Linear programming and shadow prices

The ranking rule from lesson 4 works when one resource is scarce. Real plants usually face several constraints at once: paint hours, welding hours, assembly hours, supplies of steel, minimum quantities promised to customers and maximum quantities the market will take. When more than one constraint can bind, ranking by contribution per unit of a single resource no longer guarantees the best answer. Linear programming (LP) solves this more general problem. This lesson shows how to formulate a product mix problem as a linear program, solve it graphically for two products and with software for more, and interpret the results that matter most to managers: shadow prices, slack and the ranges over which they hold. It ends with the limitations that the management accountant must explain when presenting an LP answer.

Formulating a linear program

A linear program has three parts. Decision variables are the quantities to be chosen, such as the number of each product to make. The objective function is the quantity to be maximized or minimized, such as total contribution, written as a linear combination of the decision variables. Constraints are linear inequalities that limit the decision variables, such as the hours available on a machine or the maximum demand for a product. The word linear means that every relationship is proportional: doubling output doubles the hours used and doubles the contribution.

For Northgate's Farm Systems plant, Grace defines three decision variables: the numbers of Auger A10 units (A), Conveyor C8 units (C) and Bin sweep S3 units (S) to make next year. The objective is to maximize contribution, using the contribution per unit from lesson 4. The constraints are the hours of paint, welding and assembly available, the metric tons of steel the plant's supplier has committed to deliver, minimum quantities promised to key dealers, and maximum demand.

The lower bounds on augers and conveyors reflect commitments in Northgate's dealer agreements: key dealers are guaranteed a minimum allocation. The upper bounds are the demand estimates from the sales plan. Getting the formulation right is most of the work. Each coefficient has to come from a reliable source, such as engineering standards for hours and bills of materials for steel, and each constraint must reflect a real limit rather than a convenient assumption.

The graphical method for two products

With two decision variables, an LP can be solved by drawing it. Each constraint is a straight line dividing the possible combinations into those that satisfy it and those that do not. The combinations that satisfy every constraint form the feasible region, a polygon. The objective function can be drawn as a family of parallel iso-contribution lines, each showing combinations that earn the same contribution. The optimal solution is the corner of the feasible region touched by the highest iso-contribution line. A fundamental result of linear programming is that if an optimum exists, at least one corner point is optimal, so one approach is to calculate contribution at every corner and pick the largest.

Grace fixes bin sweeps at their demand limit of 1,900, which uses 1,520 paint hours, and draws the problem for augers and conveyors. The remaining 3,680 paint hours form a line from 1,840 augers with no conveyors to 920 conveyors with no augers. The auger demand limit of 1,400 cuts across it, and so does the conveyor demand limit of 520. The corner where auger demand meets the paint constraint gives 1,400 augers and 220 conveyors, the same mix lesson 4 found by ranking. The graph also shows something the ranking could not: the steel constraint lies very close to that corner, which means a small change in paint hours might make steel binding instead.

Solving with software

Practical LP problems have many variables and constraints and are solved with software. Spreadsheet solvers handle small problems; dedicated optimization libraries handle large ones. Grace uses the open-source HiGHS solver through Python's SciPy library, which also reports the shadow prices she needs.

# Product mix by linear programming with several constraints (invented data).
import numpy as np
from scipy.optimize import linprog

names = ['Auger A10', 'Conveyor C8', 'Bin sweep S3']
cm = np.array([1590, 2700, 1060])            # contribution per unit, from lesson 4
#                 A10   C8    S3      limit
A = np.array([[2.0,  4.0,  0.8],      # paint line hours        5,200
              [3.5,  6.0,  1.5],      # welding hours          10,500
              [5.0,  9.0,  2.5],      # assembly hours         17,000
              [0.9,  1.6,  0.35]])    # steel (t)              2,300
b = np.array([5200, 10500, 17000, 2300])
labels = ['paint hours', 'welding hours', 'assembly hours', 'steel (t)']
bounds = [(150, 1400), (100, 520), (0, 1900)]   # dealer commitments and market demand

res = linprog(-cm, A_ub=A, b_ub=b, bounds=bounds, method='highs')
x = res.x
print('Optimal plan:', {n: round(v) for n, v in zip(names, x)})
print(f"Total contribution: {-res.fun:,.0f}")
for lab, used, cap, dual in zip(labels, A @ x, b, res.ineqlin.marginals):
    print(f"  {lab:15s} used {used:8,.0f} of {cap:7,}  shadow price {-dual:7.2f} per unit")
# check one shadow price by re-solving with 100 more paint hours
b2 = b.copy(); b2[0] += 100
res2 = linprog(-cm, A_ub=A, b_ub=b2, bounds=bounds, method='highs')
print(f"Extra contribution from 100 more paint hours: {(-res2.fun) - (-res.fun):,.0f}")
# reduced costs on the bounds tell us which demand limits bind
for n, lo, up in zip(names, res.lower.marginals, res.upper.marginals):
    print(f"  {n:13s} value of one more unit of demand: {-up:7.2f}")

