Econ198 Linear Programming_UP Baguio

8/14/2019 Econ198 Linear Programming_UP Baguio

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MANAGERIAL ECONOMICS 1

LINEARLINEAR

PROGRAMMINGPROGRAMMING


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LINEAR PROGRAMMINGLINEAR PROGRAMMING

A mathematical technique for solving constrained maximization andminimization problems when there are many constraints and theobjective function to be optimized, as well as the constraints faced,are linear.

1. Optimal Process SelectionGiven input prices and the quantity of the commodity that the firmwants to produce, LP can be used to determine the optimalcombination of processes needed to produce the desired level andoutput at the lowest possible cost, subject to labor, capital and

other constraints that the firm may face.

Applications of LinearProgramming

1. Optimal Product MixMost firms produce a variety of products rather than a single oneand must determine how to best use their plants, labor, andother inputs to produce the combination or mix of products that

maximizes their total profits subject to the constraints they face.

A technique that seeks to solve resource allocation problems usingthe proportional relationships between two variables.


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1. Satisfying Minimum Product Requirements

Production often requires that certain minimum productrequirements be met at minimum cost. LP specifies the totalcost function that the manager seeks to minimize and thevarious constraints that he of she must meet or satisfy.

Continued Applications of LinearProgramming

1. Long-Run Capacity Planning

Firms seek to answer how much contribution to production andprofit each unit of the various inputs make. If it exceeds the priceof the input, total profits would increase by hiring more of thatinput; if input is underused, some of it need not be hired orpurchased, or even can be sold.

1. Others

a. Least-cost routeb. Best combination of expense in advertisingc. Best routing of telephone callsd. Best portfolio of securitiese. Best allocation of personnel, etc.


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Production Process the activity through which the use ofvarious input combinations or ratios is undertaken; can be

represented by a straight line (ray) from the origin of theinput space.

Feasible Region the area of attainable input combinations;along which the best or optimal solution lies.

Objective Function the function to be optimized; refers toeither profit maximization or cost minimization.

(Inequality) Constraints the level to which the firm canuse up, but not more than, specified quantities of someinputs; or to which the firm must meet some minimumrequirements.

Non-negativity Constraint the measure that indicatesthat the firm cannot produce negative output or use anegative quantity of any input.

Decision Variables the quantities of product to produce in

Definition of Terms:


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Figure 1. The Firms Production Processes andIsoquants

The left panel shows production Process 1 using K/L = 2, Process 2 using K/L = 1, and Process 3 using K/L= 1/2 that a firm can use to produce a particular commodity. The right panel shows that 100 units ofoutputs (100Q) can be produced with 6K and 3L (point A), 4K and 4L (point B), or 6L and 3K (point C).

Joining these points, we get the isoquant for 100Q. Because of constant returns to scale, using twice asmany inputs along each production process (ray) results in twice as much output. Joining such points, we

get the isoquant for 200Q.


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Picture1

Figure 2. Feasible Region and Optimal Solution

With isocost line GH in the left panel, the feasible region is shaded triangle 0JN, and the optimalsolution is at point E where the firm uses 8L and 8K and produces 200Q. The right panel showsthat if the firm faces no cost constraint but has available only 7L and 10K, the feasible region isshaded area 0RST and the optimal solution is at point S where the firm produces 200Q. To reachpoint S, the firm produces 100Q with Process 1 (0A) and 100Q with Process 2 (0B = AS).


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P R O F I T M A X I M I Z A T I ON

Table 1. Input Requirements and Availability

forProducing Products X and Y

Quantities of InputsRequired perUnit of Output_____

Quantities ofInputs

Available perTime Period

Input Product X Product Y Total

A 1 1 7

B 0.5 1 5

C 0 0.5 2

Example.

FIRM-A produces only two products, Product X and Product Y. Each unit of

Product X contributes $30 to profit and to covering overhead (fixed)costs, and each unit of Product Y contributes $40. Suppose further that inorder to produce each unit of Product s X and Y, the firm requires inputsA, B, and C.


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Solution:

Step 1 Express objective function as an equation and the

constraints as inequalities. = $30QX + $40QY

Express the constraints of the problem as inequalities.

Input A: 1QX + 1QY 7

Input B: 0.5QX + 1QY 5

Input C: 0.5QY 2

Impose non-negativity constraints on the output ofProducts X and Y.

QX 0 QY 0

Step 2 Graph the inequality constraints and define the feasibleregion.


