Contribution per unit for model 101 = $3000
Contribution per unit for model 102 = $5000
Merton Truck Company Questions:
1. To identify the best product mix for Merton, the objective is to maximize contribution. Fixed overhead costs are not part of the objective. Once the decisions to make the products and operate the departments are taken fixed costs are sunk costs (nothing can be done about it).
Decision variables:
Let x1 be the quantity of model 101 produced and sold
Let x2 be the quantity of model 102 produced and sold.
Objective: Maximize total contribution
Max: 3000*x1 + 5000*x2
Constraints: Machine hour availabilities and machine requirements for each department are the constraints
For engine assembly – 1 machine hour is required to produce 1 unit of model 101, therefore for quantity of x1 total machine hours in engine assembly required for model 101 = 1*x1 = x1
2 machine hours are required to produce 1 unit of model 102, therefore for quantity of x2 total machine hours in engine assembly required for model 102 = 2*x2 = 2x2
Total machine hours required in engine assembly = x1 + 2x2
Total machine hours available in engine assembly = 4000
Machine hours required must not exceed machine hours available and hence we have
x1 + 2x2 <= 4000 Constraint 1
For metal stamping assembly – Total machine hours required = 2x1 + 2x2
Total availability = 6000
2x1 + 2x2 <= 6000 Constraint 2
For model 101 assembly – Total machine hours required = 2x1
Total availability = 5000
2x1 <= 5000 Constraint 3
For model 102 assembly – Total machine hours required = 3x1
Total availability = 4500
3x1 <= 4500 Constraint 4
Non-negativity constraints: x1>=0, x2>=0
Complete linear program is:
Max: 3000*x1 + 5000*x2
Subject to
x1 + 2x2 <= 4000 Constraint 1
2x1 + 2x2 <= 6000 Constraint 2
2x1 <= 5000 Constraint 3
3x1 <= 4500 Constraint 4
a. Using solver tool the best or optimal solution is:
Model 101 = 2000
Model 102 = 1000
Solution report is as follows
Total profit = Total contribution margin – total fixed overhead cost = $2,400,000
b. Sensitivity analysis table is reproduced below:
Engine assembly is a binding constraint (also indicated by positive shadow price). Shadow price of engine assembly depart is $2000, which means for 1 unit increase in capacity objective function will increase by $2000. Allowable increase in engine assembly capacity is 500, which means that the basis for optimal solution will remain the same if engine assembly capacity is increased by 500 units. Basis remains the same means that products, which are part of optimal solution, will be in the optimal solution with positive value if engine assembly capacity is increased by 500 units.
If engine assembly capacity is increased by one unit, then either produce one additional unit of model 101 or model 102. The best product mix is produce 1999 of model 101 and 1001 of model 102.
The shadow price of engine assembly capacity gives worth of it. The shadow price is $2000 and hence it is the worth of one additional unit of engine assembly capacity. This can be confirmed by increase in objective function value by 2000.
c. Each additional unit engine assembly capacity is worth of $2000. Allowable increase is 500 units, which means increase of 100 is within the limit and so calculations can be performed directly. With additional 100 units objective function will increase by 100*$2000 = $200,000.
d. The limit that can be added before worth of engine assembly capacity changes is 500 units of engine assembly capacity.
2. Renting of capacity means buying extra capacity. It means that by adding 1 unit of capacity, company will see increase in objective function value by shadow price of that capacity. Therefore, it is the maximum price that company should be willing to pay to the supplier. Any price less that shadow price will result in benefit to the company and any price less than shadow price will result in loss to the company. Renting must be done only for departments which have binding capacities.
According to sensitivity analysis assembly engine capacity is binding. Each additional unit of engine assembly capacity will result in increase in total contribution margin by $2000. In addition capacity can be increased by 500 units; therefore company must pursue this alternative. Company must be willing to pay up to $2000 for each additional unit of engine assembly capacity. It should rent up to 500 units of engine assembly capacity as anything above will change the current optimal solution.
3. Contribution margin for model 103 = $2000
Maximum of 5000 can be produced in engine assembly at full capacity, which mean per unit machine hours requirement in engine assembly for model 103 = available hours/5000 = 4000/5000 = 0.8
Similarly each unit of model 103 consumes 6000/4000 = 1.5 machine hours in metal stamping department.
Each unit of model 103 consumes 1 machine hour in model 101 assembly.
a. Model 101 assembly is not a binding constraint and therefore perform analysis for engine assembly and metal stamping department.
According to current optimal solution shadow price for engine assembly is $2000, which also means that decrease in 1 unit of assembly capacity will result in decrease in objective function by $2000. Each unit of model 103 consumes 0.8 machine hours of assembly engine capacity and hence production of 1 unit of model 103 will reduce the objective function by 0.8*$2000 = $1600
Similarly each unit of model 103 consumes 1.5 machine hours of metal stamping department. Shadow price of metal stamping capacity is $500. If 1 unit of model 103 is produced then objective function will reduce by 1.5*$500 = $750.
Total reduction in objective function due to production of 1 unit of model 103 = $1600 + $750 = $2350
Contribution per unit of model 103 = $2000
If 1 unit of model 103 is produced there will be loss of $2350 - $2000 = $350
Hence, model 103 should not be produced.
b. With the given machine hour requirements for model 103 the minimum contribution margin per unit for model 103 must be equal to reduction in objective function due to 1 unit of model 103. One unit of model 103 results in reduction profit by $2350 and hence the contribution margin per unit for model 103 must be equal to $2350 so that net benefit is $2350 - $2350 = $0. Anything above $2350, model 103 must be produced. Anything below $2350, model 103 must not produced.
4.
Direct labor cost for model 101 in engine assembly will become 1.5*$1200 = $1800
Direct labor cost for model 102 in engine assembly will become 1.5*$2400 = $3600
Contribution margin per unit for model 101 will reduce to $2400
Contribution margin per unit for model 102 will reduce to $3800
Model 101 contribution margin can reduce by 500 and model 102 contribution margin can reduce by 2000.
Due to overtime contribution margin for model 101 will decrease by $3000 - $2400 = $600 and for model 102 it will reduce by $5000 - $3800 = $1200.
These reductions are within the ranges allowed and optimal solution basis remains same.
Each additional unit of engine assembly capacity is worth $2000 and it can be increased till 500 units. For 500 units increase objective function value will increase by 500*$2000 = $1000000.
Increase in overhead costs is $0.75 million, therefore net benefit is $1 - $0.75 = $0.25 millions. This suggests that overtime option must be pursued.
5. Additional constraint:
Quantity of model 101 produced and sold = x1
Quantity of model 102 produced and sold = x2
New requirement is given below, which is called as production constraint.
Quantity of model 101 >= 3*quantity of model 102
x1 >= 3x2 Constraint 5
New linear program is
Max: 3000*x1 + 5000*x2
Subject to
x1 + 2x2 <= 4000 Constraint 1
2x1 + 2x2 <= 6000 Constraint 2
2x1 <= 5000 Constraint 3
3x1 <= 4500 Constraint 4
x1 >= 3x2 Constraint 5
New optimal product mix is
Model 101 = 2250
Model 102 = 750
Total contribution margin = $10,500,000
Total profit = $1,900,00
