Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

A phone dealer is preparing to make new orders for phone types Nokia, Samsung and LG. The unit profit for Nokia, Samsung and LG are Ghc 100, Ghc 300 and Ghc 50 respectively. The phones are kept in a specialized protective container of 10.1 square meters to prevent cracks and other malfunctions from vibration. A Nokia, Samsung and LG takes storage space of 0.05, 0.1 and 0.05 square meters

respectively. If a Nokia, Samsung and LG cost Ghc 300, Ghc 1200, and Ghc 120

respectively, and the dealer has available Ghc 93,000 to spend, what should be the

optimal mix of phone types to purchase to achieve the maximum profit possible?

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Linear Programming Problem: Computer Solution

Min 100x1+300x2+50x3 s.t

0.05x1+0.1x2+0.05x3 ≤ 10.1 30x1+1200x2+120x3 ≤ 93000

x1, x2, x3 ≥ 0

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3 Chapter 3 - Linear Programming: Computer Solution and Sensitivity Analysis

Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

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A dietician can purchase three different types of food supplement: Type A, type B and type C. each food supplement type comes in the same size bag. For one bag of type A it contains 5mg of Nutrient one, 10mg of nutrient two and 5mg of nutrient three. For a bag of type B it contains 5mg each of Nutrient one and Nutrient two and 10mg of

nutrient three. Type C contains 10mg of nutrient one, 60mg of nutrient two and 30mg of nutrient three.

For a particular patient, the dietician determines that she needs to combine the bags of food supplement to get at least 1200mg of Nutrient one, 2000mg of nutrient two and 2200mg of nutrient three. If Type A, Type B, and Type C cost Ghc 3, Ghc 4, and Ghc 10 respectively, how many bags of food supplement of types A, B and C should she purchase? Assume x1, x2, and x3 as the number of bags of type A, B, and C

respectively. Round your answer to the upper integer.

Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

Min 3Xa+4Xb+10Xc s.t

5Xa+5Xb+10Xc ≥ 1200

10Xa+5Xb+60Xc ≥ 2000

5Xa+10Xb+30Xc ≥ 2200

Xa, Xb, Xc ≥ 0

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Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

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Linear Programming Problem: Standard Form

Min x1+x2+x3 s.t

x1+x2+x3 = 1000 x1 ≥ 200 x2 ≥ 300 x3 ≥ 100 4x3 – x1 + x2 ≤ 0

x1, x2, x3 ≥ 0

Try this on your own

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Beaver Creek Pottery Example Sensitivity Analysis (1 of 4)

• Sensitivity analysis determines the effect on the optimal solution of changes in parameter values of the objective function and constraint equations.

• Changes may be reactions to anticipated uncertainties in the parameters or to

new or changed information concerning the model.

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Linear Programming Problem: Standard Form

Back to the phones problem

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Linear Programming Problem: Standard Form

Sensitivity Report for Phones problem

The sensitivity report provides information relating to:

•changing the objective function coefficient for a variable

•forcing a variable which is currently zero to be non-zero

•changing the right-hand side of a constraint.

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Sensitivity Report for Bicycle problem

Changing the objective function coefficient for a variable

Suppose we vary the coefficient of Samsung in the objective function. How will the LP optimal solution change?

Linear Programming Problem: Computer Solution

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Linear Programming Problem: Computer Solution

Sensitivity Report for the phones problem

Changing the objective function coefficient for a variable

Suppose we vary the coefficient of Samsung in the objective function. How will the LP optimal solution change?

As long as the unit profit of Samsung lies within (300-100, 300+66.6667), the values of the variables in the optimal LP solution will remain unchanged.

Note though that the actual optimal solution value will change as the objective function coefficient of Samsung is changing.

we are effectively saying that the decision to produce 94 of Nokia and 54 of Samsung

remains optimal even if the profit per unit from Samsung is not actually 300 (but lies in the

range 233.3 to 366.67). Similar conclusions can be drawn Nokia and LG.

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Linear Programming Problem: Computer Solution

Forcing a variable which is currently zero to be non-zero

For the variables, the Reduced Cost column gives us, for each variable which is currently zero

(LG), an estimate of how much the objective function will change if we make (force) that variable to be non-zero.

Sensitivity Report for the phones problem

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Linear Programming Problem: Computer Solution

Sensitivity Report for the phones problem

Forcing a variable which is currently zero to be non-zero

Reduced Cost column for a variable is often called the 'opportunity cost' for the variable.

• We ignore the sign of the reduced cost when constructing the above table.

• The objective function will always get worse if we have a maximization problem, but go up if we have a minimization problem by at least this estimate.

• The larger the increase in the variable are, the more inaccurate this estimate is of the exact

change that would occur if we were to resolve the LP with the corresponding constraint for the

new values of the variable currently with a value of zero.

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Sensitivity Report for Bicycle problem

Forcing a variable which is currently zero to be non-zero

• An alternative interpretation of the reduced cost is the amount by which the objective function coefficient for a variable needs to change before that variable will become non-zero.

• Hence for variable Propeller, the objective function needs to change by 20 (increase since we are maximising) before that variable becomes non-zero.

• In other words, referring back to our original situation, the profit per unit on Propeller would need to increase by Ghc 20 before it would be profitable to produce any Propeller.

Linear Programming Problem: Computer Solution

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Sensitivity Report for the phone problem

Changing the right-hand side of a constraint

For each constraint, the column headed Shadow Price tells us exactly how much the objective function will change if we change the right-hand side of the corresponding constraint within the limits given in the Allowable Increase/Decrease columns

Linear Programming Problem: Computer Solution

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Linear Programming Problem: Standard Form

Sensitivity Report for Bicycle problem

Changing the right-hand side of a constraint

For example for the Capital constraint, provided the right-hand side of that constraint remains between:

10.1 + 5.4 =15.5 and 10.1 – 2.35 = 7.75 the objective function change will be exactly 100*[change in right-hand side from 101].

The direction of the change in the objective function (up or down) depends upon the direction of the change in the right-hand side of the constraint and the nature of the objective (maximize or minimize).

To decide whether the objective function will go up or down use:

constraint more (less) restrictive after change in right-hand side implies objective function worse (better)

if objective is maximize (minimize) then worse means down (up), better means up (down)

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Linear Programming Problem: Standard Form

Sensitivity Report for Bicycle problem

Changing the right-hand side of a constraint

The direction of the change in the objective function (up or down) depends upon the direction of the change in the right-hand side of the constraint and the nature of the objective (maximise or minimise).

To decide whether the objective function will go up or down use:

• constraint more restrictive after change in right-hand side implies objective function worse.

• constraint less restrictive after change in right-hand side implies objective function better.

• if objective is maximize, worse means down, better means up

• if objective is minimize, worse means up, better means down

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Linear Programming Problem: Standard Form

Sensitivity Report for Bicycle problem

Changing the right-hand side of a constraint Hence,

if you had an extra 2.0 square meter of storage to which bicycle would you assign it?

if you had to take Ghc 2000 away from Samsung or LG which one would you choose?

what would the new objective function value be in these two cases?

Note: The value in the column headed Shadow Price for a constraint is often called the 'marginal

value' or 'dual value' for that constraint.