When are non-integer solutions okay? Solution is naturally divisible. Solution represents a rate. Solution only for planning purposes

Transcription

1 Session # Page 1 When are non-integer solutions okay? Solution is naturally divisible Solution represents a rate Solution only for planning purposes When is rounding okay?

2 Session # Page 2 The Challenges of Rounding Rounded solution may not be feasible Rounded solution may not be close to optimal There can be many rounded solutions Example Consider a problem with 0 variables that are non-integer in the LP solution. How many possible rounded solutions are there?

3 Session # Page How Integer Programs are Solved

4 Session # Page Applications of Binary Variables Making yes or no type decisions Set-covering problems Fixed costs

5 Session # Page Example #1 (Capital Budgeting) Mercer Development is considering the potential of four different development projects. Each project would be completed in at most three years. The required cash outflow for each project is given in the table below, along with the net present value of each project to Mercer, and the cash that is available (from previous projects) each year. Cash Outflow Required ($million) Cash Available Project 1 Project 2 Project Project ($million) Year 1 0 Year Year 0 0 NPV 1 2 1

6 Session # Page Spreadsheet Solution to Example # A B C D E F G H I Mercer Development Capital Budgeting Project 1 Project 2 Project Project NPV ($million) Cash Outflow Required ($million) Total Outflow Available Year 1 0 <= 0 Year <= 1 Year 0 0 <= Total NPV Project 1 Project 2 Project Project ($million) Undertake? B C D E F G H I Project 1 Project 2 Project Project NPV ($million) Cash Outflow Required ($million) Total Outflow Year 1 =SUMPRODUCT($C$1:$F$1,C:F) <= 0 Year 2 0 =SUMPRODUCT($C$1:$F$1,C:F) <= 1 Year 0 0 =SUMPRODUCT($C$1:$F$1,C:F) <= Available Total NPV Project 1 Project 2 Project Project ($million) Undertake? =SUMPRODUCT(C1:F1,C:F)

7 Session # Page Additional Considerations (Logic and Dependency Constraints) At least one of projects 1, 2, or : Project 2 can t be done unless project is done: Either project or project, but not both: No more than two projects total:

8 Session # Page Example #2 (Set Covering Problem) Suppose the Washington State legislature is trying to decide on locations at which to base search-and-rescue teams. The teams are expensive, and hence they would like as few as possible. However, since response time is critical, they would like every county to either have a team located in that county, or in an adjacent county. Where should the teams be located.

9 Session # Page Spreadsheet Solution to Example # A B C D E F G H I J K L M N Search & Rescue Location # Teams # Teams County Team? Nearby County Team? Nearby 1 Clallam 0 1 >= 1 1 Chelan 0 2 >= 1 2 Jefferson 1 1 >= 1 20 Douglas 0 1 >= 1 Grays Harbor 0 2 >= 1 21 Kittitas 1 1 >= 1 Pacific 0 1 >= 1 22 Grant 0 1 >= 1 Wahkiakum 0 1 >= 1 2 Yakima 0 >= 1 Kitsap 0 1 >= 1 2 Klickitat 0 1 >= 1 Mason 0 1 >= 1 2 Benton 0 1 >= 1 Thurston 0 1 >= 1 2 Ferry 0 1 >= 1 Whatcom 0 1 >= 1 2 Stevens 1 1 >= 1 Skagit 1 1 >= 1 2 Pend Oreille 0 1 >= 1 Snohomish 0 1 >= 1 2 Lincoln 0 1 >= 1 King 0 1 >= 1 0 Spokane 0 1 >= 1 1 Pierce 0 2 >= 1 1 Adams 0 1 >= 1 1 Lewis 1 2 >= 1 2 Whitman 0 2 >= 1 1 Cowlitz 0 2 >= 1 Franklin 1 1 >= 1 1 Clark 0 1 >= 1 Walla Walla 0 1 >= 1 1 Skamania 1 2 >= 1 Columbia 0 1 >= 1 1 Okanogan 0 1 >= 1 Garfield 1 1 >= 1 Asotin 0 1 >= 1 Total Teams:

