Lecture 30 — Optimization Formulations

Learning objectives

• Express problems as objectives with constraints
• Search small solution spaces exhaustively and smartly
• Recognise greedy vs search vs DP formulations

Case-based lecture: you attempt, we discuss, then we reveal.

The plan

• Four cases: cs-117, cs-118, cs-119, cs-120
• Each: 5-minute attempt → discussion → reveal → prompt
• Worked examples in between — ask anything
• Break mid-lecture · exit question at the end

Key definitions

• Decision variable — the quantity you choose (how many posters, which shifts)
• Objective function — the quantity to maximise or minimise, written in the variables
• Constraint — a limit written in the variables: budget, capacity, non-negativity
• Feasible solution — an assignment of variables satisfying every constraint
• Formulation — variables + objective + constraints, written before any algorithm
• Search space — all candidate assignments; formulation decides its size and shape

The Poster Budget: turn a wish into a formulation

best = 0
best_pf = (0, 0)
for p in range(13):
    for f in range(13):
        if 3 * p + f <= 12:
            reach = 40 * p + 10 * f
            if reach > best:
                best, best_pf = reach, (p, f)
print(best, best_pf)

Expected: maximum reach 160 with 4 posters, 0 flyers

Cost: O(budget^2) enumeration here; the formulation scales to solvers when the space explodes.

The Shift Schedule: constraints as first-class citizens

def solve():
    for s1 in ("bo",):
        for s2 in ("ada", "bo"):
            for s3 in ("ada", "bo"):
                roster = (s1, s2, s3)
                if len(set(roster)) == 3 or roster[0] == roster[1] or roster[1] == roster[2]:
                    continue
                if "ada" == roster[0]:
                    continue
                return roster
    return None

print(solve())

Expected: valid roster: shift 1 bo, shift 2 ada, shift 3 bo

Cost: Exponential worst case, but pruning makes toy instances instant - the general shape behind schedulers and solvers.

The Auction Bluff: optimise what you control

best_bid, best_val = 0, -1
for b in range(81):
    val = (100 - b) * b / 100
    if val > best_val:
        best_val, best_bid = val, b
print(best_bid, best_val)

Expected: bid 50 (expected surplus 25 under the uniform-rival assumption)

Cost: O(budget) enumeration; closed-form parabola vertex exists (enrichment: calculus).

• **Input:** budget 20; posters A price 5 reach 40 each, B price 4 reach 30; wall space 3 m, A needs 1 m, B needs 0.5 m
• **Output:** formulation: max 40a+30b s.t. 5a+4b<=20, a+0.5b<=3, integers >=0; best by enumeration: a=2,b=2 -> 140
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-117 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which constraint is cheapest to check first, and why does order matter?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-117 — not embedded in this deck.

• **Input:** 7 days, each day needs 2 staff; ada/bo/cy/dee available on subsets; max 4 days each
• **Output:** a feasible rota or proof none exists (try greedy by most-constrained day first)
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-118 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Double the input size: is this still acceptable? Show the arithmetic."**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-118 — not embedded in this deck.

• **Input:** items weight/value: (5, 30), (4, 26), (3, 14); capacity 8
• **Output:** 0/1 knapsack: best value 56 {(5,30)+(3,14)=44 no -> (4,26)+(3,14)=40? enumerate: (5,30)+(4,26) over; best = 30+14=44 or 26+14=40 -> 44; fractional would take 4/5 of first
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-119 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Which constraint is cheapest to check first, and why does order matter?"**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-119 — not embedded in this deck.

• **Input:** 4x4 grid, friends pairs (r1,c1)-(r2,c2) must not sit adjacent (share an edge); 6 students, 2 friend pairs
• **Output:** a valid seating or 'impossible' - backtracking with adjacency pruning; report the search count
• Solve for the worked instance first. Then ask what would change if one
• number, name, or condition were replaced by something stranger.

Pseudocode, flowchart or Python. State one assumption. Compare with your neighbour at minute 4.

cs-120 — discuss

• What did the deciding quantity turn out to be?
• Which assumption shaped your answer?
• After the solution is revealed, the class spends five minutes on this prompt: **"Double the input size: is this still acceptable? Show the arithmetic."**

Instructor: the reveal, the reasoning section and common wrong turns are in the instructor distribution for cs-120 — not embedded in this deck.

Common misconceptions

• Optimising an unstated objective - 'best' must be written in the variables before any algorithm.
• Forgetting integrality or non-negativity - the fractional optimum can differ from the real one.
• Confusing satisfaction with optimisation - 'find any valid roster' needs no objective, only feasibility.
• Checking expensive constraints first - order constraints by pruning power, not by narrative.

Summary & exit question

• Named technique today: Optimization Formulations
• Four cases attempted — methods before answers

Exit question (choose one, 60 seconds)

1. Name the three parts of a formulation.
2. Why is 'minimise cost' often incomplete?
3. How do you verify a claimed optimum?