DAKOTA FURNITURE COMPANY
AYMAN H. RAGAB
1. Introduction
The Dakota Furniture Company (DFC) manufactures three products, namely
desks, tables and chairs. To produce each of the items, three types of resources are
required: lumber board feet, finishing hours and carpentry hours. It is required to
determine:
• How much of each item to produce
• The corresponding resource requirements
Following the above requirements, the decision variables should be:
(1) Y = hy1 , y2, y3 iT : the amount of products to produce from items 1, . . ., 3.
(2) X = hx1 , x2, x3iT : the required amount of resources 1, . . . , 3.
where items 1,2 and 3 and resources 1,2 and 3 refer to desks, tables and chair, and
lumber board feet, finishing hours and carpentry hours correspondingly. The cost
of unit resource is given by the cost vector: C = hc1 , c2, c3i = h−2, −4, −5.2i, while
the furniture selling price per unit is given by the revenue vector: R = hr1 , r2, r3i =
h60, 40, 10i. The resource requirements per product are expressed by the matrix B
below.

 

b1,1 b1,2 b1,3
8 6
1
B = b2,1 b2,2 b2,3 = 4 2 1.5
2 1.5 0.5
b3,1 b3,2 b3,3
It is implied that the demand is realized before the production decision is made, and
therefore the amount produced is the same as that sold (since there is no advantage
in producing items and not selling them.)
2. The Deterministic Model (DM)
The demand is assumed to be deterministic and given by the demand vector D,
where DT = hd1, d2, d3i = h150, 125, 300i. The following is the formulation of the
deterministic model.
Date: September 15, 2005.
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AYMAN H. RAGAB
2.1. Formulation of Deterministic Model.
(DM) = max
− 2x1
subject to
−4x2 − 2x3
− x1
−x2
− x3
+60y1 + 40y2
+8y1 + 6y2
+2y1 + 1.5y2
+4y1 + 2y2
y1
y2
x1 ,
x2 ,
x3,
y1 ,
y2,
+10y3
+y3 ≤ 0
+0.5y3 ≤ 0
+1.5y3 ≤ 0
≤ 150
≤ 125
y3 ≤ 300
y3 , ≥ 0
2.2. Solution of Deterministic Model. The deterministic model was solved using cplex 8.0. The expected profit is $4, 165 and the values of the variables are as
indicated below:
X = h1950, 850, 487.5iT
Y = h150, 125, 0iT
2.3. Sensitivity Analysis. It is required to examine the effect of the fluctuation
of the demand on the results of the model. It turned to be that any increase or
decrease in the demand of the chairs does not affect the model anyway or the other.
As for the desks and the tables, any change in the demand is accordingly reflected
in the values of y1 and y2 . Subsequently, the values of y1 and y2 are equal to
the demand of desks and tables no matter what the demand turns to be. The
amount of acquired resources is adjusted following the values of y1 and y2 . The
sensitivity of the model to the other parameters (namely cost, revenue and resource
requirements) is not examined since the emphasis of part B of this exercise is solely
on the randomness of the demand.
3. The Stochastic Solution
The demand of desks, tables and chairs is assumed to be random and following
a multivariate distribution. The low demand, Dl = h50, 20, 200iT , happens with a
probability pl = 0.3. The medium demand, Dm = h150, 110, 225iT , happens with
a probability pm = 0.4. The high demand, Dh = h250, 250, 500iT , happens with a
probability ph = 0.3.
3.1. The Expected Value Solution. In the expected value model (EVM), the
expected value of the demand is De = h150, 125, 300iT which is equal to the deterministic demand of (DM). Both The formulation and the solution of the model are
therefore identical to those of (DM). It is obvious that:
• the solution is infeasible when low and medium demands realize
• the solution is sub-optimal when high demand realizes
• due to the combined effect of infeasibility and suboptimality, the estimated
expected profit will not be achieved in practice
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3.2. The Scenario Analysis Solution. Each of the three scenario can be formulated as the (DM) with the appropriate demand vector replacing the right hand
side of the constraints. The profit for the low demand scenario is $1124 and the
values of the associated variables are listed below:
X = h520, 240, 130iT
Y = h50, 20, 0iT
The profit for the medium demand scenario is $3982 and the values of the associated
variables are listed below:
X = h1860, 820, 465iT
Y = h150, 110, 0iT
The profit for the high demand scenario is $7450 and the values of the associated
variables are listed below:
X = h3500, 1500, 875iT
Y = h250, 250, 0iT
It is important to note, however, that this analysis doesn’t provide a mean to decide
on the quantity of resources to acquire before the realization of the demand. In
other words, the model’s value is in the insight that it provides the decision maker
and not in providing the decision maker with an optimal solution.
