Why Operations Research Turns Decisions into Mathematics
First, managers rarely lack opinions — they lack a reliable way to choose between them. Operations research gives managers that way: it turns a business decision into a mathematical model, solves the model, and reads the answer back as a decision. For example, how many chairs and tables should a workshop make, which warehouse should supply which shop, or how many checkout counters should stay open at noon? In short, operations research answers these questions with numbers instead of hunches, which is why MBA papers test it so heavily. In addition, this paper-style answer covers the OR method, linear programming, and the transportation problem, with two fully solved numerical examples: a graphical linear programming solution and a North-West Corner transportation allocation.
Furthermore, operations research grew out of wartime planning in the 1940s, when scientists used mathematics to place radar stations, route convoys and schedule aircraft. Consequently, the same scientific habit — define, model, solve, check — now serves factories, airlines, hospitals and retailers. Similarly, it connects directly with our Strategic Management paper, because OR supplies the numbers behind a strategy, and with our Business Analytics for Decision Making paper, because both disciplines turn data into decisions.
The Operations Research Method: Six Steps
First, in operations research, every problem follows the same six steps, and examiners award marks for naming them in order. First, define the problem: state the decision, the objective and the constraints in plain words. Second, build a model: translate the words into variables, an objective function and constraints. Third, solve the model: use a suitable technique (graphical, simplex, or a computer solver). Fourth, validate the solution: check it against reality — does it respect every constraint, and does it make business sense? Fifth, implement the solution: turn the numbers back into actions, people and schedules. Finally, monitor the result: compare actual outcomes with the model and revise the model when conditions change. In short, a solution that skips validation or monitoring is not operations research; it is arithmetic without responsibility.
Linear Programming in Operations Research: Formulating the Model
Indeed, linear programming is the most tested OR technique, and it rests on three ingredients. First, decision variables: the quantities the firm chooses, usually written as x and y. Second, an objective function: the quantity to maximise (profit, contribution) or minimise (cost, time), written as a linear expression such as Z = 40x + 30y. Third, constraints: the limits on resources, written as linear inequalities such as 2x + y ≤ 160. Finally, the non-negativity condition: a firm cannot make a negative number of chairs, so x ≥ 0 and y ≥ 0. Therefore, the linear programming method then finds the feasible corner point that gives the best objective value — which is exactly what Worked Example 1 does.
In particular, students often ask why linear programming assumes proportionality and additivity — that doubling inputs exactly doubles outputs, and that resources do not interact. Therefore, the model works best for short-term planning with stable prices and divisible resources, and it weakens when fixed charges, bulk discounts or integer units dominate. For example, a firm that must buy machines in whole units cannot use a fractional LP answer directly; it needs integer programming instead. However, examiners still expect the LP formulation first, because it frames the problem even when a richer method finishes it.
Worked Example 1: Solving a Linear Programme Graphically
Question (exam style): Specifically, a workshop makes chairs (x) and tables (y). Each chair earns $40 profit and each table $30. Carpentry capacity allows 2x + y ≤ 160 hours, and finishing capacity allows x + 2y ≤ 140 hours. Formulate the linear programme, find the corner points of the feasible region, and determine the profit-maximising mix.
Step 1 — Formulate. First, maximise Z = 40x + 30y, subject to 2x + y ≤ 160, x + 2y ≤ 140, and x ≥ 0, y ≥ 0. For example, the objective says every chair contributes $40 and every table $30, while each constraint says one workshop resource cannot be exceeded.
| Resource | Chairs (x) | Tables (y) | Available |
|---|---|---|---|
| Carpentry hours | 2 per chair | 1 per table | 160 |
| Finishing hours | 1 per chair | 2 per table | 140 |
| Profit per unit | $40 | $30 | — |
Step 2 — Find the corner points. Overall, the feasible region has four corners. First, (0, 0): make nothing. Second, (80, 0): use all carpentry on chairs (2 × 80 = 160, finishing 80 ≤ 140). Third, (0, 70): use all finishing on tables (2 × 70 = 140, carpentry 70 ≤ 160). Finally, the intersection of the two constraint lines: solving 2x + y = 160 and x + 2y = 140 together gives x = 60 and y = 40.
