Financial Management · Specific investment decisions (lease or buy, asset replacement, capital rationing)

Multi-Period Capital Rationing and Linear Programming

Updated 11 October 2026 · Fact-checked

Multi-period capital rationing means funds are limited in more than one year. Profitability index cannot cope, because one ranking cannot serve two limits. Instead you write a linear programming model: maximise total NPV subject to a funds constraint for each year. Solve it graphically or by simultaneous equations, then read the shadow prices.

Understand Multi-Period Capital Rationing and Linear Programming

Capital rationing means a firm has less money than it has worthwhile projects. In single-period rationing, funds are short only now (year 0). You rank projects by profitability index (NPV per $1 of the scarce funds) and fund them in order.

In multi-period rationing, funds are limited in several years, for example year 0 and year 1. A project may use a lot of year 0 money and little year 1 money, or the other way round. There is no single scarce resource to divide NPV by. Ranking by profitability index then fails, because a project that is best per $1 of year 0 funds may be poor per $1 of year 1 funds.

The answer is linear programming (LP). You define a variable for each project (the fraction or number of units undertaken). The objective is to maximise total NPV. Each year gives one constraint: the cash used by the chosen projects in that year must not exceed the funds available. If projects are divisible, variables can be fractions up to 1. If they are indivisible, you need integer programming, which is beyond what you will normally solve by hand.

The shadow price (dual value) of a constraint is the extra NPV from one more $1 of that resource. A constraint with spare funds has a shadow price of zero. A binding constraint has a positive shadow price. It tells management how much it would pay at most to relax that limit, for example by raising extra finance in that year. It holds only within a limited range of change.

LP has limits. It assumes divisible projects, certain cash flows, known and fixed limits, and a single objective of NPV. Results can be misleading if projects are really all-or-nothing.

Key rules to remember

Objective function: Maximise Σ (NPV of project j × xj); xj is the fraction (or units) of project j undertaken. For divisible, non-repeatable projects, 0 ≤ xj ≤ 1.
Funds constraint for each year t: Σ (cash outflow of project j in year t × xj) ≤ funds available in year t; Write one constraint per year in which funds are limited.
Non-negativity and upper limits: xj ≥ 0 and, if projects cannot be repeated, xj ≤ 1; Do not forget these. They often decide which corner is optimal.
Shadow price of a constraint: Shadow price = increase in maximum NPV ÷ increase in the funds limit; Zero if the constraint has slack. It is valid only within a range where the same constraints stay binding.
Profitability index (single period only): PI = NPV ÷ capital invested in the rationed period; Valid for one scarce period only. Do not use it when two or more years are limited.

How to solve Multi-Period Capital Rationing and Linear Programming questions

Use this method for any multi-period rationing question. Work in consistent units, such as $000.

1Define a variable for each project (for example x, y). State whether projects are divisible and whether they can be repeated. This sets the limits on each variable.
2Write the objective: maximise total NPV as the sum of NPV × variable.
3Write one constraint for each rationed year, using the cash outflow of each project in that year and the funds limit. Add non-negativity and upper limits (such as x ≤ 1).
4With two variables, plot each constraint line and find the feasible region. Find the corner points (or use the objective line sliding outward).
5Calculate the objective at each corner point and choose the highest. Use simultaneous equations for corners where two lines cross.
6Identify which constraints are binding at the optimum. Any constraint with spare funds has a shadow price of zero.
7Find the shadow prices of the binding constraints, either by increasing the limit slightly and re-solving, or by solving the dual equations. State the range over which each holds.
8Conclude: state the project mix, the maximum NPV, the shadow prices and what they mean for raising extra funds. Mention limits such as divisibility.

Quickest way: Corner-point check with a shadow price test

When to use it: Two projects, two rationed years, and the examiner asks for the best mix or a shadow price.

Write the constraints and the objective straight away in $000 so the numbers stay small.
List the corner points by finding where each pair of binding lines meet and where lines meet the axes or upper limits. Test each in all constraints.
Compute the NPV at each feasible corner. Pick the largest.
Mark which constraints are binding at that corner. Any constraint with slack has a shadow price of zero.
For a shadow price, raise the binding limit by a convenient amount (such as 10), re-solve the same corner and divide the NPV gain by 10. Check the answer against the original corner.

Common mistakes in Multi-Period Capital Rationing and Linear Programming

Ranking projects by profitability index when two or more years are rationed.
PI works so well in single-period questions that students apply it automatically.
Fix: Check how many years are limited. If more than one, set up LP. Mention that PI ignores the interaction between years.
Forgetting upper limits such as x ≤ 1 for projects that cannot be repeated.
Students write only the funds constraints and the answer looks too large.
Fix: Read the question for 'cannot be repeated' or 'divisible'. Write the limit down in the formulation before solving.
Using the total project cost or NPV as the objective coefficient in the wrong period, or mixing up the year columns in constraints.
Cash flow tables have many columns and time is short.
Fix: Set out a small table: project, NPV, year 0 outflow, year 1 outflow. Read each constraint straight from a column.
Giving a non-zero shadow price to a constraint that has spare funds.
Students assume every limit matters.
Fix: Substitute the optimal mix into each constraint. If it uses less than the limit, the shadow price is zero.
Stating a shadow price as valid for any change in the limit.
The figure looks like a fixed rate of return.
Fix: State the range. Beyond it another constraint becomes binding and the shadow price changes.
Rounding a fractional LP answer to a whole project without comment.
Students forget that LP assumes divisibility.
Fix: Say that the result is valid only if projects can be undertaken in part. Otherwise test whole-project combinations.

