Executive Summary

The Problem: Merton Truck Company has been experiencing difficulties related to their financial performance, and they do not know which is the optimal product mix to maximize profits and performance. Group 2 was asked to make recommendations and analyze the given situation to eliminate the difficulties and come up with the right product mix and the optimal solutions considering different alternatives and scenarios.

The Solution: Linear Programming was used to analyze the different alternatives to arrive at the optimal solution. The optimal product mix was calculated to be 2000 units of model 101 and 1000 units of model 102. In the next pages you will see the answers to the different questions asked which shows how different alternatives and interpretations could affect the optimal solution and the final decision.

Recommendations and conclusion: We recommend that Merton Truck Company adopt the suggest solutions in our analysis in order to achieve better financial results and impose new policies later on to avoid such setbacks in the future. Following our recommendations, the firm should be able to maximize the profits using the optimal product mix.
Model Description

The Problem
Merton Truck Company is not optimizing contribution from its truck lines especially model 101. The company is contemplating stopping the production of model 101 trucks. Management is in a dilemma on what approach to take in resolving this problem. In addition, two of its production lines are operating at capacity while the other two are operating below capacity.

The Solution Approach Our analysis was done using mathematical models and linear programming. The aim of the analysis was to maximize contribution and allocate truck production in an optimal manner. The mathematical models consist of variables, constraints and an objective function.

Variables
The problem has only two main variables namely number of truck model 101 and 102 to produce. A third variable in respect of the proposed model 103 truck is introduced later.

Constraints
The problem has four constraints. They reflect limitations imposed by capacity in the engine assembly line, metal stamping line, model 101 assembly line and model 102 production line.

Objective
The objective aim of the analysis is to maximize contribution/profits. The objective function calculates the profits by determining the optimal units of each truck to be produced and multiplying same with its contribution margin.

Solution Analysis
Question 1
In this section, we are calculating the optimal product mix under different scenarios.
Part A:
Fixed costs will not be affected by the mix of products so it shall be deducted after we calculate the total profit if we disregard the fixed over head we shall have the following costs (include only variable overhead): Per unit Model 101 Model 102
Direct Material 24000 20000
Direct Labor 4000 4500
Variable overhead 8000 8500
Cost: 36000 33000
Subtract from Selling Price
Selling Price 39000 38000
Cost 36000 33000
Profit (without FO) 3000 5000 Using Excel Solver we can calculate the Total production units of models 101 and 102, and the
Maximized profit. Model 101 102 (x1) (x2)
Profit per unit 3000 5000
Production 2000 1000 Total
Profit 6000000 5000000 11000000 constraints Model 101 Model 102 usage RHS Leftover
Engine Assembly 1 2 4000 TRUE 4000 0
Metal stamping 2 2 6000 TRUE 6000 0
Model 101 assembly 2 0 4000 TRUE 5000 1000
Model 102 assembly 0 3 3000 TRUE 4500 1500

