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Quantitative Analysis BA 452 Homework Questions Homework covers the theory and applications in Lessons I-1 to I-3 This document has four parts: Objectives of doing your homework Assignment of homework questions, with suggestions about which other questions may help you understand the homework questions Homework Supplemented Questions listing 58 questions: of them are your homework, others may help you understand the homework questions, and the rest may are there help just you in understand case you find fine them points.helpful Some Some supplemental supplemental questions questions refer torefer a section to a section of a chapter of a chapter in the textbook in the (for textbook example,(for Section example, 2.1 means Section Chapter 2.1 means Section Chapter of2your Section textbook) of your textbook) Homework Supplemented Answers listing answers to all 58 Homework questions excluding Supplemented your homework Answersquestions listing answers to all 58 questions including your homework questions Quantitative Analysis BA 452 Homework Questions Objectives By working through the homework questions and the supplemental questions, you will: Obtain an overview of the kinds of problems linear programming has been used to solve Learn how to develop linear programming models for simple problems Be able to identify the special features of a model that make it a linear programming model Learn how to solve two variable linear programming models by the graphical solution procedure Understand the importance of extreme points in obtaining the optimal solution Know the use and interpretation of slack and surplus variables Be able to interpret the computer solution of a linear programming problem Understand how alternative optimal solutions, infeasibility and unboundedness can occur in linear programming problems Understand the following terms: problem formulation constraint function objective function solution optimal solution nonnegativity constraints mathematical model linear program linear functions feasible solution feasible region slack variable standard form redundant constraint extreme point surplus variable alternative optimal solutions infeasibility unbounded Quantitative Analysis BA 452 Homework Questions Assignment Questions 11, 13, 27, and 53 is your homework assignment Questions 11, 13, and 27 should be answered without referring to notes or using computers (Hint: Define decision variables I = Internet fund investment in thousands, B = Blue Chip fund investment in thousands Then, the objective is to maximize the projected annual return 0.12I + 0.09B.) Question 53 can be answered with notes and computers To supplement those homework questions, you should consider (but not turn in) the following questions Questions answered without notes or computers: 1-13, 17-19, 21-26, 30-31, 34-36, 38, 42-46 Questions answered with notes and computers: 14-15, 20, 27-29, 33, 37, 39-41, 49-54 Tip: Those homework questions and supplementary questions are grouped into sets of similar type Once you have mastered the questions in a set, you can skip the rest of the questions in that set Tip: Some of your Exam questions will be variations of some of those homework questions Quantitative Analysis BA 452 Homework Questions Homework Supplemented Questions Which of the following mathematical relationships could be found in the linear programming model, and which could not? For the relationships that are unacceptable for linear programs, state why a −1𝐴𝐴 + 2𝐵𝐵 ≤ 70 b 2𝐴𝐴 − 2𝐵𝐵 = 50 c 1𝐴𝐴 − 2𝐵𝐵2 ≤ 10 d 3√𝐴𝐴 + 2𝐵𝐵 ≥ 15 e 1𝐴𝐴 + 1𝐵𝐵 = f 2𝐴𝐴 + 5𝐵𝐵 + 1𝐴𝐴𝐴𝐴 ≤ 25 a b c Find the solution that satisfy the following constraints: 4𝐴𝐴 + 2𝐵𝐵 ≤ 16 4𝐴𝐴 + 2𝐵𝐵 ≥ 16 4𝐴𝐴 + 2𝐵𝐵 = 16 Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a 3𝐴𝐴 + 2𝐵𝐵 ≤ b 12𝐴𝐴 + 8𝐵𝐵 ≥ 480 c 5𝐴𝐴 + 10𝐵𝐵 = 200 Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a 3𝐴𝐴 − 4𝐵𝐵 ≥ 60 b −6𝐴𝐴 + 5𝐵𝐵 ≤ 60 c 5𝐴𝐴 − 2𝐵𝐵 ≤ Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints: a 𝐴𝐴 ≥ 0.25(𝐴𝐴 + 𝐵𝐵) b 𝐵𝐵 ≤ 0.10(𝐴𝐴 + 𝐵𝐵) c 𝐴𝐴 ≤ 0.50(𝐴𝐴 + 𝐵𝐵) Three objective functions for linear programming problems are 7𝐴𝐴 + 10𝐵𝐵, 6𝐴𝐴 + 4𝐵𝐵, and −4𝐴𝐴 + 7𝐵𝐵 Show the graph of each for objective function values equal to 420 Identify the feasible region for the following set of constraints: 0.5𝐴𝐴 + 0.25𝐵𝐵 ≥ 30 1𝐴𝐴 + 𝐵𝐵 ≥ 250 0.25𝐴𝐴 + 0.5𝐵𝐵 ≤ 50 𝐴𝐴, 𝐵𝐵 ≥ Quantitative Analysis BA 452 Homework Questions Identify the feasible region for the following set of constraints: 2𝐴𝐴 − 1𝐵𝐵 ≤ −1𝐴𝐴 + 1.5𝐵𝐵 ≤ 200 𝐴𝐴, 𝐵𝐵 ≥ Identify the feasible region for the following set of constraints: 3𝐴𝐴 − 2𝐵𝐵 ≥ 2𝐴𝐴 − 1𝐵𝐵 ≤ 200 1𝐴𝐴 ≤ 150 𝐴𝐴, 𝐵𝐵 ≥ 10 For the linear program Max s.t 2𝐴𝐴 + 3𝐵𝐵 1𝐴𝐴 + 2𝐵𝐵 ≤ 5𝐴𝐴 + 3𝐵𝐵 ≤ 15 𝐴𝐴, 𝐵𝐵 ≥ find the optimal solution using the graphical solution procedure What is the value of the objective function at the optimal solution? 11 Solve the following linear program using the graphical solution procedure: Max 5𝐴𝐴 + 5𝐵𝐵 s.t 1𝐴𝐴 ≤ 100 1𝐵𝐵 ≤ 80 2𝐴𝐴 + 4𝐵𝐵 ≤ 400 𝐴𝐴, 𝐵𝐵 ≥ 12 Consider the following linear programming problem: Max 3𝐴𝐴 + 3𝐵𝐵 s.t 2𝐴𝐴 + 4𝐵𝐵 ≤ 12 6𝐴𝐴 + 4𝐵𝐵 ≤ 24 𝐴𝐴, 𝐵𝐵 ≥ a Find the optimal solution using the graphical solution procedure b If the objective function is changes to 2A + 6B, what will the optimal solution be? c How many extreme points are there? What are the values of A and B at each extreme point? Quantitative Analysis BA 452 Homework Questions 13 Consider the following linear program: Max 1𝐴𝐴 + 2𝐵𝐵 s.t 1𝐴𝐴 ≤5 1𝐵𝐵 ≤ 2𝐴𝐴 + 2𝐵𝐵 = 12 𝐴𝐴, 𝐵𝐵 ≥ a Show the feasible region b What are the extreme points of the feasible region? c Find the optimal solution using the graphical procedure 14 RMC, Inc., is a small firm that produces a variety of chemical products In a particular production process, three raw materials are blended (mixed together) to produce two products: a fuel additive and a solvent base Each ton of fuel additive is a mixture of 2/5 ton of material and 3/5 of material A ton of solvent base is a mixture is a mixture of ½ ton of material 1, 1/5 ton of material and 3/10 ton of material After deducting relevant costs, the profit contribution is $40 for every ton of fuel additive and $30 for every ton of solvent base RMC’s production is constrained by the limited availability of the three raw materials For the current production periods, RMC has available the following quantities of each new material: Raw Material Amount Available for Production Material 20 tons Material tons Material 21 tons a b c d Assuming the RMC is interested in maximizing the total profit contribution, answer the following: What is the linear programming model for the problem? Find the optimal solution using the graphical solution procedure How many tons of each product should be produced, and what is the projected total profit contribution? Is there any unused material? If so, how much? Are any of the constraints redundant? Is do, which ones? 15 Refer to the Par, Inc., problem described in Section 2.1 (Chapter Section of your text) Suppose that Par’s management encounters the following situations: a The accounting department revises its estimate of the profit contribution for the deluxe bag to $18 per bag b A new low-cost material is available for the standard bag, and the profit contribution per standard bag can be increased to $20 per bag (Assume that the profit contribution of the deluxe bag is the original $9 value) c New sewing equipment is available that whole increase the sewing operations capacity to 750 hours (Assume the 10A +9B is the appropriate objective function.) If each of these situations is encountered separately, what is the optimal solution and the total profit contribution? Quantitative Analysis BA 452 Homework Questions 16 a b c Refer to the feasible region for Par, Inc., problem in Figure 2.13 Develop an objective function that will make extreme point the optimal extreme point What is the optimal solution for the objective function you selected in part (a)? What are the values of the slack variables associated with this solution? 