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Chapter 7 | 
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DYNAMIC PROGRAMMING
Problems 7.1 to 7.5 | Problems 7.6 to 7.10 | Problems 7.11 to 7.18
problem7.1 | problem7.2 | problem7.3 | problem7.4 | problem7.5
7-1. For example 7-1, determine the shortest distance between the east and the west coast.
Solution:
The optimal route from San Francisco to the East coast was obtained in Example 7-1. To solve this problem, we need to complete the partial optimization at Stage 4 as shown below.
The shortest distance is 16, and the optimal paths are:
San Francisco - N3 - N2 or C2 - N1 - Boston
- C2 - N1 -
San Diego - C3 - Boston
- S2 - C1 -
7-2. Solve Example 7-5 as a network problem.
The problem statement for Example 7-5 is:
A manufacturing process is arranged as shown in Figure 7-19 along with the operating conditions and costs for each unit. Raw material which comes in two variations, A and B, is successfully heated and purified before it is sent to a reactor where it undergoes transformation into impure products. The impure products are cleaned up in a "finisher", and the final products are sold. A new heating - purifying train has just been built, and it is operated in parallel with the old one. The new unit process the same amount of material as the old unit, which is then mixed and pumped to the reactor. The impurity content is averaged in the mixer. After the reaction and finishing steps, two grades of product can be marketed, "colossal" and "stupendous". Find the maximum profit and the corresponding optimum policy for the process operations. It is not necessary for the old and new heating - purifying train to use the same row material. Solve the problems as a network problem.
Solution:
The network diagram of the solution is shown in Soln 7-2. Reading from the diagram, the optimal policy is:
Feed B to old heater. Feed A to new heater.
Old heater on high. New heater on low.
Old purifier with 0.3%.
New purifier with 0.1%.
Average impurities = 0.2%
Reactor temperature - medium
Product - colossal
7-3. In Figure 7-29, a partially completed functional diagram of a process is given that involves a diverging branch and a feed-back loop. Complete the function diagram by labelling it with the approapriate subscripts on the state and decision variables. Then give the dynamic programming algorithm, transition functions and incident identities for each stage. Also give the type of partial optimization of each stage, and describe how the feed - back loop and diverging branch are evaluated and included in the main branch.
Solution:
The completed functional diagram is shown in Soln 7-3a, and functional equations including the dynamic programming algorithms, transition functions and incident identities for each stage are given in Soln 7-3a and Soln 7-3b.
7-4. In Figure 7-30, a partially completed functional diagram of a process is given that involves a converging branch and a feed forward loop. Complete the functional diagram by labeling it with the appropriate subscripts on the state and decision variables. Then give the dynamic programming algorithm, transition functions and incident identifiers for each stage. Also, give the type of partial optimization at each stage, and describe how the feed - forward loop and converging branch are evaluated and included in the main branch.
Solution:
The completed functional diagram is shown in Soln 7-4a, and the functional equations including the dynamic programming algorithms, transition functions and incident identities for each stage are given in Soln 7-4b.
7-5. The flow diagram shown in Figure 7-31 is a simplified version of a catalytic cracking unit and associated separation facilities. Develop the dynamic programming functional equations and diagram for this process flow diagram. Define each state and decision variable, transition function and each incident identity. Describe how to calculate the return at each stage. Simplify the functional diagram where required by applying Wilde's rules, and indicate the steps to obtain the optional return and policy.
Solution:
The functional diagram is shown in Soln 7-5 for the process where every unit is a dynamic programming stage. The definition for the state and decision variables are as follows.
Stage 1:
s1 - C1, C2, C3 flow rate from proplyene splitter to fuel gas
s1 - C1, C2, C3, C3 = flow rate to splitter =
2
d1 - What is the optimal separation for propylene splitter?
R1 - Net profit from fuel gas and propylene sale
Stage 2:
s2 - C1, ...,iC4 flow rate to isobutane splitter =
d2 - What is the optimal separation for the isobutane splitter?
R2 - Net profit from iC4 sale
Stage 3:
s3 - C1, ...,C5 flow rate to pentane splitter =
d3 - What is the optimal separation for the pentane splitter?
R3 - Net profit from pentane sale
Stage 3':
s3, - Catalytic cycle oil flow rate to CCU splitter =
d3, - What is the optimal separation for the CCU splitter?
R3, - Net profit from light catalytic cycle oil sale
Stage 4:
s4 - Oil flow rate to CCU splitter =
d4 - What is the optimal separation for the CCU splitter?
R4 - Operating cost of the column
Stage 5:
s5 - Feed plus recycle rate to the catalytic cracking unit(CCU)
d5 - What is the optimal flow rate to the CCU? The feed rate is fixed but the recycle rate is a variable.
R5 - Operating cost of the CCU
To optimize the process, one would consider cutting the state of the recycle stream to the CCU (s3'). However, Wilde's rule 5 says the small loop should be optimized simultaneously. The functional diagram becomes:
The functional equations are:
The optimal solution is obtained by performing three one-state, one-decision partial optimizations and a no-state, three decision partial optimization.