Tutorial 2: Sensitivity Analysis

Problem 1 (Exam 2007): The Classic Furniture Company is trying to determinethe optimal quantities to make of six possible products: tables and chairs made ofoak, cherry, and pine. The products are to be made using the following resources:labour hours and three types of wood. The availability of each resource as well aseach items resource usage (technological coefficients) are shown in the Excellayout (Figure 1). Minimum production requirements are as follows: at least 3 eachof oak and cherry tables, at least 10 each of oak and cherry chairs, and at least 5pine chairs. The objective function coefficients in Figure 1 refer to the unit profit peritem. The LP Sensitivity Report is shown in Figure 2.

Classic Furniture Company

Oaktables

Oakchairs

Cherrytables

Cherrychairs

Pinetables

Pinechairs

Number of units 3 30 3 35.56 0 75Profit per unit () 75 35 90 60 45 20 5178.33 = 5

LHS Sign RHS

Figure 1: Excel layout

Microsoft Excel 11.0 Sensitivity Report

Adjustable CellsFinal Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease$D$6 Number of units Oak tables 3 0 75 158.33 1E+30$E$6 Number of units Oak chairs 30 0 35 1E+30 23.75$F$6 Number of units Cherry tables 3 0 90 310 1E+30$G$6 Number of units Cherry chairs 35.56 0 60 1E+30 46.5$H$6 Number of units Pine tables 0 -45 45 45 1E+30$I$6 Number of units Pine chairs 75 0 20 1E+30 10

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$J$9 Labour hours 517.83 0 1000 1E+30 482.17$J$10 Oak (pounds) 1500 1.17 1500 4132.86 600$J$11 Cherry (pounds) 2000 1.67 2000 2893 920$J$12 Pine (pounds) 3000 0.5 3000 9643.33 2800$J$13 Min oak tables 3 -158.33 3 3 3$J$14 Min cherry tables 3 -310 3 3.83 3$J$15 Min oak chairs 30 0 10 20 1E+30$J$16 Min cherry chairs 35.56 0 10 25.56 1E+30$J$17 Min pine chairs 75 0 5 70 1E+30

Figure 2: Sensitivity Report

Answer each of the following questions, each of which is independent of the others.

a) What is the optimal solution? Which constraints are binding?

b) Interpret the shadow prices of the following three constraints: labour hours, theamount of cherry wood available, and the minimum production quantity for oaktables.

c) Suppose Classic Furniture are forced to produce at least one table in pine. Whatwould be the impact on profit?

d) Classic Furniture can purchase an additional 1,000 pounds of oak at a price of0.75 per pound. Should Classic Furniture buy this wood and do they need tochange the current production plan? (Why?) What would be the impact onprofit?

e) Classic Furniture is considering the production of coffee tables. One coffee tablewould make a profit of 55 while its production requires 30 pounds of oak, 10pounds of cherry and 5 hours of labour. Should the company produce coffeetables? Why (not)?

f) It appears that Classic Furniture underestimated the profit of each item by 10%.Should Classic Furniture adapt the production plan? Why (not)?

g) Tables and chairs usually sell as sets. For each type of wood, the number ofchairs produced should not exceed 10 times the number of tables produced. Whatconstraints should be added to the LP to reflect these conditions? Does the currentsolution satisfy these constraints?

Problem 2: (Exam 2005) Steelco has received an order for 100 tonnes of steel.The order must contain at least 3.5 tonnes of nickel, at most 3 tonnes of carbon,and exactly 4 tonnes of manganese. Steelco receives 200/tonne for the order.To fill the order, Steelco can combine four alloys, whose chemical composition isgiven in Table 1. Steelco wants to maximise profit (revenues costs). FormulateSteelcos problem as an LP and solve it with Excel Solver.

Table 1: Alloy composition and cost

Alloy 1 Alloy 2 Alloy 3 Alloy 4Nickel 6% 3% 2% 1%Carbon 3% 2% 5% 6%Manganese 8% 3% 2% 1%Cost/tonne 120 100 80 60

After solving Steelcos problem with excel solver, use the sensitivity report toanswer the following questions:

a) What is the optimal solution? Which constraints are binding?

b) In what range of cost values for alloy 2 does the current solution remainoptimal? Can you calculate the new profit if alloy 2 costs 90/tonne?

c) Interpret the shadow prices of the different constraints?

d) Suppose the cost per tonne of both alloy 2 and 4 decreases by 20. Will thischange the optimal solution? Why or why not? Can you calculate the new profit?

e) Suppose the order should contain at least 3.55 tonnes of nickel. Would thischange affect the profit? What about the optimal solution?

f) Suppose that Steelco can use also alloy 5. Alloy 5 costs 100/tonne andcontains 2% of nickel, 1% of carbon and 0.3% of manganese. Would you adviseSteelco using alloy 5 in the order?

g) Suppose alloys 1, 2, 3 and 4 contain respectively 2%, 3%, 1% and 4% ofsilicon. The 100 tonne steel order should contain at most 2.5 tonnes of silicon.Should Steelco revise its production plan?

Steelco

Alloy 1 Alloy 2 Alloy 3 Alloy 4No. of tons 25 62.5 0 12.5Selling price () 200 200 200 200Cost () 120 100 85 60Profit () 80 100 115 140 10000 = 3.5Carbon constraint 0.03 0.02 0.05 0.06 2.75

Each Standard suitcase requires 0.7 hours of cutting and colouring, 0.5 hours ofassembly, 1 hour of finishing and 0.1 hours of quality and packaging. Thecorresponding numbers for each Deluxe suitcase are 1 hour, 0.83 hours, 0.67hours, and 0.25 hours, respectively. Likewise, the corresponding numbers for eachLuxury suitcase are: 1 hour, 0.67 hours, 0.9 hours, and 0.4 hours, respectively.The sales revenue for each type of suitcase is as follows: Standard 36.05, Deluxe39.5 and Luxury 43.3. The material costs are Standard 6.25, Deluxe 7.5 andLuxury 8.5.The hourly cost of labour for each department is: cutting & colouring10, assembly 6, finishing 9, and quality and packaging 8.

a) Formulate an LP to determine how much suitcases Good-to-Go should make soas to maximise profit. Enter this problem into Excel and find the optimal solution.(Use the answer and sensitivity report to answer the following questions.)

b) What is the optimal production plan? Which of the resources are scarce?

c) Suppose Good-to-Go is considering a polishing process, the cost of which wouldbe added directly to the price. Each Standard suitcase would require 10 minutes oftime of this treatment, each Deluxe suitcase would need 15 minutes and eachLuxury suitcase would need 20 minutes. Would the current production plan changeas a result of this additional process if 170 hours of polishing time were available?Explain your answer.

d) Suppose Good to-Go is considering the possible introduction of two newproducts: the Compact model and the Kiddo model (for children). Market researchsuggests that Good-to-Go can sell the Compact model for no more than 30,whereas the Kiddo model would go for as much as 37.5 to speciality toy stores.The amount of labour and the cost of raw materials for each possible new productare as follows:

Cost Category Compact KiddoCutting and colouring (hr.) 0.5 1.20

Assembly (hr.) 0.75 0.75Finishing (hr.) 0.75 0.5

Quality and packaging (hr.) 0.2 0.2Raw materials () 5 4.5

Would it be economically attractive to make any of these models? Explain youranswer.