17 Write the following linear program in standard form: Max 5𝐴𝐴 + 2𝐵𝐵 s.t 1𝐴𝐴 − 2𝐵𝐵 ≤ 420 2𝐴𝐴 + 1𝐵𝐵 ≤ 610 6𝐴𝐴 − 1𝐵𝐵 ≤ 125 𝐴𝐴, 𝐵𝐵 ≥ 18 For the linear program Max 4𝐴𝐴 + 1𝐵𝐵 s.t 10𝐴𝐴 + 2𝐵𝐵 ≤ 30 3𝐴𝐴 + 2𝐵𝐵 ≤ 12 2𝐴𝐴 + 2𝐵𝐵 ≤ 10 𝐴𝐴, 𝐵𝐵 ≥ a Write the problem in standard form b Solve the problem using the graphical solution procedure c What are the values of the three slack variables at the optimal solution? 19 Given the linear program Max 3A + 4B s.t -1A + 2B ≤ 1A + 2B ≤ 12 2A + 1B ≤ 16 A, B, ≥ a Write the problem in standard form b Solve the problem using the graphical solution procedure c What are the values of the three slack variables at the optimal solution? Quantitative Analysis BA 452 Homework Questions 20 For the linear program Max 3A + 2B s.t A+B≥4 3A + 4B ≤ 24 A≥2 A–B≤0 𝐴𝐴, 𝐵𝐵 ≥ a Write the problem in standard form b Solve the problem using the graphical solution procedure c What are the values of the three slack variables at the optimal solution? 21 Consider the following linear program: Max 2A + 3B s.t 5A + 5B ≤ 400 Constraint -1A + 1B ≤ 10 Constraint 1A + 3B ≥ 90 Constraint 𝐴𝐴, 𝐵𝐵 ≥ a Graph the constraints, and place a number (1, 2, or 3) next to each constraint line to identify which constraint it represents b Shade in the feasible region on the graph c Identify the optimal extreme point What is the optimal solution? d Which constraints are binding? Explain e How much clack or surplus is associated with the nonbinding constraint? Quantitative Analysis BA 452 Homework Questions 22 Reiser Sports Products wants to determine the number of All-Pro (A) and College (C) footballs to produce in order to maximize profit over the next four-week planning horizon Constraints affecting the production quantities are the production capacities in three departments: cutting and dyeing; sewing; and inspection and packaging For the four-week planning period, 340 hours of cutting and dyeing time, 420 hours of sewing time, and 200 hours of inspection and packaging time are available All-Pro footballs provide a profit of $5 per unit and College footballs provide a profit of $4 per unit The liner programming model with production times expressed in minutes is as follows: Max 5A + 4C s.t 12A + 6C ≤ 20,400 Cutting and dyeing 9A + 15C ≤ 25, 200 Sewing 6A + 6C ≤ 12,000 Inspection A, C ≥ a) Graph all constraints, then shade the feasible region for this problem b) Determine the coordinates of each extreme point and the corresponding profit Which extreme point generates the highest profit c) Draw the profit line corresponding to a profit of $4000 Move the profit line as far from the origin as you can in order to determine which extreme point will provide the optimal solution Compare your answer with the approach you used in part (b) d) Which constraints are binding? Explain e) Suppose that the values of the objective function coefficients are $4 for each All-Pro model produced and $5 for each college model Use the graphical solutions procedure to determine the new optimal solution and the corresponding value of profit 23 Embassy motorcycles (EM)manufactures two lightweight motorcycles designed for easy handling and safety The EZ-Rider model has a new engine and a low profile that make it easy to balance The LadySport model is slightly larger, uses a more traditional engine, and is specifically designed to appeal to women riders Embassy produces the engines for both models at its Des Moines, Iowa plant Each EZRider requires hours of manufacturing time and each Lady-Sport engine requires hours of manufacturing time The Des Moines plant has 2100 hours of engine manufacturing time available for the next production period Embassy’s motorcycle frame supplier can supply as many EZ-Rider frames as needed However, the Lady-Sport frame is more complex and the supplier can only provide up to 280 Lady-Sport frames for the next production