• Author / Uploaded
• sreedeviish

• Categories
• Economía laboral
• Inversiones
• Beneficio (Contabilidad)
• Inversor
• Bonos (Finanzas)

Citation preview

UNIVERSITY OF WATERLOO DEPARTMENT OF MANAGEMENT SCIENCES MSCI603 - Principles of Operations Research Problem set 3 Problem 1 Cornco produces two products: PS and QT. The sales prices for each product and the maximum quantity of each that can be sold during each of the next three months are given in the table below. Product PS QT

Month 1 Price ($) Demand 40 50 35 43

Month 2 Price ($) Demand 60 45 40 50

Month 3 Price ($) Demand 55 50 44 40

Each product must be processed through two assemble lines: 1 and 2. The number of hours required by each product on each assembly line is given below: Hours Product PS QT

Line 1 3 2

Line 2 2 2

The number of hours available on each assembly line during each month is given below: Line 1 2

1 1200 2140

Month 2 160 150

3 190 110

Each unit of PS requires 4 pounds of raw material; each unit of QT requires 3 pounds. As many as 710 units of raw material can be purchased at $3 per pound. At the beginning of month 1, 10 units of PS and 5 units of QT are available. It costs $10 to hold a unit of either product in inventory for a month. Solve this LP in Excel Solver and use your output to answer the following questions. (Please include a print-out of your input and the sensitivity report.) Hint: Let Pi = units of PS produced in month i, PSi = units of PS sold in month i, IPi = inventory of product P at end of month i, Qi = units of QT produced in month i, QSi = units of QT sold in month i, IQi = units of QT in inventory at end of month i, RM = pounds of raw material purchased. a) Find the new optimal solution if it costs $11 to hold a unit of PS in inventory at the end of month 1. b) Find the company’s new optimal solution if 210 hours on line 1 are available during month 1. c) Find the company’s new profit level if 109 hours are available on line 2 during month 3. d) What is the most Cornco should be willing to pay for an extra hour of line 1 time during month 2? e) What is the most Cornco should be willing to pay for an extra pound of raw material? f) What is the most Cornco should be willing to pay for an extra hour of line 1 time during month 3?

g) Find the new optimal solution if PS sells for $50 during month 2. h) Find the new optimal solution if QT sells for $50 during month 3. i) Suppose spending $20 on advertising would increase demand for QT in month 2 by 5 units. Should the advertising be done? Solution MAX 40 PS1 + 60 PS2 + 55 PS3 + 35 QS1 + 40 QS2 + 44QS3 - 3 RM 10(IP1+IP2+IP3+IQ1+IQ2+IQ3) s.t. 2) P1S ≤ 50 3) P2S ≤ 45 4) P3S ≤ 50 5) Q1S ≤ 43 6) Q2S ≤ 50 7) Q3S ≤ 40 8) 3 P1 + 2 Q1 ≤ 1200 9) 3 P2 + 2 Q2 ≤ 160 10) 3 P3 + 2 Q3 ≤ 190 11) 2 P1 + 2 Q1 ≤ 2140 12) 2 P2 + 2 Q2 ≤ 150 13) 2 P3 + 2 Q3 ≤ 110 14) PS1 + IP1 - P1 = 10 15) PS2 - IP1 + IP2 - P2 = 0 16) PS2 - IP2 + IP3 - P3 = 0 17) QS1 + IQ1 - Q1 = 5 18) QS2 - IQ1 + IQ2 - Q2 = 0 19) QS3 - IQ2 + IQ3 - Q3 = 0 20) - RM + 4 P1 + 3 Q1 + 3 Q2 + 4 P2 + 4 P3 + 3 Q3 ≤ 0 21) RM ≤ 710 All variables ≥ (See Excel file for Sensitivity and Answer Reports) a) Allowable increase for 𝐼𝑃1 is 4, so a $1 increase is within the allowable limit. Profit Down by ($1)* 𝐼𝑃1 = $25. New Profit = 7705 - 25 = $7680. b) This is a non-binding constraint, so the Shadow Price = 0 and allowable decrease is 1010.75. Constraint has positive slack. Thus new optimal solution remains the same. c) Allowable decrease is 10 hours, so this 1 hour decrease is within the allowable limit. The shadow price for the constraint is 7, so the new profit is 7705 – 7(1) = $7698. d) The most Cornco will pay is equal to the shadow price of this constraint = $3.33 e) From constraint 20, the shadow price $10. This is what we would gain if given a "free pound of raw material. Thus we would pay up to $10. f) This is a non-binding constraint, meaning there is slack in this constraint. We are not willing to pay anything for any additional time (shadow price is 0). g) A decrease of 10 is within the allowable decrease for this variable, so decision variables remain the same. New z-value = 7705 - 10(45) = $7255. h) An increase of 6 is not within the allowable increase of 1, so current basis is no longer optimal and question cannot be answered from current printout.

