Contents 1 EXPLAINING SYSTEMS ANALYSIS 1-1 2 MODELS IN CIVIL AND ENVIRONMENTAL ENGINEERING 2-1 3 A GRAPHICAL SOLUTION PROCEDURE AND FURTHER EXAMPLES 3-1 4 THE SIMPLEX ALGORITHM FOR SOLVING LINEAR PROGRAMS 4-1 5 LINEAR PROGRAMS WITH MULTIPLE OBJECTIVES 5-1 6 LINEAR PROGRAMMING MODELS OF NETWORK FLOW 6-1 7 INTEGER PROGRAMMING AND ITS APPLICATIONS 7-1 8 SCHEDULING MODELS: CRITICAL PATH METHOD 8-1 9 DECISION THEORY 9-1 10 LESSONS IN CONTEXT: SIMULATION AND THE STATISTICS OF PREDICTION 10-1 11 LESSONS IN CONTEXT: A MULTIGOAL WATER RESOURCES PROBLEM UTILIZING MULTIPLE TECHNIQUES 11-1 12 LESSONS IN CONTEXT: TRANSPORTATION SYSTEMS 12-1 13 DYNAMIC PROGRAMMING AND NONLINEAR PROGRAMMING 13-1 14 ENGINEERING ECONOMICS I: INTEREST AND EQUIVALENCE 14-1 15 ENGINEERING ECONOMICS II: CHOICE BETWEEN ALTERNATIVES 15-1 16 ENGINEERING ECONOMICS III: DEPRECIATION, TAXES, INFLATION, AND PERSONAL FINANCIAL PLANNING 16-1 Downloaded from StudyXY.com ® + StudyXY Sd Ye. o> \ | iF ’ pr E \ 3 S Stu dy Anything This ContentHas been Posted On StudyXY.com as supplementary learning material. StudyXY does not endrose any university, college or publisher. Allmaterials posted are under the liability of the contributors. wv 8) www.studyxy.com Chapter 1 HOMEWORK SOLUTIONS 1-1 Solution (a) Decision variables: Amount of sulfur dioxide to remove at each power plant in the Midwest (pounds per time period). (b) Parameters: 1. Amount of untreated sulfur dioxide emitted by each power plant — prior to removal. 2. Cost per pound for removal of sulfur dioxide at each power plant. 3. Air quality degradation at each monitoring site in the Northeast caused by a pound of sulfur dioxide emitted at each plant in the Midwest. (c) Objective function: Minimize the system-wide cost of sulfur dioxide removal at all power coal- fired power plants in the Midwest. (d) Constraints: 1. Desired air quality in the Northeast is achieved (concentrations at all monitoring sites less than or equal to an upper limit concentration). 2. Treatment scheme is seen as equitable. 1-2 Solution (a) Decision variables: Head capacity of the pump, diameter (and material) of the pipeline. (b) Parameters: 1. Flow required at the treatment plant. 2. Delivered head at the treatment plant. 3. Friction factor(s) for the pipeline. 4. Elevation difference, pipeline length. 5. Pump Characteristics. 6. Cost for each capacity of pump and each diameter of pipeline. 7. Discrete pump and pipeline size available. (c) Objective function: Minimize the total cost of pump and pipeline. 1-1 Study (d) Constraints: 1. Required flow is delivered. 2. Required head is delivered. 1-3 Solution (a) Decision variables: Number of toll booths to be installed at the exit. (b) Parameters: 1. Limit on average (across all lanes) number of cars in line. 2. Length of the rush hour period. 3. The number of arrivals at the toll exit during each two-minutes segment of the rush hour. 4. Service time per car — or cars that can be serviced by one booth during each and every two-minute segment of the rush hour. (c) Objective function: Minimize the number of toll booths at the exit. (d) Constraints: A limit on the number of cars in the line (averaged across all lanes) in any two-minute period during the rush hour. 1-4 Solution (a) Decision variables: 1. Width of beam. 2. Depth of beam. 3. Area of steel in beam. (b) Parameters: 1. Imposed moment. 2. Imposed shear. 3. Allowed deflection for given span length. 4. Unit cost of concrete. 5. Unit cost of steel. 6. Compressive strength of concrete. 7. Yield strength of steel. 1-2 Study 8. Minimum relative amount of steel required. 9. Maximum relative amount of steel required. (c) Objective function: Minimize the total cost of the beam. (d) Constraints: 1. Resisting moment of the beam is greater than moment imposed. 2. Resisting shear of the beam is greater than the shear imposed. 3. Deflection of the beam is less than the allowed deflection. 4. Relative amount of steel is greater than the code lower limit. 5. Relative amount of steel is /ess than the code upper limit. 