Decision Mathematics 2
A-Level Further Mathematics
Pearson Edexcel · Paper 4D · 9FM0/4D
Complete Revision Notes
Syllabus-matched · Full worked examples · TikZ diagrams · Exam technique
Topics Covered
1. Transportation Problems
2. Allocation / Assignment (Hungarian Algorithm)
3. Flows in Networks
4. Dynamic Programming
5. Game Theory
6. Recurrence Relations
7. Decision Analysis
,Decision Mathematics 2 Edexcel Further Maths · 9FM0/4D
0 1 Contents
1 Transportation Problems 2
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 North-West Corner Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Shadow Costs and Improvement Indices . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 LP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Allocation / Assignment Problems 4
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hungarian Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Flows in Networks 6
3.1 Core Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Example Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Labelling Procedure (Ford–Fulkerson) . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Multiple Sources, Sinks, and Restricted Vertices . . . . . . . . . . . . . . . . . . . . . 7
4 Dynamic Programming 8
4.1 Bellman’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Example Network for Shortest Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.4 Minimax and Maximin Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Game Theory 10
5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Mixed Strategies – 2×2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.4 Graphical Method – 2×n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.5 LP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Recurrence Relations 12
6.1 First-Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Second-Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Decision Analysis 14
7.1 Decision Tree Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.2 Example Decision Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Exam Technique Cheatsheet 15
1
, Decision Mathematics 2 Edexcel Further Maths · 9FM0/4D
1 1 Transpor
Problems
1.1 Overview
A transportation problem finds the minimum-cost plan to ship goods from m sources (supply
si ) to n destinations (demand dj ).
Key Conditions
P P
Balanced: si = dj .
Unbalanced: add a dummy row or column with zero costs to balance.
1.2 North-West Corner Method
1. Start at the top-left (north-west) cell.
2. Allocate min(row supply, col demand) to that cell.
3. Strike out the exhausted row or column; move right or down accordingly.
4. Repeat until all supply and demand is allocated.
A valid solution needs exactly m + n − 1 occupied cells.
Degeneracy: if fewer cells are occupied, place ε ≈ 0 in an independent empty cell before
computing shadow costs.
Worked Example
Setup: 3 factories F1 , F2 , F3 with supplies 30, 40, 50. Three warehouses W1 , W2 , W3 with
demands 20, 60, 40.
Unit cost matrix:
2 3 1
C = 5 4 8
5 6 8
North-West Corner allocation:
W1 (20) W2 (60) W3 (40) Supply
F1 (30) 20 10 – 30
F2 (40) – 40 – 40
F3 (50) – 10 40 50
Demand 20 60 40 120
Occupied cells = 5 = 3 + 3 − 1 ✓ Not degenerate.
Initial cost = 20(2) + 10(3) + 40(4) + 10(6) + 40(8) = 40 + 30 + 160 + 60 + 320 = 610
1.3 Shadow Costs and Improvement Indices
For each occupied cell (i, j):
Ri + Kj = Cij , set R1 = 0
2
A-Level Further Mathematics
Pearson Edexcel · Paper 4D · 9FM0/4D
Complete Revision Notes
Syllabus-matched · Full worked examples · TikZ diagrams · Exam technique
Topics Covered
1. Transportation Problems
2. Allocation / Assignment (Hungarian Algorithm)
3. Flows in Networks
4. Dynamic Programming
5. Game Theory
6. Recurrence Relations
7. Decision Analysis
,Decision Mathematics 2 Edexcel Further Maths · 9FM0/4D
0 1 Contents
1 Transportation Problems 2
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 North-West Corner Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Shadow Costs and Improvement Indices . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 LP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Allocation / Assignment Problems 4
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hungarian Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Flows in Networks 6
3.1 Core Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Example Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 Labelling Procedure (Ford–Fulkerson) . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.4 Multiple Sources, Sinks, and Restricted Vertices . . . . . . . . . . . . . . . . . . . . . 7
4 Dynamic Programming 8
4.1 Bellman’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Example Network for Shortest Path . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.4 Minimax and Maximin Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Game Theory 10
5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.3 Mixed Strategies – 2×2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5.4 Graphical Method – 2×n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.5 LP Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6 Recurrence Relations 12
6.1 First-Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.2 Second-Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7 Decision Analysis 14
7.1 Decision Tree Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.2 Example Decision Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.3 Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Exam Technique Cheatsheet 15
1
, Decision Mathematics 2 Edexcel Further Maths · 9FM0/4D
1 1 Transpor
Problems
1.1 Overview
A transportation problem finds the minimum-cost plan to ship goods from m sources (supply
si ) to n destinations (demand dj ).
Key Conditions
P P
Balanced: si = dj .
Unbalanced: add a dummy row or column with zero costs to balance.
1.2 North-West Corner Method
1. Start at the top-left (north-west) cell.
2. Allocate min(row supply, col demand) to that cell.
3. Strike out the exhausted row or column; move right or down accordingly.
4. Repeat until all supply and demand is allocated.
A valid solution needs exactly m + n − 1 occupied cells.
Degeneracy: if fewer cells are occupied, place ε ≈ 0 in an independent empty cell before
computing shadow costs.
Worked Example
Setup: 3 factories F1 , F2 , F3 with supplies 30, 40, 50. Three warehouses W1 , W2 , W3 with
demands 20, 60, 40.
Unit cost matrix:
2 3 1
C = 5 4 8
5 6 8
North-West Corner allocation:
W1 (20) W2 (60) W3 (40) Supply
F1 (30) 20 10 – 30
F2 (40) – 40 – 40
F3 (50) – 10 40 50
Demand 20 60 40 120
Occupied cells = 5 = 3 + 3 − 1 ✓ Not degenerate.
Initial cost = 20(2) + 10(3) + 40(4) + 10(6) + 40(8) = 40 + 30 + 160 + 60 + 320 = 610
1.3 Shadow Costs and Improvement Indices
For each occupied cell (i, j):
Ri + Kj = Cij , set R1 = 0
2
