Question 1

Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations:
 -homewark can account far up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade;
- the praject can account for up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade;
- the mid-term and final must each accaunt far betwen $10 \%$ and $40 \%$ of the grade but cannot accaunt far mare than $7 \%$ of the grade when the percentages are cambined; and
- the project and final exam grades may not collectively constitute more than $50 \%$ of the Iratade. The following LP model allows Robert to maximize his numerical grade.
 $\begin{array} { l } 
\text { Let } \quad W _ { 1 } = \text { weight assigned to hamewark } \
 \quad \quad  \quad W _ { \mathbf { 2 } } = \text { waight assigned to the praject } \
\quad \quad \quad W _ { 3 } = \text { weight assigned to the midi-term } \
 \quad \quad \quad W _ { 4 } = \text { waight assigned to the final } \
\
\text { MAX: } \quad 75 \mathrm {~W} _ { 1 } + 94 \mathrm {~W} _ { 2 } + 85 \mathrm {~W} _ { 3 } + 92 \mathrm {~W} _ { 4 } \
\text { Subject to: } \quad W _ { 1 } + W _ { 2 } + W _ { 3 } + W _ { 4 } = 1 \
 \quad \quad \quad \quad \quad  \quad W _ { 3 } + W _ { 4 } \leq0 .70 \
 \quad \quad \quad \quad \quad \quad W _ { 3 } + W _ { 4 } \geq 0.50 \
 \quad \quad \quad \quad \quad \quad\text { 0. } 05 \leq W _ { 1 } \leq 0.25 \
 \quad \quad \quad \quad \quad \quad\text { 0. } 05 \leq W _ { 2 } \leq 0.25 \
 \quad \quad \quad \quad \quad \quad\text { 0.10 } \leq W _ { 3 } \leq 0 .4 0 \
 \quad \quad \quad \quad \quad \quad0 .10 \leq W _ { 4 } \leq 0.40 \
\end{array}$   What values would you enter in the Risk Solver Platform (RSP) task pane for the cells in this Excel spreadsheet implementation of this problem?
Objective Cell:
Variables Cells:
Constraints Cells:

Accepted Answer

The answer of Robert Hope received a welcome surprise in...

Question 2

Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
 $\begin{array}{cccc}
\text { Week } & \text { Trucking Limits } & \text { Railway Limits } & \text { Air Cargo Limits } \
\hline 1 & 45 & 60 & 15 \
2 & 50 & 55 & 10 \
3 & 55 & 45 & 5 \
\hline \text { Costs (\$ per } 1000 \text { tons) } & \$ 200 & \$ 140 & \$ 400
\end{array}$
 The following is the LP model for this logistics problem.
   What formula goes in cells F10, E3, E4, E5, and B6 of this Excel spreadsheet?

Accepted Answer

The answer of Carlton construction is supplying building materials for...

Question 3

Project 3.1- The Diet Problem: Ordering Meals from McDonald's Based on: Robert A. Bosch, "Big Mac Attack: The Diet Problem revisited, Eating at McDonald's," OR/MS Today, August 1993, pp 30-31. Tina Simpson is a new fourth-grade teacher at Forest Ridge Elementary. The first teacher workshop for the upcoming school year is next Monday and by majority vote, McDonald's was selected as the food of choice. As the new person, Tina is tasked with developing the meal for the workshop. McDonald's has graciously offered to deliver whatever food Tina decides to order, along with a variety of condiments applicable to whatever is ordered. Rather than offer a menu choice, Tina has decided to simply order the same meal for each person in the workshop. To get started, Tina took a trip to McDonald's and obtained their published information on the nutritional content of their food. That data is summarized in the table below. Tina wants the meal to be nutritionally complete. The National Research Council publishes their Recommended Daily Allowances. In this publication, they contend that a diet (in this case the meal) should provide at least 100 percent of the U.S. RDA of numerous nutrients. The specific amount of the RDA depends on such factors as age, weight and gender. In addition, the council recommends daily sodium and cholesterol intakes be kept to at most 2.4 grams of sodium and 300 milligrams of cholesterol. Further, at most 30 percent of the calories consumed should come from fat, and at most 10 percent from saturated fat. Each gram of fat contains 9 calories. Based on the above information, Tina wants to design a least-cost meal that provides at least 100% of the U.S. RDA of vitamins A, C, B₁, B₂, niacin, calcium, and iron; supplies at least 55 grams of protein; contains at most 3 grams of sodium; and contains at most 30 percent of its calories from fat. Only those foods list in the table above are available for the meal. Formulate the LP model for Tina's problem. Develop a spreadsheet model of the problem and use Excel Solver to determine the least-cost meal that meets all the stated requirements. What is the recommended meal? Is this meal reasonable? If not, modify the model to obtain what you believe to be a reasonable meal that meets the stated requirements.

Accepted Answer

The answer of Project 3.1- The Diet Problem: Ordering Meals...

