Bid Markup Decision and Resource Allocation

for Cost Estimation in Competitive Bidding

Yuichi Takano (corresponding author) ytakano@isc.senshu-u.ac.jpSchool of Network and Information, Senshu University2-1-1 Higashimita, Tama-ku, Kawasaki-shi, Kanagawa 214-8580, Japan

Nobuaki Ishii ishii@shonan.bunkyo.ac.jpFaculty of Information and Communications, Bunkyo University1100 Namegaya, Chigasaki-shi, Kanagawa 253-8550, Japan

Masaaki Muraki m.muraki8511@gmail.comDepartment of Industrial Engineering and ManagementGraduate School of Decision Science and Technology, Tokyo Institute of Technology2-12-1-W9-73 Ookayama, Meguro-ku, Tokyo 152-8552, Japan

Abstract

To receive a project contract through competitive bidding, contractors submit a bid pricedetermined by putting a markup on the estimated project cost. Since a bid is inevitablyaffected by an inaccurate cost estimate, sufficient resources should be allocated to cost es-timation. This paper develops a novel optimization model for determining the bid markupand the resource allocation for cost estimation simultaneously. We derive optimality condi-tions based on some assumptions, and we report computational results demonstrating theeffectiveness of our model.

Keywords: Bidding, Optimization, Resource management, Project management

1 Introduction

Competitive bidding is widely used to choose contractors. A client needing to find a contractor

who will carry out a certain project invites potential contractors to submit bid prices, which

are closed to competing contractors. The lowest bid is the chief determinant of the winning

contractor, who is paid the bid price and executes the project as the client specified. In this

process the contractors’ profits are highly dependent on their bidding strategies.

Because a contractor determines a bid price by putting a markup on the estimated project

cost, the bid price is markedly affected by the accuracy of the estimated cost. The lack of

information in the early stages of a project makes it hard to estimate its cost accurately [20], and

the difficulty of estimating the cost of software projects increases with the size and importance

of the project [1].

The accuracy of a cost estimate is positively correlated with the man-hours (MHs) spent

making the estimate [17, 32], and the cost estimate classification matrix created by Christensen

and Dysert [4] shows a clear relationship between the accuracy of the estimate and the amount

of preparation. These studies indicate that the accuracy of an estimated cost can be improved

by increasing the amount of resources allocated to its estimation. Moreover, an appropriate

markup should depend on the accuracy of the contractor’s cost estimate. As a result, in order

1

for contractors to improve their profits, it is essential to determine the bid markup and the

resource allocation for cost estimation simultaneously.

Since the seminal work by Friedman [12], a considerable number of studies have pursued

effective models of the determination of bid markups (or bid prices) [10, 21, 27, 29]. These

models are divided into three main categories [25]: statistical models, artificial intelligence

(AI) based models, and multi-criteria utility models. Statistical models decide the optimal

markup according to the statistical bidding behavior of each competitor [22, 26, 30], and recently

an entropy metric [5] and Monte-Carlo simulation [16] were used effectively in the statistical

models. AI-based models estimate an appropriate markup by using case-based reasoning [7, 8]

and artificial neural networks [15, 23, 24]. Multi-criteria utility models include many practical

factors in the bid markup estimation [2, 3, 6, 9, 33]. These models, however, do not deal with

the resource allocation for cost estimation.

Project resource/budget allocation has been a subject of active research [11, 13, 14, 28].

To the best of our knowledge, however, only two studies have addressed the resource allocation

problem for estimating project costs in competitive bidding: Ishii et al. [19] implement a two-step

heuristic algorithm in which the first step allocates the resources preferentially to estimate the

costs of profit-making projects and the second step decides the bid prices for them, and Takano

et al. [31] build a multi-period resource allocation model for cost estimation in a sequential

competitive bidding situation. It is noteworthy, nonetheless, that these studies [19, 31] consider

the resource allocation separately from the bid markup decision. None of the previous studies

on competitive bidding have investigated the interaction between the bid markup decision and

resource allocation for cost estimation.

The purpose of this paper is to devise a novel optimization model for determining the bid

markup and resource allocation for cost estimation simultaneously. Specifically, we revise the

model developed by King and Mercer [22] so that it can incorporate the relationship between

the accuracy of the cost estimate and the amount of resources invested in making the estimate.

