Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 1 CHAPTER 12 Selection from Independent Projects Under Budget Limitation McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 2 CHAPTER 12 Learning Objectives McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University3 Learning Objectives 1. Capital Rationing rationale; 2. Use of PW analysis in capital rationing among independent projects; 3. Use of PW to select from several unequal-life independent projects; 4. Application of Linear Programming to the solution of capital budgeting problems.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 4 CHAPTER An Overview of Capital Rationing Among Projects McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Capital Budgeting Overview Investment capital represents a scarce resource; Some projects may be funded and some may not! We have, then, the “independent project selection” problem.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Projects Project: An investment opportunity for the firm; Generally been evaluated and found to be acceptable given that funds are or will be available to fund (execute) the project.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Independent Projects Independent Projects: The cash flows of one project do not in anyway impact the cash flows of any other project in the set. Selection of one project to accept or reject does not impact any other project in the set.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Project Bundles A “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.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Capital Budgeting – Characteristics Identify independent projects and their estimated cash flows; Each project is selected entirely, or it is not selected at all; Objective: maximize the return on investment

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Selection Guidelines Accept projects with the best PW values determined at the MARR; Provided the investment capital limit is not exceeded.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University The Capital Budgeting Problem (A) $$ (B) $$ (C) $$ Cash Flow Profile $$ Budget Objective: Max. PW of the selected bundles

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Max Present Worth Previous Assumption of equal Life for the alternatives is no longer valid for capital budgeting; No life cycle beyond the estimated life of each bundle; PW is over the respective life of each independent project.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Reinvestment Assumption The following is assumed for the capital budgeting problem All positive net cash flows of a project (bundle) are reinvested at the MARR from the time they are realized until the END of the LONGEST-LIVED project! With this assumption, projects (bundles) with unequal lives can be accommodated in the analysis.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Flexibility Issue Given a budget of, say, $b: This constraint may marginally disallow an acceptable project that is next in line for acceptance! How is this situation handled in practice?

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Marginally Exceeding the Budget Assume project A has a PW(i%) > 0. If the addition of A to the selected set will cause the budget to be overspent by, say, $1,000 – should A be included? Mathematically – NO! In practice – A might be added and the budget limit “b” slightly readjusted to accommodate A’s addition.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Suggested Objective Function ROR can be used to select projects; but must apply the incremental ROR method; It is suggested that PV be used as the criteria in the associated objective function. It is easier to apply PV at the MARR to all of the projects.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 17 CHAPTER Capital Rationing Using PW Analysis of Equal-Life Projects McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Rationing for Equal-Life Projects Given a set of candidate projects whose lives are all equal; Calculate the PV(MARR) for each project; Formulate all of the mutually exclusive bundles from the set;

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Mutually Exclusive Bundles Assume you have 4 projects having equal lives; Candidate set = { A, B, C, D}; The Do-Nothing (DN) alternative is also an option: Set = { DN, A, B, C, D }; Given 4 projects, how many mutually exclusive bundles can be formed?

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 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).

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Number of Bundles Manual approaches are not well suited for “large” numbers of candidate projects. If m = 30 then 2 30 bundles to evaluate; Equal 1,073,741,824 bundles! Require a more sophisticated approach other than a manual analysis.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example of Bundling: m =4 Assume: ProjectInvestment $ A$10,000 B 5,000 C 8,000 D 15,000 $38,000 Total Assume b = $25,000 (The budget max.) What, then, is the optimal combination of projects?

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example of Bundling: m =4 Assume: ProjectInvestment $ A$10,000 B 5,000 C 8,000 D 15,000 $38,000 Total Assume a b = $25,000 (The budget max.) Feasible bundles must have Positive PV and a total Budget that does not Exceed $25,000/

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Possible Combinations Where m = 4 1. Do Nothing; 13. ABD 2. A 14. BCD 3. B 15. CD 4. C 16. ACD 5. D 6. AB 7. AC 8. AD 9. ABC 10. ABCD 11. BC 12. BD TOTAL ENUMERATION OF ALL 16 POSSIBLE MUTUALLY EXCLUSIVE COMBINATIONS

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Ordering the Combinations Order the bundles from low to high based upon the total budget requirement of the combination.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Rank-Ordered Bundles: Total Investment Eliminate Those Mutually Exclusive Bundles That Exceed the $25,000 Budget Limitation.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Reduced Budget – Feasible Set Four bundles are infeasible: they exceed the budget amt. dropped from further consideration.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University The Feasible Set The feasible set of mutually exclusive bundles.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Bundle Selection The previous slide shows the feasible set; None of the combinations exceed the budget limitation; If one has the PV of each bundle, then pick the bundle with the maximum present value.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University General Solution Technique 1. Develop all mutually exclusive bundles. 2. Eliminate those bundles whose total investment requirement exceeds the budget amount. 3. Within each bundle, sum the NCF’s for all projects in that bundle and compute the PV of the bundle at the MARR.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University General Solution Technique Let “j” equal the bundle number; PW j = PW of bundles net cash flows – the initial investment. Select the bundle with the largest PW j value.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.1 (P407) Assume “b” = 20 million; Number of candidate projects = 5 Set = {DN, A,B,C,D,E} No. of bundles = 2 5 = 32 possible combinations.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example for “m” = 5 Projects 2 5 Possible Bundles:”E” is removed $21 Million > 20 Million Amounts are in units of $1,000.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.1: Feasible Bundles

