Exeter Mines produces iron ore at four different mines; however, the ores extracted at each mine are different in their iron content. Mine 1 produces magnetite ore, which has a 70% iron content; mine 2 produces limonite ore, which has a 60% iron content; mine 3 produces pyrite ore, which has a 50% iron content; and mine 4 produces taconite ore, which has only a 30% iron content. Exeter has three customers that produce steel Armco, Best, and Corcom. Armco needs 400 tons of pure (100%) iron, Best requires 250 tons of pure iron, and Corcom requires 290 tons. It costs $37 to extract and process 1 ton of magnetite ore at mine 1, $46 to produce 1 ton of limonite ore at mine 2, $50 per ton of pyrite ore at mine 3, and $42 per ton of taconite ore at mine 4. Exeter can extract 350 tons of ore at mine 1; 530 tons at mine 2; 610 tons at mine 3; and 490 tons at mine 4. The company wants to know how much ore to produce at each mine in order to minimize cost and meet its customers' demand for pure (100%) iron.
a. Formulate a linear programming model for this problem.
b. Do any of the mines have slack capacity? If yes, which one(s)?
c. If Exeter Mines could increase production capacity at any one of its mines, which should it be? Why?
d. If Exeter were to decide to increase capacity at the mine identified in (b), how much could it increase capacity before the optimal solution point (i.e., the optimal set of variables) would change?
e. If Exeter were to determine that it could increase production capacity at mine 1 from 350 tons to 500 tons, at an increase in production costs to $43 per ton, should it do so?
Unit Costs:
Magnetite: $37 => 0.7 ton of iron =>$52.86/ton
limonite: $46 => 0.6 ton => $76.67/ton
pyrite: $50 => 0.5 ton => $100/ton
taconite: $42=> 0.3 ton => $140/ton
Production:
magnetite: 350*0.7=245 tons
limonite: 0.6*530=318 tons
pyrite: 0.5*610=305 tons
taconite: 0.3*490=147 tons
Total = 1015 tons
Demand:
400+250+290=940 tons
Therefore there is excess capacity.
The given data does not give transportation costs to supply which customer, so if they are all equal, the least expensive unit price would prevail.
If further data are given, the answer may change.
a. The linear programming model for this problem can be formulated as follows:
Let x1, x2, x3, x4 be the quantities of ore produced at mines 1, 2, 3, 4 respectively.
The objective is to minimize the total cost, which can be expressed as:
Minimize 37x1 + 46x2 + 50x3 + 42x4
Subject to the following constraints:
- Iron content constraint for Armco: 0.7x1 + 0.6x2 + 0.5x3 + 0.3x4 >= 400
- Iron content constraint for Best: 0.7x1 + 0.6x2 + 0.5x3 + 0.3x4 >= 250
- Iron content constraint for Corcom: 0.7x1 + 0.6x2 + 0.5x3 + 0.3x4 >= 290
- Mine production constraints: x1 <= 350, x2 <= 530, x3 <= 610, x4 <= 490
- Non-negativity constraints: x1, x2, x3, x4 >= 0
b. To determine if any of the mines have slack capacity, we need to check if any of the mine production constraints are non-binding, i.e., if any of the variables are not at their upper limits. In this case, mine 1 has 350 tons of ore production capacity, which is its upper limit. Therefore, mine 1 does not have slack capacity.
c. To determine which mine Exeter Mines should increase production capacity at, we need to analyze the objective function coefficients for each mine. Since the objective is to minimize cost, we need to identify the mine with the lowest cost per ton of ore. Comparing the costs:
- Mine 1: $37 per ton
- Mine 2: $46 per ton
- Mine 3: $50 per ton
- Mine 4: $42 per ton
Mine 1 has the lowest cost per ton of ore, therefore Exeter Mines should increase production capacity at mine 1.
d. To determine the maximum capacity increase at mine 1 before the optimal solution point changes, we can perform sensitivity analysis on the shadow price of the iron content constraint for Armco, as that constraint is binding. The shadow price represents the increase in the objective function value for a one-unit increase in the right-hand side of the constraint.
e. To determine if Exeter Mines should increase production capacity at mine 1 from 350 tons to 500 tons, at an increase in production costs to $43 per ton, we need to compare the increase in revenue from meeting the additional demand to the increase in production costs.
We can perform sensitivity analysis on the shadow price of the iron content constraint for Armco to determine the increase in revenue for the additional 150 tons of ore. Additionally, we need to calculate the increase in production costs for 150 tons at $43 per ton.
If the increase in revenue exceeds the increase in production costs, then Exeter Mines should increase production capacity at mine 1. Otherwise, it may not be cost-effective.
a. To formulate a linear programming model for this problem, we need to define the decision variables, objective function, and constraints.
Decision variables:
Let x1, x2, x3, and x4 represent the amount of ore produced at mines 1, 2, 3, and 4, respectively.
Objective function:
Minimize the total cost of production. The cost includes the extraction and processing costs for each type of ore produced at each mine.
Minimize: 37x1 + 46x2 + 50x3 + 42x4
Constraints:
1. Iron content constraint: The total amount of iron produced from each type of ore should be sufficient to meet the demand of each steel company, considering the iron content of each ore type.
70% x1 + 60% x2 + 50% x3 + 30% x4 >= 400 tons (Armco)
70% x1 + 60% x2 + 50% x3 + 30% x4 >= 250 tons (Best)
70% x1 + 60% x2 + 50% x3 + 30% x4 >= 290 tons (Corcom)
2. Capacity constraint: The total amount of ore produced at each mine should not exceed its production capacity.
x1 <= 350 tons (Mine 1)
x2 <= 530 tons (Mine 2)
x3 <= 610 tons (Mine 3)
x4 <= 490 tons (Mine 4)
3. Non-negativity constraint: The amount of ore produced at each mine should be non-negative.
x1 >= 0
x2 >= 0
x3 >= 0
x4 >= 0
b. To determine if any of the mines have slack capacity, we need to check if the production capacity constraint is binding or not.
For mine 1:
Capacity constraint: x1 <= 350 tons
If the optimal solution satisfies x1 = 350 tons, there is no slack capacity. If x1 < 350 tons, there is slack capacity.
For mine 2, 3, and 4:
Capacity constraints: x2 <= 530 tons, x3 <= 610 tons, x4 <= 490 tons
We perform similar checks for each mine.
c. To determine which mine Exeter Mines should increase production capacity at, we need to calculate the shadow price for each constraint. The shadow price represents the marginal value of increasing the production capacity by one unit at each mine.
To find the shadow price, we perform sensitivity analysis on the iron content constraint for each steel company, while keeping the other constraints unchanged. The shadow price for an iron content constraint represents the cost reduction per ton of iron content required by that steel company.
d. To determine how much the capacity at the identified mine can be increased before the optimal solution changes, we perform sensitivity analysis on the production capacity constraint for that mine, while keeping the other constraints and objective function unchanged. We increase the production capacity constraint gradually and observe the change in the optimal solution until it changes.
e. To determine if Exeter Mines should increase production capacity at mine 1 from 350 tons to 500 tons, at an increase in production costs to $43 per ton, we need to compare the increase in revenue due to the increased production with the additional cost incurred.
We can calculate the new objective function value (total cost) by substituting the new production capacity and the increased production cost for mine 1 into the objective function. If the decrease in cost due to increased production at mine 1 is greater than or equal to the increased production cost, Exeter Mines should increase the production capacity. Otherwise, it may not be economically beneficial to do so.