1-5 Solution (a) Decision variable: The amount that is made of each class of concrete. (b) Parameters: 1. Price for each class of concrete. 2. Percentage by weight of cement, sand, and gravel in each class of concrete with ranges on those percentages. 3. Unit cost to contractor for cement, sand, and gravel. 4. Amounts of cement, sand, and gravel available. 5. Limits on the amount of each class of concrete that can be sold. (c) Objective function: . Maximize profit — which is the sum overall classes of concrete of the products of price and the amount of concrete in each class. (d) Constraints: 1. Percentages by weight of cement, sand, and gravel in each class of concrete are within allowable range. 2. Amounts used of cement, sand, and gravel are less than amounts available. 3. Amounts of each class of concrete sold are less than given limits. 1-3 Study 1-6 Solution (a) Decision variable: Dates on which activities will take place on each project. (b) Parameters: 1. Number of teams of workers of each skill category available to the firm. 2. Number of units of equipment of each category available to the firm. 3. Promised completion dates of each project. 4. Resources in terms of teams of workers and equipment needed for each activity on each project. 5. Cost of temporary workers and rental equipment. 6. Payment for completing each project — not really needed, exogenous to the problem. (c) Objective function: Minimize the total cost of completing all projects. (d) Constraints: 1. Number of teams of workers of each skill category needed on any day is less than or equal to the number available within the firm plus any temporary teams. 2. Number of units of equipment of each type is less than or equal to the number available within the firm plus rented. 3. Each project is completed on or before its promised completion date. 1-7 Solution (a) Decision variables: The instrument types to be placed aboard the satellite. (d) Parameters: 1. Cost (price) of each instrument. 2. Weight of each instrument. 3. Volume of each instrument. 4. Quality or value of information that each instrument gathers on each earth feature. 5. Total budget allowed. 6. Total weight allowed. 7. Total volume allowed. 1-4 Study (c) Objective function: Maximize the total value of information gathered. (d) Constraints: 1. Total cost of the package is less than or equal to budget allowed. 2. Total weight of the package is less than or equal to weight allowed. 3. Total volume of the package is less than or equal to volume allowed. 1-8 Solution (a) Decision variables: Which soil tests to run, and foundation type and size to build. (b) Parameters: 1. Damage cost for various degrees of settlement. 2. Cost of each possible soil test. 3. Engineer's estimate of the probability of each soil strength (no testing). 4. Probability of having a particular soil strength, given an experimental test results from a particular test. (c) Objective function: Minimize the expected value of total cost, including cost of experimental tests performed, foundation design and construction cost, and building damage cost (it is assumed that the cost of the building itself is a constant and can be ignored). (d) Constraints: None. 1-5 Study Chapter 2 HOMEWORK SOLUTIONS The problems in this chapter are intended to facilitate discussion in a very general sense about modeling, and engineering decision making. These problems do not have exact solutions. Those solutions provided are meant as examples. 