10 Session # Page Spreadsheet Solution to Example #2 (Formulas) B C D E F G # Teams County Team? Nearby 1 Clallam 0 =D+D >= 1 2 Jefferson 1 =D+D+D+D+D >= 1 Grays Harbor 0 =D+D+D+D+D+D1 >= 1 Pacific 0 =D+D+D+D1 >= 1 Wahkiakum 0 =D+D+D1+D1 >= 1 Kitsap 0 =D+D+D+D1+D1+D1 >= 1 Mason 0 =D+D+D+D+D+D1 >= 1 Thurston 0 =D+D+D+D1+D1 >= 1 Whatcom 0 =D1+D1+D22 >= 1 Skagit 1 =D1+D1+D1+D22+K >= 1 Snohomish 0 =D1+D1+D1+K >= 1 King 0 =D+D1+D1+D1+K+K >= 1 1 Pierce 0 =D+D+D+D1+D1+D1+K+K >= 1 1 Lewis 1 =D+D+D+D+D1+D1+D1+D21+K >= 1 1 Cowlitz 0 =D+D1+D1+D20+D21 >= 1 1 Clark 0 =D1+D20+D21 >= 1 1 Skamania 1 =D1+D1+D20+D21+K+K >= 1 1 Okanogan 0 =D1+D1+D22+K+K+K+K1 >= 1 Total Teams: =SUM(D:D22,K:K2) I J K L M N # Teams County Team? Nearby 1 Chelan 0 =D1+D1+D1+D22+K+K+K >= 1 20 Douglas 0 =D22+K+K+K+K >= 1 21 Kittitas 1 =D1+D1+K+K+K+K+K >= 1 22 Grant 0 =K+K+K+K+K+K1+K1+K1 >= 1 2 Yakima 0 =D1+D1+D21+K+K+K+K+K >= 1 2 Klickitat 0 =D21+K+K+K >= 1 2 Benton 0 =K+K+K+K+K1+K20 >= 1 2 Ferry 0 =D22+K+K1+K1 >= 1 2 Stevens 1 =K+K1+K1+K1+K1 >= 1 2 Pend Oreille 0 =K1+K1+K1 >= 1 2 Lincoln 0 =D22+K+K+K1+K1+K1+K1+K1 >= 1 0 Spokane 0 =K1+K1+K1+K1 >= 1 1 Adams 0 =K+K1+K1+K1+K1 >= 1 2 Whitman 0 =K1+K1+K1+K1+K1+K21+K22+K2 >= 1 Franklin 1 =K+K+K1+K1+K1+K20 >= 1 Walla Walla 0 =K+K1+K20+K21 >= 1 Columbia 0 =K1+K20+K21+K22 >= 1 Garfield 1 =K1+K21+K22+K2 >= 1 Asotin 0 =K1+K22+K2 >= 1

11 Session # Page Example # (Fixed Costs) Gandy Cloth Company is capable of manufacturing three types of clothing: shirts, shorts, and pants. The manufacture of each type of clothing requires that Gandy have the appropriate type of machinery available. This machinery can be rented at the following rates: shirt machinery, $200/week; shorts machinery, $/week; pants machinery, $0/week. Each type of clothing requires the amount of cloth and labor given in the table below. The sales price and variable cost are also given in the table. Labor Hours Cloth (sq. yds.) Sales Price Variable Cost Shirts $ $ Shorts 2 $ $ Pants $1 $ Available: hours sq. yds.

12 Session # Page Spreadsheet Solution to Example # A B C D E F G H Gandy Cloth Company Shirt Shorts Pants Sales Price $ $ $1 Variable Cost $ $ $ Fixed Cost $200 $ $0 Constraint Usage (per unit produced) Total Available Labor (hrs.) 2 <= Cloth (sq. yds.) 0 <= Lease Equipment? Revenue $.00 Production Quantity Variable Cost $ <= <= <= Fixed Cost $0.00 Produce only if Lease Profit $.00 Big Number B C D E F G H Shirt Shorts Pants Sales Price 1 Variable Cost Fixed Cost Constraint e (per unit prod Total Available Labor (hrs.) 2 =SUMPRODUCT($C$1:$E$1,C:E) <= Cloth (sq. yds.) =SUMPRODUCT($C$1:$E$1,C:E) <= Lease Equipment? Revenue =SUMPRODUCT(C1:E1,C:E) Production Quantity Variable Cost =SUMPRODUCT(C:E,C1:E1) <= <= <= Fixed Cost =SUMPRODUCT(C:E,C:E) Produce only if Lease =$C$1*C =$C$1*D =$C$1*E Profit =H1-H1-H1 Big Number 0