Exam 21: Dynamic Programming

The subscripts used in dynamic programming notation refer to states.

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

Stages of a dynamic programming solution procedure

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

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.

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

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

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

State variables in a shortest-route problem represent

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.​

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? ​

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.

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. ​

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

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?

The stage transformation function

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

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

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

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.