| Exam Board | Edexcel |
|---|---|
| Module | D2 (Decision Mathematics 2) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Network Flows |
| Type | Transportation LP formulation |
| Difficulty | Moderate -0.5 This is a standard textbook transportation problem requiring straightforward LP formulation with clear structure. Students must define 9 decision variables, write a linear objective function summing costs, and state supply/demand constraints—all routine steps with no problem-solving insight needed. Slightly easier than average due to its mechanical nature and small problem size (3×3). |
| Spec | 7.06a LP formulation: variables, constraints, objective function |
| \(J\) | \(K\) | \(L\) | |
| \(W\) | 3 | 6 | 3 |
| \(X\) | 5 | 8 | 4 |
| \(Y\) | 2 | 5 | 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| Let \(x_{ij}\) be number of units transported from \(i\) to \(j\), where \(i \in \{W, X, Y\}\) and \(j \in \{J, K, L\}\) | B1 | |
| Minimise \(c = 3x_{WJ} + 6x_{WK} + 3x_{WL} + 5x_{XJ} + 8x_{XK} + 4x_{XL} + 2x_{YJ} + 5x_{YK} + 7x_{YL}\) | B1 | |
| \(x_{WJ} + x_{WK} + x_{WL} = 34\) | M1 A1 | |
| \(x_{XJ} + x_{XK} + x_{XL} = 57\) | ||
| \(x_{YJ} + x_{YK} + x_{YL} = 25\) | A1 | |
| \(x_{WJ} + x_{XJ} + x_{YJ} = 20\) | ||
| \(x_{WK} + x_{XK} + x_{YK} = 56\) | ||
| \(x_{WL} + x_{XL} + x_{YL} = 40\) | ||
| \(x_{ij} \geq 0 \quad \forall\ i \in \{W,X,Y\},\ j \in \{J,K,L\}\) | B1 |
# Question 5:
| Answer/Working | Marks | Notes |
|---|---|---|
| Let $x_{ij}$ be number of units transported from $i$ to $j$, where $i \in \{W, X, Y\}$ and $j \in \{J, K, L\}$ | B1 | |
| Minimise $c = 3x_{WJ} + 6x_{WK} + 3x_{WL} + 5x_{XJ} + 8x_{XK} + 4x_{XL} + 2x_{YJ} + 5x_{YK} + 7x_{YL}$ | B1 | |
| $x_{WJ} + x_{WK} + x_{WL} = 34$ | M1 A1 | |
| $x_{XJ} + x_{XK} + x_{XL} = 57$ | | |
| $x_{YJ} + x_{YK} + x_{YL} = 25$ | A1 | |
| $x_{WJ} + x_{XJ} + x_{YJ} = 20$ | | |
| $x_{WK} + x_{XK} + x_{YK} = 56$ | | |
| $x_{WL} + x_{XL} + x_{YL} = 40$ | | |
| $x_{ij} \geq 0 \quad \forall\ i \in \{W,X,Y\},\ j \in \{J,K,L\}$ | B1 | |
**Total: 6 marks**
---5. Three warehouses $W , X$ and $Y$ supply televisions to three supermarkets $J , K$ and $L$. The table gives the cost, in pounds, of transporting a television from each warehouse to each supermarket. The warehouses have stocks of 34,57 and 25 televisions respectively, and the supermarkets require 20, 56 and 40 televisions respectively. The total cost of transporting the televisions is to be minimised.
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
& $J$ & $K$ & $L$ \\
\hline
$W$ & 3 & 6 & 3 \\
\hline
$X$ & 5 & 8 & 4 \\
\hline
$Y$ & 2 & 5 & 7 \\
\hline
\end{tabular}
\end{center}
Formulate this transportation problem as a linear programming problem. Make clear your decision variables, objective function and constraints.\\
\hfill \mbox{\textit{Edexcel D2 Q5 [6]}}