For our simultaneous optimization model, we first derive optimality conditions based on some

assumptions and illustrate them with numerical examples. We next perform computational

experiments to examine the contractor’s expected profit with respect to the bid markup and

resource allocation. We also evaluate the sensitivity of the computational solutions in relation

to competitors’ bidding behaviors.

The rest of this paper is organized as follows. Section 2 presents our simultaneous optimiza-

tion model for the bid markup decision and resource allocation for cost estimation, Section 3

shows analytical results of the model, and Section 4 reports our computational results. Section

5 concludes the paper with a brief summary of our work and a discussion of future research

directions.

2

2 Competitive Bidding Models

This section first describes the competitive bidding model formulated by King and Mercer [22]

and then describes our simultaneous optimization model.

2.1 King-Mercer model

To receive a project contract through competitive bidding, a contractor begins by estimating the

cost of completing the project. Since the estimated cost is subject to an unavoidable estimation

error, it is reasonable to treat it as a random variable. Therefore, we denote the estimated cost

(1 + e)C, where C is the true project cost and e is a random estimation error.

The contractor determines a bid price by putting a markup, m, on the estimated cost.

Accordingly, the bid price is written as follows:

B(m, e) := (1 +m) (1 + e)C. (1)

Let P (b) be the probability of winning the contract when the bid price is b. If the contractor

wins the contract, the eventual profit will be the difference between the bid price and the true

project cost, i.e., B(m, e) − C. As a result, King and Mercer [22] formulate the contractor’s

expected profit as follows:

R(m) :=

∫(B(m, e)− C)P (B(m, e))ϕ(e) de, (2)

where ϕ(e) is a probability density function of the estimation error, e.

When the estimated cost contains no estimation error (i.e., e = 0), it follows that B(m, e) =

(1 +m)C and the contractor’s expected profit is

R(m) = mC P ((1 +m)C).

As pointed out by King and Mercer [22], this model is equivalent to Friedman’s well-known

model [12]. So we can see that Friedman’s model ignores the effects of an inaccurate cost

estimate on the contractor’s expected profit.

2.2 Simultaneous optimization model

In the King-Mercer model (2), the probability distribution of estimation error is fixed. As

mentioned in the introduction, however, the accuracy of the cost estimate can be controlled

by adjusting the amount of resources allocated to cost estimation. Thus we assume that the

variation of estimation error depends on the amount of resources used for estimating cost. More

precisely, the probability density function of estimation error is defined as ϕ(e | w), where w is

the amount of resources used for estimating cost. This definition implies that the variance of

estimation error can be decreased by allocating a large amount of resources to cost estimation.

3

Following Christensen and Dysert [4], we represent the amount of resources as a percentage

of the project cost, C. In other words, allocating the amount w of resources to cost estimation

corresponds to investing the amount C w of money in it. Our simultaneous optimization model

determines the decision variables, m and w, simultaneously in such a way that the expected

profit is maximized. Consequently, our model is posed as follows:

maximize R(m,w) :=

∫(B(m, e)− C)P (B(m, e))ϕ(e | w) de− C w. (3)

For the sake of completeness, we define the winning probability, P (B(m, e)), in a way similar

to that in which King and Mercer [22] did. Let us denote by mk and ek the bid markup and

estimation error of each competitor k = 1, 2, . . . , n. It is also assumed that there are no cost

competitiveness gaps between contractors; that is, the project cost is C for all the competitors.

Our contractor will win the contract if his/her bid price is the lowest or, equivalently, if B(m, e) ≤B(mk, ek) for all k = 1, 2, . . . , n. It then follows from the definition (1) that

B(m, e) ≤ B(mk, ek) ⇐⇒ E(m,mk, e) ≤ ek,

where

E(m,mk, e) :=B(m, e)

(1 +mk)C− 1 =

(1 +m) (1 + e)

1 +mk− 1. (4)

Therefore the probability of winning the contract is expressed as follows:

P (B(m, e)) =

n∏k=1

∫ ∞

E(m,mk,e)ϕk(ek) dek, (5)

where ϕk(ek) is the probability density function of each competitor’s estimation error.

3 Analytical Results

This section provides optimality conditions of the simultaneous optimization model (3) based

on some assumptions. Numerical examples are also offered to illustrate closed-form solutions.

