You sleep 8 hours a day. Every day you spend hour eating breakfast and generally

Question

You sleep 8 hours a day. Every day you spend hour eating breakfast and generally taking care of your physical appearance. This leaves 15 hours of freedom each day. Note, we are only dealing with the 5 days of the week. The goal is to have weekends off so you can go vacation or do something cool. The question is how best to spend those hours. As with anything, there is a diminishing returns on activities. The more time you spend, the less benefit you receive. The data is given in the following table Over the five days of a week, you must have at least one hour in each category and work at least 40 hours. Write and solve an integer program that maximizes your benefit of the way you spend your hours during a M-F schedule. You should assume that the benefits are cumulative and that you get the lower benefit for the lower times (similar to taxes). For instance if I worked 12 hours and exercised for 3 hours on Monday, then my total benefit for Monday would be Working benefit is 9*6+2 * 4+1 *2 = 64 Exercising Monday is 6(.5)+ 4*.5+2*1 = 7. Thus, Monday would contribute 71 units towards my benefit for the week.

Explanation / Answer

Maximize benefit Z = Summation or Sigma

Let x1 to x8 represent each activity from exercise to Home maintenance respectively.

Hence x1 = exercise, x2 = TV/Video games etc;

Z = low level + mid level + high level

Maximize benefit Z = (6x11 + 4x21 + 3x31 + 8x41+4x51+6x61+7x71+6x81) + (4x12+2x22+x32+4x42+2x52+4x62+3x72+2x82)+(x13+x23+x33+2x43+x53+2x63+x73+x83)

Subject to constraints:

Low level:

0.5 <= x11 <= 0.5

0.5 <= x21 <= 1

0.5 <= x31 <= 1

0.5 <= x41 <= 1

0.5 <= x51 <= 0.5

6 <= x61 <= 9

1 <= x71 <= 1

1 <= x81 <= 1

Medium level:

0.5 <= x12 <= 1

1<= x22 <= 2

1<= x32 <= 2

1<= x42 <= 3

0.5<= x52 <= 1

9<= x62 <= 11

1<= x72 <= 2

1<= x82 <= 3

High Level:

1<= x13 <= 4

2<= x23 <= 6

2<= x33 <= 6

3<= x43 <= 6

1<= x53 <= 5

11<= x63 <= 13

2<= x73 <= 3

3<= x83 <= 5

Solving:

Solving: (can use Excel Solver as well) or

Can use simplex method – intial table au:

Constant Value (CV)

New Value (NV)

Coeffcient Ci

New Objective Function (NOF)

S1

S2 etc Ratio

A variable (is it binary or ?) to indicate which level of the 3 – low, medium or high?

There is a combination of 8 * 3 = 24 permutations and combinations or simply 24 possibilities.

Let i represet the activities from 1 to 8 like Exercise to home maintenance, and j for the level

Hence when i=1, j=1, xij = x11 means activity 1(Exercise) on low level

x12 means exercise on medium level

x13 = exercise on high level

x21 = TV / Video games at low level

x22 = TV / Video games at medium level

x23 = TV / Video games at hi level

etc up

x83 = home maintenance at high level

But beware we do not have 83 variables but just 24 variables – it is just the numbering for easy understanding – that’s all!

Diminishing marginal Utility (DMU) – we get bored with the same thing over time – hence products lose their demand in time – this is similar to diminishing returns but slightly different

Constant Value (CV)

New Value (NV)

Coeffcient Ci

New Objective Function (NOF)

S1

S2 etc Ratio

Related Questions

Navigate

• Browse (All)
• Browse Y
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.