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What is the primary characteristic of dynamic programming (DP) that distinguishes it from linear programming (LP)?
What is the primary characteristic of dynamic programming (DP) that distinguishes it from linear programming (LP)?
- DP relies on a fixed mathematical formalism, unlike LP.
- DP requires adaptation to each specific problem, lacking a universal mathematical formalism, unlike LP. (correct)
- DP is always solved using a graphical method, whereas LP is not.
- DP can only handle problems with a single variable, while LP can handle multiple variables.
How does dynamic programming (DP) reduce computational effort, especially when compared to a 'divide and conquer' approach?
How does dynamic programming (DP) reduce computational effort, especially when compared to a 'divide and conquer' approach?
- DP solves subproblems in a 'bottom-up' fashion, preserving and reusing solutions to encountered subproblems. (correct)
- DP increases calculation effort by recomputing solutions for each subproblem.
- DP avoids solving subproblems by estimating the global outcome directly.
- DP independently solves all subproblems and then combines the results.
Which of the following is a core component in the foundation of dynamic programming?
Which of the following is a core component in the foundation of dynamic programming?
- The independence of policy decisions at each stage, regardless of previous stages. (correct)
- The use of a single, overarching decision that applies to all stages.
- The absence of states within each stage, simplifying the decision-making process.
- The problem's inability to be divided into sub-problems, requiring a holistic solution.
In the context of dynamic programming, what does the 'backward method' involve?
In the context of dynamic programming, what does the 'backward method' involve?
In a dynamic programming problem, what does the notation $X_i$ typically represent?
In a dynamic programming problem, what does the notation $X_i$ typically represent?
What does $F_i(S_i, X_i)$ represent in the context of the backward approach in dynamic programming?
What does $F_i(S_i, X_i)$ represent in the context of the backward approach in dynamic programming?
In a shortest path problem solved with dynamic programming, if $R_i(S_i, X_i)$ represents the distance between cities $S_i$ and $X_i$, what does $F_i(S_i, X_i)$ represent?
In a shortest path problem solved with dynamic programming, if $R_i(S_i, X_i)$ represents the distance between cities $S_i$ and $X_i$, what does $F_i(S_i, X_i)$ represent?
In dynamic programming for optimization problems, what is the significance of calculating $F_i^*(S_i)$?
In dynamic programming for optimization problems, what is the significance of calculating $F_i^*(S_i)$?
What is the difference in the definition of $S_i$ between the forward and backward approaches in dynamic programming?
What is the difference in the definition of $S_i$ between the forward and backward approaches in dynamic programming?
When should you use the forward method in dynamic programming instead of the backward method?
When should you use the forward method in dynamic programming instead of the backward method?
Which of the following statements best describes a characteristic of Dynamic Programming?
Which of the following statements best describes a characteristic of Dynamic Programming?
How does the recursive formula in dynamic programming account for past and future decisions?
How does the recursive formula in dynamic programming account for past and future decisions?
In the context of dynamic programming, what does the Bellman Principle (or principle of optimality) state?
In the context of dynamic programming, what does the Bellman Principle (or principle of optimality) state?
An industrial firm uses dynamic programming to optimally allocate a budget of 60,000 TD between three plants. What constraint would NOT typically apply in this scenario?
An industrial firm uses dynamic programming to optimally allocate a budget of 60,000 TD between three plants. What constraint would NOT typically apply in this scenario?
In a dynamic programming problem for budget allocation, what does $R_i(S_i, X_i)$ represent?
In a dynamic programming problem for budget allocation, what does $R_i(S_i, X_i)$ represent?
In the context of dynamic programming for budget allocation, what does the expression $F_i^*(S_i) = Max_{Xi} F_i(S_i, X_i)$ signify?
In the context of dynamic programming for budget allocation, what does the expression $F_i^*(S_i) = Max_{Xi} F_i(S_i, X_i)$ signify?
In a production planning problem using dynamic programming, what does $X_i$ represent?
In a production planning problem using dynamic programming, what does $X_i$ represent?
In a dynamic programming model for production planning, the cost at month i, $R_i(S_i, X_i)$, is defined as Production cost + holding cost. If $X_i$ are the number of units produced, which of the following represents the holding cost?
In a dynamic programming model for production planning, the cost at month i, $R_i(S_i, X_i)$, is defined as Production cost + holding cost. If $X_i$ are the number of units produced, which of the following represents the holding cost?