3.1 Assumptions

We begin with introducing some assumptions to derive optimality conditions. First we make

the following assumption about our contractor’s estimation error:

Assumption 3.1 Our contractor’s estimation error follows a uniform distribution on the in-

terval [−D(w), D(w)].

It is noteworthy that King and Mercer [22] assume in deriving their analytical solutions that

the estimation errors of the decision maker (our contractor) and competitors are all independent

and identically distributed. By contrast, Assumption 3.1 implies that our contractor can control

the magnitude of his/her estimation error because the corresponding interval [−D(w), D(w)] is

dependent on the amount, w, of resources used for estimating the project cost.

We make the following assumption about competitors:

4

Assumption 3.2 Only one competitor participates in the competitive bidding (i.e., n = 1), and

his/her estimation error follows a uniform distribution on the interval [−D1, D1].

From Assumptions 3.1 and 3.2, the probability density function of our contractor’s estimation

error is defined as follows:

ϕ(e | w) =

1

2D(w), if −D(w) ≤ e ≤ D(w),

0, otherwise,

and that of the competitor’s estimation error is written as follows:

ϕ1(e1) =

1

2D1, if −D1 ≤ e1 ≤ D1,

0, otherwise.

Considering the definition (4), the probability of winning the contract is expressed as follows:

P̂ (B(m, e)) :=

∫ ∞

E(m,m1,e)ϕ1(e1) de1

=

1, if E(m,m1, e) ≤ −D1,D1 − E(m,m1, e)

2D1, if −D1 ≤ E(m,m1, e) ≤ D1,

0, if D1 ≤ E(m,m1, e).

=

1, if e ≤ E−(m),

− B(m, e)

2 (1 +m1)D1C+

1 +D1

2D1, if E−(m) ≤ e ≤ E+(m),

0, if E+(m) ≤ e,

(6)

where

E−(m) :=1 +m1

1 +m(1−D1)− 1, E+(m) :=

1 +m1

1 +m(1 +D1)− 1. (7)

One can regard [E−(m), E+(m)] as a range of a competitor’s relative estimation error.

As a result, the simultaneous optimization model (3) is reformulated as follows:

maximize R(m,w) =

∫ D(w)

−D(w)(B(m, e)− C) P̂ (B(m, e))

1

2D(w)de− C w. (8)

3.2 Optimality conditions

In what follows, to evaluate the integral in the optimization model (8), we shall consider the

following two situations depending on the ranges, [−D(w), D(w)] and [E−(m), E+(m)], of the

estimation errors:

Case 1 −D(w) ≤ E−(m) ≤ E+(m) ≤ D(w),

Case 2 E−(m) ≤ −D(w) ≤ D(w) ≤ E+(m).

5

In Case 1 our contractor’s estimation error is larger than the competitor’s in the sense

that [E−(m), E+(m)] ⊆ [−D(w), D(w)]. In this case we can obtain the following optimality

conditions:

Proposition 3.1 Let (m∗, w∗) be an optimal solution to the optimization model (8). If −D(w∗) ≤E−(m∗) ≤ E+(m∗) ≤ D(w∗), then (m∗, w∗) solves the following system of equations:

m =

√3 (1−m2

1)− (1 +m1)2D21√

3 |1−D(w)|− 1, (9)(

3 (m21 −m2) + (1 +m1)

2D21

)D′(w) + 3 (1 +m)2D(w)2D′(w) + 12 (1 +m)D(w)2 = 0.

(10)

Proof. See Appendix A.1. ■

In Case 2 our contractor’s estimation error is smaller than the competitor’s in the sense

that [−D(w), D(w)] ⊆ [E−(m), E+(m)]. In this case we can obtain the following optimality

conditions:

Proposition 3.2 Let (m∗, w∗) be an optimal solution to the optimization model (8). If E−(m∗) ≤−D(w∗) ≤ D(w∗) ≤ E+(m∗), then (m∗, w∗) solves the following system of equations:

m =3 ((1 +m1) (1 +D1) + 1)

2 (D(w)2 + 3)− 1, (11)

(1 +m)2D(w)D′(w) + 3 (1 +m1)D1 = 0. (12)

Proof. See Appendix A.2. ■

3.3 Numerical examples

This subsection illustrates the closed-form solutions to the simultaneous optimization model (8)

with numerical examples. Here, it was supposed that (m1, D1) = (0.2, 0.1), (0.2, 0.3) in relation

to the competitor’s bid price.

