Selection from Independent Projects Under Budget Limitation ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 Selection from Independent Projects Under Budget Limitation Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Mc Graw Hill CHAPTER 12 Learning Objectives ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 Learning Objectives Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Learning Objectives Capital Rationing rationale; Use of PW analysis in capital rationing among independent projects; Use of PW to select from several unequal-life independent projects; Application of Linear Programming to the solution of capital budgeting problems. Chapter Summary Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 An Overview of Capital Rationing Among Projects ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 12.1 An Overview of Capital Rationing Among Projects Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 Capital Budgeting Overview Investment capital represents a scarce resource; Generally more projects for funding consideration than there are funds available to fund. Some projects may be funded and some may not! We have, then, the “independent project selection” problem. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

An investment opportunity for the firm; 12.1 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Independent Projects: A set of projects ( two or more) are independent if: 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 in the set does not impact to accept or reject any other project in the set. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

A “bundle” is a collection of independent projects. 12.1 Project Bundles A “bundle” is a collection of independent projects. Independent-type projects tend to be quite different from each other. The candidate set of projects is all projects under consideration for funding from a given budget amount. Not all projects can be selected – budget constraints may exist. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 Capital Budgeting – Characteristics Identify independent projects and their estimated cash flows; Each project is selected entirely, or it is not selected at all; Partial projects are not permitted; A given budget constraint restricts the total amount available for investment; Objective: maximize the return on investment using a measure of worth. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Accept projects with the best PW values determined at the MARR; 12.1 Selection Guidelines Accept projects with the best PW values determined at the MARR; Over the project life; Provided the investment capital limit is not exceeded. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 The Capital Budgeting Problem Cash Flow Profile (A) $$ (B) $$ (C) $$ $$ Budget Objective: Max. PW of the selected bundles Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 Max Present Worth Previous Assumption: Equal Life for the alternatives; No longer valid for capital budgeting; No life cycle beyond the estimated life of each bundle; PW over the respective life of each independent project; Implicit Reinvestment Assumption is in place. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 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? Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 Use of ROR – To Select ROR can be used to select projects; However – must apply the incremental ROR method (cumbersome); Recall, ROR and NPV may not rank (select) projects the same; Different reinvestment assumptions within the two methods; Use incremental ROR to select! Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 Suggested Objective Function It is suggested that PV be used as the criteria in the associated objective function as opposed to ROR. Probably best to not use the incremental ROR ranking; Easier to apply PV at the MARR to all of the projects. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2. Capital Rationing Using PW Analysis of Equal-Life Projects ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 12.2. Capital Rationing Using PW Analysis of Equal-Life Projects Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 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; Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 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? Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Given “m” projects (independent), how many possible bundles are there? 12.2 Number of Bundles Given “m” projects (independent), how many possible bundles are there? Rule: Total no. of bundles = 2m; 2m – 1 bundles if you cut out the DN option; If m = 4 then 24 – 1 = 15 bundles (excluding the DN option). Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

If m = 6 then 26 = 64 bundles to evaluate; 12.2 Number of Bundles Manual approaches are not well suited for “large” numbers of candidate projects. If m = 6 then 26 = 64 bundles to evaluate; If m = 30 then 230 bundles to evaluate; Equal 1,073,741,824 bundles! Require a more sophisticated approach other than a manual analysis. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Example of Bundling: m =4 Assume: Project Investment $ A $10,000 B 5,000 C 8,000 D 15,000 $38,000 Total Assume a b = $25,000 (The budget max.) One cannot have all 4 projects because of the budget limitation. What, then, is the optimal combination of projects? Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Example of Bundling: m =4 Assume: Project Investment $ A $10,000 B 5,000 C 8,000 D 15,000 $38,000 Total Assume a b = $25,000 (The budget max.) From the 15 possible combinations, which one bundle of projects will maximize the present worth of the selected bundle? Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Example of Bundling: m =4 Assume: Project Investment $ 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/ Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Steps for a Manual Analysis 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 one at a time, two at a time, etc., and listing. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Possible Combinations Where m = 4 1. Do Nothing; 14. BCD 2. A 15. CD 3. B 16. 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 Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Ordering the Combinations Order the bundles from low to high based upon the total budget requirement of the combination. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Rank-Ordered Bundles: Total Investment Eliminate Those Mutually Exclusive Bundles That Exceed the $25,000 Budget Limitation. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Reduced Budget – Feasible Set Four bundles are infeasible: they exceed the budget amt. dropped from further consideration. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.1 The Feasible Set The feasible set of mutually exclusive bundles. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

