6103500 Econ198 Linear Programming Up Baguio

8/14/2019 6103500 Econ198 Linear Programming Up Baguio

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MANAGERIAL ECONOMICS1

LINEAR PROGRAMMINGLINEAR PROGRAMMING




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 Proc ess S electionGiven input prices and the quantity of the commodity that the firm wants to

produce, LP can be used to determine the optimal combination of processesneeded to produce the desired level and output at the lowest possible cost,subject to labor, capital and other constraints that the firm may face.

Ap pli cati ons of Li nearProgr ammi ng

1. Opt imal Prod uct M ixMost firms produce a variety of products rather than a single one and

must determine how to best use their plants, labor, and other inputs toproduce the combination or mix of products that maximizes their totalprofits subject to the constraints they face.

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




1. Satis fying M inimu m Prod uct R eq uireme nts

Production often requires that certain minimum product requirementsbe met at minimum cost. LP specifies the total cost function that themanager seeks to minimize and the various constraints that he of shemust meet or satisfy.

Continued… Ap pli cat io ns o f Linear Pro gr amm ing

1. Lon g- Ru n Cap ac ity Plan ning

Firms seek to answer how much contribution to production and profit eachunit of the various inputs make. If it exceeds the price of the input, totalprofits would increase by hiring more of that input; if input is underused,some of it need not be hired or purchased, 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.




Prod ucti on Proces s – the activity through which the use of variousinput combinations or ratios is undertaken; can be represented by

a straight line (ray) from the origin of the input space.

Fea sib le Regi on – the area of attainable input combinations; alongwhich the best or optimal solution lies.

Objectiv e Fu ncti on – the function to be optimized; refers to either

profit maximization or cost minimization.

(I nequality) C ons tr aints – the level to which the firm can use up,but not more than, specified quantities of some inputs; or to whichthe firm must meet some minimum requirements.

Non -neg ati vi ty C ons tr aint – the measure that indicates that thefirm cannot produce negative output or use a negative quantity of any input.

Deci sion Vari ables – the quantities of product to produce in order tomaximize profits or inputs to use to minimize costs.

De fini ti on of Te rms:




Figure 1. The Firm’ s Prod uction Proc esse s andIsoq uan ts

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 afirm can use to produce a particular commodity. The right panel shows that 100 units of outputs (100Q) can be producedwith 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 as many inputs along each production process (ray) results in twice asmuch output. Joining such points, we get the isoquant for 200Q.




Picture1

Figure 2. Fea sib le Region an d Optimal Solu tion

With isocost line GH in the left panel, the feasible region is shaded triangle 0JN, and the optimal solution is atpoint E where the firm uses 8L and 8K and produces 200Q. The right panel shows that if the firm faces nocost constraint but has available only 7L and 10K, the feasible region is shaded area 0RST and the optimalsolution is at point S where the firm produces 200Q. To reach point 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. Inp ut Requ ire ments and Avail ab il ity for

Producin g Pro du cts X and Y

Quant ities of Inp ut sReq ui red per

Uni t of Output _____

Quant ities ofInp ut s A vail ab leper T ime Peri od

Inp ut Pr od uc t X Prod uct Y Tot al

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.



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 of Products X and Y.

QX ≥ 0 QY ≥ 0

Step 2 Graph the inequality constraints and define the feasibleregion.



Figure 3. Feas ib le Re gio n, Isopr ofit Li nes and Pr ofi tMaxi miz at ion



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 Q x = 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. Alg ebrai c De termi nat ion of the Corne rsof the Feasi ble Re gion

The quantity of Products X and Y (QX and QY) at corner point D is obtained by substituting QY = 0 (along the QXaxis) into the constraint equation for input A. QX and QY at corner point E are obtained by solving simultaneously theconstraint equations for inputs A and B. QX and QY at point F are obtained by solving simultaneously the equations forconstraints 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 and results)



Table 2. Outpu ts of Prod uct s X and Y, and Profits atEa ch C orner of the Fea sib le Reg ion

Corn erPoin t

QX QY $30QX+ $40QY Profit

0 0 0 $30 (0) + $40 (0) $ 0

D 7 0 $30 (7) + $40 (0) $210

E 4 3 $30 (4) + $40 (3) $240

F 2 4 $30 (2) + $40 (4) $220

G 0 4 $30 (0) + $40 (4) $160

Step 3 – Continuation… (algebraic solutions and results)


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

Table 3.Sum ma ry 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 of meals. 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



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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:

14MANAGERIAL ECONOMICS1

Figure 5. Feas ibl e Re gion and Cos t Min im izati on



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Table 3. Us e of Food s X and Y, and Co sts at Ea chCorn er o f the Fe as ibl e Regi on

Corn erPoin t

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:



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LINEAR PROGRAM MING AND LOGIST ICSIN THE GLOB AL ECONOMY

Logistics Management refers to the merging at the corporate

level of the purchasing, transportation, warehousing, distribution, and customer services functions, rather than dealing with each of themseparately at division levels.

Fac tors That Le ad t o the Rapid Sp re ad of Log ist ics:• Just-i n-Tim e Inv entor y Manag ement makes the buying of inputsand the selling of the product much more tricky and more closelyintegrated with all other functions of the firm.

• Incr eas in g Trend Tow ard s Global izati on of Pro du ction andDis tr ib ution . With production, distribution, marketing and financingactivities of the leading corporations scattered around the world, theneed for logistics management becomes even more important andbeneficial.



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LINEA R PRO GRAMMIN G:THE U SE OF COMPU TER PROG RA M/S

Case

Name 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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Microsoft Excel SOLVER

























LINDO

Othe r Comp ut er Sof tware :

GIPALS

GULF

http://www.msubillings.edu/BusinessFaculty/Harris/LP_Problem1.htmTutor ial 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)

Maraming salamat po…

6103500 Econ198 Linear Programming Up Baguio

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