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ICAR AICE-JRF/SRF Optimization Techniques for Statisticians: practice questions

ICAR AICE-JRF/SRF - Agricultural Statistics, Computer Application, Bioinformatics (PhD admission and fellowship) · India · National Testing Agency for ICAR

Original practice questions written to the published syllabus · not past papers
10worked here, free
64written to this exam in this subject
10syllabus units

What this subject covers

The optimisation theory statisticians use: linear programming with simplex and duality, transportation and assignment, integer and goal programming for farm and crop planning, non-linear optimisation with Kuhn-Tucker conditions, and the numerical optimisers inside likelihood estimation. The Agribusiness track teaches operations research for managers; this goes to the theory.

In the exams: AICE-JRF Ag Stat Unit 2 'linear programming: formulation, graphical, simplex, duality, transportation and assignment' (heavy); UPSC ISS Statistics III operations research (heavy, reported); ASRB NET Ag Stat (present)

What ICAR AICE-JRF/SRF asks from it

ICAR AICE-JRF/SRF Agricultural Statistics (Unit 2)

LP formulation, graphical solution, simplex (including Big-M and two-phase), duality and shadow prices, transportation (initial solutions, MODI) and assignment (Hungarian method).

Where the 64 questions fall

How our questions for ICAR AICE-JRF/SRF spread across the units of Optimization Techniques for Statisticians. Difficulty: 14 easy, 32 medium, 18 hard.

Transportation and assignment problems10
Linear programming: formulation and graphical solution9
Simplex method, Big-M and two-phase methods8
Duality and sensitivity analysis8
Numerical optimisation: gradient, Newton and quasi-Newton methods6
Integer programming: branch and bound5
Non-linear programming: Lagrange multipliers and Kuhn-Tucker conditions5
Quadratic programming and risk programming (MOTAD)5
Goal programming for multi-objective farm planning4
Dynamic programming4

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Simplex method, Big-M and two-phase methods · easy

Which variable converts a ≤ constraint into an equation in the simplex method?

  1. A dual variable
  2. A surplus variable
  3. A slack variable
  4. An artificial variable
Answer and working

C. A slack variable

Slack = unused resource, added to ≤ constraints; surplus is subtracted from ≥ constraints.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

2 · Duality and sensitivity analysis · medium

Land has a shadow price of 20 thousand currency units per hectare. Renting more land costs 15 per hectare. What should the farmer do?

  1. Sell land at 15
  2. Rent none, since the price is positive
  3. Rent more, within the RHS range
  4. Rent unlimited land
Answer and working

C. Rent more, within the RHS range

Each extra hectare adds 20 at a cost of 15, but only within the range over which the shadow price stays valid.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

3 · Transportation and assignment problems · easy

A balanced transportation problem has 3 sources and 4 destinations. How many basic (allocated) cells does a non-degenerate basic solution have?

  1. 7
  2. 6
  3. 12
  4. 4
Answer and working

B. 6

m + n − 1 = 3 + 4 − 1 = 6.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

4 · Transportation and assignment problems · medium

An assignment problem maximises yields of varieties at sites. How is it converted for the Hungarian method?

  1. Sort each row in descending order
  2. Subtract entries from the maximum
  3. Take reciprocals of all entries
  4. Multiply all entries by −1 and add n
Answer and working

B. Subtract entries from the maximum

Regret = maximum − entry turns maximisation into minimisation; negating also works but the standard device is subtraction from the maximum.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

5 · Goal programming for multi-objective farm planning · hard

With 12 ha and a 20-day labour limit, wheat (40/ha, 2 days) and mustard (30/ha, 1 day), can an income goal of 450 be met?

  1. No; the maximum is 440
  2. No; the maximum is 400
  3. Yes, with 4 ha wheat and 8 ha mustard
  4. Yes, with 12 ha wheat
Answer and working

A. No; the maximum is 440

Maximising 40x + 30y with x + y ≤ 12 and 2x + y ≤ 20 gives (8, 4) and 440 < 450, so the goals conflict and weights decide the compromise.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

6 · Dynamic programming · hard

Which methods solve a Markov decision process?

  1. Branch and bound only
  2. The simplex and MODI methods
  3. Value and policy iteration
  4. Newton-Raphson only
Answer and working

C. Value and policy iteration

Both iterate the Bellman equation for expected values.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

7 · Numerical optimisation: gradient, Newton and quasi-Newton methods · medium

A mixed model reports non-convergence. Which cause is most common?

  1. A bug in every optimiser
  2. Over-parameterisation
  3. Too many observations
  4. Normal residuals
Answer and working

B. Over-parameterisation

Unidentifiable or boundary parameters make the objective flat or constrained; simplifying or reparameterising usually helps.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

8 · Numerical optimisation: gradient, Newton and quasi-Newton methods · hard

After an MLE optimiser converges, why check that the Hessian is negative definite?

  1. To make the gradient zero
  2. To speed up the next run
  3. To confirm a true maximum
  4. To transform the parameters
Answer and working

C. To confirm a true maximum

A stationary point may be a saddle or minimum; the inverse of −H gives the standard errors only at a proper maximum.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

9 · Linear programming: formulation and graphical solution · easy

In the farm LP with 12 ha land and 16 labour-days, which constraints bind at the optimum (4, 8)?

  1. Land only
  2. Labour only
  3. Neither
  4. Both land and labour
Answer and working

D. Both land and labour

4 + 8 = 12 ha and 2 × 4 + 8 = 16 days: both resources are fully used, so both have positive shadow prices.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

10 · Duality and sensitivity analysis · easy

What is the dual of the dual of an LP?

  1. A transportation problem
  2. An integer programme
  3. The original primal
  4. The dual itself
Answer and working

C. The original primal

Duality is symmetric: dualising twice returns the primal.

Further reading: Taha, Operations Research: An Introduction; Hazell and Norton, Mathematical Programming for Economic Analysis in Agriculture; Nocedal and Wright, Numerical Optimization

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Optimization Techniques for StatisticiansThe full study manual, free ICAR AICE-JRF/SRFFormat, sections and how it has changed

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