However now we need to deduct the fixed overhead costs which are $8.6 million from the $11.00 million. Maximum profit after deducting the fixed overhead costs is $2.4 million
Part B:
When there are potential changes in the parameter of LP models, Sensitivity Analysis is required. Here we have an increase in the RHS value of the Engine assembly constraint from 4000 to 4001. Each extra unit of capacity of Engine Assembly is worth $2000 (shadow price of engine assembly capacity) because all the hours were used up.
The contribution per unit or the objective coefficient for model 102 is 5000 $ so to maximize the profit we should produce one more unit of model 102
Part C:
Each extra unit of capacity of engine assembly is worth $2000 (shadow price of engine assembly capacity).
100 extra units of capacity of engine will be worth $ 200,000 which is 100 times the value of one extra unit of capacity ($2000)
Part D:
The allowable increase is 500 units so it is safe to say that a change might occur in the value of any additional unit in the capacity only if we exceed that limit. See appendix A for more details
Question 2
Under this question Merton Trucks management is considering the option of outsourcing engine assembly in order to maximize returns. In order to relieve the capacity problem in the engine assembly department, Merton’s production manager was contemplating whether to purchase 101 or 102 engines from an outside supplier. By doing so, Merton will basically be renting out capacity to a third party, which in other words means that the company will supply the required materials and engine components to the third party, while reimbursing the third party for labor and overhead. Merton should only adopt the alternative of purchasing from an outside supplier only if the find a supplier that charges $2000 (the shadow price from the analysis in question 1) or less per machine hour used. In the context of business, the shadow price is the maximum price that a manager is willing to pay for an additional unit of a limited resource. An example of shadow price could be the price that a manager would be willing to pay for keeping a production line operational for an additional hour, beyond its maximum 40 hour limit, as determined from the benefits that the company would derive from this change. Furthermore, if Merton chooses to go ahead with the option to purchase from a third party, the maximum rent the company should be willing to pay is $2000 per machine hour (again the shadow price). Finally, the maximum number of machine hours that Merton should consider renting out is 500, which is the allowable increase (as indicated in question 1). The allowable increase represents the level to which we can increase production hours while retaining the current profit margin. The shadow price is applicable up to the allowable increase or decrease in the constraint.
Question 3
Introduction of truck model 103
An analysis of the objective of the linear programming equation reveals that Merton should not produce any model 103 trucks at the current contribution of $2000. This is because the contribution of $2000 is far below contributions of $3000 and $5000 derived from trucks 101 and 102 and thus not worthwhile to deploy resources into its production compared to other truck models.
In analyzing this problem the objective function seeks to maximize contribution and is stated as
3000X1 + 5000X2 +2000X3 = P where
X1 = No. of truck 101 produced
X2= No. of truck 102 produced
X3= No. of truck 103 produced
The constraints for this objective function are stated as follows:
Engine assembly - 1X1 + 2X2+ 0.8X3=4000
Metal stamping - 2X1 + 2X2 + 1.5X3 = 6000
Model 101 assembly- 2X1 + 0X2 + 1X3 = 5000
Model 102 assembly - 0X1 + 3X2 + 0X3 =4500
Given this production level(2000 units of truck 101 and 1000 units of truck 102) all resources are used in engine assembly and metal stamping departments while a slack of 1000 and 1500 machine hours occurs in model 101 assembly and model 102 assembly plants respectively. At this level of production, contribution is maximized at $11,000,000. 2000 units of truck 101 and 1000 units of truck 102 are recommended for production.

Table indicating machine hours usage when production is at 2000 units of 101 trucks and 1000 units of 102 trucks

production usage Total available Slack/Unused
Engine assembly 4000 4000 0
Metal stamping 6000 6000 0
101 /103 assembly 4000 5000 1000
102 assembly 3000 4500 1500

Contribution needed to introduce truck model 103
If Merton desires to introduce model 103 trucks all hope is not lost. The model can successfully be introduced if its contribution margin increases from $2000 to $2350.5. At this point, 2857 units of the truck 103 and 857 units of truck 102 will be produced. However, no unit of truck 101 is recommended for production. Contribution is maximized at $11,001,429. Under this scenario, use of machine hours in production is as follows:
Table indicating machine hours usage when production is at 857 units of truck 102 and 2857 units of truck 103 production usage Total available Slack/Unused
Engine assembly 4000 4000 0
Metal stamping 6000 6000 0
101 /103 assembly 2857.14 5000 2142.85
102 assembly 2571.42 4500 1928.57
See appendix for B for other details.
Question 4
Here Management is considering utilizing overtime in the engine assembly department.
When given 2000 machine hours in overtime to allocate to the engine assembly department the production manager should consider whether this is profitable for the company. Would this increase the efficiency of the production process? When producing in overtime the company increases labor costs by 50% during this period which is translated to $600 additional dollars in labor costs for engine model 101 and $1200 for engine model 102. Furthermore production in overtime increases fixed overheads by $0.75 million per month. Now the question is how does this affect Merton Trucks’ profitability?
Taking the additional cost into consideration we conclude that at the optimal production mix including overtime engine production the firms profitability would drop from $2.4 million to $2.35 million. From this we conclude that the company should not assemble engines in overtime.
The details of how we came to this conclusion are provided in appendix D.
Question 5
Here management wants to maximize contribution but needs to produce 3 truck 101 models for every truck 102 model produced. They want to know the optimal mix to pursue.
The objective function for this problem is stated as 9000X1 + 5000X2 = P
The optimal result corresponds to producing 2500 units of truck 101 and 500 units of truck 102. At this point profits are maximized at $25million. Truck 101 produces $22.5 million while truck 102 produces $2.5 million.
The constraints at this point remain as initially given. However, engine assembly and model 102 assembly lines produce slack of 500 and 3000 machine hours respectively.
See appendix D for more details.