MCDONALD\'S FAMILY PRODUCTION PLANNING PROBLEM McDonald\'s family owns five parc

ID: 448024 • Letter: M

Question

MCDONALD'S FAMILY PRODUCTION PLANNING PROBLEM

McDonald's family owns five parcels of farmland broken into a southeast sector, north sector, northwest sector, west sector, and southwest sector. Family is involved primarily in growing wheat, alfalfa, and barley crops and is currently preparing production plan for next year. The Pennsylvania Water Authority has just announced its yearly water allotment, with McDonald farm receiving 7,400 acre-feet. Each parcel can only tolerate a specified amount of irrigation per growing season, as specified in the following table:

Parcel

Acres

Water irrigation limit (acre-feet)

Southeast

2,000

3,200

North

2,300

3,400

Northwest

600

800

West

1,100

500

Southwest

500

600

Each crop needs a minimum amount of water per acre, and there is a projected limit on sales of each crop. Crop data is shown below:

Crop

Maximum Sales

Water needed per acre (acre-feet)

Wheat

110,000 bushels

1.6

Alfa;fa

1,800 tons

2.9

Barley

2,200 tons

3.5

McDonald's family best estimate is that they can sell wheat at a net profit of $2 per bushel, alfalfa at $40 per ton, and barley at $50 per ton. One acre of land yields an average of 1.5 tons of alfalfa and 2.2 tons of barley. The wheat yield is approximately 50 bushels per acre.

LP was formulated to develop a production plan for McDonald's family.

The following 15 variables were used:

Xij= acres of crop i planted on parcel j

where i = 1 for wheat, 2 for alfalfa, 3 for barley

j = 1 to 5 for SE, N, NW, W, and SW parcels

OBJECTIVE FUNCTION IS TO MAXIMIZE PROFIT:

MAX ($2*50bushels)X11+ 100X12 + 100X13 + 100X14 + 100X15+ ($40*1.5tons)X21+ 60X22 + 60X23 + 60X24 + 60X25+ ($50*2.2 tons)X31+ 110X32 + 110X33 + 110X34 + 110X35 <ART FILE="08_14eq02.EPS" W="183.121pt" H="29.114pt" XS="100%" YS="100%"/>

S.T

Irrigation limits constraints:

1.6X11 + 2.9X21 + 3.5X31 £ 3,200 acre-feet in SE

1.6X12 + 2.9X22 + 3.5X32 £ 3,400 acre-feet in N

1.6X13 + 2.9X23 + 3.5X33 £ 800 acre-feet in NW

1.6X14 + 2.9X24 + 3.5X34 £ 500 acre-feet in W

1.6X15 + 2.9X25 + 3.5X35 £ 600 acre-feet in SW

Total water acre-feet constraint

1.6X11+ 1.6X12 + 1.6X13 + 1.6X14 + 1.6X15+ 2.9X21+ 2.9X22 + 2.9X23 + 2.9X24 + 2.9X25+ 3.5X31+ 3.5X32 + 3.5X33 + 3.5X34 + 3.5X35 <= 7,400 water acre-feet total

Sales limits Constraints:

X11 + X12 + X13 + X14 + X15 £ 2,200 wheat in acres (= 110,000 bushels/50 bushels per acre)

X21 + X22 + X23 + X24 + X25 £ 1,200 alfalfa in acres (= 1,800 tons/1.5 tons per acre)

X31 + X32 + X33 + X34 + X35 £ 1,000 barley in acres (= 2,200 tons/2.2 tons per acre)

Acreage availability constraints:

X11 + X21 + X31 £ 2,000 acres in SE parcel

X12 + X22 + X32 £ 2,300 acres in N parcel

X13 + X23 + X33 £ 600 acres in NW parcel

X14 + X24 + X34 £ 1,100 acres in W parcel

X15 + X25 + X35 £ 500 acres in SW parcel

LIDO software was used to solve the above problem. All of the output is shown below:

MAX 100X11+ 100X12 + 100X13 + 100X14 + 100X15+ 60X21+ 60X22 + 60X23 + 60X24 + 60X25+ 110X31+ 110X32 + 110X33 + 110X34 + 110X35

S.T.