2-1 Solution Students might be asked to discuss the conflict that might exist between objectives, rank objectives, etc. Services Objectives Trash Collection - Maximize quality of service (equity, regularity, etc.) - Maximize usage of capital equipment. - Minimize labor disputes. - Minimize cost excesses and overruns. Municipal Water - Maximize equity in provision of water to community. Supply - Maximize public confidence in the quality and reliability of water supply. - Minimize risk of water contamination. - Minimize the risk of supply shortfalls during peak demand periods. Fire Protection - Maximize residential protection coverage. - Maximize commercial protection coverage. - Maximize readiness of personnel and equipment. - Maximize coverage for co-located ambulance services. - Maximize efficiency in scheduling personnel. Swimming & - Maximize citizen comfort and safety. Recreation - Maximize equity in service among different user groups. - Maximize maintenance effectiveness. Street Cleaning - Maximize equity of service across service area. - Minimize deadhead travel (vehicle miles without performing service). - Minimize interference with traffic. - Minimize objectionable side effects of service (noise, dust, etc.). Sewage Collection ~~ - Maximize treatment efficiency. & Treatment - Minimize risk to the environment and to people. - Minimize objectionable side effects (odor, etc.). - Minimize risk of collection system failure. 2-1 Study 2-2 Solution Scheduling of classrooms is different at different universities. Students may be interested in knowing, or discovering how scheduling is handled at your institution. Possible Objectives - Maximize the accommodation of most important classes. ) - Minimize the number of unused seats during any class period. - Minimize the distance that old professors must walk to class. - Minimize the number of back-to-back courses for as many students as possible. - Minimize distance between locations of back-to-back courses for a given student. Possible Constraints - University has a finite number of classrooms available. - Classrooms have fixed capacities and locations on campus. - Type of seating in each classroom may be fixed. - Students may not take more than one class at a time. - Instructors may not teach more than one class at a time. - All required courses must be scheduled first. 2-3 Solution Here are a few suggestions. Decision Objective(s) Constraints 1. What to eat - Maximize health. - Choice of location to dine may be - Maximize limited. enjoyment. - Selection of food items to purchase or - Minimize cost. prepare. - Minimize time - Time available for eating my be needed. limited. - Funds available for acquiring foods may be limited. - Quality of food (nutrition, taste, etc.) may be limited. 2. How to get to work - Maximize comfort. - Modes of travel might be limited. - Minimize time - Choice of route might be constrained. required. - Time available for travel might be - Minimize cost. limited. 2-2 Study 3. What/when to study ~~ - Maximize grades. - Available time may be - Maximize time. limited. - Minimize time. - Minimum amount of time may be necessary. - Subjects may be of different importance. 4. How/when to - Maximize health. - Need to coordinate with exercise - Maximize enjoyment. others (team, opponent, etc.). - Cost. - Time. 5. When to sleep - Maximize rest. - Amount of sleep time required. - Need to awaken by a specific time. - Time required for non- sleeping activities. - Only 24 hours in a day. 2-4 Solution [Objecives —Wiayor] Chamber Revidens | Refators [Merchant] Maximize residential coverage. | 3 | 2 | 1 | 5 | Maximize commercial coverage. 4 1 | 6 4 1 Minimize cost of acquiring land. 1 3 2 | 5 | 2 | Minimize land development costs. 2 4 3 6 4 | Minimize amount of land required. 5 6 | 4 2 | 3 | Minimize value of land required. 6 5 5 1 6 | Students might be asked to consider other objectives or constituents, or to discuss their own rankings for objectives. 2-3 Study 2-5 Solution Possible Objective - Maximize equity in the distribution of routes across the population. - Maximize resource usage. - Minimize total lane-miles serviced within the community. - Minimize the maximum customer wait time during peak demand periods. - Minimum overlap in services. Possible Constraints - All residents must live within N blocks of a transit stop. - No resident must have to wait more than M minutes for a bus during peak periods. - Important routes must overlap at key transfer points. - Drivers must be available to staff all scheduled routes. - Maintenance funds may be limited. 