First, we examine the optimal markups (9) and (11) when the amount of resources used

for cost estimation is given. Figure 1 shows the ranges, [−D(w), D(w)] and [E−(m), E+(m)],

of the estimation errors as functions of D(w), where the range [E−(m), E+(m)] was calculated

by substituting the markup (9) into equation (7). The graphs of E−(m) and E+(m) may look

straight lines but are actually curves. Proposition 3.1 states that the markup (9) is optimal when

−D(w) ≤ E−(m) ≤ E+(m) ≤ D(w). Accordingly, in Figure 1 the ranges were represented by

dashed curves when this condition was not satisfied.

6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 1: Ranges of estimation errors based on equation (9) in Case 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.1 0.2 0.3

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 2: Ranges of estimation errors based on equation (11) in Case 2

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

Case 1

Case 2

-0.1

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3

Case 1

Case 2

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 3: Optimal markup

Figure 2 shows the ranges of the estimation errors, where the range [E−(m), E+(m)] was

calculated by substituting the markup (11) into equation (7). Similarly to Figure 1, these ranges

were represented by dashed curves when the condition E−(m) ≤ −D(w) ≤ D(w) ≤ E+(m) in

Proposition 3.2 was not satisfied.

7

The optimal markup is represented as a solid curve in Figure 3, where Case 1 and Case 2

correspond to the markups (9) and (11), respectively. We can see in Figure 3a that although

the graph of optimal markup is unconnected, it starts to rise sharply when D(w) exceeds a

certain level. In other words, our contractor should set a high markup when the accuracy of

his/her cost estimate is very low. Indeed, if the cost estimate is inaccurate, the contractor is at

a risk of submitting a bid price lower than the true project cost. In this case, s/he will win the

contract but run a large deficit. Hence, the bid markup should be set high to mitigate such a

risk. Meanwhile, in Figure 3b the optimal markup was fairly constant but relatively high. The

main reason is that the competitor’s cost estimate was inaccurate (i.e., D1 = 0.3) in Figure 3b.

As a result, the competitor is likely to overestimate the project cost, and in such a case our

contractor’s high markup produces a large profit.

Next we investigate the optimal amount of resources when the bid markup is given. For this

purpose, it was supposed that D(w) = δ/w. This implies that the range of our contractor’s

estimation error becomes narrower when the amount of allocated resources increases. The

parameter δ was set to 0.005, i.e., D(w) = 0.1 for w = 0.05. In this case, we can obtain

closed-form solutions to equations (10) and (12) as follows:

w =

√3 δ (1 +m)√

12 δ (1 +m) + 3 (m2 −m21)− (1 +m1)2D2

1

(13)

in Case 1, and

w = 3

√δ2 (1 +m)2

3 (1 +m1)D1(14)

in Case 2.

Figures 4 and 5 show the ranges of the estimation errors as functions of markup, m, in

Cases 1 and 2. Here the range [−D(w), D(w)] was calculated by substituting the amounts of

resources (13) and (14) into D(w) = δ/w. The condition in Proposition 3.1 was satisfied only

in Figure 4a, and that in Proposition 3.2 was satisfied only in Figure 5b.

The optimal amount of resources is represented as a solid curve in Figure 6, where Case 1

and Case 2 correspond to the amounts of resources (13) and (14), respectively. We can see in

Figure 6a that the optimal amount of resources decreased as our contractor’s markup rose, while

in Figure 6b it was nearly independent of the markup. Note that the competitor’s estimation

error was relatively small (i.e., D1 = 0.1) in Figure 6a. Now, let us assume that our contractor’s

estimation error is also very small, and his/her bid markup is higher than the competitor’s.

In this case our contractor has little chance of winning the contract because his/her bid price

certainly exceeds the competitor’s. Thus, to increase the probability of winning due to estimation

error, our contractor should reduce the amount of resources used for cost estimation as shown

in Figure 6a.