period Final assembly and testing requires hours for each EZ-Rider model and 2.5 hours for each Lady-Sport model A maximum of 1000 hours of assembly and testing time are available for the next production period The company’s accounting department projects a profit contribution of $2400 for each EZ-Rider produced and $1800 for each Lady-Sport produced a Formulate a linear programming model that can be used to determine the number of units of each model that should be produced in order to maximize the total contribution to profit b Solve the problem graphically What is the optimal solution? c Which constraints are binding? Quantitative Analysis BA 452 Homework Questions 24 Kelson Sporting Equipment, Inc., makes two different types of baseball gloves: A regular model and a catcher’s model The firm has 900 hours of production time available in its cutting and sewing department, 300 hours available in its finishing department, and 100 hours available in its packaging and shipping department The production time requirements and the profit contribution per glove are given in the following table: Model Regular model Catcher’s model Cutting and Sewing 3/2 Production Time (hours) Finishing Packaging and Shipping ẵ 1/8 1/3 ẳ Profit/Glove $5 $8 Assuming that the company is interesting in maximizing the total profit contribution, answer the following a What is the linear programming model for this problem? b Find the optimal solution using the graphical solution procedure How many gloves of each model should Kelson manufacture? c What is the total profit contribution Kelson can earn with the given production quantities? d How many hours of production time will be scheduled in each department? e What is the slack time in each department? 25 George Johnson recently inherited a large sum of money; he want to use a portion of this money to set up a trust fund for his two children The trust fund has two investment opportunities: (1) a bond fund and (2) a stock fund The projected returns over the life of the investments are 6% for the bond fund and 10% for the stock fund Whatever portion of the inheritance he finally decides to commit to the trust fund, he wants to invest at least 30% of that amount in the bond fund In addition, he want to select a mix that will enable him to obtain a total return of at least 7.5% a Formulate a linear programming model that can be used to determine the percentage that should be allocated to each of the possible investment alternatives b Solve the problem using the graphical solution procedure 26 The Sea Warf Restaurant would like to determine the best way to allocate a monthly advertising budget of $1000 between newspaper advertising and radio advertising Management decided that at least 25% of the budget must be spent on each type of media, and that the amount of money spent on local newspaper advertising must be at least twice the amount of money spent on radio advertising A marketing consultant developed an index that measure the audience exposure per dollar of advertising on a scale from to 100, with higher values implying greater audience exposure If the value of the index for local newspaper advertising is 50 and the value of the index for spot radio adverting is 80, how should the restaurant allocate its advertising budget in order to maximize the value of total audience exposure? a Formulate a linear programming model that can be used to determine how the restaurant should allocate its advertising budget in order to maximize the value of the total audience exposure b Solve the problem using the graphical solution procedure 10 Quantitative Analysis BA 452 Homework Questions 41 a Let R = number of gallons of regular gasoline produced P = number of gallons of premium gasoline produced Max 0.30R + 0.50P 0.30R + 0.60P ≤ 18,000 Grade A crude oil available 1R + 1P ≤ 50,000 Production capacity 1P ≤ 20,000 Demand for premium s.t R, P ≥ b Optimal Solution: 40,000 gallons of regular