The previous slide shows the feasible set; 12.2 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 General Solution Technique Develop all mutually exclusive bundles. Eliminate those bundles whose total investment requirement exceeds the budget amount. Within each bundle, sum the NCF’s for all projects in that bundle and compute the PV of the bundle at the MARR. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 General Solution Technique Let “j” equal the bundle number; PWj = PW of bundles net cash flows – the initial investment. Select the bundle with the largest PWj value. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Number of candidate projects = 5 Set = {DN, A,B,C,D,E} 12.2 Example 1 Assume “b” = 20 million; Number of candidate projects = 5 Set = {DN, A,B,C,D,E} No. of bundles = 25 = 32 possible combinations. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Example for “m” = 5 Projects Amounts are in units of $1,000. 25 Possible Bundles:”E” is removed $21 Million > 20 Million Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.2 Example 1: Feasible Bundles Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Left over budget = 6 million – assumed to be invested at the MARR 12.2 Max Bundle is { CD }; Left over budget = 6 million – assumed to be invested at the MARR Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Capital Rationing Using PW Analysis of Unequal-Life Projects ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 12.3 Capital Rationing Using PW Analysis of Unequal-Life Projects Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Example 2: Unequal Lives Project Initial Investment Ann. Net Cash Flows Life – Yrs A -$8,000 $3,870 6 B -15,000 2,930 9 C -8,000 2,680 5 D 2,540 4 24 = 16 Bundles to evaluate: 8 are feasible! Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Select { AC } for $16,000: $4,000 assumed invested at the MARR 12.3 Example 2: PV Summary Bundle Project(s) PV Comments 1 A $+6,646 2 B -1,019 Reject 3 C 984 4 D -748 5 AC 7,630 Max Bundle 6 AD 5,898 7 CD 235 8 Do Nothing Select { AC } for $16,000: $4,000 assumed invested at the MARR Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Two Independent Bundles A and B Assume two independent projects, A and B; Life of A = nA; Life of B = nB; A B (unequal lives). Assume A and B have the same net cash flow each time period. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Notation for Unequal Life Problem nL = life of the longer lived project; nj = life of the shorter lived project; nA = Life of A nB = Life of B Diagram the two cash flows on the next slide. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Unequal-Life Projects; A and B FWB PWB nB = nL B Longer life Project: i = MARR Investment For B FWA PWA FW Period of reinvestment @ MARR A nL nA Investment For A Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Shorter Project: A with Reinvestment FWA PWA FW A Period of reinvestment @ MARR nL nA Investment For A Compute the FW from nA out to nL of A. Assumed to be reinvested at the MARR rate! Yield FWA given reinvestment at the MARR rate. Then, find PWA from FWA at the MARR rate. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 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; PWBundle = PWA + PWB Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Project Init. Inv. ANCF Life C -$8,000 $2,680 5 D -8,000 2,540 4 12.3 C and D in Example 2 Project Init. Inv. ANCF Life C -$8,000 $2,680 5 D -8,000 2,540 4 Find the PW of the bundle { C,D }. Unequal life situation. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 Bundle { C, D }. Over 9 years Bundle Cash Flow: FW = $57,111 $2,540/yr (D) $2,680 (C) 0 1 2 3 4 5 6 7 8 9 FW(C,D, @ 15%) of + CF’s = +$57,111. PW(C,D @ 15%) = -$16,000 + 57,111(P/F,15%,9) = +$235.00 -$16,000 Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.3 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.4 Capital Budgeting Problem Formulation Using Linear Programming ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 12.4 Capital Budgeting Problem Formulation Using Linear Programming Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.4 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.4 Notation for ILP Formulation “b” = capital budget limitation for the time period; xk = The decision variable for project k; m = the number of projects; xk  { 0, 1 }; If xk = 1 then all of the project is accepted; If xk = 0 then none of the project is accepted. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12. Integer LP Formulation Model Objective Function: Budget Constraint: Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

The Present Worth for each project is calculated as: 12. 4 PW for Each Project: The Present Worth for each project is calculated as: Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Objective Function (Max): 12. ILP for Example 2 Objective Function (Max): Budget Constraint; Decision Variables; Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

12.4 Solution from Spreadsheet X1 = 1 X2 = 0 X3 = 1 X4 = 0 Objective Function Value = $7,630 $16,000 spent; Leaving $4,000 unspent, but assumed to be invested at the 15% rate. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Solver is an add-in optimization tool to Excel; 12.4 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Mc Graw Hill CHAPTER 12 Chapter Summary ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 Chapter Summary Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Capital represents a scarce resource; 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; Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

For “m” projects there are 2m possible combinations or bundles; Chapter 12 Summary For “m” projects there are 2m possible combinations or bundles; Manual solutions work only for very small problems; Larger problems require a mathematical programming formulation; Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

One calculates the PW of the j-th bundle at the firm’s MARR; 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. Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University

Mc Graw Hill CHAPTER 12 End of Slide Set ENGINEERING ECONOMY Fifth Edition Mc Graw Hill Blank and Tarquin CHAPTER 12 End of Slide Set Blank & Tarquin: 5th edition. Ch.12 Authored by Dr. Don Smith, Texas A&M University