Linear Programming: Slack, Shadow Prices and the Maximum Price for Extra Resources

A plastics factory in Ikeja has finished its production plan with a linear programme. The plan says: make 400 of one product and 200 of the other, and use every kilogram of material and every labour hour. The production manager then asks two questions that the plan itself does not answer. If the supplier offers ten more kilograms of material, how much is that worth to us?

And the machine shop shows spare hours on the plan: why are they not used? The answers are the shadow price and the slack of each limit.

This topic explains both, how to find them by hand with simultaneous equations, how to use a shadow price to set the most that should be paid for an extra unit of a scarce resource, the range over which a shadow price holds, and the limits of the method. It builds on the earlier topic on linear programming: formulation and graphical solution.

The next topic covers price elasticity of demand and the factors that influence price.

Binding limits, slack and shadow prices

A linear programme maximises contribution (the objective function) subject to limits on materials, labour, machine time and so on. At the optimum, two ideas describe each limit.

• Binding limit. The optimum plan uses all of the resource. The limit is part of the corner point that gives the best answer. Relaxing it would raise contribution; tightening it would lower contribution.
• Non-binding limit. The optimum plan leaves some of the resource unused. The unused amount is the slack. The limit is not touching the optimum corner, so changing it a little changes nothing.
• Shadow price. The change in the best contribution from one more unit (or one less) of a binding resource. For a non-binding limit it is zero.

The shadow price is a marginal idea. It tells management what the last unit of the resource is worth, in contribution, when the plan is re-optimised. It is also the opportunity cost of using that unit for something else, and so it is the amount to add to the normal price of the resource when a decision uses up that scarce resource.

Why a non-binding limit has no shadow price

If 100 machine hours are idle, a further machine hour adds nothing to contribution. The shadow price of machine time is zero, and a manager should not pay anything extra for it.

Using the shadow price

• Maximum price for extra resource. If a resource normally costs ₦150 an hour, and its shadow price is ₦66.67, then up to ₦216.67 an hour can be paid for extra hours without making the company worse off. Above that price, the extra hours cost more than the contribution they bring.
• Relevant cost of using a scarce resource. The relevant cost is the cash cost plus the shadow price, because using it for one job gives up the contribution it would earn elsewhere.
• Transfer pricing and capital rationing. The shadow price supports an opportunity cost for a supplying division that is short of capacity.
• Valuing a variance. A material usage variance is worth the standard cost of the material plus its shadow price, if the material is scarce.

Range of validity, usefulness and limits

A shadow price is valid only while the same limits stay binding. If a limit is relaxed far enough, a new limit becomes binding and the shadow price changes. The range is found by changing the limit as a symbol and finding the values at which another limit binds or a product falls to zero.

Beyond the range, the shadow price is worth less, and in the end falls to zero.

Uses: it values scarce resources, guides how much to pay for more, and shows where it is worth spending to remove a bottleneck.

Limits: it is valid only for small changes within a range; it assumes that all the figures, including contribution per unit, are certain and linear; it concerns one limit at a time; it is only as good as the model; and it does not show other effects of getting more of the resource, such as extra overheads or quality problems.

Dual prices of demand limits and non-limiting resources

Every constraint has a dual price, not only the resource limits. A sales demand limit has a dual price too: the extra contribution from being able to sell one more unit. A non-limiting constraint has a dual price of zero, because adding more of something already left unused changes nothing.

The dual price is always expressed in the unit of its own constraint, so a labour limit gives a figure per hour (or per minute) and a demand limit a figure per unit. Slack is read the same way: positive slack means the resource is not fully used, zero slack means the limit binds, and a negative figure would mean that the plan breaks the limit and is not feasible.

In a maximisation problem relaxing a “less than or equal to” limit can never reduce contribution, so these dual prices are never negative.

1. Binding limits.
   The demand limit y = 20 and skilled labour 20x + 30y = 900 bind.
   Then 20x + 600 = 900, so x = 15.
   Contribution = 10 × 15 + 30 × 20 = ₦750 a day.
2. Slack.
   Machine 1 used = 30 × 15 + 10 × 20 = 650, slack 250 minutes.
   Machine 2 used = 600 + 900 = 1,500, slack 300 minutes.
   Both dual prices are zero.
3. Skilled labour.
   With 901 minutes: 20x + 600 = 901, so x = 15.05
   contribution = 150.50 + 600 = 750.50.
   Dual price = ₦0.50 a minute, ₦30 an hour.
4. Demand for C.
   With y = 21: 20x = 900 − 630 = 270, so x = 13.5
   contribution = 135 + 630 = 765.
   Dual price = ₦15 per extra unit of demand, because one more C earns ₦30 but displaces 1.5 units of B worth ₦15.
5. Use. Marketing that created one extra unit of daily demand for C would be worth up to ₦15 a unit, and skilled labour up to ₦30 an hour above its normal rate. Additional machine time is worth nothing.

