73-220Quantitative Decision Models I

Assignment #22006 Fall Term

Due Date: Monday, October 23, 200610:00am for Section 17:00pm for Section 30

Instructions: 1. For the first three questions, linear programming mathematical formulations are to be

provided in terms of decision variables (including units), objective function (max or min), and constraints (preferably with appropriate labels). Either Excel solver or Management Scientist can be used to solve these models. You are required to submit the SOLVED sheets (either Excel or Management Scientist) to support your answers.

2. For the last question, all changes are non-cumulative and refer to the base model. You are expected to answer these questions without re-solving the model by plugging in the new parameters. If the impact on the optimal solution and/or objective function value can be evaluated without re-solving the model in Excel, provide the new optimal solution and/or objective function values and explain your reasons. If the impact on the optimal solution cannot be assessed without re-solving it, please clearly state your reasoning.

3. All the question numbers refer to our required textbook, Anderson, Sweeney, and Williams, An Introduction to Management Science: Quantitative Approaches to Decision Making, 11th Edition, South-Western, 2005. If you use an earlier edition, it is your responsibility to make sure that you work on the right questions.

4. Assignments are to be collected during the first 10 minutes of the class on October 23, 2006, i.e., up to 10:10am for the morning class or 7:10pm for the evening class. Late assignments up to the end of the class will be accepted with a 30% late penalty. Any submission beyond this time window will not be accepted. You must hand in your assignment to the section that you officially registered.

5. Assignments must be stapled and put inside a sufficiently large envelope with your student ID number, name, course title and number, and assignment number on it.

6. Only hard copies are accepted. Any faxed or electronic copy is not accepted.

Q3.28 on page 143Q3.35 on page 147 (Please work on part a only).Q4.5 on page 206.

Q4. Sensitivity Analysis

Note: This question is adapted from the midterm examination for the Winter 2006 term. One reason of including this question is to familiarize you with the format of sensitivity analysis questions that you will face in your midterm. In the examination, all sensitivity analysis questions will be based on Excel Solver report instead of Management Scientist report and the Excel solver sensitivity report will be provided to you. Please provide your brief explanations or calculations to support your answers.

Royal Kona Coffee manufactures a coffee product by blending three types of coffee beans. The cost per pound and the available pounds of each bean are as follows:

Bean Cost per pound Available pounds 1 $0.50 500 2 $0.70 600 3 $0.45 400

Consumer tests with coffee products were used to provide ratings on a scale of 0-100, with higher ratings indicating higher quality. Product standards for the blended coffee require a consumer rating for aroma to be at least 75 and a consumer rating for taste to be at least 80. The individual ratings of the aroma and taste for coffee made from 100% of each bean are as follows:

Bean Aroma rating Taste rating 1 75 86 2 85 88 3 60 75

Assume that the aroma and taste attributes of the coffee blend will be a weighted average of the attributes of the beans used in the blend. What is the minimum cost blend that will meet the quality standards and provide 1000 pounds of the blended coffee products?

LP Formulation:

Let x1 = pounds of bean 1 in the blended coffee products.x2 = pounds of bean 2 in the blended coffee products.x3 = pounds of bean 3 in the blended coffee products.

Min z = 0.50x1 + 0.70x2 + 0.45x3 s.t. x1 + x2 + x3 = 1000 (Total weight)

x1 ≤ 500 (Bean 1 availability)x2 < 600 (Bean 2 availability)x3 ≤ 400 (Bean 3 availability)

1 2 375 85 60 751000

x x x+ + ≥ (Aroma rating)

1 2 386 88 75 801000

x x x+ + ≥ (Taste rating)

x1, x2, x3 > 0

The Excel sensitivity report for this linear model is provided at the top of the next page.

SENSITIVITY REPORTAdjustable Cells

Final Reduced Objective Allowable AllowableCell Name Value Cost Coefficient Increase Decrease

$B$13 pounds of Bean 1 500 0 0.5 0.1 1E+30$C$13 pounds of Bean 2 300 0 0.7 1E+30 0.166666667$D$13 pounds of Bean 3 200 0 0.45 0.25 0.25

ConstraintsFinal Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease$B$20 Total weight LHS 1000 -0.15 1000 58.82352941 58.82352941$B$21 Available B1 LHS 500 -0.1 500 500 500$B$22 Available B2 LHS 300 0 600 1E+30 300$B$23 Available B3 LHS 200 0 400 1E+30 200$B$24 Aroma LHS 75 10 75 5 5$B$25 Taste LHS 84.4 0 80 4.4 1E+30

(a) Identify the optimal solution and its objective function value?The optimal solution is: x1 = x2= x3=

The objective function value is: (b) If the price of Bean 2 has been increased to $2.7 per pound, what is the impact on the objective function value?(c) Due to the limited supply from local farms, the available Bean 2 decreases to 400 pounds. What is the impact on the objective function value?(d) Royal Kona believes that at least 300 pounds of Bean 3 have to be used in the final blend. What are the optimal solution and new objective function value?(e) If Royal Kona finds that another type of bean, Bean 4, may add additional varieties of aroma and taste to the coffee blend, and Bean 4 costs $0.80 per pound, with an aroma rating of 78 and a taste rating of 90. What is the new objective function value?(f) If there exists an error in Bean 2’s aroma rating, it is actually 87 instead of 85. What is impact of this mistake on the objective function value?(g) It is found that the actual availability of Bean 2 and Bean 3 is different from original projections: Royal Kona has only 350 pounds Bean 2 but 1000 pounds of Bean 3. What is the new objective function value?(h) If Bean 3’s price increases to $0.50 per pound and the consumer rating for aroma decreases by 1 (from 75 to 74), what is the impact on the objective function value?(i) When John first formulated this model, he accidentally omitted the total weight constraint. What would you expect the objective function value that John would have obtained?(j) Recent years have witnessed significant moves in a wide variety of commodity prices. Royal Kona is not excluded from this general trend. Royal Kona notices that both Bean 2 and Bean 3 appreciate since last season. Instead of paying $0.70 per pound for Bean 2 and $0.45 per pound for Bean 3, Royal Kona now has to pay $3.70 per pound for Bean 2 and $0.60 per pound for Bean 3. What is the new objective function value given this new pricing information?