Question 4

The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
 $\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text { Hospital }$
$\begin{array}{lccc}
&\text { New Hope } & \text { Countv General } & \text { City East }\
\hline \text { Input Measures } & & & \
\text { Bed days unused (1000s) } & 83.0 & 105.0 & 104.1 \
\text { Supply expense }(\$ 1000 \mathrm{~s}) & 123.8 & 162.3 & 154.0 \
\text { Full-time staff } & 225.0 & 200.0 & 231.0 \
\text { Output Measures } & & & \
\text { Patient-days (1000s) } & 105.0 & 71.0 & 82.7 \
\text { Nurses qualified } & 253.0 & 92.0 & 175.0 \
\text { Assistants on staff } & 125.0 & 45.0 & 65.0 \
\text { satiefartion } & 98.0 & 88.0 & 83.0
\end{array}$

What are the key formulas for this Excel spreadsheet implementation of the following formulation?
 Let $\quad w _ { i } =$ werght as5igned to dutput $j , j = 1 , \ldots , 4$
$\mathbf { v } _ { i } =$ waight as5igned to ingut $i , i = 1 , \ldots , 3$
MAX: $0 0. \quad 82.7 w _ { 1 } + 175 .0 w _ {2 } + 65.0 w _ { 3 } + 83 .0 w _ { 4 }$
Subject ta:
$105.1 w _ { 1 } + 253.0 w _ { 2} + 125.0 w _ { 3 } + 98.0 w _ { 4 } - 83.0 v _ { 1 } - 123.8 v _ { 2 } - 225.0 v _ { 3 } \leq 0$
$71.0 w _ { 1 } + 92.0 w _ { 2 } + 45.0 w _ { 3 } + 88.0 w _ { 4 } - 105.0 v _ { 1 } - 162.3 v _ { 2 } - 200 v _ { 3 } \leq 0$
$82.7 w _ { 1 } + 175.0 w _ { 2} + 65.0 w _ { 3 } + 83.0 w _ { 4 } - 104.1 v _ { 1 } - 154.0 v _ { 2 } - 231.0 v _ { 3 } \leq0$
$104.1 \mathbf { v } _ { 1 } + 154.0 \mathbf { v } _ { \mathbf { 2 } } + 231.0 \mathbf { v } _ { \mathbf { 3 } } = 1$
$w _ { 1 , } w _ { 2 } w _ { 3 , } w _ { 4 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \geq0$

Accepted Answer

The answer of The hospital administrators at New Hope, County...

Question 5

The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
 $\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text { Hospital}$
$\begin{array}{lccc}
&\text { New Hope } &\text { County General }&\text {City East }\
\hline \text {Input Measures}\
 \text { Bed days unused (1000s) } & 83.0 & 105.0 & 104.1 \
\text { Supply expense }(\$ 1000 s) & 123.8 & 162.3 & 154.0 \
\text { Full-time staff } & 225.0 & 200.0 & 231.0 \
\text { Output Measures } & & & 82.7 \
\text { Patient-days (1000s) } & 105.0 & 71.0 & 175.0 \
\text { Nurses qualified } & 253.0 & 92.0 & 65.0 \
\text { Assistants on staff } & 125.0 & 45.0 & 830
\end{array}$ Based on the following formulation, is City East efficient? If not, what input and output values should they aspire to in order to become efficient?
 Let $\quad w _ { i } =$ weight assigned to output $j , j = 1 , \ldots , 4$
$\mathbf { v } _ { i } =$ weight assigned to input $i , i = 1 , \ldots , 3$
MAX: $ \quad 82.7 w _ { 1 } + 175 .0 w _ { 2 } + 65.0 w _ { 3 } + 83 .0 w _ { 4 }$
Subject to:
$105.1 w _ { 1 } + 253.0 w _ { 2} + 125.0 w _ { 3 } + 98.0 w _ { 4 } - 83.0 v _ { 1 } - 123.8 v _ { 2 } - 225.0 v _ { 3 } \leq 0$
$71.0 w _ { 1 } + 92.0 w _ { 2 } + 45.0 w _ { 3 } + 88.0 w _ { 4 } - 105.0 v _ { 1 } - 162.3 v _ { 2 } - 200 v _ { 3 } \leq 0$
$82.7 w _ { 1 } + 175.0 w _ { 2 } + 65.0 w _ { 3 } + 83.0 w _ { 4 } - 104.1 v _ { 1 } - 154.0 v _ { 2 } - 231.0 v _ { 3 } \leq 0$
$104.1 \mathbf { v } _ { 1 } + 154.0 \mathbf { v } _ { \mathbf { 2 } } + 231.0 \mathbf { v } _ { \mathbf { 3 } } = 1$
$w _ { 1 , } w _ { 2 } w _ { 3 , } w _ { 4 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \geq 0$

Accepted Answer

The answer of The hospital administrators at New Hope, County...

