Providing Software as a Service: a design decision(s) model

Abstract

We examine how Software as a Service (SaaS) providers make different design decisions using a theoretical model. We consider two non-functional attributes: modularity of the software architecture and the architectural performance of the software. We model the relationship of these two attributes with factors such as user preferences, user demand, and the price of the service. In a significant departure from traditional models of IS product development, we considered marginal cost and maintenance cost of providing SaaS service to recognize that the SaaS service has characteristics of both a product and a service. We show how to find the optimal values of design attributes that maximize SaaS provider’s profits, taking into account relevant factors such as user preferences, user demand and service price. Our research provides one of the first analytical models of optimal design decision making by SaaS providers. We use the model to further show how the SaaS providers should adjust the service design in response to changes in user preferences, associated costs and other relevant factors.

Appendices

Appendix 1: Glossary of terms

d:

Demand of the service amortized over the lifetime of the service

p:

Price of the service amortized over the lifetime of the service

${{\uppi}}$ :

Profit amortized over the lifetime of the service

m:

Modularity level of the service

s:

Architectural performance level of the service

β:

Decrease in demand d from increase in price p (sensitivity to price)

$\gamma$ :

Increase in demand from increase in modularity (sensitivity to modularity)

δ:

Increase in demand from increase in architectural performance (sensitivity to architectural performance)

${\text{A}}_{1}$ :

Fixed cost for developing the product arising from factors other than modularity and architectural performance

${\text{A}}_{2}$ :

Amortized maintenance cost over the lifetime of the product

A = A₁ + A₂ :

Total fixed cost part that includes amortized maintenance cost for providing the service

B:

Parameter related to modularity showing the saving in amortized maintenance cost arising from modular design; it is also amortized over the lifetime of the service

Z:

Marginal cost of providing service per unit amortized over the lifetime of the service

C:

Parameter related to modularity during design and development of the service

D:

Parameter related to performance during design and development of the service

θ:

Total cost amortized over the lifetime of the product. θ = c₁ + c₂ + c₃

$\varvec{c}_{1} = A_{1} + C m^{2} + Ds^{2}$ :

Fixed Cost for developing and setting up service

${\text{c}}_{2} = {\text{A}}_{2} - {\text{Bm}}$ :

Maintenance cost amortized over the lifetime of the product

c ₃ = Zd:

Marginal cost amortized over the lifetime of the product

X:

Maximum attainable architectural performance level of a service when the service is not at all modular

Y:

Ratio of change in architectural performance level s to change in modularity level m of the service

Appendix 2: Proofs of propositions

Proof of Proposition A1

Taking the partial derivative of optimal $p^{*}$ with respect to γ, we obtain

$$\frac{{\partial p^{ *}}}{\partial \gamma } = \frac{{ D ( { B (4 C D \beta + D\gamma^{2} - C\delta^{2} } ) + 4 C D( {\alpha -Z\beta } ))}}{{(4 C D \beta - D \gamma^{2} - C \delta^{2} )^{2}}}$$

The denominator of the above equation is always positive as it is a square of an expression. From Lemma 1 and boundary condition of Eq. (7), we find $4 D \beta - \delta^{2} > 0$ and $( {\alpha - Z \beta } )$ > 0 respectively. Hence the numerator is always positive.

Therefore: $\frac{{\partial p^{ *}}}{\partial \gamma } > 0$.□

Proof of Proposition A2

Taking the partial derivative of optimal $m^{*}$ with respect to γ we obtain

$$\frac{{\partial m^{*}}}{\partial \gamma } = \frac{{ D ( {(4 D \beta - \delta^{2} } )( {C ( {\alpha - Z \beta } ) + B\gamma } ) + D\gamma^{2} ( {\alpha -Z\beta } ))}}{{(4 C D \beta - D \gamma^{2} - C \delta^{2} )^{2}}}$$

The denominator of the above equation is always positive as it is a square of an expression. From Lemma 1, since $4 D \beta - \delta^{2} > \frac{{D \gamma^{2}}}{C}$, the numerator can be rewritten to

$$D\left( {\left( {\frac{{D \gamma^{2}}}{C}} \right)( {C ( {\alpha - Z \beta } ) + B\gamma } ) + D\gamma^{2} ( {\alpha - Z \beta } )} \right).$$

From Lemma 1 we find $( {\alpha - Z \beta } )$ is positive; hence the numerator is always positive.

