Design a linear programming model

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Reference no: EM13153906

Jim and Suzie Scott purchased a property in northern Victoria which has the capacity to run a variety of grazing and cropping activities.

670 ha of the property is suited to grazing and would be suitable for cattle, Merino, first-cross lamb, prime lamb or Dorper production. The market for livestock is very dynamic. Their research suggests that a cattle enterprise could generate \$325/hd, Merinos \$48/hd, first-cross lambs \$56/hd, prime lambs \$63/hd and Dorpers \$47/hd.

To intensify their grazing activities, the Scotts' have the option of improving the pasture. They have been advised that this would cost \$75/ha per year and it will take approximately 1 hour to spread fertilizer across 4 ha. The energy provided by each pasture type is shown in the table below.

Feed energy (MJ/ha)

Season                               native pasture                               improved pasture

Spring                                  8,376                                             82,800

Summer                                6,830                                             14,280

Autumn  6,149                                    9,030

Winter    4,091                                    21,000

The energy and labour requirements of each grazing activity are shown in the following table:

Cattle              Merino            first-cross lambs         Prime lambs         Dorpers

Labour (hr/ha)

Spring              0.393              0.181                   0.181                            0.173                  0.183

Summer           0.353              0.126               0.126                                0.118                  0.113

Autumn           0.368              0.121               0.121                                0.117                  0.087

Winter              0.393             0.1110.111       0.101                               0.061

Feed required (MJ/hd)

Spring             9,720                    1,449                       1,278                        1,608                1,854

Summer          5,850                    915                          748                           865                    963

Autumn          7,670                    758                          788                           931                   1,021

Winter            10,130                  817                          1,180                        1,299                1,547

The property also consists of an additional 350 ha ideal for growing wheat, barley, triticale, lupins or canola. The following table provides yields, prices, costs and the labour requirements for each crop per season.

Wheat            barley         triticale        lupins           canola

Yield (t/ha)                  2.88                 2.98             3.06             1.83              1.79

Price (\$/t)                     195                  165              135              185               370

Cost (\$/ha)                   122                  113              92                131               237

Labour (hr/ha)

Spring                           0                     0                        0                0                   0

Summer                        0.286              0.286                 0.286         0.5                0.5

Autumn                        0.893              0.893                 0.752         1.022            1.793

winter                           0.09                0.09                   0.09           0                   0.18

Constraints

Jim thinks the family can work a combined total of 650 hours per season. To maintain soil quality the Scotts would like to have at least one year of lupins for every 2 years of other crops.

Design a linear programming (LP) model to help the Scotts determine the activity mix that will maximise their profits while satisfying all their constraints. Answer the following questions.

1. Report the total gross margin, and optimal values for each activity and constraint in the base scenario. Report and interpret the shadow prices.

2. In the base scenario the Scotts produce crops only for sale. Introduce the possibility of feeding barley or triticale to livestock in Autumn. Assume it will take 15 minutes to feed one tonne of these grains. Each tonne of barley contains 12,000 MJ of energy and each tonne of triticale contains 13,000 MJ of energy  Explain results following the same guidelines as in question.

3. Take the model from question 2 and introduce the possibility of hiring labour at a cost of \$25 per hour. Select the seasons for labour hire based on your results from question 2. Justify your choice. Explain results following the same guidelines as in question 2.

Show the models from questions 1, 2 and 3 as appendices in your assignment. Create tables that summarise the main results and present those in the body of the paper.

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