Merton Trucks Case Note

Abstract

We discuss Merton Trucks [Dhe90a] as a case to introduce linearprogramming in the MBA program. This case adapted from ShermanMotor Company case, was used to introduce Linear Programmingformulations as well as duality. Refer to the teaching note [Dhe90b].

Our approach differs from the approach suggested byDhebar [Dhe90b]. First, our audience consists pre-dominantly of en-gineers with not too much work experience. As a result, handlingmath and algebra is relatively easy. Explaining the algebraic formula-tion, graphical approach and using the Excel solver do not consumethat much time. Second, because this case is used during the firstweek of the MBA program, students are still unfamiliar with the casemethodology and we spend significant time in understanding casefacts. The circular logic used in allocating fixed costs based on theproduct mix that in turn is used in deciding the product mix takessome time to understand. Third, because of the participant back-ground, they have difficulty in translating the model to the specificbusiness situation and interpreting the trade-offs involved in variouswhat-if analyses that are prompted by the case questions.

We return to the case when we teach duality. After explainingduality, we analyze the case to show how some of the questions andwhat-if analyses can be simplified using duality.

This note is based on our experiences with teaching three largebatches of students in our MBA programs.

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1 Without Duality: (2 × 70) minutes

We use the case to illustrate the following issues:

• Analyze case facts:

– What is the factual basis for some of the opinions expressed?

– Where did the overhead cost numbers shown in Table B comefrom?

– Is it appropriate to allocate fixed costs as done in Table C?

• Decision making alternatives:

– Evaluate one product suggestions and current product-mix.

– Evaluate trade-offs and how to improve a solution manuallywith its limitations.

– Algebraic formulation of the linear program.

– Graphical solution method and its limitations.

– Solution using Excel solver.

1.1 Objective: 5 minutes

We ask the students to articulate in plain English answers to the followingquestions without going into the low level details:

• What is Merton deciding?

• On what should this decision be based?

• What constrains Merton?

1.2 Roles and Rationales: 5 minutes

We would like the students to understand the data that supports the opin-ions expressed in the case. This is a prelude to analyzing validity of thedata.

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• Why do the company president and sales manager feel that 101’sare making a loss and hence 101’s production should be stopped?From Table B, it costs $40,205 to produce a 101-truck while it sells for$39,000.

• Why do the president and the production manager feel outsourcingengine assembly will help? No slack in engine assembly currently.Show “Resource Usage” worksheet of merton facts.xls.

• Why does the controller feel that cutting back on 102s is an answer?Overheads are the answer and they are dealt next.

1.3 Understanding the Exhibits: 15 minutes

The numbers in Table A are straight forward resource utilization num-bers. The numbers in Table B show total costs including fixed overheadsallocated based on Table C. Allocation of fixed costs are done based onthe ratio of resource usage. The fallacy in allocating fixed costs based onproduct mix to decide the product mix can be illustrated by using the “Ex-hibits” worksheet of merton facts.xls. The grayed out areas represent datathat does not change with product mix. Try the following combinations asa basis for the controller’s comments:

• (1000, 1500) for negative contributions of 101s and to explain currentexhibit values in Table B,

• (2500, 500) for both positive contributions,

• (2500, 125) for negative contributions of 102s and

• (500, 125) for both negative contributions.

1.4 Relevance of Overheads: 5 minutes

Allocated fixed costs should not be included in deciding the contributions.The actual costs for (101s, 102s) are:

• direct materials - ($24,000, $20,000) from Table B,

• direct labor - ($4,000, $4,500) from Table B and,

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• variable overhead - ($8,000, $8,500) from Table C.

What are the contributions of 101s and 102s? ($3000, $5000).

1.5 Evaluate current plan: 15 minutes

We use the “Evaluate” worksheet of merton facts.xls to set up the modelclose to the algebraic formulation discussed next.

Analyzing the president and sales manager’s decision to stop 101s iseasy. The constraints imply that the number of 102s produced should begiven by min[4000/2, 6000/2, 4500/3]. This gives $7.5 million, worse thanthe current plan contributions of $10.5 million.

Current plan for productions is (101s, 102s) = (1000, 1500). Is this thebest?

Reduce one model 102 and see how many extra 101s can be produced.Reducing one 102 frees (2, 2, -, 3) resources in (engine assembly, metalsstamping, 101-assembly, 102-assembly) that can be used to produce two101s. Show that this increases contribution by $1000 (2 × $3000 − $5000).

How long can you do this? Net resource effect of the substitution ofeach 102 is (-, +2, +4, -3) on remaining resources (-, 1000, 3000, -). Thisimplies that we can change up to a minimum of (-, 1000/2, 3000/4, -) andnon-negativity requirement on number of 102s produced. Explain that itis tedious to do this kind of analysis, and more than two product-linesimplies many trade-offs.

1.6 Algebraic Model: 5 minutes

How do we describe the model algebraically? Decisions are coded usingvariables called decision variables. How do we express the objective andconstraints using these variables?

Decisions variables:

t101 = number of model 101 trucks produced,

t102 = number of model 102 trucks produced.

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The algebraic formulation is:

max 3000t101 + 5000t102 Objective: contributions,1t101 + 2t102 ≤ 4000 Engine Assembly,

2t101 + 2t102 ≤ 6000 Metal Stamping,

2t101 ≤ 5000 101 Assembly,

3t102 ≤ 4500 102 Assembly,

t101 ≥ 0 Non-negativity,

t102 ≥ 0 Non-negativity.

1.7 Develop graphical model: 10 minutes

Show merton trucks.ppt, slides 2 through 13. We show how the trade offthat we talked about in section 1.5 moves along the engine assembly ca-pacity constraint line. In our first 70 minute session in 4 sections of roughly75 students each, we reached this point.

