Hard Further Maths Dynamic programming Questions

Identify the stages, states and arcs of the network

Dynamic programming labels each state by the stage it belongs to; a route uses exactly one arc per stage from the source to the sink.

Give the sink its initial value

The route ends at $J$, so from $J$ there is no further value to add.

Stage 3: optimise state $H$

The optimal value at state $H$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 3: optimise state $I$

The optimal value at state $I$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $E$

The optimal value at state $E$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $F$

The optimal value at state $F$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $G$

The optimal value at state $G$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $B$

The optimal value at state $B$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $C$

The optimal value at state $C$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $D$

The optimal value at state $D$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 0: optimise state $A$

The optimal value at state $A$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Trace the optimal route forwards from the source

Following the optimal choice recorded at each state gives the route, whose arc values add up to the optimal total.

State Bellman's principle of optimality

Any part of an optimal route is itself optimal for the sub-problem that starts where that part begins, so each state need only be optimised once.

Record the optimal value at every state

Each label is the best value obtainable on a route from that state to the sink.

Select the route of greatest total value

This route attains the maximum total value of $152$ and no other route matches it.

Identify the stages, states and arcs of the network

Dynamic programming labels each state by the stage it belongs to; a route uses exactly one arc per stage from the source to the sink.

Give the sink its initial value

The route ends at $J$, so from $J$ there is no further value to add.

Stage 3: optimise state $H$

The optimal value at state $H$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 3: optimise state $I$

The optimal value at state $I$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $E$

The optimal value at state $E$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $F$

The optimal value at state $F$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $G$

The optimal value at state $G$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $B$

The optimal value at state $B$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $C$

The optimal value at state $C$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $D$

The optimal value at state $D$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 0: optimise state $A$

The optimal value at state $A$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Trace the optimal route forwards from the source

Following the optimal choice recorded at each state gives the route, whose arc values add up to the optimal total.

State Bellman's principle of optimality

Any part of an optimal route is itself optimal for the sub-problem that starts where that part begins, so each state need only be optimised once.

Record the optimal value at every state

Each label is the best value obtainable on a route from that state to the sink.

Select the route of greatest total value

This route attains the maximum total value of $124$ and no other route matches it.

Identify the stages, states and arcs of the network

Dynamic programming labels each state by the stage it belongs to; a route uses exactly one arc per stage from the source to the sink.

Give the sink its initial value

The route ends at $J$, so from $J$ there is no further value to add.

Stage 3: optimise state $H$

The optimal value at state $H$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 3: optimise state $I$

The optimal value at state $I$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $E$

The optimal value at state $E$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $F$

The optimal value at state $F$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 2: optimise state $G$

The optimal value at state $G$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $B$

The optimal value at state $B$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $C$

The optimal value at state $C$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 1: optimise state $D$

The optimal value at state $D$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Stage 0: optimise state $A$

The optimal value at state $A$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the largest total.

Trace the optimal route forwards from the source

Following the optimal choice recorded at each state gives the route, whose arc values add up to the optimal total.

State Bellman's principle of optimality

Any part of an optimal route is itself optimal for the sub-problem that starts where that part begins, so each state need only be optimised once.

Record the optimal value at every state

Each label is the best value obtainable on a route from that state to the sink.

Select the route of greatest total value

This route attains the maximum total value of $135$ and no other route matches it.

Identify the stages, states and arcs of the network

Dynamic programming labels each state by the stage it belongs to; a route uses exactly one arc per stage from the source to the sink.

Give the sink its initial value

The route ends at $J$, so from $J$ there is no further value to add.

Stage 3: optimise state $H$

The optimal value at state $H$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 3: optimise state $I$

The optimal value at state $I$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $E$

The optimal value at state $E$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $F$

The optimal value at state $F$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $G$

The optimal value at state $G$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $B$

The optimal value at state $B$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $C$

The optimal value at state $C$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $D$

The optimal value at state $D$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 0: optimise state $A$

The optimal value at state $A$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Trace the optimal route forwards from the source

Following the optimal choice recorded at each state gives the route, whose arc values add up to the optimal total.

State Bellman's principle of optimality

Any part of an optimal route is itself optimal for the sub-problem that starts where that part begins, so each state need only be optimised once.

Record the optimal value at every state

Each label is the best value obtainable on a route from that state to the sink.

Select the route of least total cost

This route attains the minimum total cost of $68$ and no other route matches it.

Identify the stages, states and arcs of the network

Dynamic programming labels each state by the stage it belongs to; a route uses exactly one arc per stage from the source to the sink.

Give the sink its initial value

The route ends at $J$, so from $J$ there is no further value to add.

Stage 3: optimise state $H$

The optimal value at state $H$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 3: optimise state $I$

The optimal value at state $I$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $E$

The optimal value at state $E$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $F$

The optimal value at state $F$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 2: optimise state $G$

The optimal value at state $G$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $B$

The optimal value at state $B$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $C$

The optimal value at state $C$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 1: optimise state $D$

The optimal value at state $D$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Stage 0: optimise state $A$

The optimal value at state $A$ combines each outgoing arc with the value already fixed at the state it leads to, and keeps the smallest total.

Trace the optimal route forwards from the source

Following the optimal choice recorded at each state gives the route, whose arc values add up to the optimal total.

State Bellman's principle of optimality

Any part of an optimal route is itself optimal for the sub-problem that starts where that part begins, so each state need only be optimised once.

Record the optimal value at every state

Each label is the best value obtainable on a route from that state to the sink.

Select the route of least total cost

This route attains the minimum total cost of $60$ and no other route matches it.