3.3. The Fat Solution. The fat solution is intended to avoid the shortcomings of
both the expected value solution and the scenario analysis by providing the decision
maker with a solution that is always feasible. Below is the formulation of the fat
model (FM).
(FM) = max
− 2x1
subject to
−4x2 − 2x3
− x1
−x2
− x3
+60y1 + 40y2
+8y1 + 6y2
+2y1 + 1.5y2
+4y1 + 2y2
y1
y2
y1
y2
y1
y2
x1 ,
x2 ,
x3 ,
y1 ,
y2 ,
+10y3
+y3 ≤ 0
+0.5y3 ≤ 0
+1.5y3 ≤ 0
≤ 50
≤ 20
y3 ≤ 200
≤ 150
≤ 110
y3 ≤ 225
≤ 250
≤ 250
y3 ≤ 500
y3 , ≥ 0
Needless to mention, the medium and hight demand sets of constraints above are
redundant and can be omitted without loss of information. As a result, the model
produces results identical to that of the low demand scenario. Although the solution
is always feasible, it is apparently too conservative in the sense that in two thirds of
the times DFC is missing the opportunity to produce and sell more products and
hence achieve higher profits.
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AYMAN H. RAGAB
3.4. The Two-stage SLP Recourse Model Solution. In the two-stage SLP
recourse model (TRM), the amount of items to be produced and the demand are
scenario dependent. The model can be formulated as follows.
X
(TRM) = max
− 2x1
−4x2 − 2x3
+
pω̃ [60y1(ω̃) + 40y2(ω̃) +10y3 (ω̃)]
ω̃
subject to
− x1
+8y1 (ω̃) + 6y2 (ω̃)
−x2
− x3
+y3 (ω̃) ≤ 0
+2y1 (ω̃) + 1.5y2(ω̃)
+0.5y3(ω̃) ≤ 0
+4y1 (ω̃) + 2y2 (ω̃)
+1.5y3(ω̃) ≤ 0
y1 (ω̃)
≤ d1 (ω̃)
y2 (ω̃)
≤ d2 (ω̃)
y3 (ω̃) ≤ d3 (ω̃)
x1 ,
x2 ,
x3 ,
y1 (ω̃),
y2 (ω̃),
y3 (ω̃), ≥ 0
The expected profit for the model is $1, 730, and the value of the decision variables
are as follows:
First-stage variables:
X = h1300, 540, 325iT
Second-stage variables:
for low demand Yl = h50, 20, 200iT
for medium demand Ym = h80, 110, 0iT
for high demand Yh = h80, 110, 0iT
4. Solution Analysis
The deterministic case is suitable when the values of the demand are known
with absolute certainty (e.g. engineer to order situation). However, when the
values of the demand are merely an approximation for the uncertain demand (e.g.
expected value), the solution provided can be misleading. For example, in the
Dakota Furnishing Company’s (DFC) situation, it turns to be that the expected
value solution is infeasible for two of the scenarios. Hence, on the long run, the
expected profit is not going to be achieved. Furthermore, the sensitivity analysis
sometimes fails to alarm the decision maker to the need for further analysis, as
was clearly the case in the DFC situation where the sensitivity analysis concluded
that under no circumstance the production of the chairs is beneficial. The scenario
analysis solution can provide a good deal of intuition, yet it does not provide a
solution. Taking the expected value for the different scenario is just another way
to achieve the same quality of results as achieved by the expected value solution.
The fat solution resolve the problem of the possible infeasibility of the expected
value solution under some scenarios, but the most conservative scenarios have an
undue influence over the solution that in the case of the DFC the fat solution was
identical to the low demand scenario solution. Finally, the two stage SLP recourse
model provides the best results in terms:
(1) the solution is always feasible under any scenario (it is more effective than
the expected value solution)
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(2) the solution is not unduly driven by the most conservative scenario (it
performs considerably better than the fat solution because it takes into
consideration reacting differently to different scenarios)
(3) the solution is implementable and not informative (unlike the scenario analysis solution)