gasoline, and 10,000 gallons of premium gasoline Total profit contribution = $17,000 c Constraint Value of Slack Variable 0 10,000 Interpretation All available grade A crude oil is used Total production capacity is used Premium gasoline production is 10,000 gallons less than the maximum demand 56 Quantitative Analysis BA 452 Homework Questions d Grade A crude oil and production capacity are the binding constraints 42 x2 B 14 Satisfies Constraint #2 12 10 Infeasibility Satisfies Constraint #1 12 10 x1 A 43 x2 B Unbounded 44 x1 a x2 B Objective Function Optimal Solution (30/16, 30/16) Value = 60/16 x1 A b New optimal solution is A = 0, B = 3, value = 57 A R Quantitative Analysis BA 452 Homework Questions 45 a B A A B b Feasible region is unbounded c Optimal Solution: A = 3, B = 0, z = d An unbounded feasible region does not imply the problem is unbounded This will only be the case when it is unbounded in the direction of improvement for the objective function 58 Quantitative Analysis BA 452 Homework Questions 46 Let N = number of sq ft for national brands G = number of sq ft for generic brands Problem Constraints: N + G N G Extreme Point ≤ 200 Space available ≥ 120 National brands ≥ 20 N 120 180 120 Generic G 20 20 80 a Optimal solution is extreme point 2; 180 sq ft for the national brand and 20 sq ft for the generic brand b Alternative optimal solutions Any point on the line segment joining extreme point and extreme point is optimal c Optimal solution is extreme point 3; 120 sq ft for the national brand and 80 sq ft for the generic brand 59 Quantitative Analysis BA 452 Homework Questions 47 B x2 s ce s P ro 600 e Tim ing 500 400 300 Alternate optima (125,225) 200 100 (250,100) x1 100 200 300 A 400 Alternative optimal solutions exist at extreme points (A = 125, B = 225) and (A = 250, B = 100) Cost = 3(125) + 3(225) = 1050 or Cost = 3(250) + 3(100) = 1050 The solution (A = 250, B = 100) uses all available processing time However, the solution (A = 125, B = 225) uses only 2(125) + 1(225) = 475 hours Thus, (A = 125, B = 225) provides 600 - 475 = 125 hours of slack processing time which may be used for other products 60 Quantitative Analysis BA 452 Homework Questions 48 Possible Actions: i Reduce total production to A = 125, B = 350 on 475 gallons ii Make solution A = 125, B = 375 which would require 2(125) + 1(375) = 625 hours of processing time This would involve 25 hours of overtime or extra processing time iii Reduce minimum A production to 100, making A = 100, B = 400 the desired solution 61 Quantitative Analysis BA 452 Homework Questions 49 a Let P = number of full-time equivalent pharmacists T = number of full-time equivalent physicians The model and the optimal solution are shown below: MIN 40P+10T S.T 1) 2) 3) P+T >=250 2P-T>=0 P>=90 Optimal Objective Value 5200.00000 Variable P T Constraint Value 90.00000 160.00000 Reduced Cost 0.00000 0.00000 Slack/Surplus 0.00000 20.00000 0.00000 Dual Value 10.00000 0.00000 30.00000 The optimal solution requires 90 full-time equivalent pharmacists and 160 full-time equivalent technicians The total cost is $5200 per hour b Attrition Optimal Values Pharmacist Current Levels 85 10 90 New Hires Required 15 Technician 175 30 160 15 s s The payroll cost using the current levels of 85 pharmacists and 175 technicians is 40(85) + 10(175) = $5150 per hour The payroll cost using the optimal solution in part (a) is $5200 per hour Thus, the payroll cost will go up by $50 62 Quantitative Analysis BA 452 Homework Questions 50 Let M = number of Mount Everest Parkas R = number of Rocky Mountain Parkas Max s.t 100M + 150R 30M + 20R ≤ 7200 Cutting time 45M + 15R ≤ 7200 Sewing time 0.8M - 0.2R ≥ % requirement Tip: Here are steps to formulating the % requirement constraint: M must be at least 20% of total production M ≥ 0.2 (total production) M ≥ 0.2 (M + R) M ≥ 0.2M + 0.2R 0.8M - 0.2R ≥ The optimal solution is M = 65.45 and R = 261.82; the value of this solution is z = 