| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Year | 2019 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Dynamic Programming |
| Type | Dynamic programming order sequencing |
| Difficulty | Moderate -0.5 This is a standard D2 dynamic programming question requiring systematic application of the minimax algorithm to find optimal construction order. While it involves multiple stages and careful bookkeeping, it follows a well-practiced algorithmic procedure with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 7.06a LP formulation: variables, constraints, objective function |
| \cline { 2 - 7 } \multicolumn{1}{c|}{} | A | B | C | D | E | F |
| A | - | 53 | 47 | 39 | 35 | 40 |
| B | 53 | - | 32 | 46 | 41 | 43 |
| C | 47 | 32 | - | 51 | 47 | 37 |
| D | 39 | 46 | 51 | - | 36 | 49 |
| E | 35 | 41 | 47 | 36 | - | 42 |
| F | 40 | 43 | 37 | 49 | 42 | - |
| 1 | 2 | 3 | 4 | Supply | |
| A | 17 | 20 | 23 | 14 | 25 |
| B | 16 | 15 | 19 | 22 | 29 |
| C | 19 | 14 | 11 | 15 | 32 |
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| 1 | 2 | 3 | 4 | Supply | |
| A | 25 | ||||
| B | 29 | ||||
| C | 32 | ||||
| Demand | 28 | 17 | 23 | 18 |
| A | B | C | D | E | |
| Frank | 5 | 0 | 7 | 3 | 4 |
| Gill | 5 | 3 | 8 | 10 | 1 |
| Harry | 4 | 3 | 7 | 9 | 0 |
| Imogen | 6 | 3 | 6 | 5 | 4 |
| Jiao | 0 | 2 | 7 | 3 | 2 |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| A | B | C | D | E | |
| F | |||||
| G | |||||
| H | |||||
| I | |||||
| J |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | |
| \(F\) | |||||
| \(G\) | |||||
| \(H\) | |||||
| \(I\) | |||||
| \(J\) |
| Stephen plays 1 | Stephen plays 2 | Stephen plays 3 | |
| Eugene plays 1 | 4 | 5 | 0 |
| Eugene plays 2 | -2 | 1 | 1 |
| Eugene plays 3 | -3 | -4 | 3 |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value |
| 30 | |||||||
| 60 | |||||||
| 80 | |||||||
| 0 |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value | Row Ops |
| \(P\) |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value | Row Ops |
| \(P\) |
| b.v. | \(x\) | \(y\) | \(z\) | \(r\) | \(s\) | \(t\) | Value | Row Ops |
| \(P\) |
| Stage | State | Action | Dest. | Value |
| 1 | ABC | D | ABCD | 65* |
| Stage | State | Action | Dest. | Value |
| Answer | Marks | Guidance |
|---|---|---|
| minimax | B1 (1) | CAO |
| Answer | Marks | Guidance |
|---|---|---|
| Stage 1 values: | At least three additional rows for first stage. Value column must contain values of 55, 45 and 55 | |
| ABC → ABCD: \(65^*\) | M1A1 | |
| ABD → ABCD: \(55^*\) | ||
| ACD → ABCD: \(45^*\) | ||
| BCD → ABCD: \(55^*\) | CAO for first stage — entries in all columns must be correct | |
| Stage 2 values: | M1 A1 A1 A1 | Second stage — at least 12 rows |
| AB via C → ABC: \(\max(40,65)=65\) | Any two states correct | |
| AB via D → ABD: \(\max(35,55)=55^*\) | Any four states correct | |
| AC via B → ABC: \(\max(55,65)=65\) | Second stage completely correct | |
| AC via D → ACD: \(\max(30,45)=45^*\) | ||
| AD via B → ABD: \(\max(55,55)=55^*\) | ||
| AD via C → ACD: \(\max(60,45)=60\) | ||
| BC via A → ABC: \(\max(45,65)=65\) | ||
| BC via D → BCD: \(\max(50,55)=55^*\) | ||
| BD via A → ABD: \(\max(65,55)=65\) | ||
| BD via C → BCD: \(\max(35,55)=55^*\) | ||
| CD via A → ACD: \(\max(55,45)=55^*\) | ||
| CD via B → BCD: \(\max(40,55)=55^*\) | ||
| Stage 3 values: | M1 A1ft A1 | Third stage — at least 12 rows |
