hour 0 hour 4 hour 2 0 hour 2 hour 12 hour 3 3 hour 2 hour 18 hour Unit Profit $300 $500 9

10 Wyndor Case Revisited Cont. Max w.r.t. D, W 300D + 500W Subject to : D 4 (Constraint 1) 2W 12(Constraint 2) 3D + 2W 18 (Constraint 3) D 0, W 0 (Constraints 4 and 5) 10

11 Changing Wyndor to Account for Setup Costs Suppose that two changes are made to the original Wyndor problem: If Wyndor chooses to produce doors, it must pay a one time set-up cost of $700, while if Wyndor produces windows it must pay a setup cost of $1,300. We want to restrict the doors and windows to be integer values. 11

12 Graphical Solution to Original Wyndor Problem Production rate for windows W 8 Optimal solution 6 (2, 6) 4 Feasible Region P = 3,600 = 300 D+ 500 W Production rate for doors 10 D 12

13 Feasible Solutions for Wyndor W with Setup Costs Production quantity for windows 8 6 (0, 6) gives P = 1700 (2, 6) gives P = = (4, 3) gives P = = 700 (0, 0) gives P = 0 (4, 0) gives P = Production quantity for doors D 13

14 Wyndor s Mathematical Model With Set-Up Costs Max w.r.t. D, W, DS, WS 300D + 500W Subject to : D 4 2W 12 3D + 2W 18 D 99*DS W 99* WS D, W 0, and are integers DS, WS {0,1} 700DS 1300WS 14

15 Changing Wyndor to Account for Mutually Exclusive Products Suppose Wyndor decides that it only wants to produce doors or windows rather than both. This implies that either doors have to be zero or windows have to be zero. 15

16 Wyndor s Mathematical Model With Mutually Exclusive Products Max w.r.t. D, W, DS, WS 300D + 500W Subject to : D 4 2W 12 3D + 2W 18 D 99*DS W 99* WS DS + WS 1 D, W 0, and are integers DS, WS {0,1} 700DS 1300WS 16

17 Changing Wyndor to Account for Either-Or Constraints Suppose the company is trying to decide whether to build a new up-to-date plant that will be used to replace plant 3. This implies that Wyndor wants to examine the profitably of using plant 4 versus plant 3. 17

18 Wyndor s Data with Either/Or Constraint Production Time Used for Each Unit Produced Plant Doors Windows Available Per Week 1 1 hour 0 hour 4 hour 2 0 hour 2 hour 12 hour 3 3 hour 2 hour 18 hour 4 2 hours 4 hours 28 hours Unit Profit $300 $500 18

19 Graphical Solution with Plant 3 or Plant 4 (a) Choose Plant 3 (b) Choose Plant 4 W W 8 8 (2, 6) gives P = 3, (4, 5) gives P = 3,700 4 Feasible region 4 Feasible region D D 19

20 Wyndor s Mathematical Model With Either/Or Constraint Max w.r.t. D, W, P3, P4 300D + 500W Subject to : D 4 2W 12 3D + 2W *P4 2D + 4W *P3 P3+ P4 1 D, W 0, and are integers P3, P4 {0,1} 20