Question 6

The hospital administrators at New Hope, County General, and City East recently received notice of an impending state inspection of their facilities. Under new guidelines established to improve the overall health care system, state inspectors will be assessing the efficiency of each hospital. The staff at New Hope has suggested a mutual assistance program in preparation for the inspections and have proposed using DEA as a means to assess the efficiency of each facility. The data collected thus far is summarized in the following table. All data reflects averages compiled over the past six months.
 $\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\quad $$\text { Hospital }$
$\begin{array}{lccc}
&\text { New Hope } & \text { Countv General } & \text { City East }\
\hline \text { Input Measures } & & & \
\text { Bed days unused (1000s) } & 83.0 & 105.0 & 104.1 \
\text { Supply expense }(\$ 1000 \mathrm{~s}) & 123.8 & 162.3 & 154.0 \
\text { Full-time staff } & 225.0 & 200.0 & 231.0 \
\text { Output Measures } & & & \
\text { Patient-days (1000s) } & 105.0 & 71.0 & 82.7 \
\text { Nurses qualified } & 253.0 & 92.0 & 175.0 \
\text { Assistants on staff } & 125.0 & 45.0 & 65.0 \
\text { Custnmer satisfactinn } & 98.0 & 88.0 & 83.0
\end{array}$
 Enter the numbers in the appropriate cells of ranges B4:H6 in the Excel spreadsheet to solve this problem based on the following formulation.
 Let $\quad w _ { i } =$ werght as5igned to dutput $j , j = 1 , \ldots , 4$
$\mathbf { v } _ { i } =$ waight as5igned to ingut $i , i = 1 , \ldots , 3$

MAX: $\left. \left. \quad 82.7 w _ { 1 } + 175 .\right] w _ { 2 } + 65.0 w _ { 3 } + 83 .\right] w _ { 4 }$
Subject to:
$105.1 w _ { 1 } + 253.0 w _ {2 } + 125.0 w _ { 3 } + 98.0 w _ { 4 } - 83.0 v _ { 1 } - 123.8 v _ { 2 } - 225.0 v _ { 3 } \leq 0$
$71.0 w _ { 1 } + 92.0 w _ { 2 } + 45.0 w _ { 3 } + 88.0 w _ { 4 } - 105.0 v _ { 1 } - 162.3 v _ { 2 } - 200 v _ { 3 } \leq 0$
$82.7 w _ { 1 } + 175.0 w _ { 2 } + 65.0 w _ { 3 } + 83.0 w _ { 4 } - 104.1 v _ { 1 } - 154.0 v _ { 2 } - 231.0 v _ { 3 } \leq 0$
$104.1 \mathbf { v } _ { 1 } + 154.0 \mathbf { v } _ { \mathbf { 2 } } + 231.0 \mathbf { v } _ { \mathbf { 3 } } = 1$
$w _ { 1 , } w _ { 2 } w _ { 3 , } w _ { 4 } , \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } \geq0$

Accepted Answer

The answer of The hospital administrators at New Hope, County...

Question 7

Carlton construction is supplying building materials for a new mall construction project in Kansas. Their contract calls for a total of 250,000 tons of material to be delivered over a three-week period. Carlton's supply depot has access to three modes of transportation: a trucking fleet, railway delivery, and air cargo transport. Their contract calls for 120,000 tons delivered by the end of week one, 80% of the total delivered by the end of week two, and the entire amount delivered by the end of week three. Contracts in place with the transportation companies call for at least 45% of the total delivered be delivered by trucking, at least 40% of the total delivered be delivered by railway, and up to 15% of the total delivered be delivered by air cargo. Unfortunately, competing demands limit the availability of each mode of transportation each of the three weeks to the following levels (all in thousands of tons):
 $\begin{array}{cccc}
\text { Week } & \text { Trucking Limits } & \text { Railway Limits } & \text { Air Cargo Limits } \
\hline 1 & 45 & 60 & 15 \
2 & 50 & 55 & 10 \
3 & 55 & 45 & 5 \
\hline \text { Costs (\$ per } 1000 \text { tons) } & \$ 200 & \$ 140 & \$ 400
\end{array}$
 The following is the LP model for this logistics problem.
   What values would you enter in the Risk Solver Platform (RSP) task pane for the cells in this Excel spreadsheet implementation of this problem?
Objective Cell:
Variables Cells:
Constraints Cells:

Accepted Answer

The answer of Carlton construction is supplying building materials for...

Question 8

Robert Hope received a welcome surprise in this management science class; the instructor has decided to let each person define the percentage contribution to their grade for each of the graded instruments used in the class. These instruments were: homework, an individual project, a mid-term exam, and a final exam. Robert's grades on these instruments were 75, 94, 85, and 92, respectively. However, the instructor complicated Robert's task somewhat by adding the following stipulations:
 -homewark can account far up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade; 
-the praject can account for up to $25 \%$ of the grade, but must be at least $5 \%$ af the grade;
-the mid-term and final must each accaunt far betwen $10 \%$ and $40 \%$ of the grade but cannot accaunt far mare than $7 \%$ of the grade when the percentages are cambined; and 
-the project and final exam grades may not collectively constitute more than $50 \%$ of the Iratade. Formulate an LP model for Robert to maximize his numerical grade.

Accepted Answer

The answer of Robert Hope received a welcome surprise in...

Quiz 3: Modeling and Solving Lp Problems in a Spreadsheet