Deck 21: Dynamic Programming

Question 1

In a knapsack problem,if one adds another item,one must completely resolve the problem in order to find a new optimal solution.

Answer 1

False

Question 2

In a production and inventory control problem,the states can correspond to the amount of inventory on hand at the beginning of each period.

Answer 2

True

Question 3

The stage transformation function

A)transforms the input into the output.
B)transforms a stage into a state.
C)is a different function for each stage.
D)None of these are correct.

Answer 3

A

Question 4

The output of stage k is the input for stage k−1.

Question 5

Finding the optimal solution to each stage of a dynamic programming problem will always lead to an optimal solution to the total problem.

Question 6

In order to use dynamic programming,one must be able to view the problem as a multistage decision problem.

Question 7

In solving a shortest-route problem using dynamic programming,the stages represent how many arcs you are from the terminal node.

Question 8

The stage transformation function identifies which state one reaches at the next stage for a given decision.

Question 9

As opposed to a specific technique such as linear programming,dynamic programming is considered a general approach.

Question 10

State variables are a function of a state variable and a decision.

Question 11

Dynamic programming must only involve a finite number of decision alternatives and a finite number of stages.

Question 12

The return function for a shortest-route problem refers to two directional arcs between nodes.

Question 13

Dynamic programming is a general approach used when it is possible to break a large problem into interrelated smaller problems,with stage decisions proceeding recursively,solving one of the smaller problems at each stage.

Question 14

The solution of stage k of a dynamic programming problem is dependent upon the solution of stage k−1.

Question 15

The subscripts used in dynamic programming notation refer to states.

Question 16

Dynamic programming requires that its subproblems be independent of one another.

Question 17

Dynamic programming,when used for the shortest-route problem,requires complete enumeration of paths from the beginning to ending node.

Question 18

An input state variable and an output state variable together define the condition of the process at the beginning and end of a stage.

Question 19

Stages of a dynamic programming solution procedure

A)represent parts of a large mathematical model.
B)often represent a sequence of decisions made over time.
C)are usually not independent of each other.
D)All of these are correct.

Question 20

A knapsack problem is so named because the objective is to find the number of N items,each of which has a different weight and value,that can be placed in a knapsack with limited weight capacity so as to maximize the total value of the items placed in the knapsack.

Question 21

Mission Bay Development Corporation is developing an exclusive 25-acre parcel of property.It has received bids from five builders to purchase construction lots.Because of the different nature of the builders,they desire different sized lots.The data are as follows:
​ ​
How should Mission Bay sell the land to maximize total sales revenue?

Question 22

Consider the following integer linear program: ​
a.Set up the network that represents the dynamic programming formulation.
b.Solve the problem using dynamic programming.

Question 23

Find the shortest path through the following network using dynamic programming.​

Question 24

Unidyne Corporation is currently planning the production of red dye number 56 for the next four months.Production and handling costs,as well as production and storage capacity,vary from month to month.These data are given in the table below.Production and holding costs are in $1000s per batch,and production levels and storage capacities are in batches.Holding costs are based on inventory on hand at the end of the month.The number of orders for batches the sales department has received over the four-month period is also given.​ ​
Unidyne does not wish to have any inventory of the dye at the end of May.Its current inventory is two batches.Determine a production schedule for the next four months.​

Question 25

Marvelous Marvin is planning his annual "Almost Everything Must Go" inventory clearance sale.Marvin has decided to allocate 18 shelf feet to the cooking section.He is considering offering up to five items for sale in this category.​ ​
If Marvin wants at least one item A and one item B on sale,what stock should he have on sale,and what is the total expected profit?

Question 26

If x₃ = t₄(x₄,d₄)= x₄ − 2d₄ and r₄(x₄,d₄)= 16d₄,the stage transformation function is

A)t.
B)x.
C)r.
D)
D)

Question 27

The function that "transforms" the input of the stage into the output of the stage is referred to as a stage transformation function and

A)is always linear.
B)calculates the return of the transformation.
C)determines the output of the stage.
D)All of these are correct.