1.6 X11 + 2.9 X21 + 3.5 X31<=3200

1.6X12 + 2.9X22 + 3.5X32<=3400

1.6X13 + 2.9X23 + 3.5X33<=800

1.6X14 + 2.9X24 + 3.5X34<=500

1.6X15 + 2.9X25 + 3.5X35<=600

1.6X11+ 1.6X12 + 1.6X13 + 1.6X14 + 1.6X15+ 2.9X21+ 2.9X22 + 2.9X23 + 2.9X24 + 2.9X25+ 3.5X31+ 3.5X32 + 3.5X33 + 3.5X34 + 3.5X35 <= 7400

X11 + X12 + X13 + X14 + X15<=2200

X21 + X22 + X23 + X24 + X25<=1200

X31 + X32 + X33 + X34 + X35<=1000

X11 + X21 + X31<=2000

X12 + X22 + X32<=2300

X13 + X23 + X33<=600

X14 + X24 + X34<=1100

X15 + X25 + X35<=500

LP OPTIMUM FOUND AT STEP 6

OBJECTIVE FUNCTION VALUE

1) 337862.1

VARIABLE VALUE REDUCED COST

X11 187.500000 0.000000

X12 1887.500000 0.000000

X13 125.000000 0.000000

X14 0.000000 0.000000

X15 0.000000 0.000000

X21 0.000000 0.000002

X22 131.034485 0.000000

X23 0.000000 0.000002

X24 0.000000 0.000002

X25 0.000000 0.000002

X31 828.571411 0.000000

X32 0.000000 0.000000

X33 0.000000 0.000000

X34 0.000000 0.000000

X35 171.428574 0.000000

ROW SLACK OR SURPLUS DUAL PRICES

2) 0.000000 0.000000

3) 0.000000 0.000000

4) 600.000000 0.000000

5) 500.000000 0.000000

6) 0.000000 0.000000

7) 0.000000 20.689655

8) 0.000000 66.896553

9) 1068.965576 0.000000

10) 0.000000 37.586208

11) 983.928589 0.000000

12) 281.465515 0.000000

13) 475.000000 0.000000

14) 1100.000000 0.000000

15) 328.571442 0.000000

NO. ITERATIONS= 6

RANGES IN WHICH THE BASIS IS UNCHANGED:

OBJ COEFFICIENT RANGES

VARIABLE CURRENT ALLOWABLE ALLOWABLE

COEF INCREASE DECREASE

X11 100.000000 0.000000 0.000000

X12 100.000000 0.000001 0.000000

X13 100.000000 0.000000 0.000000

X14 100.000000 0.000002 INFINITY

X15 100.000000 0.000002 INFINITY

X21 60.000000 0.000002 INFINITY

X22 60.000000 31.142859 0.000002

X23 60.000000 0.000002 INFINITY

X24 60.000000 0.000002 INFINITY

X25 60.000000 0.000002 INFINITY

X31 110.000000 0.000000 0.000000

X32 110.000000 0.000002 INFINITY

X33 110.000000 0.000002 INFINITY

X34 110.000000 0.000002 INFINITY

X35 110.000000 INFINITY 0.000000

RIGHTHAND SIDE RANGES

ROW CURRENT ALLOWABLE ALLOWABLE

RHS INCREASE DECREASE

2 3200.000000 200.000000 300.000000

3 3400.000000 200.000000 600.000000

4 800.000000 INFINITY 600.000000

5 500.000000 INFINITY 500.000000

6 600.000000 200.000000 300.000000

7 7400.000000 600.000000 200.000000

8 2200.000000 237.500000 1887.500000

9 1200.000000 INFINITY 1068.965576

10 1000.000000 85.714287 828.571411

11 2000.000000 INFINITY 983.928589

12 2300.000000 INFINITY 281.465515

13 600.000000 INFINITY 475.000000

14 1100.000000 INFINITY 1100.000000

15 500.000000 INFINITY 328.571442