Given that $R_i(S_i, X_i)$ represents the cost at month i, and $F_i(S_i, X_i)$ represents the total minimum cost, what would $F_1(S_1, X_1)$ signify in a 4-month production planning dynamic programming problem?
Given that $R_i(S_i, X_i)$ represents the cost at month i, and $F_i(S_i, X_i)$ represents the total minimum cost, what would $F_1(S_1, X_1)$ signify in a 4-month production planning dynamic programming problem?
In dynamic programming, what does it mean for a problem to be decomposed into 'stages'?
In dynamic programming, what does it mean for a problem to be decomposed into 'stages'?
What best describes the term 'state' in the context of dynamic programming (DP)?
What best describes the term 'state' in the context of dynamic programming (DP)?
How does Dynamic Programming approach interdependent decisions?
How does Dynamic Programming approach interdependent decisions?
What is the relationship between sub-problems in Dynamic Programming?
What is the relationship between sub-problems in Dynamic Programming?
What feature of Dynamic Programming improves the calculation effort?
What feature of Dynamic Programming improves the calculation effort?
In the budget allocation Dynamic Programming problem, what does Xi represent?
In the budget allocation Dynamic Programming problem, what does Xi represent?
What does Si represent in the budget allocation Dynamic Programming problem?
What does Si represent in the budget allocation Dynamic Programming problem?
Given the recursive relationship used in DP: $F_i(S_i,X_i) = R_i(S_i,X_i) + F_{i+1}^(S_i + X_i – D_i)$, what is the purpose of $F_{i+1}^(S_i + X_i – D_i)$?
Given the recursive relationship used in DP: $F_i(S_i,X_i) = R_i(S_i,X_i) + F_{i+1}^(S_i + X_i – D_i)$, what is the purpose of $F_{i+1}^(S_i + X_i – D_i)$?
What factors are considered in the definition of $R_i(S_i,X_i)$ given the production Dynamic Programming Problem?
What factors are considered in the definition of $R_i(S_i,X_i)$ given the production Dynamic Programming Problem?
Consider a multi-stage dynamic programming model of inventory management. At stage i, a decision $X_i$ which is quantity to produce at month i, is made. Suppose the storage constraint is 2 units. Which one of the following is NOT considered a valid stock level?
Consider a multi-stage dynamic programming model of inventory management. At stage i, a decision $X_i$ which is quantity to produce at month i, is made. Suppose the storage constraint is 2 units. Which one of the following is NOT considered a valid stock level?
Let $g(R_i(S_i,X_i), F_{i+1}^*(S_{i+1}))$ be the relationship between the cost and the subsequent reward as defined by the Dynamic Programming relationship. What types of relationships can function g represent? (Select all that apply.)
Let $g(R_i(S_i,X_i), F_{i+1}^*(S_{i+1}))$ be the relationship between the cost and the subsequent reward as defined by the Dynamic Programming relationship. What types of relationships can function g represent? (Select all that apply.)
In a shortest path problem, the goal is to find the path with the LOWEST total cost. With that in mind, what is the purpose of $Min_{x_i} F_i(S_i, X_i)$?
In a shortest path problem, the goal is to find the path with the LOWEST total cost. With that in mind, what is the purpose of $Min_{x_i} F_i(S_i, X_i)$?
Which of the following is not a key Dynamic Programming step?
Which of the following is not a key Dynamic Programming step?
Flashcards
Dynamic programming (DP)
Dynamic programming (DP)
A recursive optimization approach for interdependent, sequential decisions.
Backward method (DP)
Backward method (DP)
Progresses from the last step back to the first.
Forward method (DP)
Forward method (DP)
Begins at the first stage and moves sequentially to the last.
