OIDD 101 Spring 2019 Assignment 5 PDF

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OIDD 101 - 2019 - Introduction to Operations, Information and Decisions
Professors: Gérard Cachon/Sergei Savin, Santiago Gallino...

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OIDD 101

Assignment 5

Spring 2019

Problem 1: Midwest Designs. The Midwest Designs (MD) is a small bicycle manufacturing company that assembles and sells highend mountain bikes for the US market. Most of the bikes that MD makes fall into four product lines: Carbon H, Titanium XL, Epic Jumper, and Force Extreme. The company’s products are in such high demand that it can sell all the products it produces. The profit contribution and resource requirements for the four products are as follows: Carbon H Profit Contribution Machining time used per item produced (hrs) Finishing used per item produced (hrs) Assembly time used per item produced (hrs)

Titanium XL

$1000 0.9

$1650 1.3

Epic Jumper $1800 1.9

0.6 0.3

0.7 0.5

1.3 0.6

Force Extreme $2000 2.7 1.5 0.9

During the next week, there are 1,200 hours of machining time available, 539 hours of finishing time available, and 321 hours of assembly time available. During the next week, the company has orders from customers that it has already agreed to fill: orders for 160 Carbon H, 90 Titanium XL, 80 Epic Jumper, and 70 Force Extreme. Consider a problem of establishing the number of units of each product line that MD should produce next week to maximize its profit without exceeding the limits on available resources. The linear model formulation for this problem is shown below in algebraic form. Decision Variables (allowed to take fractional values): C = number of units of Carbon H to produce next week T = number of units of Titanium XL to produce next week E = number of units of Epic Jumper to produce next week F = number of units of Force Extreme to produce next week Objective function (to be maximized): Total profit (in 000’s $): 1*C + 1.65*T + 1.8*E + 2*F Constraints:

0.9*C + 1.3*T + 1.9*E + 2.7*F ≤ 1,200 0.6*C + 0.7*T + 1.3*E + 1.5*F ≤ 539 0.3*C + 0.5*T + 0.6*E + 0.9*F ≤ 321 C ≥ 160, T ≥ 90, E ≥ 80, F ≥ 70

(machining time) (finishing time) (assembly time) (order constraints)

Below are the optimized spreadsheet, the corresponding Solver parameters, and the Sensitivity Report, with some values hidden.

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OIDD 101

Assignment 5

Spring 2019

Variable Cells Cell $B$4 $C$4 $D$4 $E$4

Name Units to produce C Units to produce T Units to produce E Units to produce F

Final Value 200 300 80 70

Reduced Cost 0 0 -0.21 -0.98

Objective Coefficient 1 1.65 1.8 2

Allowable Increase 0.41 0.02 0.21 0.98

Allowable Decrease 0.01 0.48 1E+30 1E+30

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OIDD 101

Assignment 5

Spring 2019

Constraints Cell $F$10 $F$11 $F$12

Name Machining Time (hr) Used Finishing Time (hr) Used Assembly Time (hr) Used

Final Value 911 539 321

Shadow Price 0.00 0.06 3.22

Constraint R.H. Side 1200 539 321

Allowable Increase 1E+30 63 5.14

Allowable Decrease 289 7.2 31.5

Answer each of the following questions using the model formulation and the optimization output presented above. Each question is independent of the others.

Q1. What is the optimal number of units of Epic Jumper to produce next week? Choose the closest answer among the variants below.

(a) (b) (c) (d) (e) (f)

300 200 160 90 80 70

Q2. What is the number of assembly hours used by the optimal solution (i.e., what is the hidden value in cell F12)? Round your answer to the closest integer value, if necessary (e.g., write “34” instead of 34.21).

Q3. What is the hidden value of the “Allowable Increase” in the RHS value of the constraint “Machining Time (hr) Used”? Choose the closest answer among the variants below.

(a) (b) (c) (d) (e)

1E+30 (infinity) 1200 911 289 0 3

OIDD 101

Assignment 5

Spring 2019

Q4. What is the hidden value of the “Allowable Decrease” in the RHS value of the constraint “Machining Time (hr) Used”? Choose the closest answer among the variants below.