Worked examples

Example 1

A company has two divisible projects that cannot be repeated. Project X has NPV $60,000, year 0 outlay $100,000 and year 1 outlay $50,000. Project Y has NPV $50,000, year 0 outlay $80,000 and year 1 outlay $100,000. Funds are limited to $150,000 in year 0 and $100,000 in year 1. Formulate the problem, find the best mix and the shadow price of year 1 funds.

Show the solution

Let x and y be the fractions of X and Y undertaken. Work in $000.
Maximise 60x + 50y.
Year 0: 100x + 80y ≤ 150. Year 1: 50x + 100y ≤ 100. Also 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
Corner points: (0, 0) gives NPV 0. (1, 0) gives 60. (0, 1) gives 50: year 0 use 80 and year 1 use 100, both feasible.
At x = 1, year 1 allows y ≤ 0.5 and year 0 allows y ≤ 0.625. So year 1 binds and the corner is (1, 0.5). Year 0 use is 100 + 40 = 140, which is feasible. NPV = 60 + 25 = 85.
The best corner is (1, 0.5) with NPV $85,000. Year 1 is binding (50 + 50 = 100). Year 0 has $10,000 spare, so its shadow price is zero.
Shadow price of year 1: raise the limit by 1 and keep x = 1. Then y rises by 0.01, so NPV rises by 0.5 per 1. That is $0.50 per $1 of year 1 funds.
Range: year 0 allows y up to 0.625, so year 1 funds can rise to 50 + 100 × 0.625 = 112.5. The shadow price holds up to $112,500.

Answer: Undertake all of X and half of Y for a maximum NPV of $85,000. Year 0 funds have a shadow price of zero. Year 1 funds have a shadow price of $0.50 per $1, valid up to a year 1 limit of $112,500.

Example 2

A firm can undertake any amount of projects A and B (both divisible and repeatable). Per unit, A has NPV $30,000, year 0 outlay $10,000 and year 1 outlay $30,000. B has NPV $40,000, year 0 outlay $20,000 and year 1 outlay $10,000. Funds available are $100,000 in year 0 and $150,000 in year 1. Find the best plan and the shadow price of each year's funds.

Show the solution

Let a and b be units of A and B. Work in $000. Maximise 30a + 40b.
Constraints: 10a + 20b ≤ 100 and 30a + 10b ≤ 150, with a, b ≥ 0.
Corner points: (0, 0) gives 0. (5, 0) uses year 1 funds of 150 and gives NPV 150. (0, 5) uses year 0 funds of 100 and year 1 funds of 50 and gives NPV 200.
Both constraints crossing: from the first, a = 10 − 2b. Substitute: 30(10 − 2b) + 10b = 150, so 300 − 50b = 150 and b = 3. Then a = 4. NPV = 120 + 120 = 240.
The best corner is a = 4, b = 3 with NPV $240,000. Both constraints are binding.
Shadow prices: let p0 and p1 be the prices. For A: 10p0 + 30p1 = 30. For B: 20p0 + 10p1 = 40.
From the second, p1 = 4 − 2p0. Substitute: 10p0 + 120 − 60p0 = 30, so p0 = 1.8 and p1 = 0.4.
Check: raise year 0 funds to 110. Then a = 3.8, b = 3.6 and NPV = 114 + 144 = 258. The gain of 18 for 10 more funds confirms 1.8.

Answer: Undertake 4 units of A and 3 units of B for a maximum NPV of $240,000. The shadow price is $1.80 per $1 of year 0 funds and $0.40 per $1 of year 1 funds. Each holds only for small changes in the limit.

Exam tips

In Section C, set out the formulation first (variables, objective, constraints). Marks are usually given for the model even if the arithmetic goes wrong.
Write a small table of NPV and yearly outflows before building constraints. It prevents column mix-ups.
In objective test questions, shadow price and slack questions are quick. Test the optimal mix in each constraint. Any constraint with spare funds has a shadow price of zero.
Always state the assumptions and limits of LP: divisibility, certainty, fixed limits and a single objective. Written parts often ask for them.
If asked why profitability index fails, say that it ranks on one scarce resource, while multi-period rationing has several limits that interact.

Practice questions from Specific investment decisions (lease or buy, asset replacement, capital rationing)

Multi-Period Capital Rationing and Linear Programming in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Multi-Period Capital Rationing and Linear Programming: frequently asked questions

What is multi-period capital rationing?

It is a situation where funds are limited in more than one year, such as year 0 and year 1. Each project uses different amounts of each year's funds. You choose the project mix that maximises total NPV within every limit, usually with linear programming.

Why can't profitability index be used for multi-period rationing?

Profitability index divides NPV by one scarce amount of capital. With two or more limited years there is no single amount to divide by, and the project ranking can differ for each year. It also cannot show how the limits interact.

What does a shadow price mean in capital rationing?

It is the extra NPV from one more $1 of a limited resource, such as year 1 funds. It is zero if the funds are not fully used. It shows the most the firm should pay to relax that limit, and it holds only within a limited range.

How do you use linear programming for capital rationing?

Define a variable for each project, maximise total NPV and write one funds constraint per rationed year. Add limits such as x ≤ 1. Solve graphically or with simultaneous equations, then read off the best mix and shadow prices.

Do I need to solve integer programming in ACCA FM?

Not by hand. If projects are indivisible, the LP answer may show fractions. You should say that the result assumes divisibility and, if needed, test whole-project combinations.