Question 28

The knapsack problem is to determine how many units of each item to place in the knapsack to

A)minimize total value.
B)maximize total value.
C)minimize the number of items in the knapsack.
D)maximize the number of items in the knapsack.

Question 29

A driver wants to make a trip from city 1 to city 7.The road mileage between cities is given below.Find the shortest route.

Question 30

A return function is a value such as profit or loss associated with making decision d_n at

A)stage n for a specific value of output variable x_n.
B)stage n for a specific value of input variable x_n.
C)stage n for a specific value of stage m.
D)input n for a specific value of output variable x_n.

Question 31

A cargo company has a set of delivery patterns for its goods from its location at city 1 to a series of cities 2,3,4,5,and 6.The delivery times between cities are given in hours below.Find the shortest route.

Question 32

State variables in a shortest-route problem represent

A)decisions.
B)locations in the network.
C)the minimum distance between nodes.
D)None of these are correct.

Question 33

If x₃ = t₄(x₄,d₄)= x₄ − 2d₄ and r₄(x₄,d₄)= 16d₄,the subscripts refer to

A)state.
B)stage.
C)transformation.
D)return.

Question 34

Audio Disks will be opening outlets in the greater Phoenix area.The estimated sales at each store are dependent not only on the store location,but also on the number of sales personnel,as presented in the table below ($1000s/year).Each store requires at least two salespeople,and a pool of nine salespeople is available.​
a.What would the states be in the dynamic programming formulation?
b.Draw the network that represents the dynamic programming formulation.
c.​
Given the above network,solve the sales personnel allocation problem by finding the longest path.

Question 35

Franklin Plate Company manufactures commemorative plates for holidays.The company is currently planning its production schedule for this year's Thanksgiving plate.Discussions with the sales manager have indicated that sales of the plate will run from August through November.The production manager has determined the manufacturing cost per plate for each of these months as well as the sales demand (in 1000s)and manufacturing capacity (in 1000s).The data are as follows:
​ ​
The maximum inventory level possible at the end of any month is 5000 plates.The storage cost per plate for each plate in inventory at the end of the month is $0.50.​
Franklin has no inventory on hand at the beginning of August and wishes to have no inventory left after November.Use dynamic programming to determine an optimal production schedule.

Question 36

If x₃ = t₄(x₄,d₄)= x₄ − 2d₄ and r₄(x₄,d₄)= 16d₄,the state variable is

A)t.
B)x.
C)r.
D)
D)

Question 37

The owner of a small construction firm is excavating at three sites.He wishes to assign his five additional trucks in such a way as to minimize his total costs.Each site can use zero to three additional trucks; no site can use more than three trucks efficiently.The following site total costs are known. ​
a.​
Use dynamic programming to find the assignment of the additional trucks that minimizes total cost.
b.​
If the owner had only four trucks to assign,what would be the optimal assignment and total cost?

Question 38

Walt's Custom Boats manufactures luxury yachts.Walt is phasing out production of his 42-foot Sportfisher that he plans to replace with a 44-foot Sportfisher.Walt currently has orders for the next four months for the 42-foot model after which he will cease production.The following data (with costs in $1000s)are for the next four months:
​ ​
If Walt's current inventory of 42-foot Sportfishers is one boat,how should Walt schedule production of the 42-foot Sportfisher over the next four months to minimize the total production and inventory holding costs?
​

Question 39

A number of types of items are to be shipped as cargo.The total available weight in the truck is 10 tons.Determine the number of units of items to be shipped to maximize profit.

Question 40

Solutions in dynamic programming

A)are not optimal.
B)are unique.
C)represent each stage.
D)All of these are correct.

Question 41

In the following shortest-route problem,travel is only permitted from left to right.Use dynamic programming to determine the shortest path from node 1 to:
​
a.node 15.
b.node 14.
​