2-6 Solution Possible Constraints - Students living more than N miles from school must be bussed. - Not more than M buses are available to the school district. - Classrooms (grade levels) must be balanced in each school remaining open. - Cultural diversity must be preserved in schools remaining open. Possible Objectives - Maximize student safety (minimize total walking distance along hazardous streets). - Minimize total student-miles traveled. - Minimize the maximum distance traveled by the student traveling the furthest. Possible Data Needs - Location of each school. - Location of each student or students by block group. - Grade and cultural distribution by block group. - Distance traveled (bus or walking) by each student or student group. - Configuration of the community transportation network. - Long-term demographic trend date. 2-4 2-7 Solution Possible Constraints ~~ - Land available for purchase or lease is limited (possible zoning restrictions, etc.). - Total volume of generated waste now and in the future must be accommodated. - Cost considerations may be restrictive. - Equity considerations among user groups may be important. Possible Objectives - Maximize equity in quality of service and distribution of costs. - Maximize impact on overall economic well-being of the community. - Minimize cost of collection, treatment, or disposal (possibly separate costs). - Minimize environmental impact (including nuisance impacts). Possible Data Needs ~~ - Location of available land for purchase or lease. - Costs for collection, transport, treatment, etc. by each alternative considered. - Demographic trends within the community. 2-5 Study Chapter 3 HOMEWORK SOLUTIONS 3-1 Solution Feasible Region in Decision Space Optimal solution is a unique . , optima: Z' = 84 {x=4,x=2} ss 5 45 4 3s X; 4B « N fre. " 294 Sn J s h 0 K NOR > . K 05 F G 1 [] . oc 1 2 3 4 5 8 7 8 x, ® Optimal Solution © Feasible extreme points Infeasible extreme points 3-2 Solution Feasible Region in Decision Space This problem is infeasible po G 7 6 ; K 5 L c Xx, N ch 0 N H 3 2 1 A F D 0 o 1 2 3 a 5s & 71 8 x, No feasible solution! Infeasible extreme points 3-1 Study 3-3 Solution Feasible Region in Decision Space This problem has a unique optimal L solution: 7 Z=-28{x=4,x=2} . . P ° 3 F Students might consider how this ! problem and its solution compare : with Problem 3-1. 4 Xx, 2,48 < C 208 . L E 0K 1 ©, Jo < G ° 1 co 1 2 3 a4 5s & 71 8 x ® Optimal Solution © Feasible extreme points Infeasible extreme points 3-4 Solution Feasible Region in Decision Space This problem has a unique « optimal solution: ‘@ Z'=1{x,=0,x,= 1} as 3 25 B xX, 2@ © 15 J Xv) OF u TN [3 02 : G H 0 L J 0 1 2 3 4 5 6 7 8 x @® Optimal Solution © Feasible extreme points Infeasible extreme points 3-2 Study 3-5 Solution Feasible Region in Decision Space This problem has a unique optimal . solution: 7 R Z'=236{x,=32,x,= 1.8} PY. SE SN 4 0 1 5 ry of 5% ACOH NX B 3 > N / 2 y oF > 1 oF ! K o 1 2 3 a4 5s 6 7 8 x, ® Optimal Solution © Feasible extreme points Infeasible extreme points 3-6 Solution Feasible Region in Decision Space 8 This problem is infeasible. It is also, unfortunately, not a very interesting ’ problem due to poor choice of Yes Rr s J coefficients. - 5M X Fe : 3 2 ! N 1 0h HD F oc 1 2 3 a 5 & 7 8 xX, No feasible solution! Infeasible extreme points 3-3 Study 3-7 Solution Feasible Region in Decision Space This problem is unbounded. This 5 i problem uses the same constraint set re as Problem 3-8, though that problem CK has alternate optimal solutions. ss] 3 a xX 25 \ 23 & 15 OVE gh N ' 05 V 0 EB 0 1 2 3 4 5 6 7 8 x Unbounded problem! © Feasible extreme points Infeasible extreme points 3-8 Solution Feasible Region in Decision Space This problem has alternate optimal . M solutions: Z* = 16 along the line & segment between (1.25, 1.5) and (1, Ae 2). The constraint set for this ho” problem is identical to that of od pt Problem 3-7, yet this problem has a gsi optimal solution while Problem 3-7 *2 | # is unbounded. 150 & oP N 1 0s Q ocE BM 0 1 2 3 4 5 85 7 8 XxX, = Optimal Solution © Feasible extreme points Infeasible extreme points 3-4 Study