Represent the graphical solution to the problem in a diagram

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Reference no: EM132369772

Assessment : Optimization

Question 1 – Linear Programming I

All parts of question 1 relate to the following example. A profit maximizing farmer, Mary, has 100 ha of crop land, 200 days of labour and a requirement to deliver at least 20 tonnes of barley to a local brewery. The profitability of barley is 1.1 thousand dollars per ha planted and the profitability of wheat is 1.3 thousand dollars. Resource use per ha is given in the following table

Resource	Barley	Wheat
Profit per ha $ thousands	1.1	1.3
Land	1	1
Labour	1.5	2.5
Barley contract	4 (barley yield tonnes per ha)

Question 1a:

State the problem as a mathematical programming problem (objective function, constraints and non-negativity). Find the optimal solution using Solver and give the optimal solution in a table.

Question 1b:

Represent the graphical solution to the problem in a diagram. Include all constraints and the isoprofit line for the optimal solution. The diagram should identify the feasible set, possible optimal solutions and the optimal solution. The diagram should be fully labelled and discussed in a comment.

Question 1c:

Give the Lagrangean multipliers (shadow prices) for all three constraints and interpret each intuitively. The farmer can rent land at $0.7 (thousand per ha). Would you advise Mary to rent more land or not. How much should she rent if her other resources and constraints are fixed? (Hint: How do the Lagrange multipliers change as the land area is increased.)

Question 1d:

Over what range of wheat profit does the optimal solution (in terms of the area of wheat and barley) remain constant (Hint find out about LP sensitivity analysis). Linear programming generates sensitivity analysis solutions automatically, explain the property of LP that makes this possible.

Question 2 – Dynamics

Locusts are a pest on Rottgut Island because they eat sheep fodder. A control scheme proposed which involves spraying locusts measured by control effort. The details of the problem are as follows.

The locust population grows according to the difference equation:
x_(t+1)=x_t+γ (x_t - θ x_t h_t)

Where x_t the population in month t is, γ is the growth rate h_t is locust control effort and a catchability parameter θ.

The cost of locust control effort is the quadratic function:
c^h (h_t)=c₀ h_t²

The damage caused by locusts is a linear function:
c^d (x_t)=c₁ x_t

If there are any locusts left after the end of the planning period they have a terminal cost of, c_T.

The objective function is to minimize the cost of the incursion over time (we are ignoring discounting):
Minimize∑_(t=1)^T (c^h (h_t )+c^d (x_t)) +c_T x_T

Parameters

Parameter	Value	Description
x₀	1000	Initial locust population
γ	0.2	Growth parameter
Θ	0.06	Catchability coefficient
c₀	200	Effort cost parameter
c₁	5	Damage cost parameter to the sheep fodder
c_r	500	Terminal cost parameter for any remaining population

Question 2a:

Find the optimal dynamic solution to this problem over a 24 month planning horizon using Solver. Give the answer as a fully labelled table and a graph.

Question 2b:

How does the solution change if the cost of effort varies over a range from 100 to 300? Present your solutions as a fully labelled graph and discuss your results in full.

Attachment:- Module.rar

Reference no: EM132369772

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