8

-0.4

-0.2

0

0.2

0.4

0.6

0.1 0.15 0.2 0.25 0.3

-0.4

-0.2

0

0.2

0.4

0.6

0.1 0.15 0.2 0.25 0.3

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 4: Ranges of estimation errors based on equation (13) in Case 1

-0.4

-0.2

0

0.2

0.4

0.6

0.1 0.15 0.2 0.25 0.3

-0.4

-0.2

0

0.2

0.4

0.6

0.1 0.15 0.2 0.25 0.3

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 5: Ranges of estimation errors based on equation (14) in Case 2

0

0.05

0.1

0.15

0.2

0.25

0.1 0.15 0.2 0.25 0.3

Case 1

Case 2

0

0.05

0.1

0.15

0.2

0.25

0.1 0.15 0.2 0.25 0.3

Case 1

Case 2

(a) (m1, D1) = (0.2, 0.1) (b) (m1, D1) = (0.2, 0.3)

Figure 6: Optimal amount of resources

4 Computational Results

This section reports computational results examining the expected profit of our simultaneous

optimization model with respect to the bid markup and resource allocation. It also evaluates

9

Table 1: Range of our contractor’s estimation error

Case A w 0.015% 0.025% 0.050% 0.100% 0.500%

Case B w 0.15% 0.25% 0.50% 1.00% 5.00%

L(w) −0.3 −0.15 −0.10 −0.05 −0.005

U(w) 0.6 0.30 0.20 0.10 0.010

-0.4

-0.2

0

0.2

0.4

0.6

0 0.1 0.2 0.3 0.4 0.5

Amount of Resources [%]

Figure 7: Piecewise linear functions L(w) and U(w) in Case A

the sensitivity of the computational solutions in relation to competitors’ bidding behaviors.

4.1 Experimental design

It was supposed that the estimation errors follow triangular distributions. More precisely, our

contractor’s estimation error follows a triangular distribution with mode zero on the interval

[L(w), U(w)]. Therefore, its probability density function was defined as follows:

ϕ(e | w) =

2 (e− L(w))

−L(w) (U(w)− L(w)), if L(w) ≤ e ≤ 0,

2 (U(w)− e)

U(w) (U(w)− L(w)), if 0 ≤ e ≤ U(w),

0, otherwise.

The requisite amount of resources for cost estimation may vary by industry sector. Accord-

ingly, as shown in Table 1 we considered two cases for the range [L(w), U(w)]. Case A is based

on the cost estimate classification matrix [4]. In Case B, our contractor needs ten times the

amount of resources of Case A to estimate project costs with the same accuracy. We considered

that L(w) and U(w) were piecewise linear functions interpolating linearly between the points

shown in Table 1. For instance in Case A, L(w) and U(w) were piecewise linear functions of

the amount of resources as shown in Figure 7. The range, [L(w), U(w)], of our contractor’s

estimation error is narrowed by increasing the amount of resources used for cost estimation.

10

-6

-3

0

3

6

9

12

0 0.05 0.1 0.15 0.2 0.25 0.3

E

x

p

e

c

t

e

d

P

r

o

f

i

t

Markup

w=0.015% w=0.025% w=0.100% w=0.500%

-6

-3

0

3

6

9

12

0 0.05 0.1 0.15 0.2 0.25 0.3

E

x

p

e

c

t

e

d

P

r

o

f

i

t

Markup

w=0.15% w=0.25% w=1.00% w=5.00%

(a) Case A (b) Case B

Figure 8: Expected profit as a function of bid markup

-6

-3

0

3

6

9

12

0 0.1 0.2 0.3 0.4

E

x

p

e

c

t

e

d

P

r

o

f

i

t

Amount of Resources [%]

m=0.05 m=0.10 m=0.20 m=0.30

-6

-3

0

3

6

9

12

0 1 2 3 4

E

x

p

e

c

t

e

d

P

r

o

f

i

t

Amount of Resources [%]

m=0.05 m=0.10 m=0.20 m=0.30

(a) Case A (b) Case B

Figure 9: Expected profit as a function of amount of resources used for cost estimation

It was also supposed that the estimation error ek of each competitor k = 1, 2, . . . , n follows

a triangular distribution with mode zero on the fixed interval [Lk, Uk]. The true project cost,

C, was set to 100, and unless otherwise noted the parameters of competitors were set as follows.

Three competitors participated in the bidding (i.e., n = 3). Their bid markups, mk, were 0.2,

and the ranges of their estimation errors were [Lk, Uk] = [−0.1, 0.2] for all k = 1, 2, . . . , n. The

optimization model (3) was solved with the FindMaximum function implemented in Mathematica

10.0 (http://www.wolfram.com/mathematica/).