| Corner point (x, y) | Calculation | Profit Z = 40x + 30y |
|---|---|---|
| (0, 0) | Make nothing | $0 |
| (80, 0) | 40 × 80 | $3,200 |
| (0, 70) | 30 × 70 | $2,100 |
| (60, 40) | 40 × 60 + 30 × 40 | $3,600 |
What the Numbers Mean
Interpretation for the answer. In short, the profit-maximising mix is 60 chairs and 40 tables, earning $3,600 — better than chairs alone ($3,200) or tables alone ($2,100). However, a top-mark answer also states why the optimum sits at a corner: in linear programming, the best feasible solution always lies at a corner point of the feasible region, so checking corners is enough. Similarly, it names the binding constraints: at (60, 40) both carpentry (2 × 60 + 40 = 160) and finishing (60 + 2 × 40 = 140) are fully used, so either resource limits further profit.
The Transportation Problem: Moving Goods at Minimum Cost
Moreover, while linear programming maximises profit, the transportation problem minimises the cost of moving goods from sources (factories, warehouses) to destinations (shops, customers). First, the problem is balanced when total supply equals total demand; if supply exceeds demand, examiners add a dummy destination with zero cost to absorb the surplus. Second, the North-West Corner Method gives a starting feasible allocation by filling the table from the top-left corner, ignoring costs. Third, Vogel’s Approximation Method usually gives a better starting allocation by penalising the rows and columns where the cheapest route matters most. Finally, the MODI (modified distribution) method or the stepping-stone method tests whether the starting allocation is optimal and improves it if it is not. In particular, a common exam trap is stopping at the North-West Corner answer and calling it optimal — it is only a starting point.
Worked Example 2: North-West Corner Method
Question (exam style): Specifically, two warehouses supply three shops. Warehouse S1 holds 30 units and S2 holds 40 units. Shops D1, D2 and D3 need 20, 25 and 25 units. Unit transport costs are: S1→D1 $4, S1→D2 $6, S1→D3 $8; S2→D1 $5, S2→D2 $3, S2→D3 $7. Find the North-West Corner allocation and its total cost.
Step 1 — Check balance. First, total supply = 30 + 40 = 70 units. Total demand = 20 + 25 + 25 = 70 units. Therefore, the problem is balanced, so no dummy row or column is needed.
| D1 (20) | D2 (25) | D3 (25) | Supply | |
|---|---|---|---|---|
| S1 (30) | $4 | $6 | $8 | 30 |
| S2 (40) | $5 | $3 | $7 | 40 |
| Demand | 20 | 25 | 25 | 70 |
Step 2 — Allocate from the north-west corner. First, S1→D1 takes 20 units (D1 satisfied, S1 has 10 left). Second, S1→D2 takes the remaining 10 units (S1 exhausted, D2 still needs 15). Third, S2→D2 takes 15 units (D2 satisfied, S2 has 25 left). Finally, S2→D3 takes 25 units (D3 satisfied, S2 exhausted). In total, four routes carry goods and two stay empty, which is normal for a starting allocation.
| Route | Units | Unit cost | Cost |
|---|---|---|---|
| S1 → D1 | 20 | $4 | $80 |
| S1 → D2 | 10 | $6 | $60 |
| S2 → D2 | 15 | $3 | $45 |
| S2 → D3 | 25 | $7 | $175 |
| Total | 70 | — | $360 |
What the Numbers Mean
Interpretation for the answer. In short, the North-West Corner allocation costs $360 (20 × $4 + 10 × $6 + 15 × $3 + 25 × $7 = 80 + 60 + 45 + 175). However, a strong answer never calls $360 optimal: the method ignores costs entirely, so it can place goods on expensive routes — here S2→D3 at $7 carries 25 units while the cheaper S2→D2 route at $3 carries only 15. Consequently, the next exam step is always an optimality test (MODI or stepping-stone) or a better starting method (VAM), which would shift units toward cheaper routes and lower the total below $360.