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.1 Max Bundle is { CD }; Left over budget = 6 million – assumed to be invested at the MARR

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 36 CHAPTER Capital Rationing Using PW Analysis of Unequal-Life Projects McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Unequal-Life Projects 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.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.2: Unequal Lives (P409) Project Initial Investment Ann. Net Cash Flows Project Life – Yrs A-$8,000$3,8706 B-15,0002,9309 C-8,0002,6805 D-8,0002, = 16 Bundles to evaluate: 8 are feasible!

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.2: PV Summary BundleProject(s)PVComments 1A$+6,646 2B-1,019Reject 3C984 4D-748Reject 5AC7,630 Max Bundle 6AD5,898 7CD235 8Do Nothing0 Select { AC } for $16,000: $4,000 assumed invested at the MARR

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Two Independent Bundles A and B Assume two independent projects, A and B; Life of A = n A ; Life of B = n B ; A  B (unequal lives). Assume both A and B have uniform series.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Notation for Unequal Life Problem n L = life of the longer lived project; n j = life of the shorter lived project; n A = Life of A n B = Life of B Diagram the two cash flows on the next slide.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Unequal-Life Projects; A and B B n B = n L FW B Investment For B PW B Longer life Project: i = MARR nAnA Investment For A FW Period of MARR FW A nLnL A PW A

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Shorter Project: A with Reinvestment nAnA Investment For A FW Period of MARR FW A nLnL A PW A Compute the FW from n A out to n L of A. Assumed to be reinvested at the MARR rate! Yield FW A given reinvestment at the MARR rate. Then, find PW A from FW A at the MARR rate.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 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

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University C and D in Example 12.2 Find the PW of the bundle { C,D }. Unequal life situation. ProjectInit. Inv.ANCFLife C-$8,000$2,6805 D-8,0002,5404

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Bundle { C, D }. Over 9 years Bundle Cash Flow: $16,000 $2,540/yr (D) $2,680 (C) FW = $57,111 15%) of + CF’s = +$57, %) = -$16, ,111(P/F,15%,9) = +$235.00

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Bundle Analysis Summary Given the life of the longest project; Find the PW(MARR) given reinvestment where required for all bundles; Throw out any bundles with negative PW’s unless other constraints require their presence.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Setting Up to Solve Complex Problems Solution of complex or large numbers of bundles is best approached by applying linear programming formulations; The next section describes a 0-1 Linear Programming formulation to the capital budgeting problem.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 49 CHAPTER Capital Budgeting Problem Formulation Using Linear Programming McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Application of Integer LP Formulate as an Integer LP model; Decision variable is a { 0, 1 } variable; Let x  {0,1}; Means: Can have all of a project or none of the project. Partial funding is not permitted! Objective Function: Max {Present worth of the selected bundles} Subject to budget and other constraints.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Notation for ILP Formulation “b” = capital budget limitation for the time period; x k = The decision variable for project k; m = the number of projects; x k  { 0, 1 }; If x k = 1 then all of the project is accepted; If x k = 0 then none of the project is accepted.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Integer LP Formulation Model Objective Function: Budget Constraint:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University PW for Each Project: The Present Worth for each project is calculated as:

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Example 12.3 (P413) Objective Function (Max): Budget Constraint; Decision Variables;

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Solution from Spreadsheet X 1 = 1 X 2 = 0 X 3 = 1 X 4 = 0 Objective Function Value = $7,630 $16,000 spent; Leaving $4,000 unspent, but assumed to be invested at the 15% rate.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University Use if Solver for ILP Solver is an add-in optimization tool to Excel; See the format in Figure 12-5; Students are encouraged to compose their own spreadsheet to evaluate this problem; Using LP analysis, perform sensitivity analysis using the reports feature of Solver.

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5 th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University 57 CHAPTER 12 Chapter Summary McMc Graw Hill ENGINEERING ECONOMY Fifth Edition Blank and Tarquin

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University58 Chapter 12 Summary Capital represents a scarce resource; Capital is limited and must be rationed; Evaluate the capital budgeting from: Equal life projects; Unequal life projects. Apply the PW method; Create mutually exclusive bundles;

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University59 Chapter 12 Summary For “m” projects there are 2 m possible combinations or bundles; Manual solutions work only for very small problems; Larger problems require a mathematical programming formulation;

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University60 Chapter 12 Summary One calculates the PW of the j-th bundle at the firm’s MARR; Determine the feasible bundles and, Select the bundle with the maximum present worth; LP methods with computer assistance, do this automatically. Other constraints may also be present.