Therefore, $\frac{{\partial m^{*}}}{\partial \gamma } > 0$.□

Proof of Proposition A3

Taking the partial derivative of optimal $p^{*}$ with respect to C we obtain

$$\frac{{\partial p^{*}}}{\partial C} = - \frac{{ D\gamma ( {4 B D \beta + 2 D\alpha \gamma - B \delta^{2} - 2DZ\beta \gamma } )}}{{(4 C D \beta - D \gamma^{2} - C \delta^{2} )^{2}}}$$

The denominator of the above equation is always positive as it is a square of an expression. Next, we can rewrite the numerator to:

$$\begin{aligned}&= - D\gamma ( {4 B D \beta - B \delta^{2} + 2 D\alpha \gamma - 2DZ\beta \gamma } )\\&= - D\gamma ( {B( {4D\beta - \delta^{2} } ) + 2 D\gamma ( {\alpha - \beta Z} )} )\end{aligned}$$

Using Lemma 1 and the boundary condition, it can be shown that the numerator will be negative. Therefore, $\frac{{\partial p^{*}}}{\partial C} < 0$.□

Proof of Proposition A4

By taking the partial derivative of optimal $m^{*}$ with respect to C we obtain

$$\frac{{\partial m^{*}}}{\partial C} = - \frac{{( {4D\beta - \delta^{2} } )( {2D( {\alpha - Z\beta } )\gamma + B( {4D\beta - \delta^{2} } )} )}}{{2( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )^{2}}} = - \frac{{( {4D\beta - \delta^{2} } ) m^{*}}}{{( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}}$$

Using Lemma 1, we observe both numerator $( {4D\beta - \delta^{2} } ) m^{*}$ and denominator $( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )$ are positive. Hence, $\frac{{\partial m^{*}}}{\partial C}$ is negative of a positive quantity.

Therefore, $\frac{{\partial m^{ *}}}{\partial C} < 0$.

By taking the partial derivative of optimal $s^{*}$ with respect to C, we obtain

$$\frac{{\partial s^{*}}}{\partial C} = - \frac{{\gamma \delta ( {2D( {\alpha - Z\beta } )\gamma + B( {4D\beta - \delta^{2} } )} )}}{{2( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )^{2}}} = - \frac{{\gamma \delta m^{*}}}{{( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}}$$

Using Lemma 1, we observe that numerator $\gamma \delta m^{*}$ and denominator $( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )$ are positive. Hence, $\frac{{\partial s^{*}}}{\partial C}$ is negative of a positive quantity.

Therefore, $\frac{{\partial s^{*}}}{\partial C} < 0$.□

Proof of Proposition A5

By taking the partial derivative of optimal $p^{*}$ with respect to Z, we obtain

$$\frac{{\partial p^{*}}}{\partial Z} = \frac{{2CD\beta - D\gamma^{2} - C\delta^{2}}}{{4CD\beta - D\gamma^{2} - C\delta^{2}}}$$

From Lemma 1, we conclude that denominator is always positive. Depending on the numerator,$\frac{{\partial p^{*}}}{\partial Z}$ becomes positive, negative or zero.

$${\text{Numerator}} = 2CD\left( {\beta - \frac{{\gamma^{2}}}{2 C} - \frac{{\delta^{2}}}{2 D}} \right)$$

• If $\beta > \frac{{\gamma^{2}}}{2 C} + \frac{{\delta^{2}}}{2 D}$, then numerator is positive.

• If $\beta < \frac{{\gamma^{2}}}{2 C} + \frac{{\delta^{2}}}{2 D}$, then numerator is negative.