Sensitivity Analysis in Operations Research: What If the Numbers Change?
Furthermore, in practice, the numbers in a linear programme are estimates, so examiners test sensitivity analysis: how far can one number move before the optimal mix changes? First, consider the profit per chair: at $40 the optimum is (60, 40), but if chair profit falls sharply, the (80, 0) corner may take over. Second, consider a resource limit: if carpentry hours rise from 160 to 180, the feasible region grows and the optimum moves outward. Consequently, sensitivity analysis does not replace the solution; it tells the firm how much confidence the solution deserves.
For example, a manager who knows the optimum survives a $5 fall in chair profit can commit to the mix confidently, while a manager whose optimum flips on a $1 change should negotiate prices or capacity first. In short, always report the optimum together with its sensitivity: the answer is stronger when it names the number that would change it.
Assignment, Queuing and Simulation in Brief
In addition, operations research covers more than linear programming and transport. First, the assignment problem gives each worker exactly one job (or each machine one task) at minimum total cost, and examiners expect the Hungarian method by name. Second, queuing theory studies waiting lines: customers arrive at some rate, servers serve at some rate, and the model predicts average waiting time and queue length — which is why it appears in bank, hospital and call-centre questions. Finally, simulation (often Monte Carlo simulation) handles problems too complex for formulas: the model runs thousands of random trials and reports the range of likely outcomes. In particular, simulation does not give one optimal answer; it gives a distribution of answers, so the exam point is risk, not precision. Consequently, these three topics also link back to our Operations Management paper, where waiting lines, staffing and scheduling appear as daily management problems.
How to Structure Your Operations Research Exam Answer
First, open with a definition (one line) and name the technique the question tests. Second, formulate the model: decision variables, objective function and constraints, with units stated. Third, show every calculation with its steps — corner points with profit values, or allocations with route costs — and add a cross-check where one exists, as in the two examples above. Finally, close with a judgement justified by your numbers: the optimal mix, the total cost, and whether the answer is a starting solution or a proven optimum. In short, answers that describe a technique without applying it to the case numbers, or that report a total without its route-by-route working, lose marks even when the final figure is right. Therefore, when a question gives both supplies and demands, always check balance before allocating anything.
FAQs
What is operations research?
In short, operations research is the scientific use of mathematical models to improve decisions: define the problem, build a model, solve it, validate the answer, implement it and monitor the result. Specifically, it covers linear programming, transportation, assignment, queuing and simulation, and it appears wherever scarce resources must serve competing demands.
How do you solve a linear programming problem graphically?
First, plot each constraint as a line and shade the feasible region where all constraints hold. Second, list the corner points of that region. Third, evaluate the objective function at each corner. Finally, choose the corner with the best value: maximum for profit, minimum for cost. For example, in Worked Example 1 the corners gave $0, $3,200, $2,100 and $3,600, so (60, 40) is optimal.
What is the difference between the North-West Corner Method and Vogel’s Approximation Method?
First, the North-West Corner Method fills the transportation table from the top-left corner without looking at costs, so it gives a feasible starting allocation that is rarely optimal. Second, Vogel’s Approximation Method looks at the penalty for missing the cheapest route in each row and column, so it usually starts much closer to the optimum. Finally, neither method proves optimality — that needs the MODI or stepping-stone test.
How should I answer an operations research case study in an MBA exam?
First, identify the technique the case tests (linear programming, transportation, assignment or queuing). Second, formulate the model with variables, objective and constraints. Third, solve it step by step with tables, and show a cross-check such as total supply equalling total demand. Finally, interpret the numbers: name the optimum, state whether it is proven or only a starting solution, and say what the firm should do next.