Finding slack and shadow prices

1. Find the optimum. Solve the two binding equations simultaneously to get the plan, and calculate the contribution.
2. Slack. For each limit, limit − use of the plan. Zero slack means binding.
3. Shadow price by a one-unit change. Add one unit to the binding limit, re-solve the two equations, calculate the new contribution and subtract the old.
4. Shadow price by the dual equations.
   For two products and two binding resources, write one equation for each product: (use of resource 1 per unit × price 1) + (use of resource 2 per unit × price 2) = contribution per unit.
   Solve for the two shadow prices.
5. Check. Total resource × shadow price, added across the binding resources, equals the total contribution.
6. Maximum rate for extra units. Normal price + shadow price.

Money is in naira, shadow prices to two decimal places, and each answer is rounded once, at the end.

Worked examples

1. Binding limits.
   Material and labour: 2A + B = 1,000 and A + 2B = 800.
   Multiply the first by 2: 4A + 2B = 2,000.
   Subtract the second: 3A = 1,200, so A = 400, and B = (800 − 400) ÷ 2 = 200.
2. Contribution.
   400 × 400 + 200 × 300 = 160,000 + 60,000 = ₦220,000.
3. Slack.
   Machine hours used = 400 + 200 = 600
   slack = 700 − 600 = 100 hours.
   Machine time is non-binding, so its shadow price is zero.
4. Dual equations. Let m = the shadow price of material and l = that of labour.
   For A: 2m + l = 400.
   For B: m + 2l = 300.
   Multiply the first by 2: 4m + 2l = 800.
   Subtract the second: 3m = 500, so m = ₦166.67 per kg and l = 400 − 333.33 = ₦66.67 per labour hour.
5. One-unit check, labour.
   With 801 hours: A + 2B = 801 and 2A + B = 1,000 give B = 200.67, A = 399.67
   contribution = 159,867 + 60,200 = 220,067, up by ₦66.67.
   Correct.
6. Check on the total.
   1,000 × 166.67 + 800 × 66.67 = 166,670 + 53,336 = ₦220,006, which agrees with ₦220,000 apart from rounding.
7. Maximum prices.
   Labour 150 + 66.67 = ₦216.67 an hour.
   Material 90 + 166.67 = ₦256.67 a kg.

1. Solve with L.
   2A + B = 1,000 and A + 2B = L give A = (2,000 − L) ÷ 3 and B = (2L − 1,000) ÷ 3.
2. Lower end. B cannot be negative: 2L − 1,000 ≥ 0, so L ≥ 500.
3. Upper end.
   Machine time must stay within 700: A + B = (1,000 + L) ÷ 3 ≤ 700, so L ≤ 1,100.
4. Value of 300 extra hours.
   From 800 to 1,100 hours, every hour is worth ₦66.67: 300 × 66.67 = ₦20,000.
   Beyond 1,100 hours, machine time binds and the price changes.
5. Comment. Overtime at up to ₦216.67 an hour is worth buying within this range, provided that the extra material is also available; labour alone cannot raise output without it.

Expect variants: a question that gives the plan and asks only for slack and the shadow price by the dual method, one with a minimum-output constraint, or a request to explain the meaning and limits.

Using shadow prices to set the highest overtime rate

The dual (shadow) price of a binding resource is found from the dual equations: for each product, the shadow prices of the resources it uses, weighted by usage, equal its contribution. Example. A earns ₦600 and B ₦400; skilled labour limit 4A + 2B = 1,000, unskilled A + 3B = 800 (both binding). Solving gives B = 220, A = 140 and contribution ₦172,000. Dual equations: 4x + y = 600; 2x + 3y = 400 gives x = ₦140 (skilled) and y = ₦40 (unskilled); check 140 × 1,000 + 40 × 800 = ₦172,000.

The shadow price is the most the firm should pay above the normal rate for an extra hour: if skilled labour normally costs ₦100 an hour, the highest overtime rate is 100 + 140 = ₦240; for unskilled labour at ₦60, 60 + 40 = ₦100. Non-binding resources have a shadow price of zero. The prices hold only for a limited range of extra hours.

Common mistakes

• Giving a shadow price to a non-binding limit. It is zero.
• Quoting the shadow price as the price to pay. The maximum price is the normal price plus the shadow price.
• Changing the wrong limit. Change only the limit whose shadow price is wanted.
• Forgetting the range. A shadow price is valid over a range only.
• Mixing up slack and shadow price. Slack is unused resource, in units; the shadow price is value per unit.
• Rounding too early. Keep decimals until the final answer.

Quick revision

• Binding: fully used, positive shadow price. Non-binding: slack, shadow price zero.
• Shadow price = change in optimum contribution for one more unit of a binding resource.
• Find by one-unit change or by the dual equations; check that resource × price adds to total contribution.
• Maximum price for extra units = normal price + shadow price.
• Valid over a range only; assumes certainty and linearity.

How the marks are earned

• Explain: binding, slack and shadow price, one mark each with a use.
• Calculate: the two equations, the plan, each shadow price and the check.
• Advise: maximum rate, range and limits.

Practice questions

These are original questions, written in the style of the Skills examination. Work each one before you open the answer.

Recall check: multiple choice

Section B style: written questions

Model answers

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