Assignment 1

Part a

Decision Variables:

Decision Variable here would be the amount of land allocated to each crop by adjusting the amount of land allocated we can optimize the revenue generated by each crop let assume land allocated to each crop

Land allocated to each crop
Corn: x1
Tomato: x2
Potato: x3
Strawberries: x4

Objective Function

Here our objective is to maximize the revenue generated by the sale of the crop. Total revenue generated.

Revenue generated by each crop would be equal to = Land allocated to crop * yield of crop in bushels * revenue per bushel of each crop

Z = 50 * 90 * x1 +38 *210 * x2 + 45 * 50 * x3 +56 * 56 * x4

Constraints

Crop / Formula / Constraints / Value
Planting time / 14planting time required*land allocated to each crop / <= / 770
Tending time / 14Tending time required*land allocated to each crop / <= / 550
Water /
14water required*land allocated to each crop / <= / 300
Fertilizer /
14Fertilizer required*land allocated to each crop / <= / 3000
Harvesting time required / 14Harvesting time required*land allocated to each crop / <= / 770
Total acres of land /
ADland allocated to each crop / = / 50

All constraints are in their respective units.

When we input all the parameters in excel solver following result is obtained for decision variables

Amount of allocated land comes to be

Land Allocated to crop / Yield/Acre (bushels) / Revenue/ Bushel / Amount allocated in Acres
Corn: x1 / 50 / $90 / 16.42857143
Tomato: x2 / 38 / $210 / 33.57142857
Potato: x3 / 45 / $50 / 0
Strawberries: x4 / 56 / $56 / 0

Optimum revenue will be generated by planting Corn on 16.43 acres and tomato on 33.57 acres

Z = Maximum total revenue generated = $ 341,828.57

Part b

Decision Variable

Since in this case we can farm only two of the given crops that is corn in strawberry our decision variable here would be the amount of land allocated to each crop in each farm by adjusting the amount of land allocated we can optimize the revenue generated by each crop let assume land allocated to each crop

Land Allocated to crop in acres
Corn on farm 1: x1
Strawberries on farm 1: x2
Corn on farm 2: x3
Strawberries on farm 2: x4

Objective Function

Here our objective is to maximize the revenue generated by the sale of the crop.

Revenue generated by each crop in each field would be equal to = Land allocated to each crop in each field * yield of each crop in bushels in each field * revenue per bushel of each crop

Z = 50 * 90 * x1 +56 *56 * x2 + 45 * 90 * x3 +50 * 56 * x4

Constraints

Crop / Formula / Constraints / Value
Planting time / 14planting time required*land allocated to each crop / <= / 1800
Tending time / 14Tending time required*land allocated to each crop / <= / 825
Water /
14water required*land allocated to each crop / <= / 510
Fertilizer /
14Fertilizer required*land allocated to each crop / <= / 7000
Harvesting time required / 14Harvesting time required*land allocated to each crop / <= / 1400
75% of each farm must be utilized / x1 + x2 / >= / 37.5
75% of each farm must be utilized / x3 +x4 / >= / 54.75
Size of farm1 / x1 + x2 / <= / 50
Size of farm 2 / x3 + x4 / <= / 73
Total acres of land /
ADland allocated to each crop / = / 123

When we input all the parameters in excel solver following result is obtained for decision variables

Amount of allocated land comes to be

Land Allocated to crop / Yield/Acre (bushels) / Revenue/ Bushel / Amount allocated in Acres
Corn on farm 1: x1 / 50 / $90 / 50
Strawberries on farm 1: x2 / 56 / $56 / 0
Corn on farm 2: x3 / 45 / $90 / 73.000001
Strawberries on farm 2: x4 / 50 / $56 / 0
Objective function: / $ 520,650.00

Maximum Revenue realized = $ 520650

Assignment 2

Here decision to minimize loss is dependent on number of bets in each game, by adjusting number of bets in each game we can minimize our loss

