1. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-1 Developed By: Dr. Don Smith, P.E. Department of Industrial Engineering Texas A&M University College Station, Texas Executive Summary Version Chapter 12 Selection from Independent Projects Under Budget Limitation

2. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-2 LEARNING OBJECTIVES 1.Capital rationing 2.Projects with equal lives 3.Projects with unequal lives 4.Linear program model This chapter introduces the concept of linear optimization via Linear Programming

3. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-3 Sct 12.1 Overview of Capital Rationing Among Projects  Investment capital represents a scarce resource ;  Generally more projects are defined for funding consideration than there are funds available  Some projects may be funded and some may not  This is “independent project selection”  Project:  An investment opportunity for the firm;  Generally been evaluated and found to be acceptable given that funds are or will be available to execute the project. Investment capital is always constrained in that there are never enough funds to execute all of the “worthy” projects

4. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-4 Independent Projects & Bundles  Independent projects:  A set of projects ( two or more) are independent if:  The cash flows of one project do not in any way impact the cash flows of any other project in the set.  Selection of one project in the set does not impact acceptance or rejection of any other project  “Bundle” is a collection of independent projects.  Independent-type projects tend to be quite different from each other.  Not all projects can be selected – budget constraints may exist.

5. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-5 Capital Budgeting - Characteristics  Identify independent projects and their estimated cash flows;  Each project is selected entirely or not selected, partial projects are not permitted;  A given budget constraint restricts the total amount available for investment;  Objective: maximize the return on investment using an economic measure of worth.  For example -- Accept all projects with the best PW values determined at the MARR until funds are depleted

6. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-6 Maximize PW to Select Projects  Equal service for the alternatives is not valid for capital budgeting;  No life cycle beyond the estimated life of each project; each project invested in only for 1 life cycle;  PW over the respective life of each independent project has an implied assumption …  Reinvestment assumption: All positive net cash flows are reinvested at the MARR from the time realized until the end of the longest lived project. This process of reinvestment is critical to the sustained success of a profitable firm.

7. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-7 Sct 12.2 Capital Rationing Using PW Analysis of Equal-Life Projects  Given a set of candidate projects whose lives are all equal  Formulate all of the mutually exclusive bundles from the set;  Selection of projects is based on the PW for each project  Assume you have 4 projects with equal lives;  Candidate set = { A, B, C, D};  The Do-Nothing (DN) project is also an option as a bundle since no projects are included  How many mutually exclusive bundles can be formed?

8. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-8 Number of Bundles  Given m projects (independent), how many possible bundles are there?  Rule: Total No. of bundles = 2 m  2 m – 1 bundles if you cut out the DN option;  If m = 4 then 2 4 – 1 = 15 bundles (excluding the DN option).  If m = 6 then 2 6 = 64 bundles to evaluate;  If m = 30 then 2 30 bundles to evaluate; equals 1,073,741,824 bundles!  Manual approaches are not well suited for “large” numbers of candidate projects.  Require a more sophisticated approach other than a manual analysis.

9. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-9 Example of Bundling: m = 4 Assume: ProjectInvestment $ A$10,000 B 5,000 C 8,000 D 15,000 $38,000 Total Assume a budget of $25,000 (budget max.) One cannot accept all 4 projects because of the budget limitation. What then is the optimal combination of projects? 2 4 -1 combinations or 15 bundles to evaluate

10. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-10 Steps for a Manual Analysis 1. Identify the investments and cash flows for all feasible combinations of the projects where each combination represents an economically mutually exclusive bundle. Consider all possible combinations by taking the projects 1 at a time, two at a time, etc and listing them. The next slide illustrates enumeration of all bundles

11. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-11 Possible Combinations for m = 4 1. Do Nothing (DN)14. BCD 2. A15. CD 3. B16. ACD 4.C 5.D 6. AB 7. AC 8. AD 9. ABC 10. ABCD 11. BC 12.BD 13.ABD TOTAL ENUMERATION OF ALL 16 POSSIBLE MUTUALLY EXCLUSIVE COMBINATIONS

12. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-12 Rank-Ordered Bundles: Total Investment Eliminate those mutually exclusive bundles that exceed the $25,000 budget limitation. Bundles 13-16 are infeasible in that they exceed the budget limitation of $25,000. Bundles 1 – 12 constitute the feasible set.

13. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-13 Step 2. Determine NCF Values  For each bundle, add the yearly net cash flow (NCF) estimates for all project in the bundle  Let j equal the bundle number  Initial year (t = 0) NCF for bundle j is referred to as NCF j0

14. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-14 Steps 3 & 4 - PW Solution Technique  3. PW j = PW of bundle net cash flows minus initial investment.  4. Select the bundle with the largest PW j value

15. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-15 Example 12.1  Assume b = $20 million;  Number of projects = 5  Set = {A,B,C,D,E}  No. of bundles = 2 5 = 32 possible combinations.  Each project has a life of 9 years  Determine the optimal bundle of projects that will maximize the PW of the total set of projects selected

16. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-16 Initial Investments for m = 5 Projects 2 5 possible bundles:”E” is removed at the very beginning since $21 million > $20 million Amounts are in units of $1,000.

17. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-17 Feasible Bundles, NCF and PW Values

18. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-18 The Maximum PW Bundle is { C,D }  Max Bundle is {CD}  Left over budget = $6 million -- assumed to be invested at the MARR = 15% per year

19. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-19 Sct 12.3 Capital Rationing Using PW Analysis of Unequal-life Projects Critical Point  It is assumed that reinvestment of positive net cash flows occurs at the MARR from the time they are realized until the end of the longest lived project.  Use of the LCM of lives is not necessary for the capital budgeting model.  See Example 12.2

20. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-20 Example 12.2: Unequal Lives Example ProjectInitial Investment, $ Annual Net Cash Flow, $/year Project Life, Years A$ - 8,000$ 3,8706 B- 15,0002,9309 C- 8,0002,6805 D- 8,0002,5404 2 4 = 16 bundles to evaluate; 8 are feasible.

21. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-21 PW Summary BundleProject(s)PWComments 1A$+ 6,646 2B- 1,019Reject 3C+ 984 4D- 748Reject 5AC+ 7,630 Max PW bundle 6AD+ 5,898 7CD+ 235 8Do Nothing0 Select {AC} for $16,000 with $4,000 assumed invested at MARR

22. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-22 Demonstration that Reinvestment Assumption is Correct  Assume two independent projects, A and B  Life of A is n A ; life of B is n B  Life of A  life of B  Assume A and B have the same net cash flow for each project for each time period.  Let n L = life of the longer lived project  and, n j = life of the shorter lived project

23. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-23 Unequal-life Projects – Cash Flows B n B = n L FW B Investment for B PW B Longer life project: i = MARR nAnA Investment for A FW Period of reinvestment at MARR FW A nLnL A PW A

24. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-24 Shorter Life Project: A with Reinvestment nAnA Investment for A FW Period of reinvestment at MARR FW A nLnL A PW A  Compute the FW from n A out to n L of A  Assume reinvestment at the MARR rate  Calculate FW A given reinvestment at the MARR rate.  Then, find PW A from FW A at the MARR rate.

25. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-25 Bundling A and B: Unequal lives  Now A and B have unequal lives;  If reinvestment at the MARR is assumed for the shorter-life project out to the life of the longer life project then:  One can create a bundle of A and B by computing;  PW Bundle = PW A + PW B

26. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-26 Illustration for C and D in Example 12.2  Find the PW of the bundle {C,D}  Unequal life situation ProjectInit. InvestmentAnnual NCFLife C-$8,000$2,6805 D-8,0002,5404

27. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-27 FW and PW of Bundle {C, D} Over 9 Years 0 1 2 3 4 5 6 7 8 9 -$16,000 $2,540/yr for D $2,680/yr for C FW = $57,111  FW(CD @ 15%) of + CF’s = +$57,111  PW(CD @ 15%) = -16,000 + 57,111(P/F,15%,9) = +$235.

28. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-28 Sct 12.4 Capital Budgeting Problem Formulation Using Linear Programming  Apply a special case of LP termed ILP 0-1 Linear Programming  Objective:  Max sum of PW of NCF’s of the selected set  Subject to:  Capital budget constraint  Each project i is either selected (x j = 1) or a given project is not selected (x i = 0)

29. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-29 Formal Statement of the Problem Maximize: Subject to: PW k of each project is calculated at the MARR rate  Bundles need not be formed

30. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-30 Notation for ILP Formulation  b = capital budget limitation for the time period  x k = decision variable for project k;  m = number of projects  x k = {0, 1}  If x k = 1 then all of s project is accepted  If x k = 0 then none of a project is accepted

31. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-31 ILP for Example 12.3  Objective Function:  Budget Constraint:  Decision Variables:

32. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-32 Solution from Spreadsheet  X 1 = 1  X 2 = 0  X 3 = 1  X 4 = 0  Objective function value Z = $7,630  $16,000 spent  Leaving $4,000 unspent but assumed to be invested at the 15% rate. Your instructor will demo the use of Excel’s LP (Solver) feature and the proper formatting.

33. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-33 Use of Solver for ILP  Solver is an add-in optimization tool to Excel  See the format in Figure 12-6  Students are encouraged to compose their own spreadsheet to evaluate this problem  Using LP analysis, perform sensitivity analysis using the reports feature of Solver.

34. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-34 Chapter Summary  Capital represents a scarce resource  Capital is limited and must be rationed  Evaluate the capital budgeting for:  Equal life projects  Unequal life projects.  Apply the PW method  Create mutually exclusive bundles

35. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-35 Summary - continued  For m projects there are 2 m possible combinations or bundles  Manual solution work only for very small problems  Larger problems require a mathematical programming formulation and spreadsheet solution

36. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-36 Summary – continued  Determine the feasible bundles  Calculates PW of the j-th bundle at the MARR  Select the bundle with the maximum present worth  LP methods with computer assistance do this automatically.  Other constraints may also be present.

37. Slide Sets to accompany Blank & Tarquin, Engineering Economy, 6 th Edition, 2005 © 2005 by McGraw-Hill, New York, N.Y All Rights Reserved 12-37 Chapter 12 End of Set