Optimal plan: {'Auger A10': 1400, 'Conveyor C8': 220, 'Bin sweep S3': 1900}
Total contribution: 4,834,000
  paint hours     used    5,200 of   5,200  shadow price  675.00 per unit
  welding hours   used    9,070 of  10,500  shadow price    0.00 per unit
  assembly hours  used   13,730 of  17,000  shadow price    0.00 per unit
  steel (t)       used    2,277 of   2,300  shadow price    0.00 per unit
Extra contribution from 100 more paint hours: 38,812
  Auger A10     value of one more unit of demand:  240.00
  Conveyor C8   value of one more unit of demand:   -0.00
  Bin sweep S3  value of one more unit of demand:  520.00

The solver confirms the mix of 1,400 augers, 220 conveyors and 1,900 bin sweeps, with contribution of 4,834,000 dollars. Only the paint line is fully used. Welding has 1,430 hours of slack, assembly 3,270 hours, and steel 23 metric tons, a narrow margin. Augers and bin sweeps are at their demand limits.

Shadow prices

The shadow price of a constraint is the change in the optimal objective value from one more unit of the constrained resource, holding everything else constant. It is also called the dual price or marginal value. Constraints with slack have a shadow price of zero: an extra welding hour is worthless when 1,430 are already unused. The binding paint constraint has a shadow price of 675 dollars an hour, the contribution per paint hour of the marginal product, the conveyor, which matches lesson 4.

Shadow prices answer practical questions directly. What is the most Northgate should pay for an extra hour of paint capacity, through overtime, a rented booth or outsourcing? Up to 675 dollars above the normal cost of that hour. Is it worth paying a premium for extra welding hours or steel at the moment? No. The reduced costs on the demand limits answer a related question: an extra unit of bin sweep demand is worth 520 dollars, and an extra auger 240 dollars, because each would displace some conveyor production on the paint line. Those values guide where the sales team should push hardest.

The range of validity

A shadow price holds only over a range. As more of a resource is added, the optimal mix changes, and eventually another constraint binds and the shadow price changes. The program checks this directly by adding 100 paint hours and solving again. If the 675-dollar shadow price held, the extra hours would add 67,500 dollars. Instead they add only 38,812 dollars. The extra paint hours allow more conveyors, but each conveyor needs 1.6 metric tons of steel, and after about 14 more conveyors the steel constraint binds. Beyond that point, extra paint hours are worth much less, because conveyors are limited by steel.

This matters for the second paint booth proposal, which would add about 2,600 hours. Valuing those hours at 675 dollars each would suggest extra contribution of about 1.75 million a year, but that ignores both the steel limit and the conveyor demand limit of 520 units. Only 300 more conveyors could be sold, using 1,200 paint hours, and they would need 480 more metric tons of steel. Unless the steel supply can be increased, the booth's value is much lower. The proper analysis re-solves the LP with the new capacity and any other changes, which is how Grace estimates the operating profit of about 0.55 million a year after depreciation that the paint booth contributes in the capstone, having negotiated additional steel with a second supplier.

Sensitivity of the objective coefficients

Solvers also report how far each product's contribution can change before the optimal mix changes. For Northgate, the conveyor's contribution could rise by a modest amount before the solver would choose to make more conveyors and fewer augers, and could fall a long way before production dropped to the dealer minimum. These ranges help answer questions such as how much a steel price increase, which affects conveyors most, would have to rise before the mix should change. They are useful for planning, but they share the limitation of shadow price ranges: they hold one coefficient changing at a time.

Assumptions and limitations

Linear programming is powerful but rests on assumptions that the accountant must check and explain.

Each assumption has a remedy. Integer programming handles indivisible items, as lesson 10's capital rationing problem shows. Non-linear relationships can sometimes be approximated with several linear pieces. Uncertainty can be explored with sensitivity analysis or simulation, as lessons 6 and 11 describe. Strategic goals can be added as constraints, as Northgate did with its dealer minimums. The most important remedy is judgment: an LP solution is a recommendation based on a model, not an instruction, and the management accountant's job is to explain what the model includes, what it leaves out and how sensitive its answer is.

Other applications

Product mix is the classic application, but LP is widely used wherever limited resources are allocated. Transportation and distribution models choose which warehouse should supply which customer to minimize freight cost. Blending models choose the cheapest mix of ingredients that meets quality specifications, a common problem in feed, fuel and chemicals. Workforce scheduling models choose shift patterns that meet demand at minimum cost. Media planning models allocate an advertising budget across channels. In each case, the formulation follows the same pattern of decision variables, a linear objective and linear constraints, and the shadow prices tell managers what an extra unit of each scarce resource is worth.