Decision Variables
Game 1 / Game 2 / Game 3 / Game 4
Number of $ 1 bet / x1 / x2 / x3 / x4
Number of $ 2 bet / y1 / y2 / y3 / y4
Number of $ 4 bet / z1 / z2 / z3 / z4
Game 1 / Game 2 / Game 3 / Game 4
Number of $ 1 bet / 0 / 20 / 6 / 0
Cost / $ 1.00 / $ 1.00 / $ 1.00 / $ 1.00
pay out / $ 12.00 / $ 8.10 / $ 4.00 / $ 1.80
Probability / $ 0.08 / $ 0.12 / $ 0.23 / $ 0.50
Expected Pay out / $ 0.92 / $ 0.93 / $ 0.92 / $ 0.90
Expected loss (cost - expected pay out ) / $ 0.08 / $ 0.07 / $ 0.08 / $ 0.10
Number of $ 2 bet / 2 / 0 / 14 / 85
Cost / $ 2.00 / $ 2.00 / $ 2.00 / $ 2.00
pay out / $ 24.55 / $ 16.35 / $ 8.15 / $ 3.80
Probability / $ 0.08 / $ 0.12 / $ 0.23 / $ 0.50
Expected Pay out / $ 1.89 / $ 1.89 / $ 1.88 / $ 1.90
Expected loss (cost - expected pay out ) / $ 0.11 / $ 0.11 / $ 0.12 / $ 0.10
Number of $ 4 bet / 18 / 0 / 0 / 0
Cost / $ 4.00 / $ 4.00 / $ 4.00 / $ 4.00
pay out / $ 49.00 / $ 32.50 / $ 16.00 / $ 7.50
Probability / $ 0.08 / $ 0.12 / $ 0.23 / $ 0.50
Expected pay-out / $ 3.77 / $ 3.75 / $ 3.69 / $ 3.75
Expected loss (cost - expected pay-out ) / $ 0.23 / $ 0.25 / $ 0.31 / $ 0.25

Objective Function

Here our objective is to minimize the overall loss from gambling. Overall less in gambling could be calculated by multiplying expected loss with number of bets of each type in each game

0.08x1+ 0.07x2+ 0.08x3+ 0.1x4+ 0.11y1+ 0.11y2+ 0.12y3+ 0.1y4+ 0.23z1+ 0.025z2+ 0.31z3+ 0.25z4

Constraints

Constraints / Formula / Value
Number of bets in game 1 / x1 + y1 +z1 / 20 / >= / 20
Number of bets in game 2 / x2 + y2 +z2 / 20 / >= / 20
Number of bets in game 3 / x3 + y3 +z3 / 20 / >= / 20
Number of bets in game 4 / x4+ y4+z4 / 85 / >= / 20
Amount spent on of $ 1 bet / (x1+x2+x3+x4) / $ 26.00 / >= / 26
Amount spent on of $ 2 bet / 2(y1+y2+y3+y4) / $ 202.00 / >= / 50
Amount spent on of $ 4 bet / 4(z1+z2+z3+z4) / $ 72.00 / >= / 72
Total amount spent / (x1+x2+x3+x4) + 2(y1+y2+y3+y4) + 4(z1+z2+z3+z4) / $ 300.00 / = / 300
Number of bets for each value in each game (all decision Variables) / integer / int

When we input all the parameters in excel solver following result is obtained for decision variables

Couple should adopt following gambling plan as per solver solutions

Game 1 / Game 2 / Game 3 / Game 4
Number of $ 1 bet / 0 / 20 / 6 / 0
Cost / $ 1.00 / $ 1.00 / $ 1.00 / $ 1.00
pay out / $ 12.00 / $ 8.10 / $ 4.00 / $ 1.80
Probability / $ 0.08 / $ 0.12 / $ 0.23 / $ 0.50
Expected Pay out / $ 0.92 / $ 0.93 / $ 0.92 / $ 0.90
Expected loss (cost - expected pay out ) / $ 0.08 / $ 0.07 / $ 0.08 / $ 0.10
Number of $ 2 bet / 2 / 0 / 14 / 85
Cost / $ 2.00 / $ 2.00 / $ 2.00 / $ 2.00
pay out / $ 24.55 / $ 16.35 / $ 8.15 / $ 3.80
Probability / $ 0.08 / $ 0.12 / $ 0.23 / $ 0.50
Expected Pay out / $ 1.89 / $ 1.89 / $ 1.88 / $ 1.90
Expected loss (cost - expected pay out ) / $ 0.11 / $ 0.11 / $ 0.12 / $ 0.10
Number of $ 4 bet / 18 / 0 / 0 / 0

Minimum Loss = $ 16.32