100(65.45) + 150(261.82) = $45,818 If we think of this situation as an on-going continuous production process, the fractional values simply represent partially completed products If this is not the case, we can approximate the optimal solution by rounding down; this yields the solution M = 65 and R = 261 with a corresponding profit of $45,650 63 Quantitative Analysis BA 452 Homework Questions 51 Let C = number sent to current customers N = number sent to new customers Note: Number of current customers that test drive = 25 C Number of new customers that test drive = 20 N Number sold = Max s.t = 12 ( 25 C ) + 20 (.20 N ) 03 C + 04 N 03C + 04N ≥ 30,000 Current Min 20 N ≥ 10,000 New Min ≥ 25 C 25 C - 40 N 4C + 6N Current vs New ≤ 1,200,00 Budget C, N, ≥ 64 Quantitative Analysis BA 452 Homework Questions 52 Let S = number of standard size rackets O = number of oversize size rackets Max s.t 10S + 15O 0.8S - 0.2O ≥ 10S 0.125S + + 12O 0.4O ≤ 4800 ≤ 80 S, O, ≥ 65 % standard Time Alloy Quantitative Analysis BA 452 Homework Questions 53 a Let R = time allocated to regular customer service N = time allocated to new customer service My answer to this homework question will Max 1.2R + N go here Tos.t.find your answer, you may want + R N ≤ 80 25R + 8N some to study the answers to ≥ 800 of the similar -0.6R + N ≥ questions R, N, ≥ b Optimal Objective Value 90.00000 Variable R N Constraint Value 50.00000 30.00000 Reduced Cost 0.00000 0.00000 Slack/Surplus 0.00000 690.00000 0.00000 Dual Value 1.12500 0.00000 -0.12500 Optimal solution: R = 50, N = 30, value = 90 HTS should allocate 50 hours to service for regular customers and 30 hours to calling on new customers 66 Quantitative Analysis BA 452 Homework Questions 54 a Let M1 M2 = number of hours spent on the M-100 machine = number of hours spent on the M-200 machine Total Cost 6(40)M1 + 6(50)M2 + 50M1 + 75M2 = 290M1 + 375M2 Total Revenue 25(18)M1 + 40(18)M2 = 450M1 + 720M2 Profit Contribution (450 - 290)M1 + (720 - 375)M2 = 160M1 + 345M2 Max s.t 160 M1 + 345M2 M1 ≤ 15 M-100 maximum 10 M-200 maximum M1 M2 ≤ ≥ M2 ≥ M-100 minimum M-200 minimum 50 M2 ≤ 1000 40 M1 + Raw material available M1, M2 ≥ b Optimal Objective Value 5450.00000 Variable M1 M2 Constraint Value 12.50000 10.00000 Reduced Cost 0.00000 145.00000 Slack/Surplus 2.50000 0.00000 7.50000 5.00000 0.00000 Dual Value 0.00000 145.00000 0.00000 0.00000 4.00000 The optimal decision is to schedule 12.5 hours on the M-100 and 10 hours on the M-200 67 Quantitative Analysis BA 452 Homework Questions 55 Mr Krtick’s solution cannot be optimal Every department has unused hours, so there are no binding constraints With unused hours in every department, clearly some more product can be made 56 No, it is not possible that the problem is now infeasible Note that the original problem was feasible (it had an optimal solution) Every solution that was feasible is still feasible when we change the constraint to less-than-or-equal-to, since the new constraint is satisfied at equality (as well as inequality) In summary, we have relaxed the constraint so that the previous solutions are feasible (and possibly more satisfying the constraint as strict inequality) 68 Quantitative Analysis BA 452 Homework Questions 57 Yes, it is possible that the modified problem is infeasible To see this, consider a redundant greater-than-or-equal to constraint as shown below Constraints 2, 3, and form the feasible region and constraint is redundant Change constraint to less-than-or-equal-to and the modified problem is infeasible Original Problem: Modified Problem: 69 Quantitative Analysis BA 452 Homework Questions 58 It makes no sense to add this constraint The objective of the problem is to minimize the number of products needed so that everyone’s top three choices are included There are only two possible outcomes relative to the boss’ new constraint First, suppose the minimum number of products is 15 Then the new constraint makes the problem infeasible 70