4.2 Graphical form of expected profit

We first graphically illustrate the expected profit of our simultaneous optimization model.

Figure 8 shows the expected profit, R(m,w), as a function of markup,m ∈ {0.01, 0.02, . . . , 0.30}.Here the amount of resources used for cost estimation was set as follows: w = 0.015%, 0.025%,

0.100%, and 0.500% in Case A; and w = 0.15%, 0.25%, 1.00%, and 5.00% in Case B.

We can see that the difference between the minimum and maximum values of the expected

profit became larger with the increasing amount of resources. In Figure 8a, for instance the

expected profit of w = 0.015% ranged from −0.11 to 2.19, whereas that of w = 0.500% ranged

from −0.12 to 10.36. This means that the optimization of bid markup has a profound effect

11

when the cost estimate is accurate. The expected profit reached its maximum with the maximal

amount of resources w = 0.500% in Case A. By contrast, since the cost estimate was more

expensive in Case B than in Case A as shown in Table 1, the expected profit in Case B was

lowered overall with the maximal amount of resources w = 5.00%.

In Figure 8a the markups optimal for maximizing the expected profits were 0.17, 0.11, 0.11,

and 0.13 for w = 0.015%, 0.025%, 0.100%, and 0.500%, respectively. This implies that when

the cost estimate is significantly inaccurate, contractors should set a high markup, e.g., 0.17,

in order to prevent a large loss due to cost estimation error. A similar result was obtained

in Figure 8b, and these observations are consistent with the numerical example of closed-form

solutions in Figure 3a. In addition, one should notice that the optimal markup depends on the

amount of resources used for estimating cost. This suggests that in the bid markup decision,

contractors need to take into account the resource allocation for cost estimation.

Figure 9 shows the expected profit, R(m,w), as a function of the amount of resources used

for cost estimation, w ∈ {0.02%, 0.04%, . . . , 0.40%} in Case A and w ∈ {0.2%, 0.4%, . . . , 4.0%}in Case B. Here the markup was set to m = 0.05, 0.10, 0.20, and 0.30 in Cases A and B.

It is found from Figure 9 that markup m = 0.10 generated high expected profits regardless of

the amount of resources used for cost estimation. The graph of the expected profit for m = 0.10

was almost flat in Figure 9a when the amount of resources was more than 0.1%. By contrast, it

had a negative slope in Figure 9b because the cost estimate in Case B was expensive.

One also sees in Figure 9 that the expected profit for m = 0.30 increased monotonically as

the amount of resources used for cost estimation decreased. This result is consistent with the

numerical example of closed-form solutions in Figure 6a. Indeed, if the contractor’s markup is

very high, s/he will have little chance to win the contract. Consequently, reducing the amount

of resources used for estimating cost leads to a high winning probability due to cost estimation

error and, accordingly, gives a large expected profit to him/her.

The optimal amounts of resources for maximizing the expected profit were 0.08%, 0.30%,

0.40%, and 0.02% for m = 0.05, 0.10, 0.20, and 0.30 in Figure 9a; and were 0.8%, 1.0%, 0.6%,

and 0.2% for m = 0.05, 0.10, 0.20, and 0.30 in Figure 9b. This means that the optimal amount

of resources for cost estimation is highly dependent on the bid markup. For this reason, it is

essential to determine the bid markup and resource allocation for cost estimation simultaneously.

Figure 10 shows contour plots of the expected profit function, R(m,w), for our simultaneous

optimization model. Here the optimal solutions (m∗, w∗) were (0.129, 0.479) in Case A and

(0.112, 1.00) in Case B. The bid markup decision had a larger impact on the expected profit

than the resource allocation did in Figure 10a. Additionally, the optimal amount of resources

was relatively low in Figure 10b.

4.3 Sensitivity analysis of optimal solutions

This subsection describes our model’s sensitivity to the number of competitors and their bidding

behaviors.

