Question

Suppose we modify the production model to obtain the following mathematical model:

Max	14x
s.t.
	ax	≤	38
	x	≥	0

where a is the number of hours of production time required for each unit produced.

With

a = 5,

the optimal solution is

x = 7.6.

If we have a stochastic model with

a = 3,

a = 4,

a = 5,

or

a = 6

as the possible values for the number of hours required per unit, what is the optimal value for x? (Round your answers to two decimal places. Let P be total profit.)

(a)

a = 3

x= P=

(b)

a = 4

x= P=

(c)

a = 5

x= 7.6P=

(d)

a = 6

x= P=

(e)

What problems does this stochastic model cause?

Since the value of a is  ---Select--- known unknown , the values of x and profit  ---Select--- cannot be is known with certainty.

Models of inventory systems frequently consider the relationships among a beginning inventory, a production quantity, a demand or sales, and an ending inventory. For a given production period j, let

sj − 1	=	ending inventory from the previous period (beginning inventory for period j)
xj	=	production quantity in period j
dj	=	demand in period j
sj	=	ending inventory for period j

(a)

Write the mathematical relationship or model that describes how these four variables are related.

  

(b)

What constraint should be added if production capacity for period j is given by

C_j?  

(c)

What constraint should be added if inventory requirements for period j mandate an ending inventory of at least

I_j?

Suppose you are going on a weekend trip to a city that is d miles away.

(a)

Develop a model that determines your round-trip gasoline costs.

Let

• d = distance,
• m = miles per gallon,
• p = price per gallon, and
• C = total cost.

C =  

2(dm​)c

  

(b)

What assumptions or approximations are necessary to treat this model as a deterministic model? (Select all that apply.)

We use the same route on the outbound and the inbound leg of the trip.m does not vary for the duration of the trip.p does not vary by location.p varies by location and due to local taxes.m varies depending on city and highway traffic conditions.

(c)

Are these assumptions or approximations acceptable?

A reasonable person would assume average values for m and p and accept the approximation.A reasonable person would conclude that the real cost would deviate widely from the estimate and reject the approximation.     A reasonable person would hedge their bets and increase the budget by 20% over the estimated cost.A reasonable person would note that neither m nor p can be known ahead of time and reject the approximation.

  

Answer 1

Given ax <= 38 and x > = 0

(a) : when a = 3 optimal value for x is x = 38/3= 12.67 and Profit = 14x = 14 * (38/3) = 177.33

(b) : when a = 4 optimal value for x is x = 38/4= 9.50 and Profit = 14x = 14 * (38/4) = 133.00

(c) : when a = 5 optimal value for x is x = 38/5= 7.60 and Profit = 14x = 14 * (38/5) = 106.40

(d) : when a = 6 optimal value for x is x = 38/6= 6.33 and Profit = 14x = 14 * (38/6) = 88.67

(e) : Since the value of a is unknown , the values of x and profit cannot be known with certainty