| A via B → AB: \(\max(45,55)=55\) | Any two states correct in third stage on ft from second stage | |
| A via C → AC: \(\max(50,45)=50^*\) | Third stage completely correct | |
| A via D → AD: \(\max(55,55)=55\) | ||
| B via A → AB: \(\max(50,55)=55^*\) | ||
| B via C → BC: \(\max(45,55)=55^*\) | ||
| B via D → BD: \(\max(55,55)=55^*\) | ||
| C via A → AC: \(\max(60,45)=60\) | ||
| C via B → BC: \(\max(45,55)=55^*\) | ||
| C via D → CD: \(\max(50,55)=55^*\) | ||
| D via A → AD: \(\max(65,55)=65\) | ||
| D via B → BD: \(\max(55,55)=55^*\) | ||
| D via C → CD: \(\max(60,55)=60\) | ||
| Stage 4 values: | M1 A1 | Fourth stage — at least 4 rows |
| None via A → A: \(\max(25,50)=50^*\) | Fourth stage completely correct | |
| None via B → B: \(\max(35,55)=55\) | ||
| None via C → C: \(\max(35,55)=55\) | ||
| None via D → D: \(\max(30,55)=55\) | (11) |
| Answer | Marks | Guidance |
|---|---|---|
| Order is A, C, D, B | DB1 | Correct order — dependent on all M marks awarded in (b) |
| Cost is \(25\,000 + 50\,000 + 30\,000 + 45\,000 = \pounds150\,000\) | M1 A1 (3) | Correct method for calculating cost for their order; CAO |
# Question 7:
## Part (a):
| minimax | B1 (1) | CAO |
## Part (b):
| **Stage 1 values:** | | At least three additional rows for first stage. Value column must contain values of 55, 45 and 55 |
| ABC → ABCD: $65^*$ | M1A1 | |
| ABD → ABCD: $55^*$ | | |
| ACD → ABCD: $45^*$ | | |
| BCD → ABCD: $55^*$ | | CAO for first stage — entries in all columns must be correct |
| **Stage 2 values:** | M1 A1 A1 A1 | Second stage — at least 12 rows |
| AB via C → ABC: $\max(40,65)=65$ | | Any two states correct |
| AB via D → ABD: $\max(35,55)=55^*$ | | Any four states correct |
| AC via B → ABC: $\max(55,65)=65$ | | Second stage completely correct |
| AC via D → ACD: $\max(30,45)=45^*$ | | |
| AD via B → ABD: $\max(55,55)=55^*$ | | |
| AD via C → ACD: $\max(60,45)=60$ | | |
| BC via A → ABC: $\max(45,65)=65$ | | |
| BC via D → BCD: $\max(50,55)=55^*$ | | |
| BD via A → ABD: $\max(65,55)=65$ | | |
| BD via C → BCD: $\max(35,55)=55^*$ | | |
| CD via A → ACD: $\max(55,45)=55^*$ | | |
| CD via B → BCD: $\max(40,55)=55^*$ | | |
| **Stage 3 values:** | M1 A1ft A1 | Third stage — at least 12 rows |
| A via B → AB: $\max(45,55)=55$ | | Any two states correct in third stage on ft from second stage |
| A via C → AC: $\max(50,45)=50^*$ | | Third stage completely correct |
| A via D → AD: $\max(55,55)=55$ | | |
| B via A → AB: $\max(50,55)=55^*$ | | |
| B via C → BC: $\max(45,55)=55^*$ | | |
| B via D → BD: $\max(55,55)=55^*$ | | |
| C via A → AC: $\max(60,45)=60$ | | |
| C via B → BC: $\max(45,55)=55^*$ | | |
| C via D → CD: $\max(50,55)=55^*$ | | |
| D via A → AD: $\max(65,55)=65$ | | |
| D via B → BD: $\max(55,55)=55^*$ | | |
| D via C → CD: $\max(60,55)=60$ | | |
| **Stage 4 values:** | M1 A1 | Fourth stage — at least 4 rows |
| None via A → A: $\max(25,50)=50^*$ | | Fourth stage completely correct |
| None via B → B: $\max(35,55)=55$ | | |
| None via C → C: $\max(35,55)=55$ | | |
| None via D → D: $\max(30,55)=55$ | (11) | |
## Part (c):
| Order is A, C, D, B | DB1 | Correct order — dependent on all M marks awarded in (b) |
| Cost is $25\,000 + 50\,000 + 30\,000 + 45\,000 = \pounds150\,000$ | M1 A1 (3) | Correct method for calculating cost for their order; CAO |7. A company has purchased a plot of land and has decided to build four holiday homes, A, B, C and D, on the land at the rate of one home per year.