12

(a) Case A (b) Case B

Figure 10: Contour plot of expected profit function

0

5

10

15

20

1 2 3 4 5 6 7 8

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(a) Maximal expected profit (b) Optimal markup

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Number of Competitors

Case A Case B

(c) Optimal amount of resources

Figure 11: Sensitivity to the number of competitors

Figure 11 shows the maximal expected profit, optimal markup, and optimal amount of re-

sources for each number of competitors, n ∈ {1, 2, . . . , 8}. It is clear that a large number

13

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Competitors' Markup

Case A Case B

(c) Optimal amount of resources

Figure 12: Sensitivity to competitors’ bid markups

of competitors intensifies the competition of the bidding and, accordingly, makes winning the

contract difficult for each contractor. For this reason, in Figure 11a our contractor’s maximal

expected profit decreased as the number of competitors increased. Moreover, Figure 11b demon-

strates that our contractor should lower the bid markup when many competitors participate in

the bidding. As shown in Figure 11c, however, the number of competitors had little effect on

the optimal amount of resources.

Figure 12 shows the maximal expected profit, optimal markup, and optimal amount of

resources for each markup of competitors, mk ∈ {0.100, 0.125, . . . , 0.300}, where m1 = m2 = m3.

High competitor markups give our contractor a chance of obtaining a large profit from the

contract. Consequently, both the maximal expected profit and optimal markup increased with

the competitors’ markups. The optimal amount of resources, however, was nearly independent

of the competitors’ markups in Figure 12c.

Figure 13 shows the maximal expected profit, optimal markup, and optimal amount of

resources with respect to the ranges of competitors’ estimation errors shown in Table 2, where

[L1, U1] = [L2, U2] = [L3, U3]. To win the contract, our contractor has to submit a bid that

is lower than any of his/her competitors’ bids. However, if the competitors’ cost estimates are

inaccurate, the lowest of their bids will be very low, and thus, it is very difficult for our contractor

to win the contract. As a result, we can see from Figure 13 that our contractor’s maximal

expected profit and the optimal markup decreased as the range of competitors’ estimation errors

became broader. Additionally in Case A of Figure 13c, the optimal amount of resources was

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Case A Case B

(c) Optimal amount of resources

Figure 13: Sensitivity to accuracy of competitors’ cost estimates (see also Table 2)

Table 2: Ranges of competitors’ estimation errors in Figure 13

1 2 3 4 5 6 7 8 9 10

Lk (k = 1, 2, 3) −0.050 −0.075 −0.100 −0.125 −0.150 −0.175 −0.200 −0.225 −0.250 −0.275

Uk (k = 1, 2, 3) 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550

lowered by competitors’ large estimation errors. This implies that when the competitors’ cost

estimates are inaccurate, our contractor’s accurate estimate is not worth the expense.

5 Conclusions

This paper devised an optimization model for determining the bid markup and the resource

allocation for cost estimation simultaneously in competitive bidding. We derived optimality

conditions for our optimization model by assuming a single competitor and uniformly distributed

estimation errors. We also conducted computational experiments to examine the graphical

form of the expected profit and analyzed the computed solutions’ sensitivity to the number of

competitors and their bidding behaviors.

The bid markup decision and resource allocation for cost estimation in competitive bid-

ding have in the past been studied independently but are closely connected. To establish a

15

profit-making bidding strategy, we developed an optimization model for determining them si-

multaneously.

The results of our computational experiments investigating the effects of the bid markup

decision and resource allocation on the contractor’s expected profit confirmed that in order

to maximize the expected profit, the bid markup and resource allocation should be considered

simultaneously. Moreover, the results of analyzing sensitivity with regard to competitors’ bidding

behaviors will be useful when developing bidding strategies in different situation. Thus our

optimization model will be of practical value for contractors who mainly receive project contracts

through competitive bidding.

Our numerical examples and computational results provided the following insights:

• Optimizing the bid markup is especially effective when the cost estimate is accurate.

• The bid markup should be set high when the cost estimate is inaccurate.

• The amount of resources used for estimating cost should be reduced when the bid markup

is high.

These insights may readily be drawn by intuition, but our research is of value in establishing

their validity through the use of mathematical modeling and analysis.

Although for simplicity triangular distributions of estimation errors were used in the com-

putational experiments, the effectiveness of our optimization model will further be supported

by using other probability distributions of estimation errors. Another direction of future re-

search will be to extend our simultaneous optimization model to a sequential competitive bid-

ding situation [30, 31]. Our optimization model is also expected to work effectively in related

decision-making problems, such as the order acceptance problem [18].

Acknowledgments

This work was supported by Grant-in-Aid for Scientific Research (C) 25350455 by the Japan

Society for the Promotion of Science.