• If $\beta = \frac{{\gamma^{2}}}{2 C} + \frac{{\delta^{2}}}{2 D}$, then numerator is zero.

Proof of Proposition B1

In order to examine the variation in the optimal decision variables with respect to $\gamma$ we differentiate $p_{ms}^{*} , m_{ms}^{*} , d_{ms}^{*}$ and ${{\uppi}}_{\text{ms}}^{ *}$ with respect to $\gamma$. We obtain

$$\frac{{\partial p_{ms}^{*}}}{\partial \gamma } = \frac{{4C( {2DXY\beta + ( {\alpha - Z\beta + X\delta } )( {\gamma - Y\delta } )} ) + B( {4C\beta + 4DY^{2} \beta + ( {\gamma - Y\delta } )^{2} } ) + 2DY( {4DXY^{2} \beta + ( {\gamma - Y\delta } )( {X\gamma + Y( {2\alpha - 2Z\beta + X\delta } )} )} )}}{{( {4\beta ( {C + DY^{2} } ) - ( {\gamma - Y\delta } )^{2} } )^{2}}}$$

$$\frac{{\partial m_{ms}^{*}}}{\partial \gamma } = \frac{{(4C( {2DXY\beta + ( {\alpha - Z\beta + X\delta } )( {\gamma - Y\delta } )} ) + B( {4C\beta + 4DY^{2} \beta + ( {\gamma - Y\delta } )^{2} } ) + 2DY( {4DXY^{2} \beta + ( {\gamma - Y\delta } )( {X\gamma + Y( {2\alpha - 2Z\beta + X\delta } )} )} ))}}{{(4\beta ( {C + DY^{2} } ) - ( {\gamma - Y\delta } )^{2} )^{2}}}$$

$$\frac{{\partial d_{ms}^{*}}}{\partial \gamma } = \frac{{\beta (2( {\gamma - Y\delta } )(2DY( {Y\alpha - YZ\beta + X\gamma } ) + 2C( {\alpha - Z\beta + X\delta } ) + B( {\gamma - Y\delta } )) + ( {B + 2DXY} )( {4C\beta + 4DY^{2} \beta - ( {\gamma - Y\delta } )^{2} } ))}}{{( {4\beta ( {C + DY^{2} } ) - ( {\gamma - Y\delta } )^{2} } )^{2}}}$$

$$\frac{{\partial \pi_{ms}^{*}}}{\partial \gamma } = \frac{{( {2DY( {Y\alpha - YZ\beta + X\gamma } ) + 2C( {\alpha - Z\beta + X\delta } ) + B( {\gamma - Y\delta } )} )* m_{ms}^{*}}}{{4\beta ( {C + DY^{2} } ) - ( {\gamma - Y\delta } )^{2}}}$$

The denominator is always positive as it is a square of an expression. By carefully examining the numerators in all the above four cases, we find that the only way the numerator could be negative in each case if ${{\upalpha}} < Z\beta$ or ${{\upgamma}} < Y\delta$ or both. According to our boundary conditions, we have ${{\upalpha}} > Z\beta$ and ${{\upgamma}} > Y\delta$. Hence, we conclude $\frac{{\partial p_{ms}^{*}}}{\partial \gamma } > 0, \frac{{\partial m_{ms}^{*}}}{\partial \gamma } > 0, \frac{{\partial d_{ms}^{*}}}{\partial \gamma } > 0,$ and $\frac{{\partial {{\uppi}}_{\text{ms}}^{ *}}}{\partial \gamma } > 0$.□

Proof of Proposition B2

By taking the partial derivative of optimal modularity level ${\text{m}}^{*}$ with respect to C (cost parameter we obtain

$$\frac{{\partial m^{ *}}}{\partial C} = - \frac{{( {4D\beta - \delta^{2} } )( {2D( {\alpha - Z\beta } )\gamma + B( {4D\beta - \delta^{2} } )} )}}{{2( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )^{2}}} = - \frac{{( {4D\beta - \delta^{2} } ) m^{ *}}}{{( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}}$$

Using Lemma 1, we observe both numerator $( {4D\beta - \delta^{2} } ) m^{ *}$ and denominator $( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )$ are positive. Hence, $\frac{{\partial m^{ *}}}{\partial C}$ is negative of a positive quantity.

