Chapter 5 Business Decision Making

Finally, we discuss operational decision-making, which, in our case, refers to finding the best design for the Chick-fil-A drive-thru that minimizes the total cost in a ten-year horizon.

Specifically, we consider the following costs: a one-time cost of a large-scale renovation, staffing costs for recruiting workers, delay cost of customers, and congestion cost for long system carline. With the last two costs captured by two hard constraints, the optimization problem is given by

\[\begin{align} \min_{I^{reno}, N^{cashier},N^{cook}}\quad &I^{reno}\cdot c^{reno}+N^{cashier}\cdot c^{cashier}+N^{cook}\cdot c^{cook}\\ \text{subject to: } &\text{Average flow time}\leq 15 \text{ minutes}\\ &\text{Average system carline}\leq 8\text{ cars} \end{align}\]

where

$I^{reno}=1$ if rebuilding he facility (with a cost of $c^{reno}$), and 0 otherwise.
$N^{cashier}$ and $N^{cook}$ are the number of workers at the ordering station and the pickup station, respectively.
$c^{cashier}$ and $c^{cook}$ are the costs of one cashier and one cook over ten years, respectively.
A design of the drive-thru corresponds to a set of values for the decision variables concerning the renovation ($I^{reno}$) and the staffing level ($N^{cashier}$, $N^{cook}$).

According to the insights from the bottleneck analysis, students can propose various designs and use the DES simulator functions to evaluate each one by checking the optimization constraints and computing the total cost.

Note that whether or not renovation takes place will affect which simulator we apply to evaluate system performance. If $I^{reno}=0$ (1), we shall run the simulator of the “old” (“new”) Chick-fil-A.

For example, we set the cost parameters as follows, with all numbers in thousands.

Assume rebuilding Chick-fil-A costs one million dollars, that is, $c^{reno}=1000$.
At an hourly rate of $15 per worker and 260 workdays per year, the cost of one worker over ten years is 312.

We next evaluate various designs with simulation.

Designs=data.frame(I=rep(0,3),Ncashier=rep(2,3),Ncook=3:5,
                   flow_time=rep(0,3),system_carline=rep(0,3))

Designs=rbind(Designs,Designs)
Designs[4:6,'I']=1

for(ii in 1:nrow(Designs)){
  if(Designs[ii,'I']==0){ # before renovation
    out=DES_before(num_cashier = Designs[ii,'Ncashier'],num_cook = Designs[ii,'Ncook'],
                   space_pickup = 6)
  }else{ # after renovation
    out=DES_after(num_cashier = Designs[ii,'Ncashier'],num_cook = Designs[ii,'Ncook'],
                   space_pickup = 10)
  }
  
  Designs[ii,'flow_time']=out$flow_time
  Designs[ii,'system_carline']=out$system_carline
}

The following table summarizes all the designs and their performance metrics.

Designs$Meet_Constraints=1*(Designs$flow_time<=15)*(Designs$system_carline<=8)
Designs$Cost=1000*Designs$I+15*8*260*10/1000*(Designs$Ncashier+Designs$Ncook)

Designs

##   I Ncashier Ncook flow_time system_carline Meet_Constraints Cost
## 1 0        2     3 18.563590      10.469911                0 1560
## 2 0        2     4  6.389379       4.466745                1 1872
## 3 0        2     5  5.467007       3.900928                1 2184
## 4 1        2     3 19.073694       8.819207                0 2560
## 5 1        2     4  6.439617       3.361279                1 2872
## 6 1        2     5  5.455356       2.889992                1 3184

5.1 Discussion

If the objective is to minimize ten-year cost while meeting the flow time and carline constraints, the best design is given by $I^{reno}=0,N^{cashier}=2,N^{cook}=4$, that is, there is no need to rebuild Chick-fil-A with considerable cost. This conclusion highlights the value of using DES to predict system performance and prescribe the best cost-effective decision.
The previous bottleneck analysis provides important insights that guide the selection of potential designs. In other words, with an understanding of constraining resources, we can restrict the search space of the optimization problem, which in practice would significantly reduce the computational cost and is a fundamental technique in operations research.
The conclusion we draw depends on the optimization problem defined by the objective function and the constraints. If we only require the flow time to be under 20 and the system carline to be under 9, then the design $I^{reno}=1,N^{cashier}=2,N^{cook}=3$ also meets the constraints. This particular design can be optimal if the unit staffing cost $c^{cook}$ is sufficiently high. Hence, collaborating with practitioners and identifying objectives and constraints play a key role in deriving relevant solutions based on a comprehensive understanding of the business operations.