Edexcel D2 2006 June — Question 5

Exam BoardEdexcel
ModuleD2 (Decision Mathematics 2)
Year2006
SessionJune
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDynamic Programming
TypeDynamic programming shortest/longest path
DifficultyModerate -0.5 This is a standard dynamic programming/network optimization problem typical of D2 module. It requires systematic application of a forward or backward pass algorithm through the network to find the maximum profit path. While it has 12 marks indicating multiple steps, the technique is routine and well-practiced - students simply need to apply the algorithm methodically without requiring novel insight or complex problem-solving beyond the standard procedure.
Spec7.05a Critical path analysis: activity on arc networks

Victor owns some kiosks selling ice cream, hot dogs and soft drinks. The network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. The negative numbers indicate losses due to the purchases of new kiosks. \includegraphics{figure_5} Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit. (Total 12 marks)

AnswerMarks Guidance
StageState Action
1H HT
IIT \(3^*\)
JJT \(12^*\)
KKT \(20^*\)
2D DH
DI\(4 + 3 = 7^*\)
EEH \(3 + 4 = 7^*\)
EI\(4 + 3 = 7^*\)
FFJ \(10 + 12 = 22^*\)
FK\(-8 + 20 = 12\)
GGJ \(10 + 12 = 22\)
GK\(17 + 20 = 37^*\)
3A AD
AE\(2 + 7 = 9\)
AF\(-5 + 22 = 17^*\)
BBD \(3 + 7 = 10\)
BE\(2 + 7 = 9\)
BF\(-6 + 22 = 16^*\)
CCF \(8 + 22 = 30^*\)
CG\(-15 + 37 = 22\)
4S SA
SB\(3 + 16 = 19\)
SC\(-10 + 30 = 20^*\)
Answer: Route S C F J T £20 000M1 A1 2
| Stage | State | Action | Value | | |
|---|---|---|---|---|---|
| 1 | H | HT | $4^*$ | M1 A1 | 2 |
| | I | IT | $3^*$ | | |
| | J | JT | $12^*$ | | |
| | K | KT | $20^*$ | | |
| 2 | D | DH | $2 + 4 = 6$ | M1 A1 | |
| | | DI | $4 + 3 = 7^*$ | | |
| | E | EH | $3 + 4 = 7^*$ | | |
| | | EI | $4 + 3 = 7^*$ | | |
| | F | FJ | $10 + 12 = 22^*$ | | |
| | | FK | $-8 + 20 = 12$ | | |
| | G | GJ | $10 + 12 = 22$ | | |
| | | GK | $17 + 20 = 37^*$ | | A1 | 3
| 3 | A | AD | $3 + 7 = 10$ | M1 A1ft | |
| | | AE | $2 + 7 = 9$ | | |
| | | AF | $-5 + 22 = 17^*$ | | |
| | B | BD | $3 + 7 = 10$ | | |
| | | BE | $2 + 7 = 9$ | | |
| | | BF | $-6 + 22 = 16^*$ | | |
| | C | CF | $8 + 22 = 30^*$ | A1 ft | 3 |
| | | CG | $-15 + 37 = 22$ | | |
| 4 | S | SA | $2 + 17 = 19$ | M1 A1ft | 2 |
| | | SB | $3 + 16 = 19$ | | |
| | | SC | $-10 + 30 = 20^*$ | | |

**Answer:** Route S C F J T £20 000 | M1 A1 | 2

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Victor owns some kiosks selling ice cream, hot dogs and soft drinks.

The network below shows the choices of action and the profits, in thousands of pounds, they generate over the next four years. The negative numbers indicate losses due to the purchases of new kiosks.

\includegraphics{figure_5}

Use a suitable algorithm to determine the sequence of actions so that the profit over the four years is maximised and state this maximum profit.
(Total 12 marks)

\hfill \mbox{\textit{Edexcel D2 2006 Q5}}