Fall 2022-Exam 1-ANSWERS
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OR6205 - Fall 2022
Exam 1: ANSWERS

Problem 1 (5 points)
A saw mill makes two products for log-home kits: fir logs and spruce logs which can be sold at
profits of $16 and $20, respectively.
Each spruce log requires 0.4 hours of processing time on the bark peeler and 1 hour on a slab
saw.
Each fir log requires 0.5 hours on the peeler and 0.7 hours on the slab saw.
Each log then, regardless of type, requires 0.3 hours on the planer.
Because of maintenance requirements and labor restrictions, the bark peeler is available 10 hours
per day, the slab saw 12 hours per day, and the planer 14 hours per day.
Bark and sawdust are by-products of these operations. All the bark can be put through a chipper
and sold in unlimited quantities to a nearby horticulture supplier. Dried fir sawdust can be
directed to a similar market, at a net profit of $1.40 per processed log. Limited amounts of the
spruce sawdust can be made into marketable pressed-wood products, but the rest must be
destroyed. For each spruce log produced, enough sawdust (5 pounds) is generated to make 3
pressed-wood products, which after manufacturing can be sold at a unit profit of $2 per pressed-
wood product. However, the market can absorb only 60 of the pressed-wood products per day
and the remaining spruce sawdust must be destroyed at a cost of $0.60 per pound.
The saw mill wishes to make the largest possible profit, considering the cost of destroying the
unusable spruce sawdust.

Formulate a linear program for this problem. Do not solve. Define your variables before you use
them. Also describe in a few words what your objective function and each of your constraints
represents.

Solution:

Let f and s be the number of fir logs and spruce logs produced and sold per day, respectively.

Let prwood be the number of pressed-wood products made and sold per day

Let waste (lbs) be the spruce sawdust to be destroyed each day

Max = 16*f + 20*s + 1.40*f + 2*prwood - 0.60*waste; maximize profit minus cost

0.5*f + 0.4*s <= 10; bark peeler daily availability
0.7*f + 1*s <= 12; slab saw daily availability
0.3*f + 0.3*s <= 14; planer daily availability
5*s = (5/3)*prwood + waste; spruce sawdust produced daily = sawdust used for
making pressed wood products + waste.

Note that (5/3) lbs of sawdust makes one pressed-wood product; prwood <= 60;

All variables are >=0
Fall 2022-Exam 1-ANSWERS
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Problem 2 (15 points)
Consider the following linear programming problem P:

Minimize: z = x1 -2 x2
Subject to: x1 +x2 ≥ 2 (1)
-x1 +x2 ≥ 1 (2)
x2 ≤ 3 (3)
____________________________________
x2 ≥ 0
x1 unconstrained in sign

Let the surplus of constraint (1) and (2) be x3 and x4, respectively, and the slack of constraint (3)
be x5. Answer the independent questions below (write the solutions after each question, in red).

1. Solve the problem graphically: Identify the feasible region by its corner points (coordinates x1
and x2 ) and shade it. Find the optimal point on the graph and write the optimal values of the
variables and z, below. 


x1 = -1 x2 = 3 x3 = 0 x4 = 3 x5 = 0 z = -7



2. Determine the optimal solution, if instead of minimization the objective was maximization.
A (1/2,3/2) 


3. Write an objective function that has multiple optima on the feasible region of problem P. 

min x1 + x2, min -x1 + x2 , max x2; or any positive multiple of them. 

4. Determine a right hand side value of constraint (3) that renders the problem infeasible. Note that
there are many values with that property.
 

Any value less than 3/2; for example, 1.2. 

Fall 2022-Exam 1-ANSWERS
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5. What is the optimal solution if constraint (3) is removed from the formulation?
The problem has no optimal solution, or it has an unbounded z-value. 

6. Construct the initial basic solution by adding artificial variables and making the necessary
variable transformations so that you can apply the Big M method to Problem P. Set up Big M
method iteration (0) tableau. Indicate the entering and leaving variable and perform a single
iteration. Write the resulting basic solution of iteration (1) and indicate whether it is feasible or
infeasible to Problem P. Indicate on the graph the point this basic solution corresponds to and the
constraints (if any) that are violated. 

Solution: see next page.
Fall 2022-Exam 1-ANSWERS