# Mixed-Integer Linear Programming Basics: Solver-Based

This example shows how to solve a mixed-integer linear problem. Although not complex, the example shows the typical steps in formulating a problem using the syntax for `intlinprog`

.

For the problem-based approach to this problem, see Mixed-Integer Linear Programming Basics: Problem-Based.

### Problem Description

You want to blend steels with various chemical compositions to obtain 25 tons of steel with a specific chemical composition. The result should have 5% carbon and 5% molybdenum by weight, meaning 25 tons*5% = 1.25 tons of carbon and 1.25 tons of molybdenum. The objective is to minimize the cost for blending the steel.

This problem is taken from Carl-Henrik Westerberg, Bengt Bjorklund, and Eskil Hultman, “*An Application of Mixed Integer Programming in a Swedish Steel Mill*.” Interfaces February 1977 Vol. 7, No. 2 pp. 39–43, whose abstract is at https://doi.org/10.1287/inte.7.2.39.

Four ingots of steel are available for purchase. Only one of each ingot is available.

$$\begin{array}{ccccc}Ingot& Weight\phantom{\rule{0.2777777777777778em}{0ex}}in\phantom{\rule{0.2777777777777778em}{0ex}}Tons& \%\phantom{\rule{0.2777777777777778em}{0ex}}Carbon& \%\phantom{\rule{0.2777777777777778em}{0ex}}Molybdenum& \frac{Cost}{Ton}\\ 1& 5& 5& 3& \$350\\ 2& 3& 4& 3& \$330\\ 3& 4& 5& 4& \$310\\ 4& 6& 3& 4& \$280\end{array}$$

Three grades of alloy steel and one grade of scrap steel are available for purchase. Alloy and scrap steels can be purchased in fractional amounts.

$$\begin{array}{cccc}Alloy& \%\phantom{\rule{0.2777777777777778em}{0ex}}Carbon& \%\phantom{\rule{0.2777777777777778em}{0ex}}Molybdenum& \frac{Cost}{Ton}\\ 1& 8& 6& \$500\\ 2& 7& 7& \$450\\ 3& 6& 8& \$400\\ Scrap& 3& 9& \$100\end{array}$$

To formulate the problem, first decide on the control variables. Take variable `x(1) = 1`

to mean you purchase ingot **1**, and `x(1) = 0`

to mean you do not purchase the ingot. Similarly, variables `x(2)`

through `x(4)`

are binary variables indicating whether you purchase ingots **2** through **4**.

Variables `x(5)`

through `x(7)`

are the quantities in tons of alloys **1**, **2**, and **3** that you purchase, and `x(8)`

is the quantity of scrap steel that you purchase.

### MATLAB® Formulation

Formulate the problem by specifying the inputs for `intlinprog`

. The relevant `intlinprog`

syntax is:

`[x,fval] = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub)`

Create the inputs for `intlinprog`

from the first (`f`

) through the last (`ub`

).

`f`

is the vector of cost coefficients. The coefficients representing the costs of ingots are the ingot weights times their cost per ton.

f = [350*5,330*3,310*4,280*6,500,450,400,100];

The integer variables are the first four.

intcon = 1:4;

**Tip:** To specify binary variables, set the variables to be integers in `intcon`

, and give them a lower bound of `0`

and an upper bound of `1`

.

The problem has no linear inequality constraints, so `A`

and `b`

are empty matrices (`[]`

).

A = []; b = [];

The problem has three equality constraints. The first is that the total weight is 25 tons.

`5*x(1) + 3*x(2) + 4*x(3) + 6*x(4) + x(5) + x(6) + x(7) + x(8) = 25`

The second constraint is that the weight of carbon is 5% of 25 tons, or 1.25 tons.

`5*0.05*x(1) + 3*0.04*x(2) + 4*0.05*x(3) + 6*0.03*x(4)`

` + 0.08*x(5) + 0.07*x(6) + 0.06*x(7) + 0.03*x(8) = 1.25`

The third constraint is that the weight of molybdenum is 1.25 tons.

`5*0.03*x(1) + 3*0.03*x(2) + 4*0.04*x(3) + 6*0.04*x(4)`

` + 0.06*x(5) + 0.07*x(6) + 0.08*x(7) + 0.09*x(8) = 1.25`

Specify the constraints, which are Aeq*x = beq in matrix form.

Aeq = [5,3,4,6,1,1,1,1; 5*0.05,3*0.04,4*0.05,6*0.03,0.08,0.07,0.06,0.03; 5*0.03,3*0.03,4*0.04,6*0.04,0.06,0.07,0.08,0.09]; beq = [25;1.25;1.25];

Each variable is bounded below by zero. The integer variables are bounded above by one.

```
lb = zeros(8,1);
ub = ones(8,1);
ub(5:end) = Inf; % No upper bound on noninteger variables
```

### Solve Problem

Now that you have all the inputs, call the solver.

[x,fval] = intlinprog(f,intcon,A,b,Aeq,beq,lb,ub);

Running HiGHS 1.7.0: Copyright (c) 2024 HiGHS under MIT licence terms Coefficient ranges: Matrix [3e-02, 6e+00] Cost [1e+02, 2e+03] Bound [1e+00, 1e+00] RHS [1e+00, 2e+01] Presolving model 3 rows, 8 cols, 24 nonzeros 0s 3 rows, 8 cols, 18 nonzeros 0s Solving MIP model with: 3 rows 8 cols (4 binary, 0 integer, 0 implied int., 4 continuous) 18 nonzeros Nodes | B&B Tree | Objective Bounds | Dynamic Constraints | Work Proc. InQueue | Leaves Expl. | BestBound BestSol Gap | Cuts InLp Confl. | LpIters Time 0 0 0 0.00% 0 inf inf 0 0 0 0 0.0s 0 0 0 0.00% 8125.6 inf inf 0 0 0 4 0.0s R 0 0 0 0.00% 8495 8495 0.00% 5 0 0 5 0.0s Solving report Status Optimal Primal bound 8495 Dual bound 8495 Gap 0% (tolerance: 0.01%) Solution status feasible 8495 (objective) 0 (bound viol.) 0 (int. viol.) 0 (row viol.) Timing 0.01 (total) 0.00 (presolve) 0.00 (postsolve) Nodes 1 LP iterations 5 (total) 0 (strong br.) 1 (separation) 0 (heuristics) Optimal solution found. Intlinprog stopped at the root node because the objective value is within a gap tolerance of the optimal value, options.AbsoluteGapTolerance = 1e-06. The intcon variables are integer within tolerance, options.ConstraintTolerance = 1e-06.

View the solution.

x,fval

`x = `*8×1*
1.0000
1.0000
0
1.0000
7.2500
0
0.2500
3.5000

fval = 8495

The optimal purchase costs $8,495. Buy ingots **1**, **2**, and **4**, but not **3**, and buy 7.25 tons of alloy **1**, 0.25 ton of alloy **3**, and 3.5 tons of scrap steel.

Set `intcon = []`

to see the effect of solving the problem without integer constraints. The solution is different, and is not realistic, because you cannot purchase a fraction of an ingot.