AC 303 Lecture 6 & 7 Linear Programming and Theory of Constraints

8/4/2019 AC 303 Lecture 6 & 7 Linear Programming and Theory of Constraints

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ADVANCED MANAGEMENTACCOUNTING

Lecture (6)

Linear Programming



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Cost Behaviour Summary The use of Least-Squares Regression

Method to analyse mixed costs andestimate cost functions. It is objectiveand incorporate all of the availableobservations into the cost estimate. Simple linear regression (manual and

computer solutions ± Excel & SPSS)

Multiple linear regression

Non-linear regression (the learning-curve-effect)



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Briefly Reviewing Cost Classifications

for Decision Making

For decision-making classified into:

Differential Cost,

Opportunity Cost, Sunk Cost.



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Differential Costs and Revenues Costs and revenues that differ among

alternatives.

Example: You have a job paying £1,500 per month in your hometown. You have a job offer in

a neighbouring city that pays £2,000 per month.

The commuting cost to the city is £300 per

month.

Differential revenue is:

£2,000 ± £1,500 = £500

Differential cost is: £300



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Opportunity Costs The potential benefit that is given up

when one alternative is selected over

another.

Example: If you were not attending university,

you could be earning £15,000 per year. Your

opportunity cost of attending university for one

year is £15,000.



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Sunk Costs Sunk costs cannot be changed by any

decision. They are not differential

costs and should be ignored whenmaking decisions.

Example: Example: You bought a car that cost

£10,000 two years ago. The £10,000 cost is sunkbecause whether you drive it, park it, trade it, or

sell it, you cannot change the £10,000 cost.



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What is Linear Programming?

Managers are routinely faced with the problem of deciding how constrained resources are going to beutilised. There is an opportunity cost for scareresources that should be included in the relevant cost

calculation for decision making. For example, a firm may have:

limited raw materials,

limited direct labour hours available, and

limited floor space.

How should the firm proceed to find the rightcombination of products to produce?

The proper combination or µmix¶ can be found by use of a quantitative method known as linear programming.



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What is Linear Programming? (cont«)

Linear Programming (LP) is a powerfulmathematical technique that can be appliedto the problem of allocating constrained

(scare) resources among many alternativeuses in such a way that the optimumbenefits (total contribution margin) can bederived from their utilisation in the short

run. (Notice: fixed costs are usually unaffected

by such choices).



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What is meant by a linear programming

problem? A linear programming problem consists of

three parts: 1) a linear function (the objective function) of

decision variables (say, x1, x2, «, xn) that is to

be maximized or minimized. 2) a set of constraints (each of which must be

a linear equality or linear inequality) that restrictthe values that may be assumed by the decisionvariables.

3) the sign restrictions, which specify for eachdecision variable xj either (1) variable xj mustbe nonnegative ± xj >= 0; or (2) variable xjmay be positive, zero, or negative ± xj isunrestricted in sign



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A single-resource constraint

problem Where a scare resource exists, that

has alternative uses, the contributionper unit should be calculated for eachof these uses.

The available capacity for this resourceis then allocated to the alternative

uses on the basis of contribution perscare resource.



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Example from Seal et al. (2006),

Chapter (9), pp.366-

367

Mountain Goat Cycles makes a line of panniers (saddlebags for bicycles).

There are two models of panniers: A touring model, and

A mountain model

Cost and revenue data for the two

models are given below:



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Mountain Goat Cycles Example (cont«)

ModelMountain

Pannier

Touring

Pannier

Selling price per unit £25 £30

Variable cost per unit 10 18Contribution margin perunit

15 12

Contribution margin (CM)

ratio

60% 40%

According to this information, the mountain

pannier appears to be much more profitable than

the touring pannier.



