Production problems
Production Problems
Production problems decide how much of each product or activity to produce.
They are among the most common linear programming models.
Basic Structure
A production problem usually has:
- products or activities
- profit per unit
- resources consumed per unit
- limited resource capacities
- nonnegative production quantities
Decision Variables
If a firm produces $n$ products, define:
\[x_j=\text{quantity of product }j\text{ produced}\]for:
\[j=1,\ldots,n\]Then:
\[x=(x_1,\ldots,x_n)\]is the production plan.
Objective Function
If product $j$ gives profit $p_j$, then total profit is:
\[p_1x_1+p_2x_2+\cdots+p_nx_n\]The objective is usually:
\[\max \sum_{j=1}^n p_jx_j\]If the problem gives cost instead of profit, the objective may be:
\[\min \sum_{j=1}^n c_jx_j\]Resource Constraints
If product $j$ uses $a_{ij}$ units of resource $i$, and resource $i$ has capacity $b_i$, then:
\[a_{i1}x_1+a_{i2}x_2+\cdots+a_{in}x_n\le b_i\]for each resource $i$.
General Production LP
The standard production model is:
\[\max \sum_{j=1}^n p_jx_j\]subject to:
\[\sum_{j=1}^n a_{ij}x_j\le b_i, \qquad i=1,\ldots,m\] \[x_j\ge 0, \qquad j=1,\ldots,n\]Matrix Form
Let:
- $p$ be the profit vector
- $A$ be the resource-consumption matrix
- $b$ be the capacity vector
Then:
\[\max p^Tx\]subject to:
\[Ax\le b\] \[x\ge 0\]Example: Two Products
A firm produces two products.
Let:
\[x_1=\text{units of product A}\] \[x_2=\text{units of product B}\]Suppose:
| Resource | Product A | Product B | Available |
|---|---|---|---|
| Machine time | 2 | 3 | 30 |
| Craftsman time | 5 | 5 | 60 |
Profits are:
\[60\]for product A and:
\[84\]for product B.
The LP is:
\[\max 60x_1+84x_2\]subject to:
\[2x_1+3x_2\le 30\] \[5x_1+5x_2\le 60\] \[x_1,x_2\ge 0\]The second constraint can also be simplified:
\[x_1+x_2\le 12\]Example: Production with Departments
Suppose a plant produces three vehicle models $A,B,C$.
Let:
\[x_1=\text{vehicles of model A}\] \[x_2=\text{vehicles of model B}\] \[x_3=\text{vehicles of model C}\]If the profit per vehicle is:
\[840,\quad 1120,\quad 1200\]then the objective is:
\[\max 840x_1+1120x_2+1200x_3\]Each department gives one capacity constraint.
For example, if motor hours are limited to $120$ and the models use $3,2,1$ hours, then:
\[3x_1+2x_2+x_3\le 120\]If bodywork hours are limited to $80$ and the models use $1,2,3$ hours, then:
\[x_1+2x_2+3x_3\le 80\]Model-specific finishing departments add constraints such as:
\[2x_1\le 96\] \[3x_2\le 102\] \[2x_3\le 40\]Minimum Production Requirements
Sometimes the problem gives minimum demand.
Example:
\[x_1\ge 100\]means at least $100$ units of product $1$ must be produced.
If products contribute to a shared demand requirement:
\[2x_1+3x_2\ge 500\]This is still linear.
Production vs Resource Allocation
Production problems are a special case of resource allocation.
The resources are inputs.
The decision variables are activity levels.
The objective is usually profit.
Common Mistakes
Avoid these mistakes:
- using resource availability as the objective
- forgetting product-specific capacity constraints
- mixing weekly and annual quantities
- forgetting nonnegativity
- using revenue when profit is given
- using $\ge$ for a capacity constraint
Key Takeaway
A production LP chooses activity levels to maximize profit or minimize cost while respecting limited resource capacities.
Exam checkpoint
For formulation questions, show variables, objective, constraints, sign restrictions, and units. Do not solve unless the question asks for a solution.