Therefore, $\frac{{\partial m^{ *}}}{\partial C} < 0$.

By taking the partial derivative of optimal ${\text{s}}^{*}$ with respect to C, we obtain

$$\frac{{\partial s^{ *}}}{\partial C} = - \frac{{\gamma \delta ( {2D( {\alpha - Z\beta } )\gamma + B( {4D\beta - \delta^{2} } )} )}}{{2( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )^{2}}} = - \frac{{\gamma \delta m^{ *}}}{{( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}}$$

Using Lemma 1, we observe that numerator $\gamma \delta m^{ *}$ and denominator $( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )$ are positive. Hence, $\frac{{\partial s^{ *}}}{\partial C}$ is negative of a positive quantity.

Therefore,

$$\frac{{\partial s^{ *}}}{\partial C} < 0.$$

Taking the partial derivative of optimal ${\text{p}}^{*}$ with respect to C we obtain

$$\frac{{\partial p^{ *}}}{\partial C} = - \frac{{ D\gamma ( {4 B D \beta + 2 D\alpha \gamma - B \delta^{2} - 2DZ\beta \gamma } )}}{{(4 C D \beta - D \gamma^{2} - C \delta^{2} )^{2}}}$$

The denominator of the above equation is always positive as it is a square of an expression. Next, we can rewrite the numerator to:

$$= - D\gamma ( {4 B D \beta - B \delta^{2} + 2 D\alpha \gamma - 2DZ\beta \gamma } )$$

$$= - D\gamma ( {B( {4D\beta - \delta^{2} } ) + 2 D\gamma ( {\alpha - \beta Z} )} )$$

Using Lemma 1 and the boundary condition, it can be shown that the numerator will be negative. Therefore, $\frac{{\partial p^{ *}}}{\partial C} < 0$.□

Proof of Proposition B3

By taking the partial derivative of optimal $p^{*}$ with respect to Z, we obtain

$$\frac{{\partial p^{*}}}{\partial Z} = \frac{{2C\beta + DY^{2} \beta - ( {\gamma - Y {{\updelta}}} )^{2}}}{{4C\beta + 4 DY^{2} \beta - ( {\gamma - Y {{\updelta}}} )^{2}}}$$

From Lemma 2, we conclude that denominator is always positive. Depending on the numerator,$\frac{{\partial p^{*}}}{\partial Z}$ becomes positive, negative or zero.

• $${\text{Numerator}} = 2C\beta + DY^{2} \beta - ( {\gamma - Y {{\updelta}}} )^{2}$$

• If ${{\upbeta}}$ $> \frac{{( {\gamma - Y {{\updelta}}} )^{2}}}{{2( { C + D Y^{2} } )}}$, then numerator is positive.

• If $\beta < \frac{{( {\gamma - Y {{\updelta}}} )^{2}}}{{2( { C + D Y^{2} } )}}$, then numerator is negative.

• If ${{\upbeta}} = \frac{{( {\gamma - Y {{\updelta}}} )^{2}}}{{2( { C + D Y^{2} } )}}$, then numerator is zero.□

Proof of Proposition B4

By taking the partial derivative of optimal ${{\uppi}}_{\text{ms}}^{ *}$ with respect to X, we obtain

$$\frac{{\partial (\pi_{ms}^{*} )}}{\partial X} = \frac{{2( {\alpha - Z\beta } )( {YD\gamma + C\delta } ) + B( {4DY\beta + \delta ( {\gamma - Y\delta } )} ) + 2X( {C\delta^{2} + D\gamma^{2} - 4CD\beta } )}}{{4C\beta + 4DY^{2} \beta - ( {\gamma - Y\delta } )^{2}}}$$