Appendix A Derivation of Optimality Conditions

This appendix gives proofs of Propositions 3.1 and 3.2 based on Assumptions 3.1 and 3.2.

16

Appendix A.1 Proof of Proposition 3.1

It follows from (6) that the expected profit (8) in Case 1 can be transformed as follows:

R(m,w) =

∫ E−(m)

−D(w)(B(m, e)− C)

1

2D(w)de

+

∫ E+(m)

E−(m)(B(m, e)− C)

(− B(m, e)

2 (1 +m1)D1C+

1 +D1

2D1

)1

2D(w)de− C w

=1

2D(w)

∫ E−(m)

−D(w)(B(m, e)− C)︸ ︷︷ ︸

f(m,e)

de

+1

4D1D(w)

∫ E−(m)

E+(m)(B(m, e)− C)

(B(m, e)

(1 +m1)C− (1 +D1)

)︸ ︷︷ ︸

g(m,e)

de− C w

=1

2D(w)

[F (m, e)

]e=E−(m)

e=−D(w)+

1

4D1D(w)

[G(m, e)

]e=E−(m)

e=E+(m)− C w,

where

F (m, e) :=

∫f(m, e) de =

(1 +m) (1 + e)2C

2− C e,

G(m, e) :=

∫g(m, e) de

=(1 +m)2 (1 + e)3C

3 (1 +m1)− ((1 +m1) (1 +D1) + 1) (1 +m) (1 + e)2C

2 (1 +m1)+ (1 +D1)C e.

Therefore, we have[F (m, e)

]e=E−(m)

e=−D(w)= F (m,E−(m))− F (m,−D(w))

=− (1 +m) (1−D(w))2C

2+

(1 +m1) (1−D1) ((1 +m1) (1−D1)− 2)C

2 (1 +m)+ (1−D(w))C,

and [G(m, e)

]e=E−(m)

e=E+(m)= G(m,E−(m))−G(m,E+(m)) =

2 (1 +m1)D21 (3m1 − (1 +m1)D1)C

3 (1 +m).

The optimality condition for markup, m, is written as follows:

∂

∂mR(m,w) =

C

12 (1 +m)2D(w)

(−3 (1 +m)2 (1−D(w))2 + 3 (1−m2

1)− (1 +m1)2D2

1

)= 0.

By solving the above equation for m, we obtain the closed-form solution (9).

On the other hand, we have

1

2D(w)

[F (m, e)

]e=E−(m)

e=−D(w)

=

(−(1 +m)C

4+

(1 +m1) (1−D1) ((1 +m1) (1−D1)− 2)C

4 (1 +m)+

C

2

)1

D(w)

+mC

2− (1 +m)C

4D(w),

17

and

1

4D1D(w)

[G(m, e)

]e=E−(m)

e=E+(m)=

((1 +m1)D1 (3m1 − (1 +m1)D1)C

6 (1 +m)

)1

D(w).

Therefore we obtain the equation (10) from the following optimality condition for amount,

w, of resources:

∂

∂wR(m,w) = −

(3 (m2

1 −m2) + (1 +m1)2D2

1

)C D′(w)

12 (1 +m)D(w)2− (1 +m)C D′(w)

4− C = 0.

Appendix A.2 Proof of Proposition 3.2

As in Appendix A.1, the expected profit (8) in Case 2 can be transformed as follows:

R(m,w) =1

4D1D(w)

[G(m, e)

]e=−D(w)

e=D(w)− C w.

Therefore we have[G(m, e)

]e=−D(w)

e=D(w)= G(m,−D(w))−G(m,D(w))

=(1 +m)2 (−2D(w)3 − 6D(w))C

3 (1 +m1)+

2 ((1 +m1) (1 +D1) + 1) (1 +m)D(w)C

1 +m1

− 2 (1 +D1)D(w)C.

The optimality condition for markup, m, is written as follows:

∂

∂mR(m,w) =

C

6 (1 +m1)D1

(−2 (1 +m) (D(w)2 + 3) + 3 ((1 +m1) (1 +D1) + 1)

)= 0.

By solving the above equation for m, we obtain the closed-form solution (11).

On the other hand, we obtain the equation (12) from the following optimality condition for

amount, w, of resources:

∂

∂wR(m,w) =

−(1 +m)2C D(w)D′(w)

3 (1 +m1)D1− C = 0.

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