Stages (DP)
Stages (DP)
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States (DP)
States (DP)
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Policy decision (DP)
Policy decision (DP)
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Principle of Optimality
Principle of Optimality
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Recursive Formula
Recursive Formula
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Decomposition
Decomposition
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Candidate states
Candidate states
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Study Notes
Introduction to Dynamic Programming (DP)
- DP is a recursive optimization approach used for interdependent and sequential decisions
- Recursive optimization optimizes over several steps
- Linear Programming (LP) has no mathematical formalism that can solve DP
- DP adapts its solution method to each unique problem
DP vs Divide & Conquer
- DP combines solved sub-problems to approach global problems, similar to "divide & conquer"
- DP subproblems are not independent, unlike in "divide & conquer"
- DP reduces calculation effort by:
- Solving subproblems in a "bottom-up" fashion
- Preserving solutions to encountered subproblems in memory
- Utilizing obtained solutions only when the related subproblems are involved
Foundations of DP
- DP problems are characterized by basic features:
- Problems are divided into sub-problems, also called stages
- Each stage has a number of states
- A policy decision is identified recursively at each stage
- Policy decisions at each stage transform the current state to a state associated with the next stage
- Given the current state, the optimal policy for the remaining stages is independent of policy decisions from previous stages
- Two calculation approaches include:
- Backward method: Start from the last step and go back to the first step
- Forward method: Start from the first stage and go to the last stage
- An n-step sequential decision problem is decomposed into n steps
- Each step feeds the next so that the output of one step serves as the input to the next step
Backward Approach
- Xᵢ: The decision at step i
- Sᵢ: The input or state of the system at step i
- Rᵢ(Sᵢ, Xᵢ) : The immediate result of decision Xᵢ given that the state is Sᵢ
- Fᵢ(Sᵢ, Xᵢ): The cumulative results from i to n given that at step i decision Xᵢ is chosen, and the state is Sᵢ
- Calculations are made from step n to 1
Shortest Path Problem Example with Backward Approach
- Aims to find the shortest path from city A to city H, given several possible routes
- Xᵢ: Destination city at step i (i=1, ..., 3)
- Sᵢ: Departure city in step i, (i=1, ..., 3)
- Rᵢ(Sᵢ, Xᵢ): The distance between cities Sᵢ and Xᵢ, (i=1, ..., 3)
- Fᵢ(Sᵢ,Xᵢ): Distance traveled from city Sᵢ to city H knowing that Xᵢ is the destination at step i
- Xᵢ*: Destination that minimizes the distance Fᵢ(Sᵢ,Xᵢ)
- Fᵢ*(Sᵢ) = Min ₓᵢ Fᵢ(Sᵢ, Xᵢ) = Fᵢ(Sᵢ,Xᵢ*) = Fᵢ *(Sᵢ)
- The shortest path linking A to H is A-B-F-H with a length of 8
- The recursive function is written as follows:
- Fᵢ(Sᵢ,Xᵢ) = Rᵢ(Sᵢ,Xᵢ) + Fᵢ₊₁*(Xᵢ) (i = 1,2)
- F₃(S₃,X₃) = R₃(S₃,X₃)
Forward Approach
- Xᵢ : decision at step i
- Sᵢ : state of the system after step i
- Rᵢ(Sᵢ,Xᵢ) : immediate result of step i based on decision Xᵢ, given the state is Sᵢ
- Fᵢ(Sᵢ,Xᵢ) : cumulative result of steps 1 to i, given that at step i decision Xᵢ, is made, and the state is Sᵢ
- Unlike the backward approach, the forward method starts at step 1 and continues step n.
Shortest Path Problem Example with Forward Approach
- Xᵢ: Departure city at step i (i = 1, ..., 3)
- Sᵢ: Destination city at step i (i = 1,.., 3)
- Rᵢ(Sᵢ,Xᵢ) : distance between the cities Xᵢ and Sᵢ
- Fᵢ(Sᵢ,Xᵢ) the distance traveled from city A to city Sᵢ knowing that Xᵢ is the departure city at step i
- Xᵢ*is the departure city which minimizes the distance Fᵢ(Sᵢ,Xᵢ)
- Fᵢ*(Sᵢ) = Minₓᵢ Fᵢ(Sᵢ,Xᵢ) = Fᵢ(Sᵢ,Xᵢ*)
- The shortest path linking A to H is A-B-F-H with a length of 8
- The recursive function is written as follows:
- F₁(S₁,X₁) = R₁(S₁,X₁) + Fᵢ₋₁*(X₁) (i = 1,2)
- F₁(S₁,X₁) = R₁(S₁,X₁)
Choosing a Method
- The method depends on the availability of information on the initial or final state
- Use the backward method if you know the initial state, but not the final state
- Use the forward method if you know the final state, but not the initial state
- Both methods apply, if both states are known
Dynamic Programming (DP) Characteristics
- The problem is decomposed into a number of steps
- At each step, multiple candidate states can apply
- To each step and corresponding state, identify possible decisions to be made
- Use a recursive formula to maintain at least one decision for each state
- The recursive formula expresses the immediate consequence of the decision Rᵢ(Sᵢ,Xᵢ) combined with the best cumulative result over the various steps
- Accounted for from last to current in a forward approach, F*ᵢ₋₁(Sᵢ₋₁)
- Accounted for from first to current in a backward approach, F*ᵢ₊₁(Sᵢ₊₁)
- The general form of the recursive formula: F(Sᵢ,Xᵢ) = f(direct result, optimal cumulative result over the previous steps)
- More precisely:
- Fᵢ(Sᵢ,Xᵢ) = g(Rᵢ(Sᵢ,Xᵢ),Fᵢ₋₁*(Sᵢ₋₁)) (forward)
- Fᵢ(Sᵢ,Xᵢ) = g(Rᵢ(Sᵢ,Xᵢ),Fᵢ₊₁*(Sᵢ₊₁)) (backward)
- Function g could be additive, multiplicative, or other
- State vectors Sᵢ₋₁ and Sᵢ₊₁ are expressed in terms of Sᵢ and Xᵢ
- DP relies on the principle of optimality or Bellman Principle. That is, any sub-policy of an optimal policy is also optimal
Budget Allocation Example
- An industrial firm with a 60,000 TD budget must allocate it among its Tunis, Sousse, and Sfax plants
- Each plant can’t receive over 40,000 TD, and all amounts must be multiples of 10,000 TD
- Plants of Tunis & Sfax each receive a minimum of 10,000 dinars for the optimal budget allocation
- Define the following:
- Xᵢ: Budget allocated to plant i; i=1 for Tunis, 2 for Sousse, and 3 for Sfax
- Sᵢ: Available budget before making a decision about plants i, i+1,..., 3, i = 1, 2, 3
- Rᵢ(Sᵢ, Xᵢ): Obtained revenue of plant i resulting from a budget allocation of Xᵢ and an available budget of Sᵢ for plants i, i+1,...3, where i = 1, 2, 3.
- Fᵢ(Sᵢ,Xᵢ) is the maximum cumulative revenue of plants i,..., 3, for a total budget of Sᵢ for these plants, and the allocated budget for plant i is Xᵢ, i = 1, 2, 3
- Xᵢ* optimal budget to allocate to plant i, maximizes revenue Fᵢ(Sᵢ,Xᵢ)
- Fᵢ*(Sᵢ) = Max ₓᵢ Fᵢ(Sᵢ,Xᵢ) = Fᵢ(Sᵢ,Xᵢ*)
- The optimal allocation is 30,000 TD for Tunis, 10,000 TD for Sousse, and 20,000 TD for Sfax. The total revenue is 210,000 TD
- The recursive function is written as follows:
- F₁(S₁,X₁) = R₁(S₁,X₁) + Fᵢ₊₁*(Sᵢ - Xᵢ) (i = 1,2)
- F₃(S₃,X₃) = R₃(S₃,X₃)
Production Planning Problem Example
- Determine the production levels of a product over the course of 4 months
- A production run involves a fixed cost of 3 DT and a variable cost of 1 DT per unit
- Excess stock at the end of each month involves a holding cost of 0.5 DT per unit
- Production capacity is 4 units each month, while the storage capacity is 2 units
- Demand for the next 4 months is 1, 3, 2, and 4, respectively
- Determine the optimal production plan, given that the initial stock is empty
- n = 4
- Use the backward method because initial stock is known
- Define the following:
- Xᵢ: Quantity to produce at month i, (i = 1, ..., 4)
- Sᵢ: Stock level at the start of month i (i = 1,..., 4)
- Rᵢ (Sᵢ, Xᵢ) = Cost at month i (i = 1,..., 4) which also equals production cost + holding cost
- Production cost of Xᵢ units + holding cost of (Sᵢ + Xᵢ - Dᵢ) units
- Where Dᵢ is the demand of month i (i =1,..., 4)
- Fᵢ (Sᵢ,Xᵢ) = total minimum cost for months i, i+1,..., 4, given that at the start of month i the stock level is Sᵢ and Xᵢ units are to be produced (i =1,...,4)
- Fᵢ*(Sᵢ) = Min ₓᵢ Fᵢ (Sᵢ,Xᵢ) (i =1,..., 4)
- F₁(S₁, X) = g(R₁(S₁,X₁), *Fᵢ₊₁(Sᵢ₊₁))
- Equals g(Rᵢ(Sᵢ,Xᵢ), F*ᵢ₊₁(Sᵢ + Xᵢ – Dᵢ))
- Equals, R(S,X) + F*(S+X-D) , (i =1,..,3)
- F₄(S₄,X₄) = R₄(S₄,X₄)
- The optimal production plan is, X₁*= 2, X₂*= 4, X₃*= 0, X₄*= 4 with a minimum cost of = 20.5 dinars
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