Problem 2 A furniture company makes tables (T) and chairs (C), and sells them to customers either finished (F) or unfinished (U). The amount of wood and labor needed, and the selling price of each product are shown in the next table: Product Wood (ft2) Labor (hr) Price ($)

UT 40 2 70

FT 40 5 140

UC 30 2 60

FC 30 4 110

If 40,000 board feet of wood and 6,000 hours of skilled labor are available. Use the sensitivity analysis output report to answer the next parts, independently. a) How much improvement in the profit can be achieved by: i. increasing the quantity of wood available by 3,000 ft2, ii. increasing the number of labor hours available by 500 hours. b) How much rise in the price of finished tables is needed for the company to start producing them? c) If the company reduces the selling price of finished chair by $5, will it need to change its product mix to optimize its profit. Solution Below is a copy of the sensitivity analysis report generated by Excel Variable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$4 UT 0 -76.66666667 70 76.66666667 1E+30 $C$4 FT 0 -6.666666667 140 6.666666667 1E+30 $D$4 UC 0 -50 60 50 1E+30 $E$4 FC 1333.333333 0 110 1E+30 5

Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $F$5 Wood LHS 40000 3.666666667 40000 5000 40000 $F$6 Labor LHS 5333.333333 0 6000 1E+30 666.6666667 a) i. Allowable increase of wood quantity without changing the basis is 5000, so we can use sensitivity analysis. Shadow price of wood = 3.67, so increasing the quantity of wood by 3000 will increase the profit by $11,000. ii. The shadow price of labor hour is zero (the constraint is non-binding), so increasing the quantity of labor hours will not affect the profit. b) The reduced cost of finished tables (FT) is 6.67, so the price of finished tables must increase by this amount before the company starts producing them. c) The allowable decrease in the FC coefficient in the objective function is 5, so the company doesn’t need to change its product mix.

Problem 3 A dairy producer makes two types of cheese. The only scarce resource that is needed to produce cheese is skilled labor. The company has two specialized workers. The first (W1) is willing to work for up to 40 hours per week and is paid $25 per hour. The second (W2) is willing to work for up to 50 hours per week and is paid $30 per hour. The time for each worker and raw materials costs required to produce a unit of each type of cheese and its selling price are shown in the table below. Type 1 2

W1 (hr) 1 2

W2 (hr) 2 2

Raw material cost ($) 250 200

Selling Price ($) 400 420

a. Formulate an LP model for this problem to maximize the profit of the dairy producer. b. Solve the LP model graphically. c. From the graphical solution obtained in (b) determine the range of prices of types 1 and 2 cheese at which the current basis remain optimal. d. If W1 worker is willing to work only 30 hours per week, would the current basis remain optimal? Would the optimal solution change? e. Determine the maximum amount that should be paid to each worker for an additional hour of work every week. f. If W2 worker is willing to work only 48 hours, what would the company’s profit be? Verify your answer graphically. Solution a. x1 = the number of type 1 cheese produced x2 = the number of type 2 cheese produced max (400‐250)x1+(420‐200)x2 ‐25(x1+2x2) – 30(2x1+2x2) s.t. x1+2x2 =0 b. The optimal solution is the intersecting point of the two constraints, which is (10,15) and a profit of 2300. c. The current solution will remain optimal as long as the slope of the iso-profit line remains between the slopes of the binding constraints. For type 1 cheese: 1 c1 2 − 2 ≥ − 110 ≥ − 2 so the price of type 1 cheese can range between 390 and 445 $/unit without changing the optimal solution. For type 2 cheese: 1 65 1 − 2 ≥ − c2 ≥ − 1 so the price of type 2 cheese can range between 375 and 440 $/unit without changing the optimal solution. d. Yes. The new optimal solution will be (x1,x2) = (20,5) and a profit of 1850. e. The shadow price is (y1,y2) = (45,10) The maximum amount that should be paid for an additional hour per week to worker 1 is 45. The maximum amount that should be paid for an additional hour per week to worker 2 is 10. f. The company’s profit decreases by 10*(50‐48) = 20. The graphical solution is (x1,x2) = (8,16), and the profit is 2280.

Problem 4 A workshop makes two types of hand-made genuine leather products: wallets and belts. Each wallet requires 1 square foot of leather and 30 minutes of labor time. Each belt requires 2 square feet of leather and 20 minutes of labor time. The workshop earns $40 profit from each wallet sold and $50 from each belt sold. Each day there are 500 square feet of leather and 200 labor-hours available. a. Formulate an LP for this problem and solve it using the graphical method. b. What is the maximum amount the workshop shall be willing to pay for an extra hour of labor? c. What is the maximum amount the workshop shall be willing to pay for an extra square foot of leather? d. What is the profit range of wallets that will keep the optimal solution unchanged? e. What is the profit range of belts that will keep the optimal solution unchanged? f.

By how much the workshop can increase or decrease the quantity of leather available without changing its optimal product mix?

g. What is the impact of increasing the labor time by 10 hours daily on the workshop profit? Solution a. Let W be the number of wallets and B be the number of belts produced daily. Max 40 W + 50 B s.t. W + 2B