(a) (b) (c) (d) (e)

1E+30 (infinity) 1200 911 289 0

Q5. MD is trying to evaluate the possibility that the profit contribution of the Carbon H model drops by 10%, from the current value of $1000 to $900, while all other profit contributions remain the same. If this happens, which of the following statements is correct? (a) The optimal production plan and the optimal profit value will both change (b) The optimal production plan will change, but the optimal profit value will remain the same (c) The optimal production plan will remain the same, but the optimal profit value will change (d) The optimal production plan and the optimal profit value will both remain the same

Q6. MD is trying to evaluate the possibility that the profit contributions of both the Carbon H and the Titanium XL models drop by 10%, simultaneously. In other words, the new profit contributions for Carbon H and Titanium XL become $900 and $1,485, respectively, while the profit contributions for the other two product lines remain the same. If this happens, which of the following statements is correct? (a) The optimal profit value will increase, and the increase will be more than $69,500 (b) The optimal profit value will increase, and the increase will be $69,500 or less (c) The optimal profit value will remain the same (d) The optimal profit value will decrease, and the decrease will be $69,500 or less (e) The optimal profit value will decrease, and the decrease will be more than $69,500

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OIDD 101

Assignment 5

Spring 2019

Q7. MD has just learned that one of the company’s employees that works on bike assembly may take a day off in the coming week. As a result, the MD’s available assembly time will decrease from 321 hours to 313 hours. If this happens, which of the following statements is correct? (a) The optimal production plan and the optimal profit value will both change (b) The optimal production plan will change, but the optimal profit value will remain the same (c) The optimal production plan will remain the same, but the optimal profit value will change (d) The optimal production plan and the optimal profit value will both remain the same

Q8. Another bike manufacturer with an ample finishing capacity has approached MD with an offer of an extra 100 hours of finishing time. As a result, MD’s available finishing time would go from 539 hours to 639 hours. In exchange, MD will have to pay to that bike manufacturer a fixed sum of $3800. Which of the following statements is correct? (a) The benefit of extra finishing time is definitely lower than the fixed sum MD is asked to pay (b) The benefit of extra finishing time is definitely higher than the fixed sum MD is asked to pay (c) Without re-solving the problem, it is impossible to determine the benefit of extra finishing time is higher or lower than the fixed sum MD is asked to pay

MD is considering an option of renting additional equipment and/or using temporary hires to increase its available machining, finishing, and assembly time by exactly 200 hours for each resource. In order to model such option, the company introduces an additional set of decision variables: ZM = a binary variable equal to 1 if available machining time is increased by 200 hours, and equal to 0 if available machining time remains unchanged, ZF = a binary variable equal to 1 if available finishing time is increased by 200 hours, and equal to 0 if available finishing time remains unchanged, ZA = a binary variable equal to 1 if available assembly time is increased by 200 hours, and equal to 0 if available assembly time remains unchanged. The questions below relate to the analysis of this option and are independent of previous questions and of each other.

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OIDD 101

Assignment 5

Spring 2019

Q9. MD is considering the following restriction on the way this option is implemented: “Available machining time cannot be increased unless both available finishing and available assembly times are increased.” The linear algebraic constraint representing this restriction can be written as (a) (b) (c) (d) (e) (f)

ZM ≤ ZF*ZA ZM ≤ ZF + ZA 3*ZM ≤ 2*ZF + 2*ZA 3*ZM ≤ ZF + ZA ZM ≤ 2*ZF + ZA ZM ≤ ZF + 2*ZA

Q10. MD is considering a different restriction on the way this option is implemented: “If available finishing time is increased, then both the available machining time and the available assembly time must be increased.” The linear algebraic constraint representing this restriction can be written as (a) (b) (c) (d) (e) (f)

ZM*ZA ≥ ZF ZM ≥ 2*ZF - ZA 3*ZM ≥ 2*ZF - 2*ZA 3*ZM ≥ ZF + ZA ZM ≥ 2*ZF + ZA ZM ≥ ZF + 2*ZA

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