Answered: According to Excel's sensitivity report…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Only need help with the shadow price for the capacity constraint

$### Optimizing Beer Stock for Super Bowl Sunday Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday and must determine how much beer to stock. Betty stocks three brands of beer: Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is shown in the table below. The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of Shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer. Betty wants to stock up completely. Betty wants to decide on the number of gallons of each brand of beer to order so as to make the most profit. Formulate a linear programming model for this problem. Define $ x_1 $ as the number of gallons of Yodel to order, $ x_2 $ as the number of gallons of Shotz to order, $ x_3 $ as the number of gallons of Rainwater to order, and $ Z $ as the total profit. Which of the following model formulations is correct? #### Cost Table | Brand | Cost/Gallon | |-----------|-------------| | Yodel | $1.50 | | Shotz | $0.90 | | Rainwater | $0.50 | ### Problem Formulation **Objective:** Maximize total profit $ Z $. **Decision Variables:** - $ x_1 $: Gallons of Yodel to order - $ x_2 $: Gallons of Shotz to order - $ x_3 $: Gallons of Rainwater to order **Constraints:** 1. Budget constraint: $ 1.50x_1 + 0.90x_2 + 0.50x_3 \leq 2000 $ 2. Capacity constraint: $ x_1 + x_2 + x_3 \leq 1000 $ 3. Demand constraints: - $ x_1 \leq 400 $ (for Yodel)$

Transcribed Image Text:### Optimizing Beer Stock for Super Bowl Sunday Betty Malloy, owner of the Eagle Tavern in Pittsburgh, is preparing for Super Bowl Sunday and must determine how much beer to stock. Betty stocks three brands of beer: Yodel, Shotz, and Rainwater. The cost per gallon (to the tavern owner) of each brand is shown in the table below. The tavern has a budget of $2,000 for beer for Super Bowl Sunday. Betty sells Yodel at a rate of $3.00 per gallon, Shotz at $2.50 per gallon, and Rainwater at $1.75 per gallon. Based on past football games, Betty has determined the maximum customer demand to be 400 gallons of Yodel, 500 gallons of Shotz, and 300 gallons of Rainwater. The tavern has the capacity to stock 1,000 gallons of beer. Betty wants to stock up completely. Betty wants to decide on the number of gallons of each brand of beer to order so as to make the most profit. Formulate a linear programming model for this problem. Define $ x_1 $ as the number of gallons of Yodel to order, $ x_2 $ as the number of gallons of Shotz to order, $ x_3 $ as the number of gallons of Rainwater to order, and $ Z $ as the total profit. Which of the following model formulations is correct? #### Cost Table | Brand | Cost/Gallon | |-----------|-------------| | Yodel | $1.50 | | Shotz | $0.90 | | Rainwater | $0.50 | ### Problem Formulation **Objective:** Maximize total profit $ Z $. **Decision Variables:** - $ x_1 $: Gallons of Yodel to order - $ x_2 $: Gallons of Shotz to order - $ x_3 $: Gallons of Rainwater to order **Constraints:** 1. Budget constraint: $ 1.50x_1 + 0.90x_2 + 0.50x_3 \leq 2000 $ 2. Capacity constraint: $ x_1 + x_2 + x_3 \leq 1000 $ 3. Demand constraints: - $ x_1 \leq 400 $ (for Yodel)

### Question on Sensitivity Analysis

**Question:**
According to Excel’s sensitivity report for the previously formulated model (Eagle Tavern), the shadow price for the capacity constraint is:

- [ ] 0
- [ ] 1.25
- [ ] 1.5
- [ ] 1.6

**Explanation:**
This question assesses the understanding of sensitivity analysis, specifically the concept of shadow prices in linear programming models. The shadow price indicates the change in the objective function's value given a one-unit increase in the right-hand side of a constraint. Here, students need to identify the shadow price for the capacity constraint based on Excel's sensitivity report for the Eagle Tavern model.

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