The company expects that the construction costs each year will vary, depending on which houses have already been constructed and which house is currently under construction. The expected construction costs, in thousands of pounds, are shown in the table below.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\cline { 2 - 7 }
\multicolumn{1}{c|}{} & A & B & C & D & E & F \\
\hline
A & - & 53 & 47 & 39 & 35 & 40 \\
\hline
B & 53 & - & 32 & 46 & 41 & 43 \\
\hline
C & 47 & 32 & - & 51 & 47 & 37 \\
\hline
D & 39 & 46 & 51 & - & 36 & 49 \\
\hline
E & 35 & 41 & 47 & 36 & - & 42 \\
\hline
F & 40 & 43 & 37 & 49 & 42 & - \\
\hline
\end{tabular}
\end{center}
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & 17 & 20 & 23 & 14 & 25 \\
\hline
B & 16 & 15 & 19 & 22 & 29 \\
\hline
C & 19 & 14 & 11 & 15 & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 1}
\end{center}
\end{table}
2.
You may not need to use all of these tables
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Table 1}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& 1 & 2 & 3 & 4 & Supply \\
\hline
A & & & & & 25 \\
\hline
B & & & & & 29 \\
\hline
C & & & & & 32 \\
\hline
Demand & 28 & 17 & 23 & 18 & \\
\hline
\end{tabular}
\end{center}
3.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& A & B & C & D & E \\
\hline
Frank & 5 & 0 & 7 & 3 & 4 \\
\hline
Gill & 5 & 3 & 8 & 10 & 1 \\
\hline
Harry & 4 & 3 & 7 & 9 & 0 \\
\hline
Imogen & 6 & 3 & 6 & 5 & 4 \\
\hline
Jiao & 0 & 2 & 7 & 3 & 2 \\
\hline
\end{tabular}
\end{center}
You may not need to use all of these tables
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& A & B & C & D & E \\
\hline
F & & & & & \\
\hline
G & & & & & \\
\hline
H & & & & & \\
\hline
I & & & & & \\
\hline
J & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
& $A$ & $B$ & $C$ & $D$ & $E$ \\
\hline
$F$ & & & & & \\
\hline
$G$ & & & & & \\
\hline
$H$ & & & & & \\
\hline
$I$ & & & & & \\
\hline
$J$ & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
& Stephen plays 1 & Stephen plays 2 & Stephen plays 3 \\
\hline
Eugene plays 1 & 4 & 5 & 0 \\
\hline
Eugene plays 2 & -2 & 1 & 1 \\
\hline
Eugene plays 3 & -3 & -4 & 3 \\
\hline
\end{tabular}
\end{center}
4.
5. (a)
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value \\
\hline
& & & & & & & 30 \\
\hline
& & & & & & & 60 \\
\hline
& & & & & & & 80 \\
\hline
& & & & & & & 0 \\
\hline
\end{tabular}
\end{center}
You may not need to use all of these tableaux
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value & Row Ops \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
$P$ & & & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value & Row Ops \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
$P$ & & & & & & & & \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | }
\hline
b.v. & $x$ & $y$ & $z$ & $r$ & $s$ & $t$ & Value & Row Ops \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
& & & & & & & & \\
\hline
$P$ & & & & & & & & \\
\hline
\end{tabular}
\end{center}
6. (a) Value of initial flow\\
(b) and (c)\\
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-20_725_1251_404_349}
\section*{Diagram 1}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-20_1070_1264_1322_349}
\end{center}
\section*{Diagram 2}
(d)\\
(e)\\
\includegraphics[max width=\textwidth, alt={}, center]{e0c66144-9e34-42fc-9f40-a87a49331483-21_714_1385_1306_283}
\section*{Diagram 3}
(f)\\
7. (a)
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Stage & State & Action & Dest. & Value \\
\hline
1 & ABC & D & ABCD & 65* \\
\hline
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\hline
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\hline
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\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Stage & State & Action & Dest. & Value \\
\hline
& & & & \\
\hline
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\end{tabular}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{e0c66144-9e34-42fc-9f40-a87a49331483-24_2642_1833_118_118}
\end{center}
\hfill \mbox{\textit{Edexcel D2 2019 Q7 [15]}}