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Mountain Goat Cycles Example

(cont«) Now let us add one more piece of

information the plant that makes panniers is operating at

capacity. The bottleneck (the constraint) is a

particular stitching machine. The mountainpannier requires 2 minutes of stitchingtime, and each unit of the touring pannier

requires one minute of the stitching time. In this situation, which product is more

profitable?



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Mountain Goat Cycles Example(cont«)

ModelMountai

n

Pannier

Touring

Pannier

Contribution margin per unit(from above) (a) £15 £12Time on stitching machinerequired to produce one unit (b) 2 1Contribution margin per unit of

the constrained resource (c)=(a) ÷ (b) 7.50 12

According to this information, the touring model

provides the larger contribution margin in relation to

the constrained resource.



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(cont«) To verify that the touring model is

indeed the more profitable, suppose

an additional hour of stitching time isavailable and there are unfilled ordersfor both products.

The additional hour could be used to

make either 30 mountain panniers or60 touring panniers.



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(cont«)Model

Mountain

Pannier

Touring

Pannier

Contribution margin per unit (fromabove) (a) £15 £12Additional units that can beprocessed in one hour X 30 X 60Additional contribution margin

450 720This example clearly shows that the touring model

provides the larger contribution margin in relation to

the constrained resource.



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Then, the rule in the case of a single resource

constraint problem

Should not necessarily promotethose products that have the highest

UNIT contribution margins. Total contribution margin will be

maximised by promoting products oraccepting orders that provide the

highest unit contribution margin inrelation to the constrained resource.



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Another Example from Drury(2004), Chapter (9), PP.321-322

Components X Y Z

Contribution per unit £12 £10 £6

Machine hours per unit 6 2 1

Estimated sales demand(units) 2,000 2,000 2,000

Required machine hours12,000 4,200 2,000

Contribution per machinehour 2 5 6

Ranking per machine hr 3 2 1

Capacity for the period is restricted to 12 000 machine hours.



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Another Example from Drury

(2004), Chapter (9)- cont« Profits are maximized by allocating scarce

capacity according to ranking per machine houras follows:

Production Machine hoursused Balance of machinehours available

2,000 units of Z 2,000 10,000

2,000 units of Y 4,000 6,000

1,000 units of X 6,000 -

The production programme will result in the following:

2 000 units of Z at £6 per unit contribution £12 000

2 000 units of Y at £10 per unit contribution £20 000

1 000 units of X at £12 per unit contribution £12 000

Total contribution £44 000



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More than one scare resource

problem Graphical Method (two products +

many constraints)

Simplex Method (More than twoproducts + many constraints)

Manual (tables/ matrices)

Computer programs:

Lindo

MS Excel



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Graphical Method Any linear programming problem with only two

variables (i.e. two products) can be solvedgraphically.

We may graphically solve an LP (max problem) with

two decision variables as follows: Step (1): Graph the feasible region.

Step (2): Draw an iso-contribution line.

Step (3) Move parallel to the iso-contribution in thedirection of increasing the objective function.

The last point in the feasible region that contactsaniso-profit line is an optimal solution to the LP.



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Example from Seal et al (2006), P.893

Colnebank Ltd is a medium-sized engineering company and is one of the large number of producers in a very competitive market. Thecompany produces two products, pumps and fans that use similar rawmaterials and labour skills. The market price of a pump is £152 and thatof a fan is £118. The resource requirements of producing one unit of each of the two products are:

Material (kg) Labour hours

Pump 10 22

Fan 15 8

Material costs are £4 per kg and labour costs are £3.50 per hour

During the coming period the company will have the following resourcesavailable to it: 4,000 kg of materials and 6,000 labour hours

Required: Determine the mix of products (pumps and fans) that willmaximise Colnebank¶s profit for the coming period, using graphicalsolution.