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Figure 3. Feasible Region, Isoprofit Lines and ProfitMaximization



Step 3 Show the algebraic solution and results

At point D: where QY= 0(Input A) 1QX + 1QY 7 substituting QY = 0

Thus: QX = 7 and QY = 0

At point E: where Qx= 4

(Input A) 1QX + 1QY = 7(Input B) _0.5QX + 1QY = 5

0.5QX = 2 substituting Qx = 4


At point F: where Qy= 4(Input B) 0.5QX + 1QY = 5 substituting Qy = 4(Input C) 0.5QY = 2




Figure 4. Algebraic Determination of theCorners

of the Feasible Region

The quantity of Products X and Y (QX and QY) at corner point D is obtained by substituting QY = 0(along the QX axis) into the constraint equation for input A. QX and QY at corner point E are obtained bysolving simultaneously the constraint equations for inputs A and B. QX and QY at point F are obtained bysolving simultaneously the equations for constraints B and C. Corner point G can be dismissed outright

because it involves the same QY as at point F but has QX = 0. The origin can also be dismissed since QX= QY = = 0.

Step 3 Continuation (algebraic solutions andresults)


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C O S T M I N I M I Z A T I ON

Table 3.Summary Data

Example.

Assume that the manager of a college dining hall is required to prepare meals thatsatisfy daily requirements of protein (P), minerals (M), and vitamins (V). Suppose

that the minimum daily requirements that have been established at 14P, 10M, and6V. The manager can use two basic foods (meat and fish) in the preparation ofmeals. Meat (food X) contains 1P, 1M, and 1V per pound. Fish (food Y) contains 2P,1M, and 0.5V per pound. The price of X is $2 per pound, and the price of fish is $3.

Meat (Food X) Fish (Food Y)

Price per pound $2 $3

Units of Nutrients per Pound of Minimum Daily

RequirementNutrient Meat (Food X) Fish (Food Y) Total

Protein (P) 1 2 14

Minerals (M) 1 1 10

Vitamins (V) 1 0.5 6



Solution:

Step 1: C = $2QX + $3QY (objective function)

1QX + 2QY 14 (protein constraint)

1QX + 1QY 10 (minerals constraint)1QX + 0.5QY 6 (vitamins constraint)

QX, QY 0 (non-negativity constraint)

Step 2:

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Figure 5. Feasible Region and Cost Minimization



Table 3. Use of Foods X and Y, and Costs atEach

Corner of the Feasible RegionCornerPoint

QX QY $2QX+ $3QY Cost

D 14 0 $2 (14) + $3 (0) $28

E 6 4 $2 (6) + $3 (0) $24

F 2 8 $2 (2) + $3 (8) $28

F 0 12 $2 (0) + $3 (12) $36

Step 3:



LINEAR PROGRAMMING ANDLOGISTICS IN THE GLOBAL ECONOMY

Logistics Managementrefers to the merging at the corporatelevel of the purchasing, transportation, warehousing, distribution, andcustomer services functions, rather than dealing with each of themseparately at division levels.

Factors That Lead to the Rapid Spread of Logistics: Just-in-Time Inventory Management makes the buying ofinputs and the selling of the product much more tricky andmore closely integrated with all other functions of the firm.

Increasing Trend Towards Globalization of Productionand Distribution. With production, distribution, marketing andfinancing activities of the leading corporations scattered aroundthe world, the need for logistics management becomes evenmore important and beneficial.



LINEAR PROGRAMMING:THE USE OF COMPUTERPROGRAM/SCaseName of Business: Maximus Computer Company (MCC)

Product/Share on Profit : Computers Net Profit Starter $50 Midrange 120 Super 250

Extreme 300

Operations (Hours): Manufacture Assembly Inspection Starter 0.1 0.2 0.1 Midrange 0.2 0.5 0.2 Super 0.7 0.25 0.3

Extreme 0.8 0.2 0.5

Total Hours per Day 250 350 150

Goal and Philosophy:To ship computers with known-brand components and offer superior service, all

at a cost to consumers that is lower than that of the competition.


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MANAGERIAL ECONOMICS1

Microsoft Excel SOLVER


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LINDO

Other Computer Software:

GIPALS

GULF

http://www.msubillings.edu/BusinessFaculty/Harris/LP_Problem1.htm

Tutorial Websites:

IMPS LP, etc.

LIPSOL

http://fisher.osu.edu/~croxton_4/tutorial/

http://www.economicsnetwork.ac.uk/cheer/ch9_3/ch9_3p07.htm

http://www.lehman.com/who/

http://www.guardian.co.uk/business/2008/sep/15/lehmanbrothers.creditcrunch


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Sources: Principles of Managerial Economics (P. Keat, P. Young)

Managerial Economics in a Global Economy (D. Salvatore)

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Econ198 Linear Programming_UP Baguio

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