1.8 Develop spreadsheet model: 10 minutes

Excel formulation. Set it up on-line using the “Evaluate” worksheet ofmerton facts.xls.

1.9 Solutions to Problems: 60 minutes

We ask the students to ignore problem 3(b) since it requires reduced costs.Problem 1: (10 minutes)

a. Optimal monthly mix is (2000, 1000) with a contribution of $11 mil-lion. This part is already handled while setting up the spreadsheetoptimization model in section 1.8.

b. What is it worth to add an extra hour of engine assembly capacity?Unutilized resources (engine assembly, metal stamping, 101 assembly,102 assembly) = (0, 0, 1000, 1500). To use the extra engine assemblyhour, we can give up a 101 truck and make a 102 truck and the netincrease in contribution for the swap (-1, +1) is $2,000. Net effect onresources is (+1, 0, -2, +3) for the swap. The effect on a graph can beseen in slides 12 and 13 of merton trucks.ppt.

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c. Resolve to verify that change of engine assembly capacity from 4000 to4100 hours increases contribution by 2, 000 × 100 = 0.2million.

d. The number of swaps that we can do is 500.

min[−−, −−, 2000,

1500

3,

]= 500.

Note 101 assembly does not impose a constraint on the number ofswaps directly because each swap increases the unused resources bytwo units. But, the number of swaps is constrained by the numberof 101s being produced currently, i.e., 2000. We can not give up moreresources than what we are using.

Problem 2: (10 minutes) This problem should not take much time butstudents get confused with the unfortunate wording used in the case. Pro-duction manager is talking about reimbursing the outside supplier for ma-terials, labor and overhead. Hence the students question why are we notconsidering the fixed overheads in column two of Table C when comput-ing the results? Fixed overheads are based on Merton’s infrastructure anddo not hold for the outside supplier. Only the variable overheads are rele-vant here.

The second doubt that arises in some students, due to a mis-readingof the case text, is about what is being outsourced. They assume that aspecific model’s capacity is increased by the outsourcing. The text is clearthat we do not outsource any specific model’s engine capacity. Either 101sor 102s are expected to be made by the outside supplier.

Since we are not using duality, the answer is same as in problem 1 (b)and 1 (d). Sourcing out engine assembly is acceptable from 1 (b), and thelargest rent Merton can pay is $2000 and no more than 500 hours of engineassembly can be purchased.

Problem 3: (10 minutes) Model 103:

a. Resources required for 103 = (.8, 1.5, 1, -) and net contribution of $2000.Re-solve the problem and it is not worth producing. The optimal mixdoes not include model 103s and hence no change in contributions.

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Decisions variables:

t101 = number of model 101 trucks produced,

t102 = number of model 102 trucks produced,

t103 = number of model 103 trucks produced.

The algebraic formulation is:

max 3000t101 + 5000t102 + 2000t103,

1t101 + 2t102 + 0.8t103 ≤ 4000,

2t101 + 2t102 + 1.5t103 ≤ 6000,

2t101 + 1t103 ≤ 5000,

3t102 ≤ 4500,

t101, t102, t103 ≥ 0.

b. Ignore.

Problem 4: (20 minutes) Overtime production - re-solve. Students taketime understanding that even though the trucks are identical after pro-duction, the trucks produced during over time and regular time have tobe disambiguated in the model by using extra variables due to differingengine assembly costs and the capacity constraints of engine assembly.

Decisions variables:

t101 = number of model 101 trucks produced in regular time,t102 = number of model 102 trucks produced regular time,o101 = number of model 101 trucks produced in over time,o102 = number of model 102 trucks produced over time.

The contribution of the trucks produced during overtime reduces to(2400, 3800) from the regular time contributions of (3000, 5000) becauseof 50% extra costs for the engine assembly labor given in Table B. Somestudents multiply total direct labor by 1.5 instead of multiplying only theengine assembly labor component. The labor costs increase to (1800, 3600).In addition to the variables, a new constraint for overtime engine assem-bly capacity has to be added and the metal stamping, 101-assembly, 102-assembly constraints have to incorporate the overtime variables. The full

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algebraic formulation is:

max 3000t101 + 5000t102 + 2400o101 + 3800o102,

1t101 + 2t102 ≤ 4000,

1o101 + 2o102 ≤ 2000,

2t101 + 2t102 + 2o101 + 2o102 ≤ 6000,

2t101 + 2o101 ≤ 5000,

3t102 + 3o102 ≤ 4500,

t101, t102, o101, o102 ≥ 0.

The optimal solution yields a contribution of $11.7 million with 1500model 101 trucks and 1250 model 102s produced in regular time and 250model 102s produced in over time. Hence paying a monthly fixed over-head of $.75 million is not worth the increase of $.7 million.

Problem 5: (10 minutes) Marketing constraint - re-solve. To the basemodel add the constraint t101 − 3t102 ≥ 0. This leads a contribution of$10.5 million with a plan of (2250, 750).

Instead of requiring the number of 101s produced to be at least threetimes the number of 102s, if this constraint was altered to say that it shouldbe at least two times the number of 102s, then the optimal solution doesnot change since the original optimal solution still remains feasible for thenew constraint. Based on this, we generalize about the effect of extra con-straints on the feasible region.

2 With Duality: 30 minutes

References

[Dhe90a] Anirudh Dhebar. Merton truck company. Case 9-189-163, Har-vard Business School, HBS Publishing, Boston, MA 02163, apr1990.

[Dhe90b] Anirudh Dhebar. Merton truck company. Teaching Note 5-189-171, Harvard Business School, HBS Publishing, Boston, MA02163, may 1990.

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