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Solution: formulating a mathematical

model of the above problemPumps (£) Fans (£)

Materials 40 (10kg at £4) 60(15kg at £4)

Labour 77 (22hrs at £3.50) 28 (8 hrs at 3.50)

117 88

Selling price 152 118

Contribution 35 30

The linear programming problem is to maximise the totalcontribution subject to the constraints. If P= units of pumpsproduced and F= units of fans produced, then the problem may beset up as follows:

1. Maximise: £35P + £30F

2. Subject to: 10P + 15F < 4000 (materials)

3. 22P + 8F < 6000 (labour hours)



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Graphing the feasible region



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Steps (2) & (3): Drawing an iso-contribution line andmoving parallel to the iso-contribution in the

direction of increasing the objective function



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³Which is the best combination

of products?´



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Sensitivity Analysis Sensitivity Analysis involves asking

µwhat-if¶ questions. For example,W

hat happens if the market price of pumps falls to £145?

What will be the loss of contributionand will the revised contribution

change the optimal combination?



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Sensitivity Analysis (cont«)



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Shadow Prices Each constraint will have an opportunity

cost, which is the profit foregone by nothaving an additional unit of the resource.

In linear programming, opportunity costsare known as shadow prices.

Shadow prices are defined as the increasein value that would be created by havingone additional unit of a scare resource.

For example: ³What is the additionalcontribution if one extra labour hour isavailable?´



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Shadow Prices (cont«) If one extra labour hour is obtained, the constraints

10 P + 15 F <= 4000 and 22 P + 8F <= 6000 will stillbe binding, and the new optimum solution can bedetermined by solving the following simultaneous

equations:10 P + 15 F <= 4000 (unchanged labour constraint)

22 P + 8 F <= 6001 (revised labour constraint)

The revised optimal output when the above equationsare solved is 111.96 units of F and 232.06 units of P.

This will increase contribution by 0.90 (the shadowprice of labour hours)



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(2004), P.1110



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(2004)- cont«



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Materials constraint (8Y + 4Z � 3,440(When Y= 0, Z = 860; when Z= 0, Y =

430



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Labour constraint 6Y + 8Z � 2,880 (WhenZ = 0, Y = 480; When Y = 0, Z =360)



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Machine capacity constraint 4Y + 6Z � 2,760(When Z = 0, Y = 690; when y = 0, Z = 460)



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Sales limitation Y � 420



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O ptimum solution



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Optimum solution



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Simplex Method: Manual Solution In 1947, George Dantzig developed the simplex

algorithm for solving linear programming problems. Thesimplex algorithm proceeds as follows: Step (1): Convert the linear programming problem to

standard form. Step (2): Obtain a basic feasible solution (bfs), if

possible, from the standard form. Step (3): Determine whether the current bfs is

optimal. Step (4): If the current bfs is not optimal, then

determine which non-basic variable should become abasic variable and which basic variable should become

a non-basic variable to find a new bfs with a betterobjective function value. Step (5): Find the new bfs with the better objective

function value. Go back to step (3).



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Applying the simplexalgorithm

See the Drury¶s example, p.1110



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Simplex method: Drury¶s example



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Simplex method contd.



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Simplex method contd. Second Matrix



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Computer Programs

The use of the LINDO ComputerPackage to solve linear programmingproblems

The use of Microsoft Excel to solvelinear programming problems«



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Assumptions underlying linearprogramming

1. Linearity.

2. Divisibility of products.

3. Divisibility of resources.4. All of the available opportunities

can be included in the model.

5. Fixed costs are constant for the

period.6. Objective of the firm (maximise

short term contribution).



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The Limitations of LinearProgramming

LP ignores marketing considerations.

It has an extensive focus on the short term.

Furthermore, most production resources can bevaried even in the short term through overtime

and buying-in. The alternative to optimizing against given

constraints is to concentrate on managingconstraints.

The theory of constraints helps in managing

constraints in the short term. It is the topic of the next lecture.



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Workshop (3)

See Exercises P21-4 & C21-8(Seal et al., 2006)

Documents

AC 303 Lecture 6 & 7 Linear Programming and Theory of Constraints