By taking the partial derivative of optimal $\pi_{ms}^{*}$ with respect to Y, we obtain

$$\frac{{\partial (\pi_{ms}^{*} )}}{\partial Y} = - \frac{{( {2B\beta + 4DXY\beta + ( {\alpha - Z\beta + X\delta } )( {\gamma - Y\delta } )} )(2( {\alpha - Z\beta } )( {YD\gamma + C\delta } ) + B( {4DY\beta + \delta ( {\gamma - Y\delta } )} ) + 2X( {C\delta^{2} + D\gamma^{2} - 4CD\beta } ))}}{{( {4C\beta + 4DY^{2} \beta - ( {\gamma - Y\delta } )^{2} } )^{2}}}$$

We observe,

$$\frac{{\partial (\pi_{ms}^{*} )}}{\partial Y } = - ( {m_{ms}^{*} } ) \left( {\frac{{\partial (\pi_{ms}^{*} )}}{\partial X}} \right)$$

We note, as $m_{ms}^{*}$ is always positive, hence both $\frac{{\partial (\pi_{ms}^{*} )}}{\partial Y }$ and $\frac{{\partial (\pi_{ms}^{*} )}}{\partial X}$ always have opposite signs and if one is zero then the other is zero too.

We observe using Lemma 2, denominator of $\frac{{\partial ({{\uppi}}_{\text{ms}}^{ *} )}}{{\partial {\text{X}}}}$ is always positive. However, the numerator could be positive, negative or zero depending on the relationship between X and other parameters.

We observe, when

$$X < \frac{{2( {\alpha - Z\beta } )( {DY\gamma + C\delta } ) + B( {4DY\beta + \delta ( {\gamma - Y\delta } )} )}}{{2( {4CD\beta - D\gamma^{2} - C\delta^{2} } )}},\quad \frac{{\partial (\pi_{ms}^{*} )}}{\partial X} > 0$$

That means $\pi_{ms}^{*}$ will increase as X increases.

However, when

$$X > \frac{{2( {\alpha - Z\beta } )( {DY\gamma + C\delta } ) + B( {4DY\beta + \delta ( {\gamma - Y\delta } )} )}}{{2( {4CD\beta - D\gamma^{2} - C\delta^{2} } )}},\quad \frac{{\partial (\pi_{ms}^{*} )}}{\partial X} < 0$$

which means $\pi_{ms}^{*}$ will decrease as X increases.

Hence, $\pi_{ms}^{*}$ will be maximum with respect to X and Y, when

$$X = \frac{{2( {\alpha - Z\beta } )( {DY\gamma + C\delta } ) + B( {4DY\beta + \delta ( {\gamma - Y\delta } )} )}}{{2( {4CD\beta - D\gamma^{2} - C\delta^{2} } )}}$$

We also find from Eqs. 20–24, for the above value of X, the optimal decision variables become

$$\begin{aligned}p_{ms}^{*} &= \frac{{2CD( {\alpha - Z\beta } ) + BD\gamma}}{{4 C D \beta - D \gamma^{2} - C \delta^{2}}} - Z = p^{*}\\m_{ms}^{*} &= \frac{{4B D \beta + 2 D \alpha \gamma - 2DZ\beta \gamma - B\delta^{2}}}{{2 ( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}} = m^{*}\\s_{ms}^{*} &= \frac{{( {2 C \alpha + B \gamma - 2CZ\beta } )\delta}}{{2 ( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}} = s^{*}\\d_{ms}^{*} &= \frac{{D\beta ( {2C( {\alpha - Z\beta } ) + B\gamma - 2 C } )}}{{( {4 C D \beta - D \gamma^{2} - C \delta^{2} } )}} = d^{*}\\\pi_{ms}^{*} &= \frac{{4D( {\alpha - Z\beta } )( {B\gamma + C( {\alpha - Z\beta } )} ) + B^{2} ( {4D\beta - \delta^{2} } )}}{{4( {4 CD\beta - D\gamma^{2} - C\delta^{2} } )}} - A = \pi^{*} - A\end{aligned}$$

□