OptimizationQuestions

4.1 Two plants manufacture soybean oil. Plant A has six truckloads ready for shipment. Plant B has twelve truckloads ready for shipment. The products will be delivered to 3 warehouses: warehouse 1 needs seven truckloads; warehouse 2 needs five truck- loads; and warehouse 3 needs six truckloads.

Shipping a truckload of soybean oil from plant A to warehouse 1, 2 and 3 costs $25, $21 and $27, respectively, and shipping a truckload of soybean oil from plant B to warehouse 1, 2 and 3 costs $23, $24 and $22, respectively. The cost can be reduced by shipping more than one truckload to the same warehouse, and the discounted cost is given by:

Cn = C

n 1 3

(1)

where C is the cost for only one truckload used for shipping to a warehouse and n is the number of truckloads from a plant to the same warehouse.

A total of eighteen truckloads are available at points of origin for delivery. Determine how many truckloads to ship from each plant to each warehouse to meet the needs of each warehouse at the lowest cost.

Note: For this problem the integer variables can be included in the nonlinear equa- tions.

4.2 There are eight cities in Alloy Valley County. The county administration is planning to build fire stations. The planning target is to build the minimum number of fire stations needed to ensure that at least one fire station is within 18 minutes (driving times) of each city. The times (in minutes) required to drive between the cities in Alloy Valley county are shown in Table 1. Determine the minimum number of fire stations and also where they should be located.

Table 1: Driving Times (minutes) Between Cities in Alloy Valley County. Driving Time (minutes) from City to City

City 1 2 3 4 5 6 7 8 1 0 18 13 16 9 29 20 25 2 18 0 25 35 22 10 15 19 3 13 25 0 15 30 25 18 20 4 16 35 15 0 15 20 25 28 5 9 22 30 15 0 17 14 29 6 29 10 25 20 17 0 25 10 7 20 15 18 25 14 25 0 15 8 25 19 20 28 29 10 15 0

Hint: Form a table of all cities and how many cities are within 18 minutes of given city.

4.3 Consider the following small Mixed Integer Nonlinear Programming (MINLP) prob- lem:

Minimize z = −y + 2×1 + x2

Subject to x1 − 2exp(−x2) = 0

−x1 + x2 + y ≤ 0

0.5 ≤ x1 ≤ 1.4

yϵ[0, 1]

Solve the two NLPs by fixing y = 0 and y = 1. Locate the optimum.

For the above problem

– Eliminate x2 and write down the iterative solution procedure using OA. Write down iterative solution procedure using GBD.

– Now instead of eliminating x2, eliminate x1. Assume y = 0 as a starting point. Find the solution using OA.

4.4 Given a mixture of four components A, B, C, D for which two separation technologies (given in Table 2) are to be considered.

1. Determine the tree and network representation for all the alternative sequences.

2. Find the optimal sequence with the depth-first strategy.

3. Find the optimal sequence with the breadth-first strategy.

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Table 2: Cost of Separators in $/year. Separator Cost A/BCD 50,000 AB/CD 56,000 ABC/D 21,000 A/BC 40,000 AB/C 31,000 B/CD 38,000 BC/D 18,000 A/B 35,000 B/C 44,000 C/D 21,000

4. Heuristics has provided a good lower bound for this problem. The cost of the separation should not exceed $91,000. Use the depth-first strategy to find the solution.

5. Compare the solution obtained by using the lower bound with the solution ob- tained without the lower bound information.

4.5 The following simple cost function is derived from the Brayton Cycle example (Painton and Diwekar, 1994) to illustrate the use of Simulated Annealing and Genetic Algo- rithms.

Min Cost = N1∑ i=1

[ (N1 − 3)2 + (N2i − 3)2 + (N3i − 3)2

] where N1 is allowed to vary from one to five and both N i2 and N i3 can take any value from one to five.

1. How many total combinations are involved in the total enumeration?

2. Set up the problem using Simulated Annealing or Genetic Algorithms.

3. For Simulated Annealing, assume the initial temperature to be 50, the number of moves at each temperature to be 100, and the freezing temperature to be 0.01. Use the temperature decrement formula Tnew = αTold where α is 0.95.

4. Use the binary string representation for Genetic Algorithms and Solve the prob- lem.

5. Compare the solution obtained by the binary representation above with the solu- tion obtained using the natural representation consisting of the vector N = (N1, N2(i), i = 1, 2, . . . , 5 , N3(i) i = 1, 2, . . . , 5).

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